Subgroups — #11770864

Order 1152.

389 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#67744	—	3	1	no	maximal
5	#67668	—	3	1	no	maximal
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	2	yes	—
8	C₂×C₃×#208	C₄	1	2	yes	—
9	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	C₃×#830	C₄	1	2	yes	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₃×#803	—	2	1	no	—
14	C₃×C₄×C₃⋊C₈	—	2	1	no	—
15	C₃×C₄×C₃⋊C₈	—	2	1	no	—
16	C₃×#431	—	3	1	no	—
17	C₃×C₄×M₁₆	—	3	1	no	—
18	#4925	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₄⋊(C₃⋊C₈)	—	6	1	no	—
22	C₄×#208	D₃	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₃²×C₄²	D₄	1	1	yes	characteristic
27	C₃²×C₄²	D₄	1	1	yes	characteristic
28	C₃²×C₄²	D₄	1	1	yes	characteristic
29	C₃²×C₄²	Q₈	1	1	yes	characteristic
30	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
31	C₂×C₃⋊(C₃⋊C₈)	—	4	2	no	—
32	C₃×#208	—	2	2	no	—
33	C₃×#208	C₈	1	4	yes	complemented
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	—	2	2	no	—
36	#1283	—	9	1	no	Sylow
37	C₂×C₃×M₁₆	—	3	2	no	—
38	C₂×C₃×C₄²	—	2	1	no	—
39	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂×#208	Dic₃	1	2	yes	—
42	C₂²×C₃⋊C₈	—	3	2	no	—
43	C₂²×C₃⋊C₈	—	3	2	no	—
44	C₂²×C₃⋊C₈	—	6	2	no	—
45	C₃×C₄⋊C₈	—	6	1	no	—
46	C₃×C₂²⋊C₈	—	3	2	no	—
47	C₃×#97	—	6	1	no	—
48	C₃×C₄×C₈	—	6	1	no	—
49	C₃×C₄×C₈	—	6	1	no	—
50	#830	Dic₃	1	2	yes	—
51	C₄⋊(C₃⋊C₈)	—	2	1	no	—
52	C₄⋊(C₃⋊C₈)	—	6	2	no	—
53	C₄⋊(C₃⋊C₈)	—	6	2	no	—
54	C₄⋊(C₃⋊C₈)	—	6	1	no	—
55	C₄⋊(C₃⋊C₈)	—	6	1	no	—
56	C₄⋊(C₃⋊C₈)	—	6	1	no	—
57	C₄⋊(C₃⋊C₈)	—	6	1	no	—
58	#803	—	2	1	no	—
59	C₄×C₃⋊C₈	—	2	1	no	—
60	C₄×C₃⋊C₈	—	2	1	no	—
61	C₂³×C₃²	C₄²	1	1	yes	characteristic
62	C₂×C₃²×C₄	—	2	1	no	—
63	C₂×C₃²×C₄	—	2	1	no	—
64	C₂×C₃²×C₄	—	4	1	no	—
65	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
72	C₂×C₃²×C₄	M₁₆	1	2	yes	—
73	C₃⋊(C₃⋊C₈)	—	4	2	no	—
74	C₃×C₃⋊C₈	—	2	4	no	—
75	C₃×C₃⋊C₈	—	4	2	no	—
76	#431	—	3	1	no	—
77	C₂×C₄⋊C₈	—	9	1	no	—
78	C₄×M₁₆	—	3	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	—	2	1	no	—
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
88	C₃×M₁₆	—	3	4	no	—
89	C₃×M₁₆	—	6	2	no	—
90	C₂×C₃×C₈	—	6	2	no	—
91	C₂×C₃×C₈	—	6	2	no	—
92	C₃×C₄²	—	2	1	no	—
93	C₃×C₄²	—	2	1	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	—	2	1	no	—
96	C₃×C₄²	Dic₆	1	1	yes	characteristic
97	C₃×C₄²	Dic₆	1	1	yes	characteristic
98	C₃×C₄²	D₁₂	1	1	yes	characteristic
99	C₃×C₄²	D₁₂	1	1	yes	characteristic
100	C₃×C₄²	#67	1	1	yes	characteristic
101	C₃×C₄²	#67	1	1	yes	characteristic
102	C₃×C₄²	#67	1	1	yes	characteristic
103	C₃×C₄²	#67	1	1	yes	characteristic
104	#208	—	2	2	no	—
105	#208	C₃⋊C₈	1	4	yes	complemented
106	C₂×C₃⋊C₈	—	2	2	no	—
107	C₂×C₃⋊C₈	—	2	2	no	—
108	C₂×C₃⋊C₈	—	6	4	no	—
109	C₂×C₃⋊C₈	—	6	2	no	—
110	C₂×C₃⋊C₈	—	6	2	no	—
111	C₂×C₃⋊C₈	—	12	4	no	—
112	C₂×C₃⋊C₈	—	12	2	no	—
113	C₂×C₃⋊C₈	—	12	2	no	—
114	C₂²×C₃²	—	2	1	no	—
115	C₂²×C₃²	—	2	1	no	—
116	C₂²×C₃²	#95	1	1	yes	characteristic
117	C₂²×C₃²	C₄×C₈	1	1	yes	characteristic
118	C₂²×C₃²	#97	1	1	yes	characteristic
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	—	2	1	no	—
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
123	C₃²×C₄	#100	1	1	yes	characteristic
124	C₃²×C₄	#101	1	1	yes	characteristic
125	C₃²×C₄	C₄⋊C₈	1	2	yes	—
126	C₂×M₁₆	—	3	2	no	—
127	C₂²×C₈	—	9	2	no	—
128	C₂×C₄²	D₃²	1	1	yes	characteristic
129	C₄⋊C₈	—	6	1	no	—
130	C₄⋊C₈	—	18	1	no	—
131	C₄⋊C₈	—	18	1	no	—
132	C₂²⋊C₈	—	3	2	no	—
133	#97	—	6	1	no	—
134	C₄×C₈	—	6	1	no	—
135	C₄×C₈	—	6	1	no	—
136	C₂³×C₃	—	2	1	no	—
137	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
138	C₂³×C₃	C₄×Dic₃	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	4	1	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
157	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
158	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
159	C₂×C₃×C₄	#208	1	2	yes	—
160	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
166	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₂×C₃×C₄	#217	1	1	yes	characteristic
171	C₂×C₃×C₄	#217	1	1	yes	characteristic
172	C₃×C₈	—	6	4	no	—
173	C₃×C₈	—	12	2	no	—
174	C₃⋊C₈	—	2	4	no	—
175	C₃⋊C₈	—	4	2	no	—
176	C₃⋊C₈	—	12	4	no	—
177	C₃⋊C₈	—	12	2	no	—
178	C₃⋊C₈	—	12	2	no	—
179	C₂×C₃²	—	2	1	no	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	#336	1	1	yes	characteristic
182	C₂×C₃²	#343	1	1	yes	characteristic
183	C₂×C₃²	#344	1	1	yes	characteristic
184	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
185	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
186	C₂²×C₄	#624	1	1	yes	characteristic
187	M₁₆	—	3	4	no	—
188	M₁₆	—	6	2	no	—
189	C₂×C₈	—	6	2	no	—
190	C₂×C₈	—	6	2	no	—
191	C₂×C₈	—	18	2	no	—
192	C₂×C₈	—	36	2	no	—
193	C₄²	#625	1	1	yes	characteristic
194	C₄²	#626	1	1	yes	characteristic
195	C₄²	#626	1	1	yes	characteristic
196	C₄²	#627	1	1	yes	characteristic
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	C₄×C₃⋊C₈	1	1	yes	characteristic
207	C₂²×C₃	#803	1	1	yes	characteristic
208	C₂²×C₃	C₈×Dic₃	1	1	yes	characteristic
209	C₂²×C₃	#815	1	1	yes	characteristic
210	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
211	C₂²×C₃	C₃⋊#95	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	4	1	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	C₄⋊(C₃⋊C₈)	1	2	yes	—
226	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
227	C₃×C₄	(C₂×D₃)⋊C₈	1	2	yes	—
228	C₃×C₄	#823	1	1	yes	characteristic
229	C₃×C₄	#823	1	1	yes	characteristic
230	C₃×C₄	#824	1	1	yes	characteristic
231	C₃×C₄	#824	1	1	yes	characteristic
232	C₃×C₄	#830	1	2	yes	—
233	C₃²	#1283	1	1	yes	characteristic, complemented, Sylow
234	C₂³	Dic₃²	1	1	yes	characteristic
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
239	C₂×C₄	#3711	1	2	yes	—
240	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
241	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
242	C₂×C₄	#3722	1	1	yes	characteristic
243	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
244	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
245	C₂×C₄	#3724	1	1	yes	characteristic
246	C₈	—	6	4	no	—
247	C₈	—	12	2	no	—
248	C₈	—	36	2	no	—
249	C₂×C₃	—	2	2	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	1	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	#4524	1	1	yes	characteristic
259	C₂×C₃	#4533	1	1	yes	characteristic
260	C₂×C₃	#4550	1	1	yes	characteristic
261	C₂×C₃	#4554	1	1	yes	characteristic
262	C₂×C₃	#4554	1	1	yes	characteristic
263	C₂×C₃	#4559	1	1	yes	characteristic
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	Dic₃×C₃⋊C₈	1	1	yes	characteristic
267	C₂²	#63554	1	1	yes	characteristic
268	C₂²	#63579	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	(C₂×D₃)⋊(C₃⋊C₈)	1	2	yes	—
273	C₄	#63559	1	1	yes	characteristic
274	C₄	#63561	1	1	yes	characteristic
275	C₄	Dic₃⋊(C₃⋊C₈)	1	2	yes	—
276	C₃	—	2	1	no	—
277	C₃	#67668	1	1	yes	characteristic, complemented
278	C₃	#67744	1	1	yes	characteristic, complemented
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	#10589155	1	1	yes	characteristic
282	C₂	#10589168	1	1	yes	characteristic
283	C₂	#10589170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3722
commutator	C₂²×C₃²	C₄×C₈
Frattini, hypercenter	C₂×C₄²	D₃²
Fitting, cosocle	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄²
hypocenter	C₃²	#1283
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
33. C₃×#208	C₈	semidirect
105. #208	C₃⋊C₈	semidirect
233. C₃²	#1283	semidirect
277. C₃	#67668	semidirect
278. C₃	#67744	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
C₃×#4925	2
C₃×C₄×#208	2
#67744	3
#67668	3