Subgroups — #11777813

Order 1152.

393 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592387	C₂	1	1	yes	maximal, characteristic
2	C₃×#4679	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#3717	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589084	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589082	C₂	1	1	yes	maximal, characteristic
8	#70950	—	3	1	no	maximal
9	#68055	—	3	1	no	maximal
10	C₂×#3743	—	2	1	no	—
11	C₄×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₃×C₄×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×#3717	C₄	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#63647	—	2	1	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
17	C₃×#815	—	2	1	no	—
18	C₃×Dic₃⋊C₈	—	2	1	no	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#63573	C₄	1	1	yes	characteristic
23	#63566	C₄	1	1	yes	characteristic
24	#63564	C₄	1	1	yes	characteristic
25	C₃×#446	—	3	1	no	—
26	C₄×#215	—	3	1	no	—
27	C₄×#204	—	3	1	no	—
28	#4688	—	3	1	no	—
29	#4688	—	3	1	no	—
30	#4688	—	6	1	no	—
31	#4679	D₃	1	1	yes	characteristic, complemented
32	C₃×C₈²	—	3	1	no	—
33	#4535	—	3	1	no	—
34	#4483	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4459	—	3	1	no	—
37	#4457	—	3	1	no	—
38	#4456	—	3	1	no	—
39	C₈×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
40	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃⋊D₃	—	2	1	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
45	C₃²×C₄²	C₂³	1	1	yes	characteristic
46	#3752	—	2	1	no	—
47	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	#3743	—	4	2	no	—
50	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×Dic₃⋊C₄	—	2	1	no	—
52	C₃×C₄×Dic₃	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
57	#3717	—	2	2	no	—
58	#1590	—	9	1	no	Sylow
59	C₃×C₄×Q₈	—	3	1	no	—
60	C₂×#215	—	3	1	no	—
61	C₂×#204	—	3	1	no	—
62	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
63	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
64	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
65	C₄×D₁₂	—	3	1	no	—
66	C₄×D₁₂	—	3	1	no	—
67	C₄×D₁₂	—	6	1	no	—
68	C₄×Dic₆	C₂×D₃	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	3	1	no	—
70	C₃×C₄⋊C₈	—	6	1	no	—
71	C₃×#97	—	6	1	no	—
72	C₃×C₄×C₈	—	2	1	no	—
73	C₃×C₄×C₈	—	3	1	no	—
74	C₃×C₄×C₈	—	3	1	no	—
75	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
77	#835	—	3	1	no	—
78	#821	—	3	1	no	—
79	(C₂×D₃)⋊C₈	—	6	2	no	—
80	(C₂×D₃)⋊C₈	—	6	1	no	—
81	(C₂×D₃)⋊C₈	—	6	1	no	—
82	#817	—	3	1	no	—
83	#816	—	3	1	no	—
84	#815	—	2	1	no	—
85	Dic₃⋊C₈	—	2	1	no	—
86	#809	—	3	1	no	—
87	#808	—	3	1	no	—
88	C₄⋊(C₃⋊C₈)	—	3	1	no	—
89	C₄⋊(C₃⋊C₈)	—	3	1	no	—
90	C₄⋊(C₃⋊C₈)	—	6	1	no	—
91	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
93	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
94	C₂²×C₃⋊D₃	—	2	1	no	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	#41	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	—	2	1	no	—
99	C₄⋊(C₃⋊D₃)	—	2	1	no	—
100	C₄⋊(C₃⋊D₃)	—	2	1	no	—
101	C₄×C₃⋊D₃	—	4	1	no	—
102	C₄×C₃⋊D₃	—	4	1	no	—
103	C₂×C₃×Dic₃	—	2	1	no	—
104	C₂×C₃×Dic₃	—	2	1	no	—
105	C₃×Dic₆	—	2	1	no	—
106	C₃×Dic₆	—	2	1	no	—
107	C₃²×C₈	—	2	1	no	—
108	C₃²×C₈	SD₁₆	1	2	yes	complemented
109	C₃⋊(C₃⋊C₈)	—	4	1	no	—
110	C₃×C₃⋊C₈	—	2	2	no	—
111	C₃×C₃⋊C₈	—	2	1	no	—
112	C₃×C₃⋊C₈	—	4	1	no	—
113	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
114	#446	—	3	1	no	—
115	C₄×SD₁₆	—	9	1	no	—
116	#436	—	9	1	no	—
117	#334	—	9	1	no	—
118	Q₈⋊C₈	—	9	1	no	—
119	D₄⋊C₈	—	9	1	no	—
120	C₈²	—	3	1	no	—
121	C₂×C₃×Q₈	—	3	1	no	—
122	C₂×D₁₂	—	3	1	no	—
123	C₂×D₁₂	—	3	1	no	—
124	C₂×D₁₂	—	6	1	no	—
125	C₂×C₄×D₃	—	6	2	no	—
126	C₂×C₄×D₃	—	6	1	no	—
127	C₂×C₄×D₃	—	6	1	no	—
128	C₂×Dic₆	C₄×D₃	1	1	yes	characteristic
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	3	1	no	—
132	C₂×C₃×C₈	—	3	1	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₂×C₃×C₈	—	6	1	no	—
135	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
136	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
137	C₂×C₃×C₈	#67	1	1	yes	characteristic
138	C₂×C₃×C₈	#67	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	3	1	no	—
140	C₃×C₄⋊C₄	—	6	1	no	—
141	C₃×C₄²	—	2	1	no	—
142	C₃×C₄²	—	6	1	no	—
143	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
144	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
145	#215	—	6	2	no	—
146	(C₂×D₃)⋊C₄	—	6	2	no	—
147	(C₂×D₃)⋊C₄	—	6	1	no	—
148	(C₂×D₃)⋊C₄	—	6	1	no	—
149	C₄⋊Dic₃	—	3	1	no	—
150	C₄⋊Dic₃	—	3	1	no	—
151	C₄⋊Dic₃	—	6	1	no	—
152	C₄⋊Dic₃	C₄×D₃	1	1	yes	characteristic
153	Dic₃⋊C₄	—	2	1	no	—
154	C₄×Dic₃	—	2	1	no	—
155	C₂×C₃⋊C₈	—	2	1	no	—
156	C₂×C₃⋊C₈	—	2	1	no	—
157	C₂×C₃⋊C₈	—	6	2	no	—
158	C₂×C₃⋊C₈	—	6	1	no	—
159	C₂×C₃⋊C₈	—	6	1	no	—
160	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
161	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
162	#204	—	6	2	no	—
163	ΓL(1,𝔽₂₅)	—	12	4	no	—
164	ΓL(1,𝔽₂₅)	—	12	2	no	—
165	ΓL(1,𝔽₂₅)	—	12	2	no	—
166	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
167	C₂×C₃⋊D₃	—	4	1	no	—
168	C₂×C₃⋊D₃	—	4	1	no	—
169	C₂×C₃⋊D₃	—	4	1	no	—
170	C₃²×C₄	—	2	1	no	—
171	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
172	C₃²×C₄	#131	1	1	yes	characteristic
173	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
174	C₃²×C₄	#135	1	1	yes	characteristic
175	C₃⋊Dic₃	—	4	1	no	—
176	C₃×Dic₃	—	4	1	no	—
177	C₃×Dic₃	—	4	1	no	—
178	C₂×SD₁₆	—	9	1	no	—
179	C₂×M₁₆	—	18	1	no	—
180	C₄×Q₈	—	3	1	no	—
181	C₄×D₄	—	9	1	no	—
182	#106	—	9	1	no	—
183	C₄⋊C₈	—	3	1	no	—
184	C₄⋊C₈	—	6	1	no	—
185	C₄⋊C₈	—	9	1	no	—
186	#103	—	9	1	no	—
187	#102	—	9	1	no	—
188	C₂²⋊C₈	—	18	1	no	—
189	#97	—	6	1	no	—
190	C₄×C₈	—	3	1	no	—
191	C₄×C₈	—	3	1	no	—
192	C₄×C₈	D₃²	1	1	yes	characteristic
193	C₂²×D₃	—	6	2	no	—
194	C₂²×D₃	—	6	1	no	—
195	C₂²×D₃	—	6	1	no	—
196	C₃×Q₈	—	6	1	no	—
197	C₃×Q₈	—	6	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	6	1	no	—
202	C₂×C₃×C₄	—	6	1	no	—
203	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₃×C₄	#235	1	1	yes	characteristic
207	C₂×C₃×C₄	#235	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
209	C₂×Dic₃	—	2	1	no	—
210	C₂×Dic₃	—	2	1	no	—
211	C₂×Dic₃	—	6	2	no	—
212	C₂×Dic₃	—	6	1	no	—
213	C₂×Dic₃	—	6	1	no	—
214	D₁₂	—	6	1	no	—
215	D₁₂	—	6	1	no	—
216	D₁₂	—	6	1	no	—
217	D₁₂	—	6	1	no	—
218	D₁₂	—	12	1	no	—
219	D₁₂	—	12	1	no	—
220	C₄×D₃	—	12	2	no	—
221	C₄×D₃	—	12	2	no	—
222	C₄×D₃	—	12	1	no	—
223	C₄×D₃	—	12	1	no	—
224	C₄×D₃	—	12	1	no	—
225	C₄×D₃	—	12	1	no	—
226	Dic₆	—	2	1	no	—
227	Dic₆	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	2	no	—
231	C₃×C₈	—	3	2	no	—
232	C₃×C₈	—	4	1	no	—
233	C₃×C₈	—	6	2	no	—
234	C₃×C₈	—	6	1	no	—
235	C₃×C₈	—	12	1	no	—
236	C₃×C₈	#204	1	2	yes	complemented
237	C₃×C₈	#215	1	2	yes	complemented
238	C₃⋊C₈	—	2	2	no	—
239	C₃⋊C₈	—	2	1	no	—
240	C₃⋊C₈	—	4	1	no	—
241	C₃⋊C₈	—	12	2	no	—
242	C₃⋊C₈	—	12	1	no	—
243	C₃⋊C₈	—	12	1	no	—
244	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
245	C₂×C₃²	#436	1	1	yes	characteristic
246	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
247	C₂×C₃²	#443	1	1	yes	characteristic
248	C₃⋊D₃	—	8	1	no	—
249	C₂×Q₈	—	3	1	no	—
250	C₂×D₄	—	9	1	no	—
251	C₂²×C₄	—	18	1	no	—
252	SD₁₆	—	18	2	no	—
253	M₁₆	—	36	2	no	—
254	C₂×C₈	—	3	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	—	18	1	no	—
259	C₂×C₈	#626	1	1	yes	characteristic
260	C₂×C₈	#626	1	1	yes	characteristic
261	C₄⋊C₄	—	3	1	no	—
262	C₄⋊C₄	—	6	1	no	—
263	C₄⋊C₄	—	9	1	no	—
264	C₂²⋊C₄	—	18	1	no	—
265	C₄²	—	6	1	no	—
266	C₄²	C₂×D₃²	1	1	yes	characteristic
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
269	C₂²×C₃	C₄×#67	1	1	yes	characteristic
270	C₂×D₃	—	12	2	no	—
271	C₂×D₃	—	12	2	no	—
272	C₂×D₃	—	12	2	no	—
273	C₂×D₃	—	12	1	no	—
274	C₂×D₃	—	12	1	no	—
275	C₂×D₃	—	12	1	no	—
276	C₂×D₃	—	12	1	no	—
277	C₂×D₃	—	12	1	no	—
278	C₂×D₃	—	12	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	—	12	1	no	—
287	C₃×C₄	—	12	1	no	—
288	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
289	C₃×C₄	#901	1	1	yes	characteristic
290	C₃×C₄	C₂×#204	1	1	yes	characteristic
291	C₃×C₄	#904	1	1	yes	characteristic
292	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
293	C₃×C₄	#907	1	1	yes	characteristic
294	C₃×C₄	C₂×#215	1	1	yes	characteristic
295	C₃×C₄	#950	1	1	yes	characteristic
296	Dic₃	—	4	1	no	—
297	Dic₃	—	4	1	no	—
298	Dic₃	—	12	2	no	—
299	Dic₃	—	12	1	no	—
300	Dic₃	—	12	1	no	—
301	C₃²	#1590	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	18	1	no	—
303	Q₈	—	6	1	no	—
304	Q₈	—	6	1	no	—
305	D₄	—	18	1	no	—
306	D₄	—	18	1	no	—
307	C₂×C₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	18	1	no	—
310	C₂×C₄	—	36	1	no	—
311	C₂×C₄	—	36	1	no	—
312	C₂×C₄	#3798	1	1	yes	characteristic
313	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
314	C₂×C₄	C₂×#626	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	3	2	no	—
317	C₈	—	6	2	no	—
318	C₈	—	6	1	no	—
319	C₈	—	12	1	no	—
320	C₈	—	36	1	no	—
321	C₈	#3717	1	2	yes	complemented
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	#4688	1	1	yes	characteristic
326	C₂×C₃	C₄×#204	1	1	yes	characteristic
327	C₂×C₃	#4700	1	1	yes	characteristic
328	C₂×C₃	C₄×#215	1	1	yes	characteristic
329	C₂×C₃	#5128	1	1	yes	characteristic
330	C₂×C₃	#5144	1	1	yes	characteristic
331	D₃	—	24	2	no	—
332	D₃	—	24	1	no	—
333	D₃	—	24	1	no	—
334	C₂²	—	36	1	no	—
335	C₂²	—	36	1	no	—
336	C₂²	—	36	1	no	—
337	C₂²	C₄×#626	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	12	1	no	—
340	C₄	—	12	1	no	—
341	C₄	—	36	1	no	—
342	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
343	C₄	#63804	1	1	yes	characteristic
344	C₄	#63829	1	1	yes	characteristic
345	C₄	C₂×#3717	1	1	yes	characteristic
346	C₃	—	2	1	no	—
347	C₃	#68055	1	1	yes	characteristic, complemented
348	C₃	#70950	1	1	yes	characteristic, complemented
349	C₂	—	72	1	no	—
350	C₂	C₄×#3717	1	1	yes	characteristic
351	C₂	#10590729	1	1	yes	characteristic
352	C₂	#10590923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1590
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#4679	C₂	semidirect
4. C₃×C₈×C₃⋊C₈	C₂	semidirect
5. #10589097	C₂	semidirect
6. #10589084	C₂	semidirect
31. #4679	D₃	semidirect
39. C₈×C₃⋊C₈	D₃	semidirect
108. C₃²×C₈	SD₁₆	semidirect
113. C₃×C₃⋊C₈	M₁₆	semidirect
236. C₃×C₈	#204	semidirect
237. C₃×C₈	#215	semidirect
244. C₃⋊C₈	ΓL(1,𝔽₂₅)	semidirect
301. C₃²	#1590	semidirect
321. C₈	#3717	semidirect
347. C₃	#68055	semidirect
348. C₃	#70950	semidirect

Maximal subgroups

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

subgroup	index
#10592387	2
C₃×#4679	2
C₄×#3717	2
C₃×C₈×C₃⋊C₈	2
#10589097	2
#10589084	2
#10589082	2
#70950	3
#68055	3