Subgroups — C₃×#22462 #10641069

Order 768.

1320 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3584	C₂	1	1	yes	maximal, characteristic
2	C₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2901	C₂	1	2	yes	maximal, complemented
5	C₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1906	C₂	1	2	yes	maximal, complemented
8	C₃×#1881	C₂	1	2	yes	maximal, complemented
9	C₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#1794	C₂	1	2	yes	maximal, complemented
11	C₃×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	#22462	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#585	C₂²	1	2	yes	—
14	C₃×#578	—	2	1	no	—
15	C₃×#578	C₄	1	2	yes	complemented
16	C₃×#577	—	2	2	no	—
17	C₃×#577	C₄	1	4	yes	complemented
18	C₃×#576	—	2	1	no	—
19	C₃×#576	C₄	1	2	yes	complemented
20	C₃×#575	—	2	2	no	—
21	C₃×#575	C₂²	1	2	yes	—
22	C₂×C₃×#135	—	2	2	no	—
23	C₂×C₃×#135	C₂²	1	1	yes	characteristic
24	C₂×C₃×#135	C₂²	1	1	yes	characteristic
25	C₃×#568	C₂²	1	2	yes	complemented
26	C₂×C₃×#131	C₂²	1	1	yes	characteristic
27	C₃×C₄×#41	C₂²	1	2	yes	complemented
28	C₃×#444	—	2	2	no	—
29	C₃×#443	—	2	2	no	—
30	C₃×C₄×Q₁₆	—	2	1	no	—
31	C₃×C₄×SD₁₆	—	2	2	no	—
32	C₃×C₄×D₈	—	2	1	no	—
33	C₂×C₃×#108	C₂²	1	1	yes	characteristic, complemented
34	C₃×#427	C₂²	1	1	yes	characteristic, complemented
35	C₃×#421	—	2	4	no	—
36	C₃×#421	C₂²	1	2	yes	—
37	C₃×#421	C₂²	1	2	yes	complemented
38	C₃×#421	C₂²	1	2	yes	complemented
39	C₂×C₃×#104	—	2	4	no	—
40	C₂×C₃×#104	C₂²	1	1	yes	characteristic
41	C₂×C₃×#104	C₂²	1	2	yes	complemented
42	C₃×#416	C₂²	1	1	yes	characteristic
43	C₃×#416	C₂²	1	2	yes	complemented
44	C₃×#410	C₂²	1	2	yes	complemented
45	C₃×#408	C₂²	1	2	yes	complemented
46	C₃×#405	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
48	C₃×#344	C₂²	1	2	yes	complemented
49	C₃×#341	C₂²	1	1	yes	characteristic, complemented
50	C₃×#339	C₂²	1	1	yes	characteristic, complemented
51	#3584	C₂×C₃	1	1	yes	characteristic
52	#2957	C₂×C₃	1	1	yes	characteristic, complemented
53	C₄×#135	C₂×C₃	1	1	yes	characteristic, complemented
54	#2901	C₂×C₃	1	2	yes	complemented
55	#1976	C₂×C₃	1	1	yes	characteristic, complemented
56	#1974	C₂×C₃	1	1	yes	characteristic, complemented
57	#1906	C₂×C₃	1	2	yes	complemented
58	#1881	C₂×C₃	1	2	yes	complemented
59	#1814	C₂×C₃	1	1	yes	characteristic, complemented
60	#1794	C₂×C₃	1	2	yes	complemented
61	#1764	C₂×C₃	1	1	yes	characteristic, complemented
62	C₃×2₋¹⁺⁴	—	2	2	no	—
63	C₃×2₋¹⁺⁴	—	2	2	no	—
64	C₃×2₋¹⁺⁴	C₂×C₄	1	4	yes	complemented
65	C₃×H(5,𝔽₂)	—	2	2	no	—
66	C₃×H(5,𝔽₂)	—	2	2	no	—
67	C₃×H(5,𝔽₂)	—	4	2	no	—
68	C₃×H(5,𝔽₂)	C₂×C₄	1	4	yes	complemented
69	C₂×C₃×#41	—	2	4	no	—
70	C₂×C₃×#41	—	2	4	no	—
71	C₂×C₃×#41	—	4	2	no	—
72	C₂×C₃×#41	C₂³	1	2	yes	—
73	C₂×C₃×#41	C₂³	1	2	yes	—
74	C₂×C₃×#41	C₂³	1	1	yes	characteristic
75	C₃×#137	—	2	2	no	—
76	C₃×#137	—	2	4	no	—
77	C₃×#137	—	2	4	no	—
78	C₃×#137	C₂×C₄	1	4	yes	complemented
79	C₃×AGL(1,ℤ/8)	—	2	2	no	—
80	C₃×AGL(1,ℤ/8)	—	2	4	no	—
81	C₃×AGL(1,ℤ/8)	—	2	4	no	—
82	C₃×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	complemented
83	C₃×#135	—	2	4	no	—
84	C₃×#135	—	2	2	no	—
85	C₃×#135	—	2	4	no	—
86	C₃×#135	—	2	2	no	—
87	C₃×#135	—	2	1	no	—
88	C₃×#135	—	2	1	no	—
89	C₃×#135	—	2	1	no	—
90	C₃×#135	—	2	1	no	—
91	C₃×#135	C₂×C₄	1	2	yes	—
92	C₃×#135	C₂×C₄	1	2	yes	—
93	C₃×#135	D₄	1	2	yes	—
94	C₃×#135	D₄	1	2	yes	—
95	C₂×C₃×Q₁₆	—	2	2	no	—
96	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
97	C₂×C₃×Q₁₆	D₄	1	1	yes	characteristic
98	C₂×C₃×SD₁₆	—	2	4	no	—
99	C₂×C₃×SD₁₆	C₂×C₄	1	2	yes	complemented
100	C₂×C₃×SD₁₆	D₄	1	2	yes	—
101	C₂×C₃×D₈	—	2	2	no	—
102	C₂×C₃×D₈	C₂×C₄	1	1	yes	characteristic, complemented
103	C₂×C₃×D₈	D₄	1	1	yes	characteristic
104	C₃×#131	—	2	1	no	—
105	C₃×#131	—	2	1	no	—
106	C₃×#131	—	2	4	no	—
107	C₃×#131	—	4	2	no	—
108	C₃×#131	C₂×C₄	1	2	yes	complemented
109	C₃×#131	C₂×C₄	1	2	yes	complemented
110	C₂×C₃×M₁₆	—	2	1	no	—
111	C₂×C₃×M₁₆	—	2	4	no	—
112	C₂×C₃×M₁₆	C₂³	1	2	yes	—
113	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
114	C₂×C₃×M₁₆	C₂³	1	1	yes	characteristic
115	C₂²×C₃×C₈	—	2	1	no	—
116	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
117	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
118	C₃×C₄×Q₈	—	2	2	no	—
119	C₃×C₄×D₄	—	2	2	no	—
120	C₃×C₄×D₄	—	4	2	no	—
121	C₃×#117	—	2	2	no	—
122	C₃×#117	C₂³	1	1	yes	characteristic
123	C₃×#117	C₂³	1	2	yes	—
124	C₂×C₃×C₄²	—	2	2	no	—
125	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
126	C₃×#108	—	2	1	no	—
127	C₃×#108	—	2	1	no	—
128	C₃×#107	—	2	1	no	—
129	C₃×#106	—	2	1	no	—
130	C₃×#104	—	2	4	no	—
131	C₃×#104	—	2	4	no	—
132	C₃×#104	—	2	4	no	—
133	C₃×#104	—	2	2	no	—
134	C₃×#104	—	2	2	no	—
135	C₃×#104	D₄	1	2	yes	—
136	C₃×#104	D₄	1	2	yes	—
137	C₃×#103	—	2	2	no	—
138	C₃×#103	—	2	1	no	—
139	C₃×#102	—	2	2	no	—
140	C₃×#102	—	2	1	no	—
141	C₃×#99	—	4	2	no	—
142	C₃×C₂²⋊C₈	—	4	2	no	—
143	C₃×#97	—	2	1	no	—
144	C₃×C₄×C₈	—	2	1	no	—
145	C₃×C₄×C₈	—	2	1	no	—
146	C₃×C₄×C₈	—	2	1	no	—
147	#585	C₂²×C₃	1	2	yes	—
148	#578	—	2	1	no	—
149	#578	C₃×C₄	1	2	yes	complemented
150	#577	—	2	2	no	—
151	#577	C₃×C₄	1	4	yes	complemented
152	#576	—	2	1	no	—
153	#576	C₃×C₄	1	2	yes	complemented
154	#575	—	2	2	no	—
155	#575	C₂²×C₃	1	2	yes	—
156	C₂×#135	—	2	2	no	—
157	C₂×#135	C₂²×C₃	1	1	yes	characteristic
158	C₂×#135	C₂²×C₃	1	1	yes	characteristic
159	#568	C₂²×C₃	1	2	yes	complemented
160	C₂×#131	C₂²×C₃	1	1	yes	characteristic
161	C₄×#41	C₂²×C₃	1	2	yes	complemented
162	#444	—	2	2	no	—
163	#443	—	2	2	no	—
164	C₄×Q₁₆	—	2	1	no	—
165	C₄×SD₁₆	—	2	2	no	—
166	C₄×D₈	—	2	1	no	—
167	C₂×#108	C₂²×C₃	1	1	yes	characteristic, complemented
168	#427	C₂²×C₃	1	1	yes	characteristic, complemented
169	#421	—	2	4	no	—
170	#421	C₂²×C₃	1	2	yes	—
171	#421	C₂²×C₃	1	2	yes	complemented
172	#421	C₂²×C₃	1	2	yes	complemented
173	C₂×#104	—	2	4	no	—
174	C₂×#104	C₂²×C₃	1	1	yes	characteristic
175	C₂×#104	C₂²×C₃	1	2	yes	complemented
176	#416	C₂²×C₃	1	1	yes	characteristic
177	#416	C₂²×C₃	1	2	yes	complemented
178	#410	C₂²×C₃	1	2	yes	complemented
179	#408	C₂²×C₃	1	2	yes	complemented
180	#405	C₂²×C₃	1	1	yes	characteristic, complemented
181	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
182	#344	C₂²×C₃	1	2	yes	complemented
183	#341	C₂²×C₃	1	1	yes	characteristic, complemented
184	#339	C₂²×C₃	1	1	yes	characteristic, complemented
185	C₃×#41	—	2	1	no	—
186	C₃×#41	—	2	2	no	—
187	C₃×#41	—	2	2	no	—
188	C₃×#41	—	2	1	no	—
189	C₃×#41	—	2	4	no	—
190	C₃×#41	—	2	4	no	—
191	C₃×#41	—	2	4	no	—
192	C₃×#41	—	2	4	no	—
193	C₃×#41	—	2	2	no	—
194	C₃×#41	—	2	2	no	—
195	C₃×#41	—	2	2	no	—
196	C₃×#41	—	2	2	no	—
197	C₃×#41	—	4	4	no	—
198	C₃×#41	—	4	4	no	—
199	C₃×#41	—	4	4	no	—
200	C₃×#41	—	4	4	no	—
201	C₃×#41	—	4	2	no	—
202	C₃×#41	—	4	2	no	—
203	C₃×#41	C₂²×C₄	1	4	yes	—
204	C₃×#41	C₂²×C₄	1	2	yes	—
205	C₃×#41	C₂×D₄	1	4	yes	—
206	C₃×#41	C₂×D₄	1	2	yes	—
207	C₂×C₃×Q₈	—	2	4	no	—
208	C₂×C₃×Q₈	—	2	4	no	—
209	C₂×C₃×Q₈	—	4	2	no	—
210	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
211	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
212	C₂×C₃×Q₈	#41	1	1	yes	characteristic
213	C₂×C₃×D₄	—	2	2	no	—
214	C₂×C₃×D₄	—	2	1	no	—
215	C₂×C₃×D₄	—	2	4	no	—
216	C₂×C₃×D₄	—	2	4	no	—
217	C₂×C₃×D₄	—	2	2	no	—
218	C₂×C₃×D₄	—	4	4	no	—
219	C₂×C₃×D₄	—	4	2	no	—
220	C₂×C₃×D₄	—	4	4	no	—
221	C₂×C₃×D₄	—	4	4	no	—
222	C₂×C₃×D₄	C₂²×C₄	1	2	yes	—
223	C₂×C₃×D₄	C₂×D₄	1	2	yes	—
224	C₂×C₃×D₄	#41	1	1	yes	characteristic
225	C₂²×C₃×C₄	—	2	2	no	—
226	C₂²×C₃×C₄	—	2	1	no	—
227	C₂²×C₃×C₄	—	2	1	no	—
228	C₂²×C₃×C₄	—	2	2	no	—
229	C₂²×C₃×C₄	—	4	4	no	—
230	C₂²×C₃×C₄	—	4	2	no	—
231	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
232	C₃×Q₁₆	—	2	4	no	—
233	C₃×Q₁₆	—	2	1	no	—
234	C₃×Q₁₆	—	2	1	no	—
235	C₃×Q₁₆	C₂²⋊C₄	1	2	yes	—
236	C₃×Q₁₆	C₂²⋊C₄	1	2	yes	—
237	C₃×SD₁₆	—	2	4	no	—
238	C₃×SD₁₆	—	2	4	no	—
239	C₃×SD₁₆	—	2	2	no	—
240	C₃×SD₁₆	—	2	2	no	—
241	C₃×SD₁₆	C₂²⋊C₄	1	4	yes	—
242	C₃×SD₁₆	C₂²⋊C₄	1	4	yes	—
243	C₃×D₈	—	2	4	no	—
244	C₃×D₈	—	2	1	no	—
245	C₃×D₈	—	2	1	no	—
246	C₃×D₈	C₂²⋊C₄	1	2	yes	—
247	C₃×D₈	C₂²⋊C₄	1	2	yes	—
248	C₃×M₁₆	—	2	1	no	—
249	C₃×M₁₆	—	2	1	no	—
250	C₃×M₁₆	—	2	2	no	—
251	C₃×M₁₆	—	2	2	no	—
252	C₃×M₁₆	—	2	2	no	—
253	C₃×M₁₆	—	2	2	no	—
254	C₃×M₁₆	—	4	4	no	—
255	C₃×M₁₆	C₂²×C₄	1	2	yes	—
256	C₃×M₁₆	C₂×D₄	1	2	yes	—
257	C₂×C₃×C₈	—	2	1	no	—
258	C₂×C₃×C₈	—	2	1	no	—
259	C₂×C₃×C₈	—	2	2	no	—
260	C₂×C₃×C₈	—	2	2	no	—
261	C₂×C₃×C₈	—	2	1	no	—
262	C₂×C₃×C₈	—	2	1	no	—
263	C₂×C₃×C₈	—	2	1	no	—
264	C₂×C₃×C₈	—	2	2	no	—
265	C₂×C₃×C₈	—	4	4	no	—
266	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
267	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
268	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
269	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
270	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
271	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
272	C₃×C₄⋊C₄	—	2	2	no	—
273	C₃×C₄⋊C₄	—	2	1	no	—
274	C₃×C₄⋊C₄	—	4	2	no	—
275	C₃×C₂²⋊C₄	—	2	1	no	—
276	C₃×C₂²⋊C₄	—	2	2	no	—
277	C₃×C₂²⋊C₄	—	4	2	no	—
278	C₃×C₄²	—	2	1	no	—
279	C₃×C₄²	—	2	1	no	—
280	C₃×C₄²	—	2	2	no	—
281	C₃×C₄²	—	4	2	no	—
282	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
283	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
284	2₋¹⁺⁴	—	2	2	no	—
285	2₋¹⁺⁴	—	2	2	no	—
286	2₋¹⁺⁴	C₂×C₃×C₄	1	4	yes	complemented
287	H(5,𝔽₂)	—	2	2	no	—
288	H(5,𝔽₂)	—	2	2	no	—
289	H(5,𝔽₂)	—	4	2	no	—
290	H(5,𝔽₂)	C₂×C₃×C₄	1	4	yes	complemented
291	C₂×#41	—	2	4	no	—
292	C₂×#41	—	2	4	no	—
293	C₂×#41	—	4	2	no	—
294	C₂×#41	C₂³×C₃	1	2	yes	—
295	C₂×#41	C₂³×C₃	1	2	yes	—
296	C₂×#41	C₂³×C₃	1	1	yes	characteristic
297	#137	—	2	2	no	—
298	#137	—	2	4	no	—
299	#137	—	2	4	no	—
300	#137	C₂×C₃×C₄	1	4	yes	complemented
301	AGL(1,ℤ/8)	—	2	2	no	—
302	AGL(1,ℤ/8)	—	2	4	no	—
303	AGL(1,ℤ/8)	—	2	4	no	—
304	AGL(1,ℤ/8)	C₂×C₃×C₄	1	4	yes	complemented
305	#135	—	2	4	no	—
306	#135	—	2	2	no	—
307	#135	—	2	4	no	—
308	#135	—	2	2	no	—
309	#135	—	2	1	no	—
310	#135	—	2	1	no	—
311	#135	—	2	1	no	—
312	#135	—	2	1	no	—
313	#135	C₂×C₃×C₄	1	2	yes	—
314	#135	C₂×C₃×C₄	1	2	yes	—
315	#135	C₃×D₄	1	2	yes	—
316	#135	C₃×D₄	1	2	yes	—
317	C₂×Q₁₆	—	2	2	no	—
318	C₂×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic, complemented
319	C₂×Q₁₆	C₃×D₄	1	1	yes	characteristic
320	C₂×SD₁₆	—	2	4	no	—
321	C₂×SD₁₆	C₂×C₃×C₄	1	2	yes	complemented
322	C₂×SD₁₆	C₃×D₄	1	2	yes	—
323	C₂×D₈	—	2	2	no	—
324	C₂×D₈	C₂×C₃×C₄	1	1	yes	characteristic, complemented
325	C₂×D₈	C₃×D₄	1	1	yes	characteristic
326	#131	—	2	1	no	—
327	#131	—	2	1	no	—
328	#131	—	2	4	no	—
329	#131	—	4	2	no	—
330	#131	C₂×C₃×C₄	1	2	yes	complemented
331	#131	C₂×C₃×C₄	1	2	yes	complemented
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	4	no	—
334	C₂×M₁₆	C₂³×C₃	1	2	yes	—
335	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
336	C₂×M₁₆	C₂³×C₃	1	1	yes	characteristic
337	C₂²×C₈	—	2	1	no	—
338	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
340	C₄×Q₈	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	4	2	no	—
343	#117	—	2	2	no	—
344	#117	C₂³×C₃	1	2	yes	—
345	#117	C₂³×C₃	1	1	yes	characteristic
346	C₂×C₄²	—	2	2	no	—
347	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
348	#108	—	2	1	no	—
349	#108	—	2	1	no	—
350	#107	—	2	1	no	—
351	#106	—	2	1	no	—
352	#104	—	2	4	no	—
353	#104	—	2	4	no	—
354	#104	—	2	4	no	—
355	#104	—	2	2	no	—
356	#104	—	2	2	no	—
357	#104	C₃×D₄	1	2	yes	—
358	#104	C₃×D₄	1	2	yes	—
359	#103	—	2	2	no	—
360	#103	—	2	1	no	—
361	#102	—	2	2	no	—
362	#102	—	2	1	no	—
363	#99	—	4	2	no	—
364	C₂²⋊C₈	—	4	2	no	—
365	#97	—	2	1	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	—	2	1	no	—
368	C₄×C₈	—	2	1	no	—
369	C₂³×C₃	—	2	2	no	—
370	C₂³×C₃	—	2	1	no	—
371	C₂³×C₃	—	2	2	no	—
372	C₂³×C₃	—	4	4	no	—
373	C₂³×C₃	H(5,𝔽₂)	1	1	yes	characteristic
374	C₃×Q₈	—	2	1	no	—
375	C₃×Q₈	—	2	2	no	—
376	C₃×Q₈	—	2	2	no	—
377	C₃×Q₈	—	2	2	no	—
378	C₃×Q₈	—	4	4	no	—
379	C₃×Q₈	C₂×C₂²⋊C₄	1	4	yes	—
380	C₃×Q₈	C₄×D₄	1	2	yes	—
381	C₃×D₄	—	2	2	no	—
382	C₃×D₄	—	2	4	no	—
383	C₃×D₄	—	2	2	no	—
384	C₃×D₄	—	2	4	no	—
385	C₃×D₄	—	2	2	no	—
386	C₃×D₄	—	2	2	no	—
387	C₃×D₄	—	2	1	no	—
388	C₃×D₄	—	2	2	no	—
389	C₃×D₄	—	4	4	no	—
390	C₃×D₄	—	4	4	no	—
391	C₃×D₄	—	4	4	no	—
392	C₃×D₄	—	4	2	no	—
393	C₃×D₄	—	4	2	no	—
394	C₃×D₄	C₂×C₂²⋊C₄	1	4	yes	—
395	C₃×D₄	C₄×D₄	1	2	yes	—
396	C₂×C₃×C₄	—	2	1	no	—
397	C₂×C₃×C₄	—	2	2	no	—
398	C₂×C₃×C₄	—	2	1	no	—
399	C₂×C₃×C₄	—	2	2	no	—
400	C₂×C₃×C₄	—	2	4	no	—
401	C₂×C₃×C₄	—	2	4	no	—
402	C₂×C₃×C₄	—	2	2	no	—
403	C₂×C₃×C₄	—	2	2	no	—
404	C₂×C₃×C₄	—	2	1	no	—
405	C₂×C₃×C₄	—	4	4	no	—
406	C₂×C₃×C₄	—	4	4	no	—
407	C₂×C₃×C₄	—	4	4	no	—
408	C₂×C₃×C₄	—	4	2	no	—
409	C₂×C₃×C₄	—	4	2	no	—
410	C₂×C₃×C₄	—	4	2	no	—
411	C₂×C₃×C₄	—	4	2	no	—
412	C₂×C₃×C₄	—	4	1	no	—
413	C₂×C₃×C₄	—	8	4	no	—
414	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
415	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
416	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
417	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
418	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
419	C₂×C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
420	C₃×C₈	—	2	2	no	—
421	C₃×C₈	—	2	2	no	—
422	C₃×C₈	—	4	2	no	—
423	C₃×C₈	—	4	1	no	—
424	C₃×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
425	C₃×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
426	C₃×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
427	C₃×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
428	#41	—	2	1	no	—
429	#41	—	2	2	no	—
430	#41	—	2	2	no	—
431	#41	—	2	1	no	—
432	#41	—	2	4	no	—
433	#41	—	2	4	no	—
434	#41	—	2	4	no	—
435	#41	—	2	4	no	—
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	2	2	no	—
439	#41	—	2	2	no	—
440	#41	—	4	4	no	—
441	#41	—	4	4	no	—
442	#41	—	4	4	no	—
443	#41	—	4	4	no	—
444	#41	—	4	2	no	—
445	#41	—	4	2	no	—
446	#41	C₂²×C₃×C₄	1	4	yes	—
447	#41	C₂²×C₃×C₄	1	2	yes	—
448	#41	C₂×C₃×D₄	1	4	yes	—
449	#41	C₂×C₃×D₄	1	2	yes	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	4	2	no	—
453	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
454	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
455	C₂×Q₈	C₃×#41	1	1	yes	characteristic
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	4	4	no	—
462	C₂×D₄	—	4	2	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	C₂²×C₃×C₄	1	2	yes	—
466	C₂×D₄	C₂×C₃×D₄	1	2	yes	—
467	C₂×D₄	C₃×#41	1	1	yes	characteristic
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
475	Q₁₆	—	2	4	no	—
476	Q₁₆	—	2	1	no	—
477	Q₁₆	—	2	1	no	—
478	Q₁₆	C₃×C₂²⋊C₄	1	2	yes	—
479	Q₁₆	C₃×C₂²⋊C₄	1	2	yes	—
480	SD₁₆	—	2	4	no	—
481	SD₁₆	—	2	4	no	—
482	SD₁₆	—	2	2	no	—
483	SD₁₆	—	2	2	no	—
484	SD₁₆	C₃×C₂²⋊C₄	1	4	yes	—
485	SD₁₆	C₃×C₂²⋊C₄	1	4	yes	—
486	D₈	—	2	4	no	—
487	D₈	—	2	1	no	—
488	D₈	—	2	1	no	—
489	D₈	C₃×C₂²⋊C₄	1	2	yes	—
490	D₈	C₃×C₂²⋊C₄	1	2	yes	—
491	M₁₆	—	2	1	no	—
492	M₁₆	—	2	1	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	—	2	2	no	—
495	M₁₆	—	2	2	no	—
496	M₁₆	—	2	2	no	—
497	M₁₆	—	4	4	no	—
498	M₁₆	C₂²×C₃×C₄	1	2	yes	—
499	M₁₆	C₂×C₃×D₄	1	2	yes	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	2	no	—
508	C₂×C₈	—	4	4	no	—
509	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
510	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
511	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
512	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
513	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
514	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	4	2	no	—
518	C₂²⋊C₄	—	2	1	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	4	2	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	4	2	no	—
525	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
526	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
527	C₂²×C₃	—	2	2	no	—
528	C₂²×C₃	—	2	4	no	—
529	C₂²×C₃	—	4	1	no	—
530	C₂²×C₃	—	4	2	no	—
531	C₂²×C₃	—	8	2	no	—
532	C₂²×C₃	—	8	1	no	—
533	C₂²×C₃	—	8	2	no	—
534	C₂²×C₃	—	16	4	no	—
535	C₂²×C₃	#518	1	1	yes	characteristic
536	C₂²×C₃	D₄²	1	1	yes	characteristic
537	C₂²×C₃	#546	1	1	yes	characteristic
538	C₃×C₄	—	2	2	no	—
539	C₃×C₄	—	2	2	no	—
540	C₃×C₄	—	2	4	no	—
541	C₃×C₄	—	4	1	no	—
542	C₃×C₄	—	4	2	no	—
543	C₃×C₄	—	8	1	no	—
544	C₃×C₄	—	8	2	no	—
545	C₃×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
546	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
547	C₃×C₄	D₄²	1	1	yes	characteristic
548	C₃×C₄	#546	1	1	yes	characteristic
549	C₂³	—	2	2	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	4	4	no	—
553	C₂³	C₃×H(5,𝔽₂)	1	1	yes	characteristic
554	Q₈	—	2	1	no	—
555	Q₈	—	2	2	no	—
556	Q₈	—	2	2	no	—
557	Q₈	—	2	2	no	—
558	Q₈	—	4	4	no	—
559	Q₈	C₂×C₃×C₂²⋊C₄	1	4	yes	—
560	Q₈	C₃×C₄×D₄	1	2	yes	—
561	D₄	—	2	2	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	2	no	—
564	D₄	—	2	4	no	—
565	D₄	—	2	2	no	—
566	D₄	—	2	2	no	—
567	D₄	—	2	1	no	—
568	D₄	—	2	2	no	—
569	D₄	—	4	4	no	—
570	D₄	—	4	4	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	2	no	—
573	D₄	—	4	2	no	—
574	D₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
575	D₄	C₃×C₄×D₄	1	2	yes	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	8	4	no	—
594	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
595	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
596	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
597	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
598	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
599	C₂×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
600	C₈	—	2	2	no	—
601	C₈	—	2	2	no	—
602	C₈	—	4	2	no	—
603	C₈	—	4	1	no	—
604	C₈	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
605	C₈	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
606	C₈	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
607	C₈	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
608	C₂×C₃	—	2	1	no	—
609	C₂×C₃	—	2	1	no	—
610	C₂×C₃	—	2	1	no	—
611	C₂×C₃	—	4	4	no	—
612	C₂×C₃	—	4	2	no	—
613	C₂×C₃	—	8	2	no	—
614	C₂×C₃	D₄×C₂²⋊C₄	1	1	yes	characteristic
615	C₂²	—	2	2	no	—
616	C₂²	—	2	4	no	—
617	C₂²	—	4	1	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	8	2	no	—
620	C₂²	—	8	1	no	—
621	C₂²	—	8	2	no	—
622	C₂²	—	16	4	no	—
623	C₂²	C₃×#518	1	1	yes	characteristic
624	C₂²	C₃×D₄²	1	1	yes	characteristic
625	C₂²	C₃×#546	1	1	yes	characteristic
626	C₄	—	2	2	no	—
627	C₄	—	2	2	no	—
628	C₄	—	2	4	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	2	no	—
631	C₄	—	8	2	no	—
632	C₄	—	8	1	no	—
633	C₄	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
634	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
635	C₄	C₃×D₄²	1	1	yes	characteristic
636	C₄	C₃×#546	1	1	yes	characteristic
637	C₃	#22462	1	1	yes	characteristic, direct product factor, Sylow
638	C₂	—	2	1	no	—
639	C₂	—	2	1	no	—
640	C₂	—	2	1	no	—
641	C₂	—	4	2	no	—
642	C₂	—	4	4	no	—
643	C₂	—	8	2	no	—
644	C₂	C₃×D₄×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₄²
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	D₄×C₂²⋊C₄
prime radical	C₃×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2957	C₂	semidirect
3. C₃×C₄×#135	C₂	semidirect
4. C₃×#2901	C₂	semidirect
5. C₃×#1976	C₂	semidirect
6. C₃×#1974	C₂	semidirect
7. C₃×#1906	C₂	semidirect
8. C₃×#1881	C₂	semidirect
9. C₃×#1814	C₂	semidirect
10. C₃×#1794	C₂	semidirect
11. C₃×#1764	C₂	semidirect
12. #22462	C₃	direct (N × H)
15. C₃×#578	C₄	semidirect
17. C₃×#577	C₄	semidirect
19. C₃×#576	C₄	semidirect
25. C₃×#568	C₂²	semidirect
27. C₃×C₄×#41	C₂²	semidirect
33. C₂×C₃×#108	C₂²	semidirect
34. C₃×#427	C₂²	semidirect
37. C₃×#421	C₂²	semidirect
38. C₃×#421	C₂²	semidirect
41. C₂×C₃×#104	C₂²	semidirect
43. C₃×#416	C₂²	semidirect
44. C₃×#410	C₂²	semidirect
45. C₃×#408	C₂²	semidirect
46. C₃×#405	C₂²	semidirect
47. C₂×C₃×C₄×C₈	C₂²	semidirect
48. C₃×#344	C₂²	semidirect
49. C₃×#341	C₂²	semidirect
50. C₃×#339	C₂²	semidirect
52. #2957	C₂×C₃	semidirect
53. C₄×#135	C₂×C₃	semidirect
54. #2901	C₂×C₃	semidirect
55. #1976	C₂×C₃	semidirect
56. #1974	C₂×C₃	semidirect
57. #1906	C₂×C₃	semidirect
58. #1881	C₂×C₃	semidirect
59. #1814	C₂×C₃	semidirect
60. #1794	C₂×C₃	semidirect
61. #1764	C₂×C₃	semidirect
64. C₃×2₋¹⁺⁴	C₂×C₄	semidirect
68. C₃×H(5,𝔽₂)	C₂×C₄	semidirect
78. C₃×#137	C₂×C₄	semidirect
82. C₃×AGL(1,ℤ/8)	C₂×C₄	semidirect
96. C₂×C₃×Q₁₆	C₂×C₄	semidirect
99. C₂×C₃×SD₁₆	C₂×C₄	semidirect
102. C₂×C₃×D₈	C₂×C₄	semidirect
108. C₃×#131	C₂×C₄	semidirect
109. C₃×#131	C₂×C₄	semidirect
149. #578	C₃×C₄	semidirect
151. #577	C₃×C₄	semidirect
153. #576	C₃×C₄	semidirect
159. #568	C₂²×C₃	semidirect
161. C₄×#41	C₂²×C₃	semidirect
167. C₂×#108	C₂²×C₃	semidirect
168. #427	C₂²×C₃	semidirect
171. #421	C₂²×C₃	semidirect
172. #421	C₂²×C₃	semidirect
175. C₂×#104	C₂²×C₃	semidirect
177. #416	C₂²×C₃	semidirect
178. #410	C₂²×C₃	semidirect
179. #408	C₂²×C₃	semidirect
180. #405	C₂²×C₃	semidirect
181. C₂×C₄×C₈	C₂²×C₃	semidirect
182. #344	C₂²×C₃	semidirect
183. #341	C₂²×C₃	semidirect
184. #339	C₂²×C₃	semidirect
286. 2₋¹⁺⁴	C₂×C₃×C₄	semidirect
290. H(5,𝔽₂)	C₂×C₃×C₄	semidirect
300. #137	C₂×C₃×C₄	semidirect
304. AGL(1,ℤ/8)	C₂×C₃×C₄	semidirect
318. C₂×Q₁₆	C₂×C₃×C₄	semidirect
321. C₂×SD₁₆	C₂×C₃×C₄	semidirect
324. C₂×D₈	C₂×C₃×C₄	semidirect
330. #131	C₂×C₃×C₄	semidirect
331. #131	C₂×C₃×C₄	semidirect
637. C₃	#22462	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3584	2
C₃×#2957	2
C₃×C₄×#135	2
C₃×#2901	2
C₃×#1976	2
C₃×#1974	2
C₃×#1906	2
C₃×#1881	2
C₃×#1814	2
C₃×#1794	2
C₃×#1764	2
#22462	3

Subgroups — C3×#22462 #10641069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22462 #10641069