Subgroups — C₃×#10596246 #13179526

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64367	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64361	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64159	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#3754	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64124	C₂	1	2	yes	maximal, complemented
7	C₃×SD₁₆×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
8	C₃×#64121	C₂	1	2	yes	maximal, complemented
9	C₃×#64120	C₂	1	2	yes	maximal, complemented
10	C₃×#64118	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#64116	C₂	1	1	yes	maximal, characteristic, complemented
12	#10596246	C₃	1	1	yes	maximal, characteristic, direct product factor
13	C₃×#5775	—	3	4	no	maximal
14	C₃³×#41	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×#638	—	2	2	no	—
17	C₃×Q₈×C₃⋊D₃	C₂²	1	2	yes	complemented
18	C₃×#3830	—	2	2	no	—
19	C₃×#3830	—	2	1	no	—
20	C₃×#3830	C₂²	1	2	yes	—
21	C₃×#3830	C₂²	1	2	yes	complemented
22	C₃×D₄×C₃⋊D₃	—	2	1	no	—
23	C₃×D₄×C₃⋊D₃	C₂²	1	2	yes	—
24	C₃×#3828	—	2	1	no	—
25	C₃×#3828	C₂²	1	1	yes	characteristic
26	C₃×#3828	C₂²	1	1	yes	characteristic
27	C₂×C₃×#634	—	2	1	no	—
28	C₂×C₃×#634	C₂²	1	1	yes	characteristic, complemented
29	C₃³×SD₁₆	C₂²	1	2	yes	complemented
30	C₃³×D₈	C₂²	1	2	yes	complemented
31	C₃³×M₁₆	C₂²	1	1	yes	characteristic, complemented
32	C₃×#3756	C₂²	1	1	yes	characteristic, complemented
33	C₃×#3755	C₂²	1	1	yes	characteristic, complemented
34	C₃×#3754	C₂²	1	2	yes	complemented
35	C₃×#3754	C₂²	1	1	yes	characteristic, complemented
36	C₃×#3754	C₂²	1	2	yes	complemented
37	C₃×#3753	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
39	C₃×#3746	C₂²	1	2	yes	complemented
40	C₃×#3744	C₂²	1	2	yes	complemented
41	C₃×#3743	C₂²	1	2	yes	complemented
42	C₃×C₈×C₃⋊D₃	C₂²	1	2	yes	complemented
43	#64367	C₂×C₃	1	1	yes	characteristic, complemented
44	#64361	C₂×C₃	1	1	yes	characteristic, complemented
45	C₃×#1010	—	3	4	no	—
46	C₃×#1004	—	3	4	no	—
47	C₃²×AGL(1,ℤ/8)	—	2	4	no	—
48	C₃²×AGL(1,ℤ/8)	D₃	1	4	yes	complemented
49	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
50	#64159	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×#3754	C₂×C₃	1	1	yes	characteristic, complemented
52	#64124	C₂×C₃	1	2	yes	complemented
53	SD₁₆×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
54	#64121	C₂×C₃	1	2	yes	complemented
55	#64120	C₂×C₃	1	2	yes	complemented
56	#64118	C₂×C₃	1	1	yes	characteristic, complemented
57	#64116	C₂×C₃	1	1	yes	characteristic, complemented
58	C₃×#950	—	3	4	no	—
59	C₂×C₃×#214	—	3	4	no	—
60	C₃×#915	—	3	8	no	—
61	C₃×D₃×SD₁₆	—	3	8	no	—
62	C₃×#912	—	3	8	no	—
63	C₃×#911	—	3	8	no	—
64	C₃×#909	—	3	4	no	—
65	C₃×#907	—	3	4	no	—
66	C₂³×C₃³	—	2	1	no	—
67	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
68	C₃³×Q₈	C₂³	1	1	yes	characteristic
69	C₃³×D₄	—	2	1	no	—
70	C₃³×D₄	—	2	1	no	—
71	C₃³×D₄	C₂³	1	1	yes	characteristic
72	C₃³×D₄	C₂³	1	2	yes	—
73	C₂×C₃³×C₄	—	2	1	no	—
74	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
75	C₃×#638	—	2	1	no	—
76	C₃×#638	—	2	2	no	—
77	C₃×#638	—	2	2	no	—
78	C₃×#638	—	2	2	no	—
79	C₃×#638	D₄	1	2	yes	—
80	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
81	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
82	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
83	C₃×C₄⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
84	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	1	yes	characteristic
85	C₃×C₄×C₃⋊D₃	—	2	2	no	—
86	C₃×C₄×C₃⋊D₃	—	2	1	no	—
87	C₃×C₄×C₃⋊D₃	C₂³	1	2	yes	—
88	C₃×#634	—	2	1	no	—
89	C₃×#634	—	2	2	no	—
90	C₃×#634	D₄	1	1	yes	characteristic
91	C₃×#634	C₂³	1	2	yes	—
92	C₃×#634	C₂³	1	1	yes	characteristic
93	C₃³×C₈	C₂³	1	2	yes	—
94	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
95	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
96	C₃×#577	—	9	1	no	—
97	#5775	—	3	4	no	—
98	C₃²×#41	—	2	4	no	—
99	C₃²×#41	C₂×D₃	1	4	yes	complemented
100	C₃²×#41	C₂²×C₃	1	1	yes	characteristic, complemented
101	C₂×C₃²×D₄	—	2	4	no	—
102	C₂×C₃²×D₄	C₂×D₃	1	4	yes	—
103	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
104	C₂×#638	—	2	2	no	—
105	Q₈×C₃⋊D₃	C₂²×C₃	1	2	yes	complemented
106	#3830	—	2	2	no	—
107	#3830	—	2	1	no	—
108	#3830	C₂²×C₃	1	2	yes	—
109	#3830	C₂²×C₃	1	2	yes	complemented
110	D₄×C₃⋊D₃	—	2	1	no	—
111	D₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
112	#3828	—	2	1	no	—
113	#3828	C₂²×C₃	1	1	yes	characteristic
114	#3828	C₂²×C₃	1	1	yes	characteristic
115	C₂×#634	—	2	1	no	—
116	C₂×#634	C₂²×C₃	1	1	yes	characteristic, complemented
117	C₂×C₃×#67	—	6	8	no	—
118	C₃×D₃×Q₈	—	3	8	no	—
119	C₃×#237	—	3	8	no	—
120	C₃×#237	—	3	8	no	—
121	C₃×#237	—	6	4	no	—
122	C₃×#237	—	6	8	no	—
123	C₃×D₃×D₄	—	3	8	no	—
124	C₃×D₃×D₄	—	6	4	no	—
125	C₃×#235	—	3	4	no	—
126	C₃×#235	—	3	4	no	—
127	C₃×#235	—	6	4	no	—
128	C₂×C₃×Dic₆	—	3	4	no	—
129	C₂×C₃×Dic₆	—	6	4	no	—
130	C₃²×SD₁₆	—	2	8	no	—
131	C₃²×SD₁₆	C₂×D₃	1	8	yes	complemented
132	C₃²×SD₁₆	C₂²×C₃	1	2	yes	complemented
133	C₃²×D₈	—	2	8	no	—
134	C₃²×D₈	C₂×D₃	1	8	yes	complemented
135	C₃²×D₈	C₂²×C₃	1	2	yes	complemented
136	C₃²×M₁₆	—	2	4	no	—
137	C₃²×M₁₆	C₂×D₃	1	4	yes	complemented
138	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
139	#3756	C₂²×C₃	1	1	yes	characteristic, complemented
140	#3755	C₂²×C₃	1	1	yes	characteristic, complemented
141	#3754	C₂²×C₃	1	2	yes	complemented
142	#3754	C₂²×C₃	1	1	yes	characteristic, complemented
143	#3754	C₂²×C₃	1	2	yes	complemented
144	#3753	C₂²×C₃	1	1	yes	characteristic, complemented
145	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic, complemented
146	#3746	C₂²×C₃	1	2	yes	complemented
147	#3744	C₂²×C₃	1	2	yes	complemented
148	#3743	C₂²×C₃	1	2	yes	complemented
149	C₈×C₃⋊D₃	C₂²×C₃	1	2	yes	complemented
150	C₃×#216	—	3	4	no	—
151	C₃×#215	—	3	4	no	—
152	C₃×#214	—	3	8	no	—
153	C₃×#214	—	3	4	no	—
154	C₃×#214	—	3	8	no	—
155	C₃×#213	—	3	4	no	—
156	C₂×C₃×C₃⋊C₈	—	3	4	no	—
157	C₃×Dic₁₂	—	3	8	no	—
158	C₃×#204	—	3	8	no	—
159	C₃×ΓL(1,𝔽₂₅)	—	3	8	no	—
160	C₃×D₃×C₈	—	3	8	no	—
161	C₂²×C₃³	—	2	1	no	—
162	C₂²×C₃³	—	2	2	no	—
163	C₂²×C₃³	—	8	1	no	—
164	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×C₃⋊D₃	—	2	1	no	—
166	C₂×C₃×C₃⋊D₃	—	8	2	no	—
167	C₂×C₃×C₃⋊D₃	C₂×D₄	1	2	yes	—
168	C₃³×C₄	—	2	1	no	—
169	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
170	C₃³×C₄	C₂⁴	1	1	yes	characteristic
171	C₃×C₃⋊Dic₃	—	2	2	no	—
172	C₃×C₃⋊Dic₃	—	2	1	no	—
173	C₃×C₃⋊Dic₃	C₂×D₄	1	2	yes	—
174	C₃×2₋¹⁺⁴	—	9	1	no	—
175	C₃×H(5,𝔽₂)	—	9	1	no	—
176	#1010	—	3	4	no	—
177	#1004	—	3	4	no	—
178	C₃×#137	—	9	2	no	—
179	C₃×#137	—	9	1	no	—
180	C₃×AGL(1,ℤ/8)	—	2	4	no	—
181	C₃×AGL(1,ℤ/8)	—	9	2	no	—
182	C₃×AGL(1,ℤ/8)	C₃×D₃	1	4	yes	complemented
183	C₃×AGL(1,ℤ/8)	C₃⋊D₃	1	1	yes	characteristic, complemented
184	C₃×#135	—	9	1	no	—
185	C₃×#135	—	9	2	no	—
186	C₂×C₃×SD₁₆	—	9	1	no	—
187	C₂×C₃×SD₁₆	—	9	2	no	—
188	C₃×#131	—	9	1	no	—
189	#950	—	3	4	no	—
190	C₂×#214	—	3	4	no	—
191	#915	—	3	8	no	—
192	D₃×SD₁₆	—	3	8	no	—
193	#912	—	3	8	no	—
194	#911	—	3	8	no	—
195	#909	—	3	4	no	—
196	#907	—	3	4	no	—
197	C₂³×C₃²	—	2	1	no	—
198	C₂³×C₃²	—	2	4	no	—
199	C₂³×C₃²	—	4	4	no	—
200	C₂²×C₃⋊D₃	—	2	2	no	—
201	C₂²×C₃×D₃	—	6	8	no	—
202	C₃²×Q₈	—	2	4	no	—
203	C₃²×Q₈	C₂²×D₃	1	4	yes	—
204	C₃²×Q₈	C₂³×C₃	1	1	yes	characteristic
205	C₃²×D₄	—	2	1	no	—
206	C₃²×D₄	—	2	1	no	—
207	C₃²×D₄	—	2	4	no	—
208	C₃²×D₄	—	2	8	no	—
209	C₃²×D₄	—	2	4	no	—
210	C₃²×D₄	—	2	4	no	—
211	C₃²×D₄	—	4	4	no	—
212	C₃²×D₄	—	4	4	no	—
213	C₃²×D₄	C₂²×D₃	1	4	yes	—
214	C₃²×D₄	C₂²×D₃	1	8	yes	—
215	C₃²×D₄	C₂³×C₃	1	1	yes	characteristic
216	C₃²×D₄	C₂³×C₃	1	2	yes	—
217	C₂×C₃²×C₄	—	2	1	no	—
218	C₂×C₃²×C₄	—	2	4	no	—
219	C₂×C₃²×C₄	—	2	4	no	—
220	C₂×C₃²×C₄	—	4	4	no	—
221	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
222	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
223	#638	—	2	1	no	—
224	#638	—	2	2	no	—
225	#638	—	2	2	no	—
226	#638	—	2	2	no	—
227	#638	C₃×D₄	1	2	yes	—
228	C₂×C₃⋊Dic₃	—	2	1	no	—
229	C₂×C₃⋊Dic₃	—	2	2	no	—
230	C₂×C₃⋊Dic₃	—	2	2	no	—
231	C₄⋊(C₃⋊D₃)	C₃×D₄	1	1	yes	characteristic
232	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	1	yes	characteristic
233	C₄×C₃⋊D₃	—	2	2	no	—
234	C₄×C₃⋊D₃	—	2	1	no	—
235	C₄×C₃⋊D₃	C₂³×C₃	1	2	yes	—
236	#634	—	2	1	no	—
237	#634	—	2	2	no	—
238	#634	C₃×D₄	1	1	yes	characteristic
239	#634	C₂³×C₃	1	2	yes	—
240	#634	C₂³×C₃	1	1	yes	characteristic
241	C₃×#67	—	3	8	no	—
242	C₃×#67	—	6	8	no	—
243	C₃×#67	—	6	8	no	—
244	C₃×#67	—	6	8	no	—
245	C₃×#67	—	6	4	no	—
246	C₂×C₃×Dic₃	—	6	8	no	—
247	C₂×C₃×Dic₃	—	6	8	no	—
248	C₂×C₃×Dic₃	—	6	4	no	—
249	C₃×D₁₂	—	3	4	no	—
250	C₃×D₁₂	—	3	4	no	—
251	C₃×C₄×D₃	—	3	8	no	—
252	C₃×C₄×D₃	—	6	8	no	—
253	C₃×C₄×D₃	—	6	4	no	—
254	C₃×Dic₆	—	3	4	no	—
255	C₃×Dic₆	—	3	8	no	—
256	C₃×Dic₆	—	3	4	no	—
257	C₃×Dic₆	—	6	8	no	—
258	C₃×Dic₆	—	6	4	no	—
259	C₃²×C₈	—	2	8	no	—
260	C₃²×C₈	C₂²×D₃	1	8	yes	—
261	C₃²×C₈	C₂³×C₃	1	2	yes	—
262	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	1	yes	characteristic
263	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	1	yes	characteristic
264	C₃×C₃⋊C₈	—	3	4	no	—
265	C₃×C₃⋊C₈	—	3	4	no	—
266	#577	—	9	1	no	Sylow
267	C₂×C₃³	—	2	1	no	—
268	C₂×C₃³	—	4	2	no	—
269	C₂×C₃³	—	4	1	no	—
270	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
271	C₃×C₃⋊D₃	—	2	2	no	—
272	C₃×C₃⋊D₃	—	4	1	no	—
273	C₃×#41	—	2	4	no	—
274	C₃×#41	—	9	1	no	—
275	C₃×#41	—	9	2	no	—
276	C₃×#41	—	9	2	no	—
277	C₃×#41	—	9	1	no	—
278	C₃×#41	—	18	1	no	—
279	C₃×#41	—	18	2	no	—
280	C₃×#41	—	18	1	no	—
281	C₃×#41	C₂×C₃×D₃	1	4	yes	complemented
282	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
283	C₂×C₃×Q₈	—	9	2	no	—
284	C₂×C₃×Q₈	—	9	1	no	—
285	C₂×C₃×Q₈	—	18	1	no	—
286	C₂×C₃×D₄	—	2	4	no	—
287	C₂×C₃×D₄	—	9	2	no	—
288	C₂×C₃×D₄	—	18	1	no	—
289	C₂×C₃×D₄	—	18	2	no	—
290	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
291	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
292	C₂×#67	—	6	8	no	—
293	D₃×Q₈	—	3	8	no	—
294	#237	—	3	8	no	—
295	#237	—	3	8	no	—
296	#237	—	6	4	no	—
297	#237	—	6	8	no	—
298	D₃×D₄	—	3	8	no	—
299	D₃×D₄	—	6	4	no	—
300	#235	—	3	4	no	—
301	#235	—	3	4	no	—
302	#235	—	6	4	no	—
303	C₂×Dic₆	—	3	4	no	—
304	C₂×Dic₆	—	6	4	no	—
305	C₃×Q₁₆	—	9	1	no	—
306	C₃×Q₁₆	—	9	2	no	—
307	C₃×SD₁₆	—	2	8	no	—
308	C₃×SD₁₆	—	9	1	no	—
309	C₃×SD₁₆	—	9	2	no	—
310	C₃×SD₁₆	—	9	2	no	—
311	C₃×SD₁₆	—	9	1	no	—
312	C₃×SD₁₆	—	9	2	no	—
313	C₃×SD₁₆	C₂×C₃×D₃	1	8	yes	complemented
314	C₃×SD₁₆	C₂×C₃⋊D₃	1	2	yes	complemented
315	C₃×D₈	—	2	8	no	—
316	C₃×D₈	—	9	1	no	—
317	C₃×D₈	C₂×C₃×D₃	1	8	yes	complemented
318	C₃×D₈	C₂×C₃⋊D₃	1	2	yes	complemented
319	C₃×M₁₆	—	2	4	no	—
320	C₃×M₁₆	—	9	2	no	—
321	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	complemented
322	C₃×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
323	C₂×C₃×C₈	—	9	1	no	—
324	C₂×C₃×C₈	—	9	2	no	—
325	#216	—	3	4	no	—
326	#215	—	3	4	no	—
327	#214	—	3	8	no	—
328	#214	—	3	4	no	—
329	#214	—	3	8	no	—
330	#213	—	3	4	no	—
331	C₂×C₃⋊C₈	—	3	4	no	—
332	Dic₁₂	—	3	8	no	—
333	#204	—	3	8	no	—
334	ΓL(1,𝔽₂₅)	—	3	8	no	—
335	D₃×C₈	—	3	8	no	—
336	C₂²×C₃²	—	2	1	no	—
337	C₂²×C₃²	—	2	2	no	—
338	C₂²×C₃²	—	2	4	no	—
339	C₂²×C₃²	—	2	4	no	—
340	C₂²×C₃²	—	2	8	no	—
341	C₂²×C₃²	—	4	4	no	—
342	C₂²×C₃²	—	4	8	no	—
343	C₂²×C₃²	—	8	1	no	—
344	C₂²×C₃²	—	8	4	no	—
345	C₂²×C₃²	—	8	8	no	—
346	C₂²×C₃²	D₃×D₄	1	4	yes	—
347	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
348	C₂×C₃⋊D₃	—	2	1	no	—
349	C₂×C₃⋊D₃	—	8	2	no	—
350	C₂×C₃⋊D₃	C₂×C₃×D₄	1	2	yes	—
351	C₂×C₃×D₃	—	3	8	no	—
352	C₂×C₃×D₃	—	6	4	no	—
353	C₂×C₃×D₃	—	24	8	no	—
354	C₃²×C₄	—	2	1	no	—
355	C₃²×C₄	—	2	4	no	—
356	C₃²×C₄	—	2	4	no	—
357	C₃²×C₄	—	2	4	no	—
358	C₃²×C₄	—	4	4	no	—
359	C₃²×C₄	D₃×D₄	1	4	yes	—
360	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
361	C₃²×C₄	C₂³×D₃	1	4	yes	—
362	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
363	C₃⋊Dic₃	—	2	2	no	—
364	C₃⋊Dic₃	—	2	1	no	—
365	C₃⋊Dic₃	C₂×C₃×D₄	1	2	yes	—
366	C₃×Dic₃	—	3	8	no	—
367	C₃×Dic₃	—	6	8	no	—
368	C₃×Dic₃	—	6	4	no	—
369	2₋¹⁺⁴	—	9	1	no	—
370	H(5,𝔽₂)	—	9	1	no	—
371	#137	—	9	2	no	—
372	#137	—	9	1	no	—
373	AGL(1,ℤ/8)	—	9	2	no	—
374	AGL(1,ℤ/8)	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
375	#135	—	9	1	no	—
376	#135	—	9	2	no	—
377	C₂×SD₁₆	—	9	1	no	—
378	C₂×SD₁₆	—	9	2	no	—
379	#131	—	9	1	no	—
380	C₃³	#577	1	1	yes	characteristic, complemented, Sylow
381	C₂³×C₃	—	2	4	no	—
382	C₂³×C₃	—	2	1	no	—
383	C₂³×C₃	—	4	4	no	—
384	C₂³×C₃	—	18	2	no	—
385	C₂²×D₃	—	6	8	no	—
386	C₃×Q₈	—	2	4	no	—
387	C₃×Q₈	—	9	1	no	—
388	C₃×Q₈	—	9	2	no	—
389	C₃×Q₈	—	9	1	no	—
390	C₃×Q₈	—	18	2	no	—
391	C₃×Q₈	—	18	1	no	—
392	C₃×Q₈	C₂²×C₃×D₃	1	4	yes	—
393	C₃×Q₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
394	C₃×D₄	—	2	4	no	—
395	C₃×D₄	—	2	1	no	—
396	C₃×D₄	—	2	4	no	—
397	C₃×D₄	—	2	1	no	—
398	C₃×D₄	—	2	4	no	—
399	C₃×D₄	—	2	8	no	—
400	C₃×D₄	—	4	4	no	—
401	C₃×D₄	—	4	4	no	—
402	C₃×D₄	—	9	2	no	—
403	C₃×D₄	—	9	1	no	—
404	C₃×D₄	—	9	1	no	—
405	C₃×D₄	—	18	2	no	—
406	C₃×D₄	—	18	2	no	—
407	C₃×D₄	—	18	2	no	—
408	C₃×D₄	—	18	1	no	—
409	C₃×D₄	C₂²×C₃×D₃	1	4	yes	—
410	C₃×D₄	C₂²×C₃×D₃	1	8	yes	—
411	C₃×D₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
412	C₃×D₄	C₂²×C₃⋊D₃	1	2	yes	—
413	C₂×C₃×C₄	—	2	4	no	—
414	C₂×C₃×C₄	—	2	1	no	—
415	C₂×C₃×C₄	—	2	4	no	—
416	C₂×C₃×C₄	—	4	4	no	—
417	C₂×C₃×C₄	—	9	2	no	—
418	C₂×C₃×C₄	—	18	2	no	—
419	C₂×C₃×C₄	—	18	2	no	—
420	C₂×C₃×C₄	—	18	2	no	—
421	C₂×C₃×C₄	—	18	1	no	—
422	C₂×C₃×C₄	—	18	1	no	—
423	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
424	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
425	#67	—	3	8	no	—
426	#67	—	6	8	no	—
427	#67	—	6	8	no	—
428	#67	—	6	8	no	—
429	#67	—	6	4	no	—
430	C₂×Dic₃	—	6	8	no	—
431	C₂×Dic₃	—	6	8	no	—
432	C₂×Dic₃	—	6	4	no	—
433	D₁₂	—	3	4	no	—
434	D₁₂	—	3	4	no	—
435	C₄×D₃	—	3	8	no	—
436	C₄×D₃	—	6	8	no	—
437	C₄×D₃	—	6	4	no	—
438	Dic₆	—	3	4	no	—
439	Dic₆	—	3	8	no	—
440	Dic₆	—	3	4	no	—
441	Dic₆	—	6	8	no	—
442	Dic₆	—	6	4	no	—
443	C₃×C₈	—	2	8	no	—
444	C₃×C₈	—	9	1	no	—
445	C₃×C₈	—	9	1	no	—
446	C₃×C₈	C₂²×C₃×D₃	1	8	yes	—
447	C₃×C₈	C₂²×C₃⋊D₃	1	2	yes	—
448	C₃⋊C₈	—	3	4	no	—
449	C₃⋊C₈	—	3	4	no	—
450	C₂×C₃²	—	2	1	no	—
451	C₂×C₃²	—	2	4	no	—
452	C₂×C₃²	—	2	4	no	—
453	C₂×C₃²	—	4	4	no	—
454	C₂×C₃²	—	4	2	no	—
455	C₂×C₃²	—	4	1	no	—
456	C₂×C₃²	—	4	8	no	—
457	C₂×C₃²	—	4	4	no	—
458	C₂×C₃²	—	8	4	no	—
459	C₂×C₃²	—	8	8	no	—
460	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
461	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
462	C₃⋊D₃	—	2	2	no	—
463	C₃⋊D₃	—	4	1	no	—
464	C₃×D₃	—	6	8	no	—
465	C₃×D₃	—	12	4	no	—
466	#41	—	9	1	no	—
467	#41	—	9	2	no	—
468	#41	—	9	2	no	—
469	#41	—	9	1	no	—
470	#41	—	18	1	no	—
471	#41	—	18	2	no	—
472	#41	—	18	1	no	—
473	#41	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
474	C₂×Q₈	—	9	2	no	—
475	C₂×Q₈	—	9	1	no	—
476	C₂×Q₈	—	18	1	no	—
477	C₂×D₄	—	9	2	no	—
478	C₂×D₄	—	18	1	no	—
479	C₂×D₄	—	18	2	no	—
480	C₂×D₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
481	Q₁₆	—	9	1	no	—
482	Q₁₆	—	9	2	no	—
483	SD₁₆	—	9	1	no	—
484	SD₁₆	—	9	2	no	—
485	SD₁₆	—	9	2	no	—
486	SD₁₆	—	9	1	no	—
487	SD₁₆	—	9	2	no	—
488	SD₁₆	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
489	D₈	—	9	1	no	—
490	D₈	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
491	M₁₆	—	9	2	no	—
492	M₁₆	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
493	C₂×C₈	—	9	1	no	—
494	C₂×C₈	—	9	2	no	—
495	C₂²×C₃	—	2	4	no	—
496	C₂²×C₃	—	2	8	no	—
497	C₂²×C₃	—	2	1	no	—
498	C₂²×C₃	—	2	2	no	—
499	C₂²×C₃	—	2	4	no	—
500	C₂²×C₃	—	4	4	no	—
501	C₂²×C₃	—	4	8	no	—
502	C₂²×C₃	—	8	4	no	—
503	C₂²×C₃	—	8	1	no	—
504	C₂²×C₃	—	8	8	no	—
505	C₂²×C₃	—	9	2	no	—
506	C₂²×C₃	—	18	1	no	—
507	C₂²×C₃	—	72	2	no	—
508	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
509	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
510	C₂×D₃	—	3	8	no	—
511	C₂×D₃	—	6	4	no	—
512	C₂×D₃	—	24	8	no	—
513	C₃×C₄	—	2	4	no	—
514	C₃×C₄	—	2	1	no	—
515	C₃×C₄	—	2	4	no	—
516	C₃×C₄	—	2	4	no	—
517	C₃×C₄	—	4	4	no	—
518	C₃×C₄	—	9	2	no	—
519	C₃×C₄	—	18	1	no	—
520	C₃×C₄	—	18	2	no	—
521	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
522	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
523	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
524	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
525	Dic₃	—	3	8	no	—
526	Dic₃	—	6	8	no	—
527	Dic₃	—	6	4	no	—
528	C₃²	—	2	4	no	—
529	C₃²	#5775	1	4	yes	complemented
530	C₃²	C₃×#577	1	1	yes	characteristic, complemented
531	C₂³	—	2	1	no	—
532	C₂³	—	18	2	no	—
533	Q₈	—	9	1	no	—
534	Q₈	—	9	2	no	—
535	Q₈	—	9	1	no	—
536	Q₈	—	18	2	no	—
537	Q₈	—	18	1	no	—
538	Q₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
539	D₄	—	2	1	no	—
540	D₄	—	2	1	no	—
541	D₄	—	9	2	no	—
542	D₄	—	9	1	no	—
543	D₄	—	9	1	no	—
544	D₄	—	18	2	no	—
545	D₄	—	18	2	no	—
546	D₄	—	18	2	no	—
547	D₄	—	18	1	no	—
548	D₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
549	D₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	9	2	no	—
552	C₂×C₄	—	18	2	no	—
553	C₂×C₄	—	18	2	no	—
554	C₂×C₄	—	18	2	no	—
555	C₂×C₄	—	18	1	no	—
556	C₂×C₄	—	18	1	no	—
557	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
558	C₈	—	9	1	no	—
559	C₈	—	9	1	no	—
560	C₈	C₂²×C₃×C₃⋊D₃	1	2	yes	—
561	C₂×C₃	—	2	4	no	—
562	C₂×C₃	—	2	1	no	—
563	C₂×C₃	—	2	4	no	—
564	C₂×C₃	—	4	4	no	—
565	C₂×C₃	—	4	8	no	—
566	C₂×C₃	—	4	4	no	—
567	C₂×C₃	—	4	2	no	—
568	C₂×C₃	—	4	1	no	—
569	C₂×C₃	—	8	8	no	—
570	C₂×C₃	—	8	4	no	—
571	C₂×C₃	—	18	2	no	—
572	C₂×C₃	—	36	1	no	—
573	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
574	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
575	D₃	—	6	8	no	—
576	D₃	—	12	4	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	9	2	no	—
581	C₂²	—	18	1	no	—
582	C₂²	—	72	2	no	—
583	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
584	C₄	—	2	1	no	—
585	C₄	—	9	2	no	—
586	C₄	—	18	1	no	—
587	C₄	—	18	2	no	—
588	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
589	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
590	C₃	—	2	4	no	—
591	C₃	C₃×#5775	1	4	yes	complemented
592	C₃	#10596246	1	1	yes	characteristic, direct product factor
593	C₂	—	2	1	no	—
594	C₂	—	4	2	no	—
595	C₂	—	4	1	no	—
596	C₂	—	18	2	no	—
597	C₂	—	36	1	no	—
598	C₂	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×AGL(1,ℤ/8)	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃×AGL(1,ℤ/8)	C₃⋊D₃
hypocenter	C₃²	C₃×#577
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. C₃×#64367	C₂	semidirect
2. C₃×#64361	C₂	semidirect
3. C₃³×AGL(1,ℤ/8)	C₂	semidirect
4. C₃×#64159	C₂	semidirect
5. C₂×C₃×#3754	C₂	semidirect
6. C₃×#64124	C₂	semidirect
7. C₃×SD₁₆×C₃⋊D₃	C₂	semidirect
8. C₃×#64121	C₂	semidirect
9. C₃×#64120	C₂	semidirect
10. C₃×#64118	C₂	semidirect
11. C₃×#64116	C₂	semidirect
12. #10596246	C₃	direct (N × H)
14. C₃³×#41	C₂²	semidirect
17. C₃×Q₈×C₃⋊D₃	C₂²	semidirect
21. C₃×#3830	C₂²	semidirect
28. C₂×C₃×#634	C₂²	semidirect
29. C₃³×SD₁₆	C₂²	semidirect
30. C₃³×D₈	C₂²	semidirect
31. C₃³×M₁₆	C₂²	semidirect
32. C₃×#3756	C₂²	semidirect
33. C₃×#3755	C₂²	semidirect
34. C₃×#3754	C₂²	semidirect
35. C₃×#3754	C₂²	semidirect
36. C₃×#3754	C₂²	semidirect
37. C₃×#3753	C₂²	semidirect
38. C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	semidirect
39. C₃×#3746	C₂²	semidirect
40. C₃×#3744	C₂²	semidirect
41. C₃×#3743	C₂²	semidirect
42. C₃×C₈×C₃⋊D₃	C₂²	semidirect
43. #64367	C₂×C₃	semidirect
44. #64361	C₂×C₃	semidirect
48. C₃²×AGL(1,ℤ/8)	D₃	semidirect
49. C₃²×AGL(1,ℤ/8)	C₂×C₃	semidirect
50. #64159	C₂×C₃	semidirect
51. C₂×#3754	C₂×C₃	semidirect
52. #64124	C₂×C₃	semidirect
53. SD₁₆×C₃⋊D₃	C₂×C₃	semidirect
54. #64121	C₂×C₃	semidirect
55. #64120	C₂×C₃	semidirect
56. #64118	C₂×C₃	semidirect
57. #64116	C₂×C₃	semidirect
99. C₃²×#41	C₂×D₃	semidirect
100. C₃²×#41	C₂²×C₃	semidirect
105. Q₈×C₃⋊D₃	C₂²×C₃	semidirect
109. #3830	C₂²×C₃	semidirect
116. C₂×#634	C₂²×C₃	semidirect
131. C₃²×SD₁₆	C₂×D₃	semidirect
132. C₃²×SD₁₆	C₂²×C₃	semidirect
134. C₃²×D₈	C₂×D₃	semidirect
135. C₃²×D₈	C₂²×C₃	semidirect
137. C₃²×M₁₆	C₂×D₃	semidirect
138. C₃²×M₁₆	C₂²×C₃	semidirect
139. #3756	C₂²×C₃	semidirect
140. #3755	C₂²×C₃	semidirect
141. #3754	C₂²×C₃	semidirect
142. #3754	C₂²×C₃	semidirect
143. #3754	C₂²×C₃	semidirect
144. #3753	C₂²×C₃	semidirect
145. C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	semidirect
146. #3746	C₂²×C₃	semidirect
147. #3744	C₂²×C₃	semidirect
148. #3743	C₂²×C₃	semidirect
149. C₈×C₃⋊D₃	C₂²×C₃	semidirect
182. C₃×AGL(1,ℤ/8)	C₃×D₃	semidirect
183. C₃×AGL(1,ℤ/8)	C₃⋊D₃	semidirect
281. C₃×#41	C₂×C₃×D₃	semidirect
282. C₃×#41	C₂×C₃⋊D₃	semidirect
313. C₃×SD₁₆	C₂×C₃×D₃	semidirect
314. C₃×SD₁₆	C₂×C₃⋊D₃	semidirect
317. C₃×D₈	C₂×C₃×D₃	semidirect
318. C₃×D₈	C₂×C₃⋊D₃	semidirect
321. C₃×M₁₆	C₂×C₃×D₃	semidirect
322. C₃×M₁₆	C₂×C₃⋊D₃	semidirect
374. AGL(1,ℤ/8)	C₃×C₃⋊D₃	semidirect
380. C₃³	#577	semidirect
473. #41	C₂×C₃×C₃⋊D₃	semidirect
488. SD₁₆	C₂×C₃×C₃⋊D₃	semidirect
490. D₈	C₂×C₃×C₃⋊D₃	semidirect
492. M₁₆	C₂×C₃×C₃⋊D₃	semidirect
529. C₃²	#5775	semidirect
530. C₃²	C₃×#577	semidirect
591. C₃	C₃×#5775	semidirect
592. C₃	#10596246	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×#64367	2
C₃×#64361	2
C₃³×AGL(1,ℤ/8)	2
C₃×#64159	2
C₂×C₃×#3754	2
C₃×#64124	2
C₃×SD₁₆×C₃⋊D₃	2
C₃×#64121	2
C₃×#64120	2
C₃×#64118	2
C₃×#64116	2
#10596246	3
C₃×#5775	3

Subgroups — C3×#10596246 #13179526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596246 #13179526