Subgroups — #11777049

Order 1152.

359 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592624	C₂	1	1	yes	maximal, characteristic
2	C₃×#4926	C₂	1	1	yes	maximal, characteristic
3	C₃×#4925	C₂	1	1	yes	maximal, characteristic
4	#10589100	C₂	1	2	yes	maximal, complemented
5	#10589098	C₂	1	2	yes	maximal, complemented
6	#70796	—	3	1	no	maximal
7	#70747	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	#63657	C₄	1	2	yes	—
10	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	2	yes	—
11	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₃×#830	—	2	1	no	—
14	C₃×#830	—	2	1	no	—
15	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₃×#803	—	2	1	no	—
18	C₃×C₄×C₃⋊C₈	—	2	1	no	—
19	C₃×#432	—	3	1	no	—
20	C₃×#431	—	3	1	no	—
21	#4926	D₃	1	1	yes	characteristic
22	#4925	—	3	1	no	—
23	#4925	—	3	1	no	—
24	#4925	—	6	1	no	—
25	#4925	D₃	1	1	yes	characteristic
26	#4480	—	3	2	no	—
27	#4480	—	3	2	no	—
28	#4479	—	3	2	no	—
29	#4479	—	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)	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₃⋊C₈	—	2	2	no	—
40	C₂×C₃×C₃⋊C₈	—	2	2	no	—
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	—	6	1	no	—
48	C₃×C₂²⋊C₈	—	6	1	no	—
49	C₃×#97	—	6	1	no	—
50	C₃×C₄×C₈	—	6	1	no	—
51	#830	—	2	1	no	—
52	#830	—	2	1	no	—
53	#830	—	3	2	no	—
54	#830	—	3	2	no	—
55	#830	—	6	2	no	—
56	C₄⋊(C₃⋊C₈)	—	3	2	no	—
57	C₄⋊(C₃⋊C₈)	—	3	2	no	—
58	C₄⋊(C₃⋊C₈)	—	6	2	no	—
59	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
60	C₄⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
61	#803	—	2	1	no	—
62	C₄×C₃⋊C₈	—	2	1	no	—
63	C₄×C₃⋊C₈	—	3	1	no	—
64	C₄×C₃⋊C₈	—	3	1	no	—
65	C₄×C₃⋊C₈	—	3	1	no	—
66	C₄×C₃⋊C₈	—	3	1	no	—
67	C₄×C₃⋊C₈	—	6	1	no	—
68	C₄×C₃⋊C₈	—	6	1	no	—
69	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	4	1	no	—
73	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₃⋊(C₃⋊C₈)	—	2	2	no	—
80	C₃⋊(C₃⋊C₈)	—	2	2	no	—
81	C₃×C₃⋊C₈	—	4	2	no	—
82	C₃×C₃⋊C₈	—	4	2	no	—
83	#432	—	3	1	no	—
84	#431	—	3	1	no	—
85	#431	—	9	1	no	—
86	#335	—	9	2	no	—
87	#334	—	9	2	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	2	1	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
92	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	#67	1	1	yes	characteristic
96	C₂²×C₃×C₄	#67	1	1	yes	characteristic
97	C₂×C₃×C₈	—	6	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	C₄×D₃	1	1	yes	characteristic
107	C₃×C₄²	#67	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	—	2	2	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂×C₃⋊C₈	—	3	2	no	—
115	C₂×C₃⋊C₈	—	3	2	no	—
116	C₂×C₃⋊C₈	—	3	2	no	—
117	C₂×C₃⋊C₈	—	6	2	no	—
118	C₂×C₃⋊C₈	—	6	2	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	—	2	1	no	—
122	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	—	2	1	no	—
125	C₃²×C₄	—	2	1	no	—
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	#131	1	2	yes	—
128	C₃²×C₄	#135	1	2	yes	—
129	C₂×C₄²	D₃²	1	1	yes	characteristic
130	C₄⋊C₈	—	3	2	no	—
131	C₄⋊C₈	—	3	2	no	—
132	C₄⋊C₈	—	9	2	no	—
133	C₂²⋊C₈	—	6	1	no	—
134	C₂²⋊C₈	—	6	1	no	—
135	C₂²⋊C₈	—	9	2	no	—
136	#97	—	6	1	no	—
137	C₄×C₈	—	6	1	no	—
138	C₄×C₈	—	9	1	no	—
139	C₄×C₈	—	9	1	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
142	C₂³×C₃	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	2	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	—	2	1	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	4	1	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	Dic₃⋊C₄	1	1	yes	characteristic
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	C₂×Dic₆	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
170	C₃×C₈	—	12	2	no	—
171	C₃×C₈	—	12	2	no	—
172	C₃⋊C₈	—	4	2	no	—
173	C₃⋊C₈	—	4	2	no	—
174	C₃⋊C₈	—	6	2	no	—
175	C₃⋊C₈	—	6	2	no	—
176	C₃⋊C₈	—	6	2	no	—
177	C₃⋊C₈	—	6	2	no	—
178	C₃⋊C₈	—	12	2	no	—
179	C₃⋊C₈	—	12	2	no	—
180	C₂×C₃²	—	4	1	no	—
181	C₂×C₃²	#424	1	1	yes	characteristic
182	C₂×C₃²	#427	1	1	yes	characteristic
183	C₂×C₃²	#430	1	1	yes	characteristic
184	C₂²×C₄	#624	1	1	yes	characteristic
185	C₂²×C₄	#625	1	1	yes	characteristic
186	C₂²×C₄	#627	1	1	yes	characteristic
187	C₂×C₈	—	6	2	no	—
188	C₂×C₈	—	6	2	no	—
189	C₂×C₈	—	9	2	no	—
190	C₂×C₈	—	9	2	no	—
191	C₄²	#624	1	1	yes	characteristic
192	C₄²	#625	1	1	yes	characteristic
193	C₄²	#627	1	1	yes	characteristic
194	C₄²	C₂×D₃²	1	1	yes	characteristic
195	C₂²×C₃	—	2	2	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	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
206	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	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	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	1	no	—
220	C₃×C₄	—	4	1	no	—
221	C₃×C₄	#901	1	2	yes	—
222	C₃×C₄	#907	1	2	yes	—
223	C₃×C₄	#950	1	2	yes	—
224	C₃×C₄	#950	1	2	yes	—
225	C₃²	#1572	1	1	yes	characteristic, complemented, Sylow
226	C₂³	#3724	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	#3724	1	1	yes	characteristic
231	C₂×C₄	#3724	1	1	yes	characteristic
232	C₂×C₄	#3724	1	1	yes	characteristic
233	C₂×C₄	C₂×#624	1	1	yes	characteristic
234	C₂×C₄	C₂×#625	1	1	yes	characteristic
235	C₂×C₄	C₂×#627	1	1	yes	characteristic
236	C₈	—	12	2	no	—
237	C₈	—	12	2	no	—
238	C₈	—	18	2	no	—
239	C₈	—	18	2	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	2	1	no	—
243	C₂×C₃	—	4	2	no	—
244	C₂×C₃	—	4	1	no	—
245	C₂×C₃	—	4	1	no	—
246	C₂×C₃	#4998	1	1	yes	characteristic
247	C₂×C₃	#4998	1	1	yes	characteristic
248	C₂×C₃	#5102	1	1	yes	characteristic
249	C₂×C₃	#5119	1	1	yes	characteristic
250	C₂×C₃	#5124	1	1	yes	characteristic
251	C₂×C₃	#5124	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂×#3724	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	#63817	1	2	yes	—
261	C₄	#63822	1	2	yes	—
262	C₃	—	2	1	no	—
263	C₃	#70747	1	1	yes	characteristic, complemented
264	C₃	#70796	1	1	yes	characteristic, complemented
265	C₂	—	4	1	no	—
266	C₂	#10591119	1	1	yes	characteristic
267	C₂	#10591138	1	1	yes	characteristic
268	C₂	#10591143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#625
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. #10589100	C₂	semidirect
5. #10589098	C₂	semidirect
225. C₃²	#1572	semidirect
263. C₃	#70747	semidirect
264. C₃	#70796	semidirect

Maximal subgroups

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

subgroup	index
#10592624	2
C₃×#4926	2
C₃×#4925	2
#10589100	2
#10589098	2
#70796	3
#70747	3