Subgroups — #11777050

Order 1152.

355 conjugacy classes of subgroups.

The subgroup lattice has 267 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592625	C₂	1	1	yes	maximal, characteristic
2	C₃×#4925	C₂	1	1	yes	maximal, characteristic
3	C₃×#4925	C₂	1	1	yes	maximal, characteristic
4	#10589099	C₂	1	2	yes	maximal, complemented
5	#10589097	C₂	1	2	yes	maximal, complemented
6	#70751	—	3	1	no	maximal
7	#70747	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	#63657	—	2	1	no	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
11	#63630	—	2	1	no	—
12	C₃×#830	—	2	1	no	—
13	C₃×#830	C₄	1	2	yes	—
14	C₃×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
15	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	—	2	1	no	—
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#431	—	3	1	no	—
20	C₃×#431	—	3	1	no	—
21	#4926	—	3	1	no	—
22	#4926	—	3	1	no	—
23	#4926	—	6	1	no	—
24	#4925	D₃	1	1	yes	characteristic
25	#4925	D₃	1	1	yes	characteristic
26	#4480	—	3	2	no	—
27	#4479	—	3	2	no	—
28	#4458	—	3	2	no	—
29	#4457	—	3	2	no	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
33	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄²	D₄	1	1	yes	characteristic
35	C₃²×C₄²	Q₈	1	1	yes	characteristic
36	C₃²×C₄²	C₂³	1	1	yes	characteristic
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	#1572	—	9	1	no	Sylow
42	C₂×C₃×C₄²	—	2	1	no	—
43	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₃×C₄⋊C₈	—	3	2	no	—
46	C₃×C₄⋊C₈	—	3	2	no	—
47	C₃×C₂²⋊C₈	—	3	2	no	—
48	C₃×C₂²⋊C₈	—	6	1	no	—
49	C₃×C₄×C₈	—	3	1	no	—
50	C₃×C₄×C₈	—	3	1	no	—
51	C₃×C₄×C₈	—	6	1	no	—
52	#830	—	2	1	no	—
53	#830	—	6	2	no	—
54	#830	—	6	1	no	—
55	#830	—	6	1	no	—
56	#830	Dic₃	1	2	yes	—
57	C₄⋊(C₃⋊C₈)	—	3	2	no	—
58	C₄⋊(C₃⋊C₈)	—	3	2	no	—
59	C₄⋊(C₃⋊C₈)	—	6	2	no	—
60	C₄⋊(C₃⋊C₈)	Dic₃	1	2	yes	—
61	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
62	#803	—	6	2	no	—
63	#803	—	6	1	no	—
64	#803	—	6	1	no	—
65	C₄×C₃⋊C₈	—	2	1	no	—
66	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
67	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
68	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	—	2	2	no	—
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	—	4	1	no	—
72	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₃⋊(C₃⋊C₈)	—	4	2	no	—
79	C₃×C₃⋊C₈	—	2	2	no	—
80	C₃×C₃⋊C₈	—	2	2	no	—
81	C₃×C₃⋊C₈	—	4	2	no	—
82	#432	—	9	1	no	—
83	#431	—	3	1	no	—
84	#431	—	3	1	no	—
85	#335	—	9	2	no	—
86	#334	—	9	2	no	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
91	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	#67	1	1	yes	characteristic
96	C₂×C₃×C₈	—	3	2	no	—
97	C₂×C₃×C₈	—	3	2	no	—
98	C₂×C₃×C₈	—	6	2	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	Dic₆	1	1	yes	characteristic
104	C₃×C₄²	Dic₆	1	1	yes	characteristic
105	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
106	C₃×C₄²	D₁₂	1	1	yes	characteristic
107	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
108	C₃×C₄²	#67	1	1	yes	characteristic
109	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
110	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₃⋊C₈	—	2	2	no	—
112	C₂×C₃⋊C₈	—	6	4	no	—
113	C₂×C₃⋊C₈	—	6	2	no	—
114	C₂×C₃⋊C₈	—	6	2	no	—
115	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
116	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	—	2	1	no	—
125	C₃²×C₄	#131	1	2	yes	—
126	C₃²×C₄	#135	1	2	yes	—
127	C₂×C₄²	D₃²	1	1	yes	characteristic
128	C₄⋊C₈	—	3	2	no	—
129	C₄⋊C₈	—	3	2	no	—
130	C₄⋊C₈	—	9	2	no	—
131	C₂²⋊C₈	—	3	2	no	—
132	C₂²⋊C₈	—	6	1	no	—
133	C₂²⋊C₈	—	18	1	no	—
134	#97	—	18	1	no	—
135	C₄×C₈	—	3	1	no	—
136	C₄×C₈	—	3	1	no	—
137	C₄×C₈	—	6	1	no	—
138	C₂³×C₃	—	2	1	no	—
139	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
140	C₂³×C₃	C₄⋊Dic₃	1	1	yes	characteristic
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	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	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	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	Dic₃⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
158	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
161	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
168	C₃×C₈	—	6	2	no	—
169	C₃×C₈	—	6	2	no	—
170	C₃×C₈	—	12	2	no	—
171	C₃⋊C₈	—	2	2	no	—
172	C₃⋊C₈	—	2	2	no	—
173	C₃⋊C₈	—	4	2	no	—
174	C₃⋊C₈	—	12	4	no	—
175	C₃⋊C₈	—	12	2	no	—
176	C₃⋊C₈	—	12	2	no	—
177	C₂×C₃²	—	4	1	no	—
178	C₂×C₃²	#424	1	1	yes	characteristic
179	C₂×C₃²	#427	1	1	yes	characteristic
180	C₂×C₃²	#430	1	1	yes	characteristic
181	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
182	C₂²×C₄	#626	1	1	yes	characteristic
183	C₂²×C₄	#627	1	1	yes	characteristic
184	C₂×C₈	—	3	2	no	—
185	C₂×C₈	—	3	2	no	—
186	C₂×C₈	—	6	2	no	—
187	C₂×C₈	—	18	2	no	—
188	C₄²	D₃×Dic₃	1	1	yes	characteristic
189	C₄²	#626	1	1	yes	characteristic
190	C₄²	#627	1	1	yes	characteristic
191	C₄²	C₂×D₃²	1	1	yes	characteristic
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	4	1	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
203	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	1	no	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	4	1	no	—
218	C₃×C₄	#901	1	2	yes	—
219	C₃×C₄	#904	1	2	yes	—
220	C₃×C₄	#948	1	2	yes	—
221	C₃×C₄	#950	1	2	yes	—
222	C₃²	#1572	1	1	yes	characteristic, complemented, Sylow
223	C₂³	Dic₃⋊Dic₃	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
228	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
229	C₂×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
230	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
231	C₂×C₄	C₂×#626	1	1	yes	characteristic
232	C₂×C₄	C₂×#627	1	1	yes	characteristic
233	C₈	—	6	2	no	—
234	C₈	—	6	2	no	—
235	C₈	—	12	2	no	—
236	C₈	—	36	2	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	2	1	no	—
240	C₂×C₃	—	4	2	no	—
241	C₂×C₃	—	4	1	no	—
242	C₂×C₃	—	4	1	no	—
243	C₂×C₃	#4998	1	1	yes	characteristic
244	C₂×C₃	#4999	1	1	yes	characteristic
245	C₂×C₃	#5102	1	1	yes	characteristic
246	C₂×C₃	#5106	1	1	yes	characteristic
247	C₂×C₃	#5124	1	1	yes	characteristic
248	C₂×C₃	#5125	1	1	yes	characteristic
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	#63814	1	2	yes	—
258	C₄	#63829	1	2	yes	—
259	C₃	—	2	1	no	—
260	C₃	#70747	1	1	yes	characteristic, complemented
261	C₃	#70751	1	1	yes	characteristic, complemented
262	C₂	—	4	1	no	—
263	C₂	#10591116	1	1	yes	characteristic
264	C₂	#10591122	1	1	yes	characteristic
265	C₂	#10591141	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1572
LCS	G ▷ C₂×C₃²×C₄ ▷ 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
4. #10589099	C₂	semidirect
5. #10589097	C₂	semidirect
222. C₃²	#1572	semidirect
260. C₃	#70747	semidirect
261. C₃	#70751	semidirect

Maximal subgroups

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

subgroup	index
#10592625	2
C₃×#4925	2
#10589099	2
#10589097	2
#70751	3
#70747	3