Subgroups — #11776954

Order 1152.

395 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592536	C₂	1	1	yes	maximal, characteristic
2	C₄×#3712	C₂	1	1	yes	maximal, characteristic
3	#10590260	C₂	1	1	yes	maximal, characteristic
4	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4479	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589099	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
8	#68571	—	3	1	no	maximal
9	#68531	—	3	1	no	maximal
10	C₂×#3743	—	2	1	no	—
11	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₂×#3712	C₄	1	1	yes	characteristic
13	C₂×#3712	C₄	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#63641	—	2	1	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	#63557	—	2	1	no	—
22	#63555	—	2	1	no	—
23	#63555	C₄	1	1	yes	characteristic
24	#63555	C₄	1	1	yes	characteristic
25	#4837	—	3	1	no	—
26	#4837	—	3	1	no	—
27	#4837	—	6	1	no	—
28	#4835	—	3	1	no	—
29	#4835	—	3	1	no	—
30	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
31	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
32	C₃×#335	—	3	1	no	—
33	C₃×#334	—	3	1	no	—
34	#4480	—	3	1	no	—
35	#4480	D₃	1	1	yes	characteristic, complemented
36	#4479	—	3	1	no	—
37	#4479	D₃	1	1	yes	characteristic, complemented
38	#4458	—	3	1	no	—
39	#4457	—	3	1	no	—
40	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
42	C₂×C₄×C₃⋊D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₃²×C₄²	C₂³	1	1	yes	characteristic
45	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
47	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
48	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	#3743	—	4	2	no	—
50	C₂×C₃×C₃⋊C₈	—	2	1	no	—
51	C₂×C₃×C₃⋊C₈	—	2	1	no	—
52	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	#3712	—	2	2	no	—
57	#3712	—	2	2	no	—
58	#1569	—	9	1	no	Sylow
59	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
60	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
61	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
62	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
63	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
64	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
65	C₂×ΓL(1,𝔽₂₅)	—	6	2	no	—
66	C₄²×D₃	—	3	1	no	—
67	C₄²×D₃	—	3	1	no	—
68	C₄²×D₃	—	6	1	no	—
69	C₃×C₄⋊C₈	—	2	1	no	—
70	C₃×C₄⋊C₈	—	3	1	no	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
73	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄×C₈	—	3	1	no	—
75	C₃×C₄×C₈	—	3	1	no	—
76	(C₂×D₃)⋊C₈	—	6	1	no	—
77	(C₂×D₃)⋊C₈	—	6	1	no	—
78	#815	—	3	1	no	—
79	#815	—	3	1	no	—
80	#815	—	3	1	no	—
81	#815	—	3	1	no	—
82	#815	—	6	1	no	—
83	#815	—	6	1	no	—
84	Dic₃⋊C₈	—	6	1	no	—
85	Dic₃⋊C₈	—	6	1	no	—
86	Dic₃⋊C₈	—	6	2	no	—
87	C₄⋊(C₃⋊C₈)	—	3	1	no	—
88	C₄⋊(C₃⋊C₈)	—	3	1	no	—
89	C₄⋊(C₃⋊C₈)	—	6	1	no	—
90	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₃⋊D₃	—	2	1	no	—
102	C₄×C₃⋊D₃	—	2	1	no	—
103	C₄×C₃⋊D₃	—	2	1	no	—
104	C₄×C₃⋊D₃	—	2	1	no	—
105	C₄×C₃⋊D₃	—	4	1	no	—
106	C₃²×C₈	—	4	1	no	—
107	C₃⋊(C₃⋊C₈)	—	4	1	no	—
108	C₃×C₃⋊C₈	—	2	1	no	—
109	C₃×C₃⋊C₈	—	2	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	—	4	1	no	—
112	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
113	C₃×C₃⋊C₈	M₁₆	1	2	yes	complemented
114	#432	—	9	1	no	—
115	#423	—	9	1	no	—
116	C₄×M₁₆	—	9	1	no	—
117	#335	—	3	1	no	—
118	#335	—	9	1	no	—
119	#334	—	3	1	no	—
120	#334	—	9	1	no	—
121	C₂×C₄×D₃	—	3	1	no	—
122	C₂×C₄×D₃	—	3	1	no	—
123	C₂×C₄×D₃	—	3	1	no	—
124	C₂×C₄×D₃	—	3	1	no	—
125	C₂×C₄×D₃	—	3	1	no	—
126	C₂×C₄×D₃	—	3	1	no	—
127	C₂×C₄×D₃	—	6	1	no	—
128	C₂×C₄×D₃	—	6	1	no	—
129	C₂×C₄×D₃	—	6	1	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	3	1	no	—
134	C₂×C₃×C₈	—	3	1	no	—
135	C₂×C₃×C₈	—	3	1	no	—
136	C₂×C₃×C₈	—	3	1	no	—
137	C₂×C₃×C₈	—	6	1	no	—
138	C₂×C₃×C₈	—	6	1	no	—
139	C₃×C₄²	—	2	1	no	—
140	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
141	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
142	C₄×Dic₃	—	3	1	no	—
143	C₄×Dic₃	—	3	1	no	—
144	C₄×Dic₃	—	3	1	no	—
145	C₄×Dic₃	—	3	1	no	—
146	C₄×Dic₃	—	3	1	no	—
147	C₄×Dic₃	—	3	1	no	—
148	C₄×Dic₃	—	6	1	no	—
149	C₄×Dic₃	—	6	1	no	—
150	C₄×Dic₃	—	6	1	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	—	6	2	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	—	6	1	no	—
156	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
157	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
160	ΓL(1,𝔽₂₅)	—	6	2	no	—
161	ΓL(1,𝔽₂₅)	—	6	2	no	—
162	ΓL(1,𝔽₂₅)	—	6	2	no	—
163	ΓL(1,𝔽₂₅)	—	6	2	no	—
164	ΓL(1,𝔽₂₅)	—	12	4	no	—
165	ΓL(1,𝔽₂₅)	—	12	2	no	—
166	ΓL(1,𝔽₂₅)	—	12	2	no	—
167	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃⋊D₃	—	2	1	no	—
169	C₂×C₃⋊D₃	—	2	1	no	—
170	C₂×C₃⋊D₃	—	2	1	no	—
171	C₃²×C₄	—	2	1	no	—
172	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
173	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
174	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₃²×C₄	#137	1	1	yes	characteristic
176	C₃⋊Dic₃	—	2	1	no	—
177	C₃⋊Dic₃	—	2	1	no	—
178	C₃⋊Dic₃	—	2	1	no	—
179	C₂×M₁₆	—	9	1	no	—
180	C₂×M₁₆	—	9	1	no	—
181	C₂×M₁₆	—	18	1	no	—
182	C₂×C₄²	—	9	1	no	—
183	C₄⋊C₈	—	3	1	no	—
184	C₄⋊C₈	—	3	1	no	—
185	C₄⋊C₈	—	9	1	no	—
186	C₄⋊C₈	—	18	1	no	—
187	C₄⋊C₈	D₃²	1	1	yes	characteristic
188	C₂²⋊C₈	—	18	1	no	—
189	#97	—	9	1	no	—
190	#97	—	9	1	no	—
191	#97	—	18	1	no	—
192	C₄×C₈	—	3	1	no	—
193	C₄×C₈	—	3	1	no	—
194	C₂²×D₃	—	3	1	no	—
195	C₂²×D₃	—	3	1	no	—
196	C₂²×D₃	—	6	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
203	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
205	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
206	C₂×Dic₃	—	3	1	no	—
207	C₂×Dic₃	—	3	1	no	—
208	C₂×Dic₃	—	3	1	no	—
209	C₂×Dic₃	—	3	1	no	—
210	C₂×Dic₃	—	3	1	no	—
211	C₂×Dic₃	—	3	1	no	—
212	C₂×Dic₃	—	6	1	no	—
213	C₂×Dic₃	—	6	1	no	—
214	C₂×Dic₃	—	6	1	no	—
215	C₄×D₃	—	6	2	no	—
216	C₄×D₃	—	6	2	no	—
217	C₄×D₃	—	6	1	no	—
218	C₄×D₃	—	6	1	no	—
219	C₄×D₃	—	6	1	no	—
220	C₄×D₃	—	6	1	no	—
221	C₄×D₃	—	6	1	no	—
222	C₄×D₃	—	6	1	no	—
223	C₄×D₃	—	6	1	no	—
224	C₄×D₃	—	6	1	no	—
225	C₄×D₃	—	12	1	no	—
226	C₄×D₃	—	12	1	no	—
227	C₄×D₃	—	12	1	no	—
228	C₄×D₃	—	12	1	no	—
229	C₄×D₃	—	12	2	no	—
230	C₃×C₈	—	3	2	no	—
231	C₃×C₈	—	3	2	no	—
232	C₃×C₈	—	4	2	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	—	6	1	no	—
236	C₃×C₈	—	6	1	no	—
237	C₃×C₈	—	12	1	no	—
238	C₃×C₈	—	12	1	no	—
239	C₃⋊C₈	—	2	1	no	—
240	C₃⋊C₈	—	2	1	no	—
241	C₃⋊C₈	—	4	1	no	—
242	C₃⋊C₈	—	4	1	no	—
243	C₃⋊C₈	—	12	2	no	—
244	C₃⋊C₈	—	12	1	no	—
245	C₃⋊C₈	—	12	1	no	—
246	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
247	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
248	C₂×C₃²	#423	1	1	yes	characteristic
249	C₂×C₃²	#428	1	1	yes	characteristic
250	C₂×C₃²	#430	1	1	yes	characteristic
251	C₃⋊D₃	—	4	1	no	—
252	C₂²×C₄	—	9	1	no	—
253	C₂²×C₄	—	9	1	no	—
254	C₂²×C₄	—	9	1	no	—
255	M₁₆	—	18	2	no	—
256	M₁₆	—	18	2	no	—
257	M₁₆	—	36	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	3	1	no	—
260	C₂×C₈	—	3	1	no	—
261	C₂×C₈	—	3	1	no	—
262	C₂×C₈	—	3	1	no	—
263	C₂×C₈	—	6	1	no	—
264	C₂×C₈	—	6	1	no	—
265	C₂×C₈	—	18	1	no	—
266	C₄²	—	9	1	no	—
267	C₄²	—	9	1	no	—
268	C₄²	—	9	1	no	—
269	C₄²	C₂×D₃²	1	1	yes	characteristic
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
272	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
273	C₂×D₃	—	6	2	no	—
274	C₂×D₃	—	6	1	no	—
275	C₂×D₃	—	6	1	no	—
276	C₂×D₃	—	6	1	no	—
277	C₂×D₃	—	6	1	no	—
278	C₂×D₃	—	6	1	no	—
279	C₂×D₃	—	6	1	no	—
280	C₂×D₃	—	12	1	no	—
281	C₂×D₃	—	12	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	4	1	no	—
289	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
290	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
291	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
292	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
293	C₃×C₄	#911	1	1	yes	characteristic
294	C₃×C₄	#914	1	1	yes	characteristic
295	C₃×C₄	#915	1	1	yes	characteristic
296	C₃×C₄	#918	1	1	yes	characteristic
297	Dic₃	—	6	2	no	—
298	Dic₃	—	6	2	no	—
299	Dic₃	—	6	1	no	—
300	Dic₃	—	6	1	no	—
301	Dic₃	—	6	1	no	—
302	Dic₃	—	6	1	no	—
303	Dic₃	—	6	1	no	—
304	Dic₃	—	6	1	no	—
305	Dic₃	—	12	1	no	—
306	C₃²	#1569	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	9	1	no	—
308	C₂×C₄	—	9	1	no	—
309	C₂×C₄	—	9	1	no	—
310	C₂×C₄	—	9	1	no	—
311	C₂×C₄	—	18	1	no	—
312	C₂×C₄	—	18	1	no	—
313	C₂×C₄	—	18	1	no	—
314	C₂×C₄	—	18	1	no	—
315	C₂×C₄	—	36	1	no	—
316	C₂×C₄	#3797	1	1	yes	characteristic
317	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
318	C₂×C₄	#3811	1	1	yes	characteristic
319	C₈	—	3	2	no	—
320	C₈	—	3	2	no	—
321	C₈	—	4	1	no	—
322	C₈	—	6	1	no	—
323	C₈	—	6	1	no	—
324	C₈	—	12	1	no	—
325	C₈	—	12	1	no	—
326	C₈	—	36	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	#4837	1	1	yes	characteristic
331	C₂×C₃	#4837	1	1	yes	characteristic
332	C₂×C₃	#4861	1	1	yes	characteristic
333	C₂×C₃	#4881	1	1	yes	characteristic
334	C₂×C₃	#4893	1	1	yes	characteristic
335	C₂×C₃	#4893	1	1	yes	characteristic
336	D₃	—	12	2	no	—
337	D₃	—	12	1	no	—
338	D₃	—	12	1	no	—
339	C₂²	—	18	1	no	—
340	C₂²	—	18	1	no	—
341	C₂²	—	18	1	no	—
342	C₂²	#63883	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	18	1	no	—
345	C₄	—	18	1	no	—
346	C₄	—	18	1	no	—
347	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
348	C₄	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
349	C₄	#63937	1	1	yes	characteristic
350	C₄	#63951	1	1	yes	characteristic
351	C₃	—	2	1	no	—
352	C₃	#68531	1	1	yes	characteristic, complemented
353	C₃	#68571	1	1	yes	characteristic, complemented
354	C₂	—	36	1	no	—
355	C₂	#10590583	1	1	yes	characteristic
356	C₂	#10590666	1	1	yes	characteristic
357	C₂	#10590974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3811
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄⋊C₈	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄⋊C₈	D₃²
hypocenter	C₃²	#1569
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#4480	C₂	semidirect
5. C₃×#4479	C₂	semidirect
6. #10589099	C₂	semidirect
7. #10589097	C₂	semidirect
35. #4480	D₃	semidirect
37. #4479	D₃	semidirect
112. C₃×C₃⋊C₈	M₁₆	semidirect
113. C₃×C₃⋊C₈	M₁₆	semidirect
246. C₃⋊C₈	ΓL(1,𝔽₂₅)	semidirect
247. C₃⋊C₈	ΓL(1,𝔽₂₅)	semidirect
306. C₃²	#1569	semidirect
352. C₃	#68531	semidirect
353. C₃	#68571	semidirect

Maximal subgroups

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

subgroup	index
#10592536	2
C₄×#3712	2
#10590260	2
C₃×#4480	2
C₃×#4479	2
#10589099	2
#10589097	2
#68571	3
#68531	3