Subgroups — #11861499

Order 1152.

1164 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#10592723	C₂	1	1	yes	maximal, characteristic, complemented
4	#10592536	C₂	1	1	yes	maximal, characteristic
5	#10592536	C₂	1	2	yes	maximal
6	#10592529	C₂	1	1	yes	maximal, characteristic, complemented
7	#10592527	C₂	1	2	yes	maximal, complemented
8	#10592406	C₂	1	2	yes	maximal
9	#10592400	C₂	1	2	yes	maximal, complemented
10	C₄×#3743	C₂	1	1	yes	maximal, characteristic
11	C₈×#634	C₂	1	1	yes	maximal, characteristic, complemented
12	#79962	—	3	4	no	maximal
13	C₂×Q₈×C₃⋊D₃	C₄	1	1	yes	characteristic
14	C₃²×C₄×Q₈	C₂²	1	1	yes	characteristic
15	Q₈×C₃⋊Dic₃	C₄	1	1	yes	characteristic
16	C₂×#3743	C₂²	1	1	yes	characteristic
17	C₂×#3743	C₂²	1	2	yes	—
18	C₂×#3743	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃⋊D₃	C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₃⋊D₃	C₄	1	2	yes	—
21	#64096	C₄	1	2	yes	—
22	#64096	C₄	1	1	yes	characteristic
23	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
24	C₄²×C₃⋊D₃	C₂²	1	2	yes	—
25	C₄×#634	C₂²	1	2	yes	—
26	C₄×#634	C₂²	1	1	yes	characteristic
27	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
28	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃²×#97	C₂²	1	2	yes	—
30	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
31	#63642	C₂²	1	1	yes	characteristic
32	#63642	C₂²	1	1	yes	characteristic
33	#63641	C₂²	1	2	yes	—
34	#63641	C₂²	1	2	yes	—
35	#63641	C₂²	1	2	yes	—
36	C₈×C₃⋊Dic₃	C₂²	1	2	yes	—
37	C₈×C₃⋊Dic₃	C₂²	1	2	yes	—
38	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
39	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
40	#63630	C₂²	1	2	yes	—
41	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄×D₃×Q₈	—	3	4	no	—
43	C₃×#446	D₃	1	4	yes	complemented
44	#5024	—	3	4	no	—
45	#4837	—	3	4	no	—
46	#4837	—	3	8	no	—
47	#4830	—	3	4	no	—
48	#4828	—	3	8	no	—
49	#4707	—	3	8	no	—
50	#4701	—	3	8	no	—
51	C₄×ΓL(1,𝔽₂₅)	—	3	4	no	—
52	C₈×Dic₆	—	3	4	no	—
53	C₂×C₃²×Q₈	C₂×C₄	1	1	yes	characteristic
54	Q₈×C₃⋊D₃	—	2	4	no	—
55	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
57	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
60	C₂×#634	C₂×C₄	1	2	yes	—
61	C₂×#634	C₂×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₈	C₂³	1	2	yes	—
63	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
65	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
66	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₃²×C₄²	C₂³	1	1	yes	characteristic
68	C₃²×C₄²	C₂³	1	2	yes	—
69	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
70	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	1	yes	characteristic
71	#3750	C₂×C₄	1	2	yes	—
72	#3750	C₂×C₄	1	2	yes	—
73	#3750	C₂×C₄	1	2	yes	—
74	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
75	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
76	C₄×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
77	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₃⋊Dic₃	C₂³	1	2	yes	—
79	C₄×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
80	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
81	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
82	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
83	#3743	—	2	2	no	—
84	#3743	—	2	4	no	—
85	#3743	Q₈	1	4	yes	complemented
86	Q₈×M₁₆	—	9	1	no	Sylow
87	C₂×D₃×Q₈	—	3	4	no	—
88	C₃×C₄×Q₈	C₂×D₃	1	4	yes	—
89	Q₈×Dic₃	—	3	4	no	—
90	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
91	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
92	C₂×ΓL(1,𝔽₂₅)	—	3	8	no	—
93	D₃×C₄⋊C₄	—	3	8	no	—
94	D₃×C₄⋊C₄	—	3	4	no	—
95	#887	—	3	4	no	—
96	#887	—	3	8	no	—
97	C₄²×D₃	—	3	4	no	—
98	C₄²×D₃	—	3	8	no	—
99	C₄×Dic₆	—	3	4	no	—
100	C₄×Dic₆	—	3	8	no	—
101	C₃×C₄⋊C₈	C₂×D₃	1	8	yes	—
102	C₃×C₄⋊C₈	C₂×D₃	1	4	yes	—
103	C₃×#97	C₂×D₃	1	8	yes	—
104	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
105	#815	—	3	4	no	—
106	#815	—	3	4	no	—
107	Dic₃⋊C₈	—	3	8	no	—
108	Dic₃⋊C₈	—	3	8	no	—
109	Dic₃⋊C₈	—	3	8	no	—
110	C₈×Dic₃	—	3	8	no	—
111	C₈×Dic₃	—	3	8	no	—
112	C₄⋊(C₃⋊C₈)	—	3	4	no	—
113	C₄⋊(C₃⋊C₈)	—	3	8	no	—
114	#803	—	3	8	no	—
115	C₄×C₃⋊C₈	—	3	4	no	—
116	C₂²×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
117	C₃²×Q₈	—	2	2	no	—
118	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
120	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
121	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
123	C₂×C₃⋊Dic₃	C₂²×C₄	1	2	yes	—
124	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃⋊Dic₃	C₂²×C₄	1	2	yes	—
128	C₄×C₃⋊D₃	—	2	2	no	—
129	C₄×C₃⋊D₃	—	2	4	no	—
130	C₄×C₃⋊D₃	—	4	1	no	—
131	C₄×C₃⋊D₃	—	4	2	no	—
132	C₄×C₃⋊D₃	C₂×Q₈	1	2	yes	—
133	C₄×C₃⋊D₃	#41	1	2	yes	—
134	#634	—	2	4	no	—
135	#634	M₁₆	1	4	yes	complemented
136	C₃²×C₈	—	2	2	no	—
137	C₃²×C₈	—	2	1	no	—
138	C₃²×C₈	C₂×Q₈	1	2	yes	complemented
139	C₃⋊(C₃⋊C₈)	—	2	1	no	—
140	C₃⋊(C₃⋊C₈)	—	2	2	no	—
141	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	2	yes	complemented
142	C₂×C₄×Q₈	—	9	1	no	—
143	#446	—	9	1	no	—
144	#446	—	9	2	no	—
145	#446	—	9	2	no	—
146	#446	C₃⋊D₃	1	1	yes	characteristic, complemented
147	C₈×Q₈	—	9	1	no	—
148	C₈×Q₈	—	9	1	no	—
149	#423	—	9	2	no	—
150	#423	—	9	1	no	—
151	C₄×M₁₆	—	9	2	no	—
152	C₄×M₁₆	—	9	1	no	—
153	C₂×C₃×Q₈	C₄×D₃	1	4	yes	—
154	D₃×Q₈	—	6	16	no	—
155	C₂×C₄×D₃	—	3	4	no	—
156	C₂×C₄×D₃	—	3	8	no	—
157	C₂×C₄×D₃	—	3	4	no	—
158	C₂×C₄×D₃	—	3	4	no	—
159	C₂×C₄×D₃	—	3	8	no	—
160	C₂×Dic₆	—	3	4	no	—
161	C₂×Dic₆	—	3	8	no	—
162	C₂×C₃×C₈	C₂²×D₃	1	8	yes	—
163	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
164	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
165	C₃×C₄⋊C₄	C₄×D₃	1	8	yes	—
166	C₃×C₄⋊C₄	C₄×D₃	1	4	yes	—
167	C₃×C₄²	C₂²×D₃	1	4	yes	—
168	C₃×C₄²	C₂²×D₃	1	8	yes	—
169	C₄⋊Dic₃	—	3	8	no	—
170	C₄⋊Dic₃	—	3	4	no	—
171	Dic₃⋊C₄	—	3	8	no	—
172	Dic₃⋊C₄	—	3	8	no	—
173	Dic₃⋊C₄	—	3	8	no	—
174	C₄×Dic₃	—	3	8	no	—
175	C₄×Dic₃	—	3	4	no	—
176	C₄×Dic₃	—	3	8	no	—
177	C₄×Dic₃	—	3	4	no	—
178	C₄×Dic₃	—	3	8	no	—
179	C₄×Dic₃	—	3	4	no	—
180	C₂×C₃⋊C₈	—	3	4	no	—
181	C₂×C₃⋊C₈	—	3	4	no	—
182	C₂×C₃⋊C₈	—	3	8	no	—
183	ΓL(1,𝔽₂₅)	—	3	16	no	—
184	ΓL(1,𝔽₂₅)	—	6	8	no	—
185	ΓL(1,𝔽₂₅)	—	6	16	no	—
186	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₃⋊D₃	—	2	1	no	—
188	C₂×C₃⋊D₃	—	2	1	no	—
189	C₂×C₃⋊D₃	C₄×Q₈	1	2	yes	—
190	C₃²×C₄	—	2	2	no	—
191	C₃²×C₄	—	2	1	no	—
192	C₃²×C₄	—	2	2	no	—
193	C₃²×C₄	C₂×M₁₆	1	2	yes	—
194	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
195	C₃²×C₄	C₂×#41	1	1	yes	characteristic
196	C₃⋊Dic₃	—	2	1	no	—
197	C₃⋊Dic₃	—	2	2	no	—
198	C₃⋊Dic₃	—	2	1	no	—
199	C₃⋊Dic₃	C₄×Q₈	1	2	yes	—
200	C₃⋊Dic₃	C₂×M₁₆	1	4	yes	—
201	C₂²×Q₈	—	9	1	no	—
202	C₂×M₁₆	—	9	1	no	—
203	C₂×M₁₆	—	9	2	no	—
204	C₂×M₁₆	—	9	1	no	—
205	C₄×Q₈	—	9	1	no	—
206	C₄×Q₈	—	9	1	no	—
207	C₄×Q₈	—	9	2	no	—
208	C₄×Q₈	—	9	1	no	—
209	C₄×Q₈	—	9	2	no	—
210	C₄×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	9	2	no	—
212	C₂×C₄⋊C₄	—	9	1	no	—
213	C₂×C₄²	—	9	2	no	—
214	C₂×C₄²	—	9	1	no	—
215	C₄⋊C₈	—	9	2	no	—
216	C₄⋊C₈	—	9	2	no	—
217	C₄⋊C₈	—	9	2	no	—
218	C₄⋊C₈	—	9	1	no	—
219	C₄⋊C₈	—	9	2	no	—
220	C₄⋊C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
221	C₄⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
222	#97	—	9	1	no	—
223	#97	—	9	1	no	—
224	#97	—	9	2	no	—
225	#97	C₂×C₃⋊D₃	1	2	yes	—
226	C₄×C₈	—	9	2	no	—
227	C₄×C₈	—	9	2	no	—
228	C₄×C₈	—	9	1	no	—
229	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
230	C₂²×D₃	—	3	4	no	—
231	C₃×Q₈	—	2	8	no	—
232	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
233	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
234	C₂×C₃×C₄	C₂×C₄×D₃	1	8	yes	—
235	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
236	C₂×C₃×C₄	C₂³×D₃	1	4	yes	—
237	C₂×Dic₃	—	3	4	no	—
238	C₂×Dic₃	—	3	8	no	—
239	C₂×Dic₃	—	3	8	no	—
240	C₂×Dic₃	—	3	4	no	—
241	C₂×Dic₃	—	3	4	no	—
242	C₄×D₃	—	3	8	no	—
243	C₄×D₃	—	3	8	no	—
244	C₄×D₃	—	6	16	no	—
245	C₄×D₃	—	6	8	no	—
246	C₄×D₃	—	12	4	no	—
247	C₄×D₃	—	12	8	no	—
248	Dic₆	—	3	16	no	—
249	Dic₆	—	6	16	no	—
250	C₃×C₈	—	2	8	no	—
251	C₃×C₈	—	2	4	no	—
252	C₃×C₈	D₃×Q₈	1	8	yes	complemented
253	C₃⋊C₈	—	3	8	no	—
254	C₃⋊C₈	—	6	4	no	—
255	C₃⋊C₈	—	6	8	no	—
256	C₂×C₃²	C₂×C₄×Q₈	1	1	yes	characteristic
257	C₂×C₃²	C₂²×M₁₆	1	1	yes	characteristic
258	C₂×C₃²	#568	1	1	yes	characteristic
259	C₃⋊D₃	—	2	2	no	—
260	C₂×Q₈	—	9	1	no	—
261	C₂×Q₈	—	9	2	no	—
262	C₂×Q₈	—	18	4	no	—
263	C₂×Q₈	C₄×C₃⋊D₃	1	1	yes	characteristic
264	C₂²×C₄	—	9	1	no	—
265	C₂²×C₄	—	9	2	no	—
266	C₂²×C₄	—	9	1	no	—
267	C₂²×C₄	—	9	2	no	—
268	C₂²×C₄	—	9	1	no	—
269	M₁₆	—	9	4	no	—
270	M₁₆	—	18	4	no	—
271	M₁₆	—	18	2	no	—
272	C₂×C₈	—	9	1	no	—
273	C₂×C₈	—	9	2	no	—
274	C₂×C₈	—	9	1	no	—
275	C₂×C₈	C₂²×C₃⋊D₃	1	2	yes	—
276	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
277	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
278	C₄⋊C₄	—	9	2	no	—
279	C₄⋊C₄	—	9	2	no	—
280	C₄⋊C₄	—	9	2	no	—
281	C₄⋊C₄	—	9	2	no	—
282	C₄⋊C₄	—	9	1	no	—
283	C₄⋊C₄	C₄×C₃⋊D₃	1	2	yes	—
284	C₄⋊C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
285	C₄²	—	9	2	no	—
286	C₄²	—	9	1	no	—
287	C₄²	—	9	2	no	—
288	C₄²	—	9	2	no	—
289	C₄²	—	9	1	no	—
290	C₄²	—	9	1	no	—
291	C₄²	C₂²×C₃⋊D₃	1	1	yes	characteristic
292	C₄²	C₂²×C₃⋊D₃	1	2	yes	—
293	C₂²×C₃	C₂²×C₄×D₃	1	4	yes	—
294	C₂×D₃	—	3	8	no	—
295	C₂×D₃	—	6	4	no	—
296	C₂×D₃	—	6	4	no	—
297	C₃×C₄	—	2	8	no	—
298	C₃×C₄	—	2	4	no	—
299	C₃×C₄	—	2	8	no	—
300	C₃×C₄	D₃×M₁₆	1	8	yes	—
301	C₃×C₄	C₂×D₃×Q₈	1	4	yes	—
302	C₃×C₄	D₃×#41	1	4	yes	—
303	Dic₃	—	3	16	no	—
304	Dic₃	—	3	8	no	—
305	Dic₃	—	6	4	no	—
306	Dic₃	—	6	8	no	—
307	Dic₃	—	6	4	no	—
308	C₃²	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
309	C₂³	—	9	1	no	—
310	Q₈	—	2	2	no	—
311	Q₈	—	9	4	no	—
312	Q₈	—	18	4	no	—
313	C₂×C₄	—	9	1	no	—
314	C₂×C₄	—	9	2	no	—
315	C₂×C₄	—	9	2	no	—
316	C₂×C₄	—	9	2	no	—
317	C₂×C₄	—	9	2	no	—
318	C₂×C₄	—	9	1	no	—
319	C₂×C₄	—	9	1	no	—
320	C₂×C₄	—	18	4	no	—
321	C₂×C₄	—	18	2	no	—
322	C₂×C₄	—	36	1	no	—
323	C₂×C₄	—	36	2	no	—
324	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
326	C₂×C₄	C₂×C₄×C₃⋊D₃	1	2	yes	—
327	C₂×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
328	C₂×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	—	9	2	no	—
332	C₈	—	18	2	no	—
333	C₈	—	18	1	no	—
334	C₈	Q₈×C₃⋊D₃	1	2	yes	complemented
335	C₂×C₃	C₄×D₃×Q₈	1	4	yes	—
336	C₂×C₃	C₂×D₃×M₁₆	1	4	yes	—
337	C₂×C₃	#5750	1	4	yes	—
338	D₃	—	6	8	no	—
339	C₂²	—	9	2	no	—
340	C₂²	—	18	1	no	—
341	C₂²	—	18	1	no	—
342	C₂²	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	9	4	no	—
347	C₄	—	9	2	no	—
348	C₄	—	18	1	no	—
349	C₄	—	18	2	no	—
350	C₄	—	18	1	no	—
351	C₄	M₁₆×C₃⋊D₃	1	2	yes	—
352	C₄	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
353	C₄	#41×C₃⋊D₃	1	1	yes	characteristic
354	C₃	#79962	1	4	yes	complemented
355	C₂	—	18	2	no	—
356	C₂	C₄×Q₈×C₃⋊D₃	1	1	yes	characteristic
357	C₂	C₂×M₁₆×C₃⋊D₃	1	1	yes	characteristic
358	C₂	#10596221	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂³×C₄
Fitting	C₃²×#446	C₂
prime radical	C₂²×C₃⋊D₃	C₂²×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
hypercenter	#446	C₃⋊D₃
hypocenter	C₃²	Q₈×M₁₆
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₃²×#446	C₂	semidirect
3. #10592723	C₂	semidirect
6. #10592529	C₂	semidirect
7. #10592527	C₂	semidirect
9. #10592400	C₂	semidirect
11. C₈×#634	C₂	semidirect
43. C₃×#446	D₃	semidirect
85. #3743	Q₈	semidirect
135. #634	M₁₆	semidirect
138. C₃²×C₈	C₂×Q₈	semidirect
141. C₃⋊(C₃⋊C₈)	C₂×Q₈	semidirect
146. #446	C₃⋊D₃	semidirect
252. C₃×C₈	D₃×Q₈	semidirect
308. C₃²	Q₈×M₁₆	semidirect
334. C₈	Q₈×C₃⋊D₃	semidirect
354. C₃	#79962	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₃⋊D₃	2
C₃²×#446	2
#10592723	2
#10592536	2
#10592529	2
#10592527	2
#10592406	2
#10592400	2
C₄×#3743	2
C₈×#634	2
#79962	3