Subgroups — C₃×#84010 #11886896

Order 1152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#5835	C₂	1	2	yes	maximal, complemented
3	C₃×#5769	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5241	C₂	1	2	yes	maximal, complemented
5	C₃×#5199	C₂	1	2	yes	maximal, complemented
6	C₃×#5198	C₂	1	2	yes	maximal, complemented
7	C₃×#5196	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5160	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#5151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#5144	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#5142	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#3292	—	3	1	no	maximal
13	#84010	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₃²×H(5,𝔽₂)	C₂²	1	2	yes	complemented
15	C₃×#1009	C₂²	1	2	yes	complemented
16	C₃²×AGL(1,ℤ/8)	—	2	2	no	—
17	C₃²×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
18	C₃²×#135	—	2	1	no	—
19	C₃²×#135	C₂²	1	1	yes	characteristic
20	C₂×C₃²×D₈	—	2	1	no	—
21	C₂×C₃²×D₈	C₂²	1	1	yes	characteristic, complemented
22	C₃²×#131	C₂²	1	1	yes	characteristic, complemented
23	C₃×#950	—	2	2	no	—
24	C₃×#950	C₂²	1	1	yes	characteristic
25	C₃×#949	—	2	2	no	—
26	C₃×#949	C₂²	1	2	yes	complemented
27	C₃×#949	C₂²	1	2	yes	complemented
28	C₃×#948	C₂²	1	2	yes	complemented
29	C₃×#940	C₂²	1	2	yes	complemented
30	C₃×#932	—	2	2	no	—
31	C₃×#932	C₂²	1	2	yes	complemented
32	C₂×C₃×#213	—	2	2	no	—
33	C₂×C₃×#213	C₂²	1	2	yes	complemented
34	C₃×#919	—	2	1	no	—
35	C₃×#914	—	2	2	no	—
36	C₃×D₃×D₈	—	2	1	no	—
37	C₂×C₃×D₂₄	C₂²	1	1	yes	characteristic, complemented
38	C₃×#901	C₂²	1	1	yes	characteristic, complemented
39	C₃×#837	C₂²	1	2	yes	complemented
40	C₃×#837	C₂²	1	1	yes	characteristic, complemented
41	C₃×#833	C₂²	1	2	yes	complemented
42	C₃×#825	C₂²	1	1	yes	characteristic, complemented
43	C₃×#823	C₂²	1	2	yes	complemented
44	C₃×#822	C₂²	1	1	yes	characteristic, complemented
45	C₃×#819	C₂²	1	1	yes	characteristic, complemented
46	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
47	C₃×#576	—	2	1	no	—
48	C₃×#576	—	3	2	no	—
49	C₃×#576	—	3	1	no	—
50	C₃×#576	D₃	1	1	yes	characteristic, complemented
51	C₃×#576	C₂×C₃	1	1	yes	characteristic, complemented
52	#5835	C₂×C₃	1	2	yes	complemented
53	#5769	C₂×C₃	1	1	yes	characteristic, complemented
54	C₃×#493	—	3	1	no	—
55	C₃×#472	—	3	2	no	—
56	C₃×#472	—	3	1	no	—
57	C₃×#472	—	3	1	no	—
58	C₃×#453	—	3	2	no	—
59	C₃×#453	—	3	2	no	—
60	C₃×#443	—	3	1	no	—
61	C₃×#443	—	3	1	no	—
62	#5241	C₂×C₃	1	2	yes	complemented
63	#5199	C₂×C₃	1	2	yes	complemented
64	#5198	C₂×C₃	1	2	yes	complemented
65	#5196	C₂×C₃	1	1	yes	characteristic, complemented
66	#5160	C₂×C₃	1	1	yes	characteristic, complemented
67	#5151	C₂×C₃	1	1	yes	characteristic, complemented
68	#5144	C₂×C₃	1	1	yes	characteristic, complemented
69	#5142	C₂×C₃	1	1	yes	characteristic, complemented
70	C₃²×#41	—	2	2	no	—
71	C₃²×#41	—	2	2	no	—
72	C₃²×#41	C₂³	1	2	yes	—
73	C₃²×#41	C₂³	1	1	yes	characteristic
74	C₂×C₃²×D₄	—	2	2	no	—
75	C₂×C₃²×D₄	—	2	2	no	—
76	C₂×C₃²×D₄	—	4	2	no	—
77	C₂×C₃²×D₄	C₂³	1	2	yes	—
78	C₂×C₃×#67	—	4	2	no	—
79	C₃×#239	—	2	2	no	—
80	C₃×D₃×D₄	—	2	2	no	—
81	C₃×D₃×D₄	—	4	2	no	—
82	C₃×#235	—	2	2	no	—
83	C₃×#235	C₂³	1	1	yes	characteristic
84	C₂×C₃×D₁₂	—	2	2	no	—
85	C₂×C₃×D₁₂	C₂³	1	2	yes	—
86	C₃²×Q₁₆	D₄	1	1	yes	characteristic, complemented
87	C₃²×SD₁₆	—	2	2	no	—
88	C₃²×SD₁₆	D₄	1	2	yes	complemented
89	C₃²×D₈	—	2	2	no	—
90	C₃²×D₈	—	2	1	no	—
91	C₃²×D₈	—	2	1	no	—
92	C₃²×D₈	D₄	1	1	yes	characteristic, complemented
93	C₃²×M₁₆	—	2	1	no	—
94	C₃²×M₁₆	C₂³	1	1	yes	characteristic
95	C₂×C₃²×C₈	—	2	1	no	—
96	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
97	C₃×#216	D₄	1	1	yes	characteristic, complemented
98	C₃×#215	—	2	2	no	—
99	C₃×#215	—	2	2	no	—
100	C₃×#215	D₄	1	1	yes	characteristic, complemented
101	C₃×#214	—	2	2	no	—
102	C₃×#214	D₄	1	1	yes	characteristic, complemented
103	C₃×#213	—	2	2	no	—
104	C₃×#213	—	2	2	no	—
105	C₃×#213	—	2	2	no	—
106	C₃×#213	—	2	2	no	—
107	C₃×#213	—	2	2	no	—
108	C₃×#213	D₄	1	1	yes	characteristic, complemented
109	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
110	C₃×#208	—	2	2	no	—
111	C₃×#208	C₂³	1	1	yes	characteristic
112	C₃×#208	C₂³	1	2	yes	—
113	C₂×C₃×C₃⋊C₈	—	2	2	no	—
114	C₂×C₃×C₃⋊C₈	C₂³	1	1	yes	characteristic
115	C₃×D₂₄	—	2	1	no	—
116	C₃×D₂₄	—	2	1	no	—
117	C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
118	C₃×D₃×C₈	—	2	1	no	—
119	#3292	—	3	1	no	Sylow
120	C₃×H(5,𝔽₂)	—	2	2	no	—
121	C₃×H(5,𝔽₂)	—	3	2	no	—
122	C₃×H(5,𝔽₂)	C₂×D₃	1	2	yes	complemented
123	C₃×H(5,𝔽₂)	C₂²×C₃	1	2	yes	complemented
124	#1009	C₂²×C₃	1	2	yes	complemented
125	C₃×AGL(1,ℤ/8)	—	2	4	no	—
126	C₃×AGL(1,ℤ/8)	—	2	2	no	—
127	C₃×AGL(1,ℤ/8)	—	2	2	no	—
128	C₃×AGL(1,ℤ/8)	—	2	2	no	—
129	C₃×AGL(1,ℤ/8)	—	3	2	no	—
130	C₃×AGL(1,ℤ/8)	—	3	2	no	—
131	C₃×AGL(1,ℤ/8)	—	3	2	no	—
132	C₃×AGL(1,ℤ/8)	—	6	2	no	—
133	C₃×AGL(1,ℤ/8)	—	6	2	no	—
134	C₃×AGL(1,ℤ/8)	—	6	2	no	—
135	C₃×AGL(1,ℤ/8)	C₂×D₃	1	2	yes	complemented
136	C₃×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
137	C₃×#135	—	2	2	no	—
138	C₃×#135	—	2	1	no	—
139	C₃×#135	—	2	1	no	—
140	C₃×#135	—	2	1	no	—
141	C₃×#135	—	3	1	no	—
142	C₃×#135	—	6	2	no	—
143	C₃×#135	—	6	1	no	—
144	C₃×#135	C₂×D₃	1	1	yes	characteristic
145	C₃×#135	C₂²×C₃	1	1	yes	characteristic
146	C₂×C₃×D₈	—	2	2	no	—
147	C₂×C₃×D₈	—	2	1	no	—
148	C₂×C₃×D₈	—	2	1	no	—
149	C₂×C₃×D₈	—	2	1	no	—
150	C₂×C₃×D₈	—	3	2	no	—
151	C₂×C₃×D₈	—	3	1	no	—
152	C₂×C₃×D₈	—	6	2	no	—
153	C₂×C₃×D₈	—	6	1	no	—
154	C₂×C₃×D₈	C₂×D₃	1	1	yes	characteristic, complemented
155	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
156	C₃×#131	—	2	1	no	—
157	C₃×#131	—	3	2	no	—
158	C₃×#131	—	3	1	no	—
159	C₃×#131	C₂×D₃	1	1	yes	characteristic, complemented
160	C₃×#131	C₂²×C₃	1	1	yes	characteristic, complemented
161	C₃×#127	—	3	2	no	—
162	#950	—	2	2	no	—
163	#950	C₂²×C₃	1	1	yes	characteristic
164	#949	—	2	2	no	—
165	#949	C₂²×C₃	1	2	yes	complemented
166	#949	C₂²×C₃	1	2	yes	complemented
167	#948	C₂²×C₃	1	2	yes	complemented
168	#940	C₂²×C₃	1	2	yes	complemented
169	#932	—	2	2	no	—
170	#932	C₂²×C₃	1	2	yes	complemented
171	C₂×#213	—	2	2	no	—
172	C₂×#213	C₂²×C₃	1	2	yes	complemented
173	#919	—	2	1	no	—
174	#914	—	2	2	no	—
175	D₃×D₈	—	2	1	no	—
176	C₂×D₂₄	C₂²×C₃	1	1	yes	characteristic, complemented
177	#901	C₂²×C₃	1	1	yes	characteristic, complemented
178	C₃×#108	—	3	1	no	—
179	C₃×#108	—	3	1	no	—
180	C₃×#104	—	3	1	no	—
181	C₃×#104	—	3	2	no	—
182	C₃×#104	—	3	1	no	—
183	C₃×#100	—	3	2	no	—
184	C₃×#100	—	3	2	no	—
185	C₃×C₄×C₈	—	3	1	no	—
186	#837	C₂²×C₃	1	1	yes	characteristic, complemented
187	#837	C₂²×C₃	1	2	yes	complemented
188	#833	C₂²×C₃	1	2	yes	complemented
189	#825	C₂²×C₃	1	1	yes	characteristic, complemented
190	#823	C₂²×C₃	1	2	yes	complemented
191	#822	C₂²×C₃	1	1	yes	characteristic, complemented
192	#819	C₂²×C₃	1	1	yes	characteristic, complemented
193	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic, complemented
194	C₂³×C₃²	—	2	2	no	—
195	C₂³×C₃²	—	4	2	no	—
196	C₂²×C₃×D₃	—	2	2	no	—
197	C₂²×C₃×D₃	—	4	2	no	—
198	C₃²×Q₈	C₂×D₄	1	1	yes	characteristic
199	C₃²×Q₈	C₂×D₄	1	2	yes	—
200	C₃²×D₄	—	2	1	no	—
201	C₃²×D₄	—	2	2	no	—
202	C₃²×D₄	—	2	2	no	—
203	C₃²×D₄	—	2	2	no	—
204	C₃²×D₄	—	4	2	no	—
205	C₃²×D₄	—	4	2	no	—
206	C₃²×D₄	C₂×D₄	1	1	yes	characteristic
207	C₃²×D₄	C₂×D₄	1	2	yes	—
208	C₂×C₃²×C₄	—	2	1	no	—
209	C₂×C₃²×C₄	—	2	2	no	—
210	C₂×C₃²×C₄	—	4	2	no	—
211	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
212	C₃×#67	—	2	1	no	—
213	C₃×#67	—	4	2	no	—
214	C₃×#67	—	8	4	no	—
215	C₂×C₃×Dic₃	—	2	1	no	—
216	C₃×D₁₂	—	2	2	no	—
217	C₃×D₁₂	—	2	2	no	—
218	C₃×D₁₂	—	4	2	no	—
219	C₃×D₁₂	C₂×D₄	1	1	yes	characteristic
220	C₃×C₄×D₃	—	2	1	no	—
221	C₃×C₄×D₃	—	4	2	no	—
222	C₃×Dic₆	C₂×D₄	1	1	yes	characteristic
223	C₃²×C₈	—	2	1	no	—
224	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
225	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
226	C₃×C₃⋊C₈	—	2	1	no	—
227	C₃×C₃⋊C₈	—	2	2	no	—
228	C₃×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
229	C₃×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
230	#576	—	3	2	no	—
231	#576	—	3	1	no	—
232	#576	C₃×D₃	1	1	yes	characteristic, complemented
233	#493	—	3	1	no	—
234	#472	—	3	2	no	—
235	#472	—	3	1	no	—
236	#472	—	3	1	no	—
237	#453	—	3	2	no	—
238	#453	—	3	2	no	—
239	#443	—	3	1	no	—
240	#443	—	3	1	no	—
241	C₃×#41	—	2	4	no	—
242	C₃×#41	—	2	2	no	—
243	C₃×#41	—	2	2	no	—
244	C₃×#41	—	2	2	no	—
245	C₃×#41	—	2	2	no	—
246	C₃×#41	—	2	1	no	—
247	C₃×#41	—	2	2	no	—
248	C₃×#41	—	3	1	no	—
249	C₃×#41	—	4	2	no	—
250	C₃×#41	—	6	2	no	—
251	C₃×#41	—	6	2	no	—
252	C₃×#41	C₂²×D₃	1	1	yes	characteristic
253	C₃×#41	C₂²×D₃	1	2	yes	—
254	C₃×#41	C₂³×C₃	1	2	yes	—
255	C₃×#41	C₂³×C₃	1	1	yes	characteristic
256	C₂×C₃×D₄	—	2	4	no	—
257	C₂×C₃×D₄	—	2	2	no	—
258	C₂×C₃×D₄	—	2	2	no	—
259	C₂×C₃×D₄	—	2	2	no	—
260	C₂×C₃×D₄	—	2	2	no	—
261	C₂×C₃×D₄	—	2	2	no	—
262	C₂×C₃×D₄	—	3	2	no	—
263	C₂×C₃×D₄	—	4	2	no	—
264	C₂×C₃×D₄	—	4	2	no	—
265	C₂×C₃×D₄	—	4	2	no	—
266	C₂×C₃×D₄	—	4	4	no	—
267	C₂×C₃×D₄	—	6	2	no	—
268	C₂×C₃×D₄	—	6	2	no	—
269	C₂×C₃×D₄	—	12	2	no	—
270	C₂×C₃×D₄	—	12	2	no	—
271	C₂×C₃×D₄	C₂²×D₃	1	2	yes	—
272	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
273	C₂×#67	—	4	2	no	—
274	#239	—	2	2	no	—
275	D₃×D₄	—	2	2	no	—
276	D₃×D₄	—	4	2	no	—
277	#235	—	2	2	no	—
278	#235	C₂³×C₃	1	1	yes	characteristic
279	C₂×D₁₂	—	2	2	no	—
280	C₂×D₁₂	C₂³×C₃	1	2	yes	—
281	C₃×Q₁₆	—	2	1	no	—
282	C₃×Q₁₆	—	3	1	no	—
283	C₃×Q₁₆	#67	1	1	yes	characteristic, complemented
284	C₃×Q₁₆	C₃×D₄	1	1	yes	characteristic, complemented
285	C₃×SD₁₆	—	2	4	no	—
286	C₃×SD₁₆	—	2	2	no	—
287	C₃×SD₁₆	—	2	2	no	—
288	C₃×SD₁₆	—	2	2	no	—
289	C₃×SD₁₆	—	3	1	no	—
290	C₃×SD₁₆	—	3	1	no	—
291	C₃×SD₁₆	—	6	2	no	—
292	C₃×SD₁₆	—	6	2	no	—
293	C₃×SD₁₆	—	6	2	no	—
294	C₃×SD₁₆	#67	1	2	yes	complemented
295	C₃×SD₁₆	C₃×D₄	1	2	yes	complemented
296	C₃×D₈	—	2	2	no	—
297	C₃×D₈	—	2	4	no	—
298	C₃×D₈	—	2	2	no	—
299	C₃×D₈	—	2	2	no	—
300	C₃×D₈	—	2	2	no	—
301	C₃×D₈	—	2	1	no	—
302	C₃×D₈	—	2	1	no	—
303	C₃×D₈	—	2	1	no	—
304	C₃×D₈	—	2	1	no	—
305	C₃×D₈	—	2	1	no	—
306	C₃×D₈	—	3	1	no	—
307	C₃×D₈	—	6	2	no	—
308	C₃×D₈	—	6	2	no	—
309	C₃×D₈	—	6	2	no	—
310	C₃×D₈	—	6	2	no	—
311	C₃×D₈	—	6	2	no	—
312	C₃×D₈	—	6	1	no	—
313	C₃×D₈	—	6	1	no	—
314	C₃×D₈	#67	1	1	yes	characteristic, complemented
315	C₃×D₈	C₃×D₄	1	1	yes	characteristic, complemented
316	C₃×M₁₆	—	2	2	no	—
317	C₃×M₁₆	—	2	1	no	—
318	C₃×M₁₆	—	2	1	no	—
319	C₃×M₁₆	—	2	1	no	—
320	C₃×M₁₆	—	3	2	no	—
321	C₃×M₁₆	—	3	1	no	—
322	C₃×M₁₆	—	6	2	no	—
323	C₃×M₁₆	—	6	1	no	—
324	C₃×M₁₆	C₂²×D₃	1	1	yes	characteristic
325	C₃×M₁₆	C₂³×C₃	1	1	yes	characteristic
326	C₂×C₃×C₈	—	2	2	no	—
327	C₂×C₃×C₈	—	2	1	no	—
328	C₂×C₃×C₈	—	2	1	no	—
329	C₂×C₃×C₈	—	2	1	no	—
330	C₂×C₃×C₈	—	3	1	no	—
331	C₂×C₃×C₈	—	6	2	no	—
332	C₂×C₃×C₈	—	6	1	no	—
333	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
334	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
335	C₃×C₄²	—	3	1	no	—
336	#216	C₃×D₄	1	1	yes	characteristic, complemented
337	#215	—	2	2	no	—
338	#215	—	2	2	no	—
339	#215	C₃×D₄	1	1	yes	characteristic, complemented
340	#214	—	2	2	no	—
341	#214	C₃×D₄	1	1	yes	characteristic, complemented
342	#213	—	2	2	no	—
343	#213	—	2	2	no	—
344	#213	—	2	2	no	—
345	#213	—	2	2	no	—
346	#213	—	2	2	no	—
347	#213	C₃×D₄	1	1	yes	characteristic, complemented
348	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
349	#208	—	2	2	no	—
350	#208	C₂³×C₃	1	2	yes	—
351	#208	C₂³×C₃	1	1	yes	characteristic
352	C₂×C₃⋊C₈	—	2	2	no	—
353	C₂×C₃⋊C₈	C₂³×C₃	1	1	yes	characteristic
354	D₂₄	—	2	1	no	—
355	D₂₄	—	2	1	no	—
356	ΓL(1,𝔽₂₅)	—	2	1	no	—
357	D₃×C₈	—	2	1	no	—
358	C₂²×C₃²	—	2	1	no	—
359	C₂²×C₃²	—	2	2	no	—
360	C₂²×C₃²	—	4	2	no	—
361	C₂²×C₃²	—	8	2	no	—
362	C₂²×C₃²	—	16	2	no	—
363	C₂²×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
364	C₂×C₃×D₃	—	2	1	no	—
365	C₂×C₃×D₃	—	4	2	no	—
366	C₂×C₃×D₃	—	8	2	no	—
367	C₂×C₃×D₃	—	16	2	no	—
368	C₃²×C₄	—	2	1	no	—
369	C₃²×C₄	—	2	2	no	—
370	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
371	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
372	C₃×Dic₃	—	2	2	no	—
373	C₃×Dic₃	—	2	1	no	—
374	H(5,𝔽₂)	—	3	2	no	—
375	H(5,𝔽₂)	C₂×C₃×D₃	1	2	yes	complemented
376	AGL(1,ℤ/8)	—	2	2	no	—
377	AGL(1,ℤ/8)	—	3	2	no	—
378	AGL(1,ℤ/8)	—	3	2	no	—
379	AGL(1,ℤ/8)	—	3	2	no	—
380	AGL(1,ℤ/8)	—	6	2	no	—
381	AGL(1,ℤ/8)	—	6	2	no	—
382	AGL(1,ℤ/8)	—	6	2	no	—
383	AGL(1,ℤ/8)	C₂×C₃×D₃	1	2	yes	complemented
384	#135	—	2	1	no	—
385	#135	—	3	1	no	—
386	#135	—	6	2	no	—
387	#135	—	6	1	no	—
388	#135	C₂×C₃×D₃	1	1	yes	characteristic
389	C₂×D₈	—	2	1	no	—
390	C₂×D₈	—	3	2	no	—
391	C₂×D₈	—	3	1	no	—
392	C₂×D₈	—	6	2	no	—
393	C₂×D₈	—	6	1	no	—
394	C₂×D₈	C₂×C₃×D₃	1	1	yes	characteristic, complemented
395	#131	—	3	2	no	—
396	#131	—	3	1	no	—
397	#131	C₂×C₃×D₃	1	1	yes	characteristic, complemented
398	#127	—	3	2	no	—
399	#108	—	3	1	no	—
400	#108	—	3	1	no	—
401	#104	—	3	1	no	—
402	#104	—	3	2	no	—
403	#104	—	3	1	no	—
404	#100	—	3	2	no	—
405	#100	—	3	2	no	—
406	C₄×C₈	—	3	1	no	—
407	C₂³×C₃	—	2	2	no	—
408	C₂³×C₃	—	2	2	no	—
409	C₂³×C₃	—	4	2	no	—
410	C₂³×C₃	—	4	2	no	—
411	C₂³×C₃	—	4	2	no	—
412	C₂³×C₃	—	4	4	no	—
413	C₂³×C₃	—	6	2	no	—
414	C₂³×C₃	—	12	2	no	—
415	C₂²×D₃	—	2	2	no	—
416	C₂²×D₃	—	4	2	no	—
417	C₃×Q₈	—	2	1	no	—
418	C₃×Q₈	—	2	2	no	—
419	C₃×Q₈	—	3	1	no	—
420	C₃×Q₈	D₃×D₄	1	1	yes	characteristic
421	C₃×Q₈	C₂×#67	1	2	yes	—
422	C₃×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
423	C₃×Q₈	C₂×C₃×D₄	1	2	yes	—
424	C₃×D₄	—	2	2	no	—
425	C₃×D₄	—	2	1	no	—
426	C₃×D₄	—	2	2	no	—
427	C₃×D₄	—	2	2	no	—
428	C₃×D₄	—	2	1	no	—
429	C₃×D₄	—	2	4	no	—
430	C₃×D₄	—	2	2	no	—
431	C₃×D₄	—	2	2	no	—
432	C₃×D₄	—	2	2	no	—
433	C₃×D₄	—	2	2	no	—
434	C₃×D₄	—	2	1	no	—
435	C₃×D₄	—	2	2	no	—
436	C₃×D₄	—	3	1	no	—
437	C₃×D₄	—	4	2	no	—
438	C₃×D₄	—	4	2	no	—
439	C₃×D₄	—	4	2	no	—
440	C₃×D₄	—	4	2	no	—
441	C₃×D₄	—	4	2	no	—
442	C₃×D₄	—	4	2	no	—
443	C₃×D₄	—	4	4	no	—
444	C₃×D₄	—	4	4	no	—
445	C₃×D₄	—	6	1	no	—
446	C₃×D₄	—	6	2	no	—
447	C₃×D₄	—	6	2	no	—
448	C₃×D₄	—	12	2	no	—
449	C₃×D₄	—	12	2	no	—
450	C₃×D₄	—	24	4	no	—
451	C₃×D₄	D₃×D₄	1	1	yes	characteristic
452	C₃×D₄	C₂×#67	1	2	yes	—
453	C₃×D₄	C₂×C₃×D₄	1	1	yes	characteristic
454	C₃×D₄	C₂×C₃×D₄	1	2	yes	—
455	C₂×C₃×C₄	—	2	2	no	—
456	C₂×C₃×C₄	—	2	2	no	—
457	C₂×C₃×C₄	—	2	1	no	—
458	C₂×C₃×C₄	—	2	2	no	—
459	C₂×C₃×C₄	—	2	1	no	—
460	C₂×C₃×C₄	—	2	1	no	—
461	C₂×C₃×C₄	—	4	2	no	—
462	C₂×C₃×C₄	—	4	2	no	—
463	C₂×C₃×C₄	—	4	2	no	—
464	C₂×C₃×C₄	—	4	4	no	—
465	C₂×C₃×C₄	—	6	1	no	—
466	C₂×C₃×C₄	—	6	1	no	—
467	C₂×C₃×C₄	—	12	2	no	—
468	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
469	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
470	#67	—	2	1	no	—
471	#67	—	4	2	no	—
472	#67	—	8	4	no	—
473	C₂×Dic₃	—	2	1	no	—
474	D₁₂	—	2	2	no	—
475	D₁₂	—	2	2	no	—
476	D₁₂	—	4	2	no	—
477	D₁₂	C₂×C₃×D₄	1	1	yes	characteristic
478	C₄×D₃	—	2	1	no	—
479	C₄×D₃	—	4	2	no	—
480	Dic₆	C₂×C₃×D₄	1	1	yes	characteristic
481	C₃×C₈	—	2	2	no	—
482	C₃×C₈	—	2	1	no	—
483	C₃×C₈	—	2	1	no	—
484	C₃×C₈	—	2	1	no	—
485	C₃×C₈	—	2	1	no	—
486	C₃×C₈	—	3	1	no	—
487	C₃×C₈	—	3	1	no	—
488	C₃×C₈	—	6	2	no	—
489	C₃×C₈	—	6	1	no	—
490	C₃×C₈	C₂×#67	1	1	yes	characteristic
491	C₃×C₈	C₂×#67	1	1	yes	characteristic
492	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
493	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
494	C₃⋊C₈	—	2	1	no	—
495	C₃⋊C₈	—	2	2	no	—
496	C₃⋊C₈	C₂×C₃×D₄	1	1	yes	characteristic
497	C₃⋊C₈	C₂×C₃×D₄	1	1	yes	characteristic
498	C₂×C₃²	—	2	1	no	—
499	C₂×C₃²	—	4	1	no	—
500	C₂×C₃²	—	4	2	no	—
501	C₂×C₃²	—	8	2	no	—
502	C₂×C₃²	D₄²	1	1	yes	characteristic
503	C₃×D₃	—	4	1	no	—
504	C₃×D₃	—	8	2	no	—
505	#41	—	2	2	no	—
506	#41	—	2	2	no	—
507	#41	—	3	1	no	—
508	#41	—	6	2	no	—
509	#41	—	6	2	no	—
510	#41	C₂²×C₃×D₃	1	1	yes	characteristic
511	#41	C₂²×C₃×D₃	1	2	yes	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	2	no	—
514	C₂×D₄	—	3	2	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	6	2	no	—
517	C₂×D₄	—	6	2	no	—
518	C₂×D₄	—	12	2	no	—
519	C₂×D₄	—	12	2	no	—
520	C₂×D₄	C₂²×C₃×D₃	1	2	yes	—
521	Q₁₆	—	3	1	no	—
522	Q₁₆	C₃×#67	1	1	yes	characteristic, complemented
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	3	1	no	—
525	SD₁₆	—	3	1	no	—
526	SD₁₆	—	6	2	no	—
527	SD₁₆	—	6	2	no	—
528	SD₁₆	—	6	2	no	—
529	SD₁₆	C₃×#67	1	2	yes	complemented
530	D₈	—	2	2	no	—
531	D₈	—	2	1	no	—
532	D₈	—	2	1	no	—
533	D₈	—	3	1	no	—
534	D₈	—	6	2	no	—
535	D₈	—	6	2	no	—
536	D₈	—	6	2	no	—
537	D₈	—	6	2	no	—
538	D₈	—	6	2	no	—
539	D₈	—	6	1	no	—
540	D₈	—	6	1	no	—
541	D₈	C₃×#67	1	1	yes	characteristic, complemented
542	M₁₆	—	2	1	no	—
543	M₁₆	—	3	2	no	—
544	M₁₆	—	3	1	no	—
545	M₁₆	—	6	2	no	—
546	M₁₆	—	6	1	no	—
547	M₁₆	C₂²×C₃×D₃	1	1	yes	characteristic
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	3	1	no	—
550	C₂×C₈	—	6	2	no	—
551	C₂×C₈	—	6	1	no	—
552	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
553	C₄²	—	3	1	no	—
554	C₂²×C₃	—	2	2	no	—
555	C₂²×C₃	—	2	1	no	—
556	C₂²×C₃	—	2	2	no	—
557	C₂²×C₃	—	2	2	no	—
558	C₂²×C₃	—	2	1	no	—
559	C₂²×C₃	—	2	1	no	—
560	C₂²×C₃	—	4	2	no	—
561	C₂²×C₃	—	4	4	no	—
562	C₂²×C₃	—	4	2	no	—
563	C₂²×C₃	—	4	2	no	—
564	C₂²×C₃	—	6	1	no	—
565	C₂²×C₃	—	8	2	no	—
566	C₂²×C₃	—	8	2	no	—
567	C₂²×C₃	—	12	2	no	—
568	C₂²×C₃	—	16	2	no	—
569	C₂²×C₃	—	16	2	no	—
570	C₂²×C₃	—	16	2	no	—
571	C₂²×C₃	—	16	4	no	—
572	C₂²×C₃	—	24	2	no	—
573	C₂²×C₃	—	48	2	no	—
574	C₂²×C₃	#1004	1	1	yes	characteristic
575	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
576	C₂×D₃	—	2	1	no	—
577	C₂×D₃	—	4	2	no	—
578	C₂×D₃	—	8	2	no	—
579	C₂×D₃	—	16	2	no	—
580	C₃×C₄	—	2	2	no	—
581	C₃×C₄	—	2	1	no	—
582	C₃×C₄	—	2	2	no	—
583	C₃×C₄	—	2	2	no	—
584	C₃×C₄	—	2	1	no	—
585	C₃×C₄	—	2	1	no	—
586	C₃×C₄	—	2	1	no	—
587	C₃×C₄	—	4	2	no	—
588	C₃×C₄	—	6	2	no	—
589	C₃×C₄	—	6	1	no	—
590	C₃×C₄	C₂×D₃×D₄	1	1	yes	characteristic
591	C₃×C₄	C₂²×#67	1	1	yes	characteristic
592	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
593	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
594	Dic₃	—	2	2	no	—
595	Dic₃	—	2	1	no	—
596	C₃²	#3292	1	1	yes	characteristic, complemented, Sylow
597	C₂³	—	2	2	no	—
598	C₂³	—	4	2	no	—
599	C₂³	—	6	2	no	—
600	C₂³	—	12	2	no	—
601	Q₈	—	3	1	no	—
602	Q₈	C₃×D₃×D₄	1	1	yes	characteristic
603	Q₈	C₂×C₃×#67	1	2	yes	—
604	D₄	—	2	1	no	—
605	D₄	—	2	2	no	—
606	D₄	—	2	2	no	—
607	D₄	—	2	2	no	—
608	D₄	—	3	1	no	—
609	D₄	—	4	2	no	—
610	D₄	—	4	2	no	—
611	D₄	—	6	1	no	—
612	D₄	—	6	2	no	—
613	D₄	—	6	2	no	—
614	D₄	—	12	2	no	—
615	D₄	—	12	2	no	—
616	D₄	—	24	4	no	—
617	D₄	C₃×D₃×D₄	1	1	yes	characteristic
618	D₄	C₂×C₃×#67	1	2	yes	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	—	6	1	no	—
623	C₂×C₄	—	6	1	no	—
624	C₂×C₄	—	12	2	no	—
625	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
626	C₈	—	2	1	no	—
627	C₈	—	3	1	no	—
628	C₈	—	3	1	no	—
629	C₈	—	6	2	no	—
630	C₈	—	6	1	no	—
631	C₈	C₂×C₃×#67	1	1	yes	characteristic
632	C₈	C₂×C₃×#67	1	1	yes	characteristic
633	C₂×C₃	—	2	1	no	—
634	C₂×C₃	—	2	1	no	—
635	C₂×C₃	—	2	1	no	—
636	C₂×C₃	—	4	1	no	—
637	C₂×C₃	—	4	2	no	—
638	C₂×C₃	—	4	1	no	—
639	C₂×C₃	—	4	2	no	—
640	C₂×C₃	—	4	2	no	—
641	C₂×C₃	—	4	1	no	—
642	C₂×C₃	—	8	2	no	—
643	C₂×C₃	—	8	4	no	—
644	C₂×C₃	—	8	2	no	—
645	C₂×C₃	—	8	2	no	—
646	C₂×C₃	—	12	1	no	—
647	C₂×C₃	—	24	2	no	—
648	C₂×C₃	D₄×#67	1	1	yes	characteristic
649	C₂×C₃	C₃×D₄²	1	1	yes	characteristic
650	D₃	—	4	1	no	—
651	D₃	—	8	2	no	—
652	C₂²	—	2	1	no	—
653	C₂²	—	2	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	6	1	no	—
656	C₂²	—	8	2	no	—
657	C₂²	—	12	2	no	—
658	C₂²	—	16	2	no	—
659	C₂²	—	24	2	no	—
660	C₂²	—	48	2	no	—
661	C₂²	C₃×#1004	1	1	yes	characteristic
662	C₄	—	2	1	no	—
663	C₄	—	2	2	no	—
664	C₄	—	6	2	no	—
665	C₄	—	6	1	no	—
666	C₄	C₂×C₃×D₃×D₄	1	1	yes	characteristic
667	C₄	C₂²×C₃×#67	1	1	yes	characteristic
668	C₃	—	2	1	no	—
669	C₃	#84010	1	1	yes	characteristic, direct product factor
670	C₃	C₃×#3292	1	1	yes	characteristic, complemented
671	C₂	—	2	1	no	—
672	C₂	—	4	1	no	—
673	C₂	—	4	2	no	—
674	C₂	—	8	2	no	—
675	C₂	—	12	1	no	—
676	C₂	—	24	2	no	—
677	C₂	C₃×D₄×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	D₄×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#576	C₂
socle	C₂×C₃²	D₄²
hypercenter	C₃×#576	D₃
hypocenter	C₃	C₃×#3292
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#576

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃²×#576	C₂	semidirect
2. C₃×#5835	C₂	semidirect
3. C₃×#5769	C₂	semidirect
4. C₃×#5241	C₂	semidirect
5. C₃×#5199	C₂	semidirect
6. C₃×#5198	C₂	semidirect
7. C₃×#5196	C₂	semidirect
8. C₃×#5160	C₂	semidirect
9. C₃×#5151	C₂	semidirect
10. C₃×#5144	C₂	semidirect
11. C₃×#5142	C₂	semidirect
13. #84010	C₃	direct (N × H)
14. C₃²×H(5,𝔽₂)	C₂²	semidirect
15. C₃×#1009	C₂²	semidirect
17. C₃²×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂×C₃²×D₈	C₂²	semidirect
22. C₃²×#131	C₂²	semidirect
26. C₃×#949	C₂²	semidirect
27. C₃×#949	C₂²	semidirect
28. C₃×#948	C₂²	semidirect
29. C₃×#940	C₂²	semidirect
31. C₃×#932	C₂²	semidirect
33. C₂×C₃×#213	C₂²	semidirect
37. C₂×C₃×D₂₄	C₂²	semidirect
38. C₃×#901	C₂²	semidirect
39. C₃×#837	C₂²	semidirect
40. C₃×#837	C₂²	semidirect
41. C₃×#833	C₂²	semidirect
42. C₃×#825	C₂²	semidirect
43. C₃×#823	C₂²	semidirect
44. C₃×#822	C₂²	semidirect
45. C₃×#819	C₂²	semidirect
46. C₃×C₈×Dic₃	C₂²	semidirect
50. C₃×#576	D₃	semidirect
51. C₃×#576	C₂×C₃	semidirect
52. #5835	C₂×C₃	semidirect
53. #5769	C₂×C₃	semidirect
62. #5241	C₂×C₃	semidirect
63. #5199	C₂×C₃	semidirect
64. #5198	C₂×C₃	semidirect
65. #5196	C₂×C₃	semidirect
66. #5160	C₂×C₃	semidirect
67. #5151	C₂×C₃	semidirect
68. #5144	C₂×C₃	semidirect
69. #5142	C₂×C₃	semidirect
86. C₃²×Q₁₆	D₄	semidirect
88. C₃²×SD₁₆	D₄	semidirect
92. C₃²×D₈	D₄	semidirect
97. C₃×#216	D₄	semidirect
100. C₃×#215	D₄	semidirect
102. C₃×#214	D₄	semidirect
108. C₃×#213	D₄	semidirect
122. C₃×H(5,𝔽₂)	C₂×D₃	semidirect
123. C₃×H(5,𝔽₂)	C₂²×C₃	semidirect
124. #1009	C₂²×C₃	semidirect
135. C₃×AGL(1,ℤ/8)	C₂×D₃	semidirect
136. C₃×AGL(1,ℤ/8)	C₂²×C₃	semidirect
154. C₂×C₃×D₈	C₂×D₃	semidirect
155. C₂×C₃×D₈	C₂²×C₃	semidirect
159. C₃×#131	C₂×D₃	semidirect
160. C₃×#131	C₂²×C₃	semidirect
165. #949	C₂²×C₃	semidirect
166. #949	C₂²×C₃	semidirect
167. #948	C₂²×C₃	semidirect
168. #940	C₂²×C₃	semidirect
170. #932	C₂²×C₃	semidirect
172. C₂×#213	C₂²×C₃	semidirect
176. C₂×D₂₄	C₂²×C₃	semidirect
177. #901	C₂²×C₃	semidirect
186. #837	C₂²×C₃	semidirect
187. #837	C₂²×C₃	semidirect
188. #833	C₂²×C₃	semidirect
189. #825	C₂²×C₃	semidirect
190. #823	C₂²×C₃	semidirect
191. #822	C₂²×C₃	semidirect
192. #819	C₂²×C₃	semidirect
193. C₈×Dic₃	C₂²×C₃	semidirect
232. #576	C₃×D₃	semidirect
283. C₃×Q₁₆	#67	semidirect
284. C₃×Q₁₆	C₃×D₄	semidirect
294. C₃×SD₁₆	#67	semidirect
295. C₃×SD₁₆	C₃×D₄	semidirect
314. C₃×D₈	#67	semidirect
315. C₃×D₈	C₃×D₄	semidirect
336. #216	C₃×D₄	semidirect
339. #215	C₃×D₄	semidirect
341. #214	C₃×D₄	semidirect
347. #213	C₃×D₄	semidirect
375. H(5,𝔽₂)	C₂×C₃×D₃	semidirect
383. AGL(1,ℤ/8)	C₂×C₃×D₃	semidirect
394. C₂×D₈	C₂×C₃×D₃	semidirect
397. #131	C₂×C₃×D₃	semidirect
522. Q₁₆	C₃×#67	semidirect
529. SD₁₆	C₃×#67	semidirect
541. D₈	C₃×#67	semidirect
596. C₃²	#3292	semidirect
669. C₃	#84010	direct (N × H)
670. C₃	C₃×#3292	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×#576	2
C₃×#5835	2
C₃×#5769	2
C₃×#5241	2
C₃×#5199	2
C₃×#5198	2
C₃×#5196	2
C₃×#5160	2
C₃×#5151	2
C₃×#5144	2
C₃×#5142	2
C₃×#3292	3
#84010	3

Subgroups — C3×#84010 #11886896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#84010 #11886896