Subgroups — #11777490

Order 1152.

393 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#3755	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#10590338	C₂	1	1	yes	maximal, characteristic
4	C₃×#4535	C₂	1	1	yes	maximal, characteristic, complemented
5	#10589098	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589082	C₂	1	1	yes	maximal, characteristic
7	#10589078	C₂	1	1	yes	maximal, characteristic, complemented
8	#70933	—	3	1	no	maximal
9	#68217	—	3	1	no	maximal
10	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₂×#3755	C₄	1	1	yes	characteristic
12	C₄×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
13	C₂×#3712	—	2	1	no	—
14	#63662	C₄	1	1	yes	characteristic
15	#63636	C₄	1	1	yes	characteristic
16	#63635	C₄	1	1	yes	characteristic
17	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊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	—	2	1	no	—
22	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#63557	—	2	1	no	—
25	C₃×C₈×Q₈	—	3	1	no	—
26	C₄×#215	—	3	1	no	—
27	C₄×#215	—	3	1	no	—
28	C₄×#215	—	6	1	no	—
29	Q₈×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
30	#4838	—	3	1	no	—
31	#4688	—	3	1	no	—
32	C₃×Q₈⋊C₈	—	3	1	no	—
33	#4535	D₃	1	1	yes	characteristic, complemented
34	#4483	—	3	1	no	—
35	#4480	—	3	1	no	—
36	#4480	—	3	1	no	—
37	#4459	—	3	1	no	—
38	#4455	—	3	1	no	—
39	#4454	—	3	1	no	—
40	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃⋊D₃	—	2	1	no	—
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₃²×C₄²	—	2	1	no	—
46	C₃²×C₄²	C₂³	1	1	yes	characteristic
47	#3755	—	2	2	no	—
48	#3752	—	2	1	no	—
49	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
57	#3712	—	4	2	no	—
58	#1583	—	9	1	no	Sylow
59	C₃×C₄×Q₈	—	2	1	no	—
60	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
61	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
62	C₂×#215	—	3	1	no	—
63	C₂×#215	—	3	1	no	—
64	C₂×#215	—	6	1	no	—
65	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
66	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
67	C₄×D₁₂	—	3	1	no	—
68	C₄×D₁₂	—	3	1	no	—
69	C₄×D₁₂	—	6	1	no	—
70	C₃×C₄⋊C₈	—	3	1	no	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	—	6	1	no	—
73	C₃×C₄×C₈	—	3	1	no	—
74	C₃×C₄×C₈	—	3	1	no	—
75	C₃×C₄×C₈	—	6	1	no	—
76	#835	—	3	1	no	—
77	#835	—	3	1	no	—
78	#835	—	6	1	no	—
79	(C₂×D₃)⋊C₈	—	6	1	no	—
80	(C₂×D₃)⋊C₈	—	6	1	no	—
81	#809	—	3	1	no	—
82	#809	—	3	1	no	—
83	#809	—	6	1	no	—
84	#808	—	3	1	no	—
85	#808	—	3	1	no	—
86	#808	—	6	1	no	—
87	C₄⋊(C₃⋊C₈)	—	2	1	no	—
88	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
89	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
90	C₄×C₃⋊C₈	—	2	1	no	—
91	C₄×C₃⋊C₈	—	3	1	no	—
92	C₄×C₃⋊C₈	—	3	1	no	—
93	C₄×C₃⋊C₈	—	6	1	no	—
94	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
95	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
96	C₂²×C₃⋊D₃	—	2	1	no	—
97	C₃²×Q₈	—	2	1	no	—
98	C₃²×Q₈	M₁₆	1	2	yes	complemented
99	C₂×C₃²×C₄	—	2	1	no	—
100	C₂×C₃²×C₄	—	2	1	no	—
101	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	#41	1	1	yes	characteristic
104	C₂×C₃⋊Dic₃	—	2	1	no	—
105	C₄⋊(C₃⋊D₃)	—	2	1	no	—
106	C₄⋊(C₃⋊D₃)	—	2	1	no	—
107	C₄×C₃⋊D₃	—	4	1	no	—
108	C₄×C₃⋊D₃	—	4	1	no	—
109	C₃⋊(C₃⋊C₈)	—	2	1	no	—
110	C₃⋊(C₃⋊C₈)	—	2	1	no	—
111	C₃×C₃⋊C₈	—	2	1	no	—
112	C₃×C₃⋊C₈	—	4	1	no	—
113	C₃×C₃⋊C₈	—	4	1	no	—
114	C₃×C₃⋊C₈	—	4	1	no	—
115	C₃×C₃⋊C₈	SD₁₆	1	2	yes	complemented
116	C₈×Q₈	—	3	1	no	—
117	C₄×SD₁₆	—	9	1	no	—
118	#436	—	9	1	no	—
119	#334	—	9	1	no	—
120	Q₈⋊C₈	—	3	1	no	—
121	D₄⋊C₈	—	9	1	no	—
122	#322	—	9	1	no	—
123	C₂×C₃×Q₈	—	2	1	no	—
124	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
125	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
126	C₂×D₁₂	—	3	1	no	—
127	C₂×D₁₂	—	3	1	no	—
128	C₂×D₁₂	—	6	1	no	—
129	C₂×C₄×D₃	—	6	1	no	—
130	C₂×C₄×D₃	—	6	1	no	—
131	C₂×C₄×D₃	—	12	1	no	—
132	C₂×C₃×C₈	—	3	1	no	—
133	C₂×C₃×C₈	—	3	1	no	—
134	C₂×C₃×C₈	—	6	1	no	—
135	C₂×C₃×C₈	—	6	1	no	—
136	C₂×C₃×C₈	—	6	1	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	4	1	no	—
141	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	2	1	no	—
146	C₃×C₄²	—	4	1	no	—
147	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
148	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
149	#215	—	6	4	no	—
150	#215	—	6	2	no	—
151	#215	—	6	2	no	—
152	(C₂×D₃)⋊C₄	—	6	1	no	—
153	(C₂×D₃)⋊C₄	—	6	1	no	—
154	(C₂×D₃)⋊C₄	—	12	1	no	—
155	C₄⋊Dic₃	—	3	1	no	—
156	C₄⋊Dic₃	—	3	1	no	—
157	C₄⋊Dic₃	—	6	1	no	—
158	C₂×C₃⋊C₈	—	2	1	no	—
159	C₂×C₃⋊C₈	—	2	1	no	—
160	C₂×C₃⋊C₈	—	2	1	no	—
161	C₂×C₃⋊C₈	—	3	1	no	—
162	C₂×C₃⋊C₈	—	3	1	no	—
163	C₂×C₃⋊C₈	—	3	1	no	—
164	C₂×C₃⋊C₈	—	3	1	no	—
165	C₂×C₃⋊C₈	—	6	1	no	—
166	C₂×C₃⋊C₈	—	6	1	no	—
167	C₂×C₃⋊C₈	#67	1	1	yes	characteristic
168	C₂×C₃⋊C₈	#67	1	1	yes	characteristic
169	ΓL(1,𝔽₂₅)	—	12	2	no	—
170	ΓL(1,𝔽₂₅)	—	12	2	no	—
171	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃⋊D₃	—	4	1	no	—
173	C₂×C₃⋊D₃	—	4	1	no	—
174	C₂×C₃⋊D₃	—	4	1	no	—
175	C₃²×C₄	—	2	2	no	—
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	—	4	1	no	—
178	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
179	C₃²×C₄	#131	1	1	yes	characteristic
180	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
181	C₃²×C₄	#135	1	1	yes	characteristic
182	C₃⋊Dic₃	—	4	1	no	—
183	C₂×SD₁₆	—	9	1	no	—
184	C₂×M₁₆	—	18	1	no	—
185	C₄×Q₈	D₃²	1	1	yes	characteristic
186	C₄×D₄	—	9	1	no	—
187	#106	—	9	1	no	—
188	C₄⋊C₈	—	3	1	no	—
189	C₄⋊C₈	—	3	1	no	—
190	C₄⋊C₈	—	6	1	no	—
191	#103	—	9	1	no	—
192	#102	—	9	1	no	—
193	C₂²⋊C₈	—	18	1	no	—
194	C₄×C₈	—	3	1	no	—
195	C₄×C₈	—	3	1	no	—
196	C₄×C₈	—	6	1	no	—
197	C₄×C₈	—	9	1	no	—
198	C₂²×D₃	—	6	1	no	—
199	C₂²×D₃	—	6	1	no	—
200	C₂²×D₃	—	12	1	no	—
201	C₃×Q₈	—	2	1	no	—
202	C₃×Q₈	—	2	1	no	—
203	C₃×Q₈	—	2	2	no	—
204	C₃×Q₈	—	4	1	no	—
205	C₃×Q₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
206	C₃×Q₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
218	C₂×C₃×C₄	#235	1	1	yes	characteristic
219	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	#237	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
222	C₂×Dic₃	—	6	1	no	—
223	C₂×Dic₃	—	6	1	no	—
224	C₂×Dic₃	—	12	1	no	—
225	D₁₂	—	6	2	no	—
226	D₁₂	—	6	1	no	—
227	D₁₂	—	6	1	no	—
228	D₁₂	—	6	1	no	—
229	D₁₂	—	6	1	no	—
230	D₁₂	—	12	1	no	—
231	C₄×D₃	—	12	1	no	—
232	C₄×D₃	—	12	1	no	—
233	C₄×D₃	—	12	1	no	—
234	C₄×D₃	—	12	1	no	—
235	C₄×D₃	—	24	1	no	—
236	C₄×D₃	—	24	1	no	—
237	C₃×C₈	—	3	2	no	—
238	C₃×C₈	—	6	1	no	—
239	C₃×C₈	—	12	1	no	—
240	C₃×C₈	—	12	1	no	—
241	C₃×C₈	—	12	1	no	—
242	C₃⋊C₈	—	2	1	no	—
243	C₃⋊C₈	—	4	1	no	—
244	C₃⋊C₈	—	4	1	no	—
245	C₃⋊C₈	—	4	1	no	—
246	C₃⋊C₈	—	6	2	no	—
247	C₃⋊C₈	—	6	1	no	—
248	C₃⋊C₈	—	6	1	no	—
249	C₃⋊C₈	—	6	1	no	—
250	C₃⋊C₈	—	6	1	no	—
251	C₃⋊C₈	—	12	1	no	—
252	C₃⋊C₈	#215	1	2	yes	complemented
253	C₂×C₃²	#435	1	1	yes	characteristic
254	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
255	C₂×C₃²	#444	1	1	yes	characteristic
256	C₃⋊D₃	—	8	1	no	—
257	C₂×Q₈	#624	1	1	yes	characteristic
258	C₂×D₄	—	9	1	no	—
259	C₂²×C₄	—	18	1	no	—
260	SD₁₆	—	18	2	no	—
261	M₁₆	—	36	2	no	—
262	C₂×C₈	—	3	1	no	—
263	C₂×C₈	—	3	1	no	—
264	C₂×C₈	—	6	1	no	—
265	C₂×C₈	—	6	1	no	—
266	C₂×C₈	—	6	1	no	—
267	C₂×C₈	—	9	1	no	—
268	C₂×C₈	—	9	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	9	1	no	—
271	C₄⋊C₄	#624	1	1	yes	characteristic
272	C₂²⋊C₄	—	18	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	C₂×D₃²	1	1	yes	characteristic
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
277	C₂²×C₃	C₄×#67	1	1	yes	characteristic
278	C₂×D₃	—	12	2	no	—
279	C₂×D₃	—	12	1	no	—
280	C₂×D₃	—	12	1	no	—
281	C₂×D₃	—	12	1	no	—
282	C₂×D₃	—	12	1	no	—
283	C₂×D₃	—	12	1	no	—
284	C₂×D₃	—	12	1	no	—
285	C₂×D₃	—	24	1	no	—
286	C₂×D₃	—	24	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	4	1	no	—
296	C₃×C₄	—	4	2	no	—
297	C₃×C₄	—	4	1	no	—
298	C₃×C₄	—	4	1	no	—
299	C₃×C₄	—	8	1	no	—
300	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
301	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
302	C₃×C₄	#901	1	1	yes	characteristic
303	C₃×C₄	#907	1	1	yes	characteristic
304	C₃×C₄	D₃×SD₁₆	1	1	yes	characteristic
305	C₃×C₄	#919	1	1	yes	characteristic
306	C₃×C₄	C₂×#215	1	1	yes	characteristic
307	C₃×C₄	#950	1	1	yes	characteristic
308	Dic₃	—	12	1	no	—
309	Dic₃	—	12	1	no	—
310	Dic₃	—	24	1	no	—
311	C₃²	#1583	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	18	1	no	—
313	Q₈	—	2	1	no	—
314	Q₈	#3712	1	2	yes	complemented
315	D₄	—	18	1	no	—
316	D₄	—	18	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	18	1	no	—
320	C₂×C₄	—	36	1	no	—
321	C₂×C₄	—	36	1	no	—
322	C₂×C₄	#3804	1	1	yes	characteristic
323	C₂×C₄	C₂×#624	1	1	yes	characteristic
324	C₂×C₄	D₃×#67	1	1	yes	characteristic
325	C₈	—	3	2	no	—
326	C₈	—	6	1	no	—
327	C₈	—	12	1	no	—
328	C₈	—	12	1	no	—
329	C₈	—	12	1	no	—
330	C₈	—	18	1	no	—
331	C₈	—	18	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	#4730	1	1	yes	characteristic
336	C₂×C₃	#4788	1	1	yes	characteristic
337	C₂×C₃	#4823	1	1	yes	characteristic
338	C₂×C₃	C₄×#215	1	1	yes	characteristic
339	C₂×C₃	#5111	1	1	yes	characteristic
340	C₂×C₃	#5145	1	1	yes	characteristic
341	D₃	—	24	2	no	—
342	D₃	—	24	1	no	—
343	D₃	—	24	1	no	—
344	C₂²	—	36	1	no	—
345	C₂²	—	36	1	no	—
346	C₂²	—	36	1	no	—
347	C₂²	#63952	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	36	1	no	—
352	C₄	#63817	1	1	yes	characteristic
353	C₄	C₂×#3712	1	1	yes	characteristic
354	C₄	D₃×#215	1	1	yes	characteristic
355	C₄	#63949	1	1	yes	characteristic
356	C₃	—	2	1	no	—
357	C₃	#68217	1	1	yes	characteristic, complemented
358	C₃	#70933	1	1	yes	characteristic, complemented
359	C₂	—	72	1	no	—
360	C₂	#10591157	1	1	yes	characteristic
361	C₂	#10591536	1	1	yes	characteristic
362	C₂	#10591663	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×#67
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄×Q₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂×C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×Q₈	D₃²
hypocenter	C₃²	#1583
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₃⋊C₈	C₂	semidirect
4. C₃×#4535	C₂	semidirect
5. #10589098	C₂	semidirect
7. #10589078	C₂	semidirect
29. Q₈×C₃⋊C₈	D₃	semidirect
33. #4535	D₃	semidirect
98. C₃²×Q₈	M₁₆	semidirect
115. C₃×C₃⋊C₈	SD₁₆	semidirect
205. C₃×Q₈	ΓL(1,𝔽₂₅)	semidirect
206. C₃×Q₈	ΓL(1,𝔽₂₅)	semidirect
252. C₃⋊C₈	#215	semidirect
311. C₃²	#1583	semidirect
314. Q₈	#3712	semidirect
357. C₃	#68217	semidirect
358. C₃	#70933	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3755	2
C₃×Q₈×C₃⋊C₈	2
#10590338	2
C₃×#4535	2
#10589098	2
#10589082	2
#10589078	2
#70933	3
#68217	3