Finite Group Databasebeta

Subgroups — #11790152

Order 1152.

286 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63769	C₂	1	1	yes	maximal, characteristic
2	C₂×#63769	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#157	C₂	1	1	yes	maximal, characteristic
4	#10593554	C₂	1	2	yes	maximal
5	#10590093	C₂	1	2	yes	maximal
6	C₂×C₄⋊#157	C₂²	1	2	yes	—
7	C₂×C₄⋊#157	C₂²	1	2	yes	—
8	C₂×C₄×#157	C₂²	1	1	yes	characteristic
9	C₂×#3787	C₂²	1	1	yes	characteristic
10	C₂×#3777	—	2	2	no	—
11	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	#63769	—	2	2	no	—
13	#63769	C₄	1	4	yes	complemented
14	C₈×#157	—	2	2	no	—
15	#63747	—	2	2	no	—
16	C₂²×#157	—	2	2	no	—
17	C₂²×#157	—	2	2	no	—
18	C₂²×#157	D₄	1	1	yes	characteristic
19	C₂²×#157	D₄	1	1	yes	characteristic
20	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
21	C₂×#622	Q₈	1	2	yes	—
22	C₄⋊#157	—	2	2	no	—
23	C₄⋊#157	—	2	2	no	—
24	C₄⋊#157	—	2	2	no	—
25	C₄⋊#157	C₂×C₄	1	4	yes	—
26	C₄×#157	—	2	1	no	—
27	C₄×#157	—	2	1	no	—
28	#3787	—	2	1	no	—
29	#3787	—	2	1	no	—
30	#3777	—	4	4	no	—
31	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
33	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
34	C₈×C₃⋊D₃	D₄	1	2	yes	—
35	#1942	—	9	1	no	maximal, Sylow
36	C₂×D₃×C₈	—	12	1	no	—
37	C₂²×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
38	C₂×#157	—	2	1	no	—
39	C₂×#157	—	2	1	no	—
40	C₂×#157	—	2	1	no	—
41	C₂×#157	—	2	1	no	—
42	C₂×#157	—	2	4	no	—
43	C₂×#157	—	4	2	no	—
44	C₂×#157	—	4	2	no	—
45	C₂×#157	—	4	2	no	—
46	C₂×#157	—	4	2	no	—
47	C₂×#157	—	4	2	no	—
48	C₂×#157	D₈	1	2	yes	—
49	C₂×#157	Q₁₆	1	2	yes	—
50	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
52	C₄×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
53	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
54	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
55	C₄×C₃⋊D₃	#41	1	1	yes	characteristic
56	#622	—	2	2	no	—
57	C₃²×C₈	C₄⋊C₄	1	2	yes	complemented
58	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	complemented
59	C₂×#107	—	9	1	no	—
60	C₂×#107	—	9	1	no	—
61	C₂×C₄×C₈	—	9	1	no	—
62	#389	—	9	2	no	—
63	#340	—	9	2	no	—
64	C₂×C₄×D₃	—	12	1	no	—
65	C₂×C₃×C₈	—	4	1	no	—
66	C₂×C₃⋊C₈	—	12	1	no	—
67	D₃×C₈	—	12	2	no	—
68	D₃×C₈	—	12	2	no	—
69	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₃⋊D₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₃⋊D₃	#121	1	1	yes	characteristic
72	C₂×C₃⋊D₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₃⋊D₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₃⋊D₃	#135	1	1	yes	characteristic
75	C₂×C₃⋊D₃	#135	1	1	yes	characteristic
76	#157	—	2	2	no	—
77	#157	—	2	2	no	—
78	#157	—	4	4	no	—
79	#157	—	8	2	no	—
80	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₃²×C₄	#125	1	1	yes	characteristic
82	C₃⋊Dic₃	C₄×Q₈	1	1	yes	characteristic
83	C₃⋊Dic₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂²×C₈	—	9	1	no	—
85	C₂²×C₈	—	9	1	no	—
86	C₂×C₄⋊C₄	—	9	2	no	—
87	C₂×C₄⋊C₄	—	9	2	no	—
88	C₂×C₄⋊C₄	—	18	2	no	—
89	C₂×C₄²	—	9	1	no	—
90	#107	—	9	4	no	—
91	#107	—	18	2	no	—
92	C₄×C₈	—	18	2	no	—
93	#95	—	18	2	no	—
94	C₂²×D₃	—	12	1	no	—
95	C₂×C₃×C₄	—	4	1	no	—
96	C₂×Dic₃	—	12	1	no	—
97	C₄×D₃	—	12	1	no	—
98	C₄×D₃	—	12	1	no	—
99	C₄×D₃	—	12	1	no	—
100	C₄×D₃	—	12	1	no	—
101	C₃×C₈	—	4	2	no	—
102	C₃⋊C₈	—	12	2	no	—
103	C₂×C₃²	#389	1	1	yes	characteristic
104	C₂×C₃²	#499	1	1	yes	characteristic
105	C₂×C₃²	#500	1	1	yes	characteristic
106	C₃⋊D₃	C₄×D₈	1	1	yes	characteristic
107	C₃⋊D₃	C₄×Q₁₆	1	1	yes	characteristic
108	C₃⋊D₃	#466	1	1	yes	characteristic
109	C₃⋊D₃	#467	1	1	yes	characteristic
110	C₂²×C₄	—	9	1	no	—
111	C₂²×C₄	—	9	1	no	—
112	C₂²×C₄	—	9	1	no	—
113	C₂²×C₄	—	18	2	no	—
114	C₂²×C₄	—	18	2	no	—
115	C₂×C₈	—	9	1	no	—
116	C₂×C₈	—	9	2	no	—
117	C₂×C₈	—	9	2	no	—
118	C₂×C₈	—	9	2	no	—
119	C₂×C₈	—	18	1	no	—
120	C₂×C₈	—	18	1	no	—
121	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
122	C₄⋊C₄	—	9	4	no	—
123	C₄⋊C₄	—	18	2	no	—
124	C₄⋊C₄	—	18	2	no	—
125	C₄⋊C₄	—	18	2	no	—
126	C₄⋊C₄	—	36	4	no	—
127	C₄²	—	18	1	no	—
128	C₄²	—	18	1	no	—
129	C₂²×C₃	—	4	1	no	—
130	C₂×D₃	—	12	1	no	—
131	C₂×D₃	—	12	1	no	—
132	C₂×D₃	—	12	1	no	—
133	C₂×D₃	—	12	1	no	—
134	C₂×D₃	—	12	1	no	—
135	C₂×D₃	—	12	1	no	—
136	C₃×C₄	—	4	1	no	—
137	C₃×C₄	—	4	1	no	—
138	Dic₃	—	12	1	no	—
139	Dic₃	—	12	1	no	—
140	C₃²	#1942	1	1	yes	characteristic, complemented, Sylow
141	C₂³	—	9	1	no	—
142	C₂×C₄	—	9	1	no	—
143	C₂×C₄	—	9	1	no	—
144	C₂×C₄	—	9	1	no	—
145	C₂×C₄	—	9	1	no	—
146	C₂×C₄	—	9	1	no	—
147	C₂×C₄	—	9	2	no	—
148	C₂×C₄	—	9	2	no	—
149	C₂×C₄	—	18	1	no	—
150	C₂×C₄	—	18	1	no	—
151	C₂×C₄	—	18	1	no	—
152	C₂×C₄	—	18	1	no	—
153	C₂×C₄	—	18	4	no	—
154	C₂×C₄	—	36	2	no	—
155	C₂×C₄	—	36	2	no	—
156	C₂×C₄	—	36	2	no	—
157	C₂×C₄	—	36	2	no	—
158	C₂×C₄	—	36	2	no	—
159	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
160	C₈	—	9	2	no	—
161	C₈	—	18	2	no	—
162	C₈	#3777	1	2	yes	complemented
163	C₂×C₃	—	4	1	no	—
164	C₂×C₃	—	4	1	no	—
165	C₂×C₃	—	4	1	no	—
166	D₃	—	12	1	no	—
167	D₃	—	12	1	no	—
168	D₃	—	12	1	no	—
169	D₃	—	12	1	no	—
170	C₂²	—	9	1	no	—
171	C₂²	—	9	1	no	—
172	C₂²	—	9	1	no	—
173	C₂²	—	9	1	no	—
174	C₂²	—	9	1	no	—
175	C₂²	—	9	1	no	—
176	C₂²	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
177	C₄	—	9	1	no	—
178	C₄	—	9	1	no	—
179	C₄	—	18	2	no	—
180	C₄	—	18	2	no	—
181	C₄	—	36	4	no	—
182	C₄	—	72	2	no	—
183	C₄	#64243	1	1	yes	characteristic
184	C₄	C₂×#3777	1	1	yes	characteristic
185	C₃	—	4	1	no	—
186	C₂	—	9	1	no	—
187	C₂	—	9	1	no	—
188	C₂	—	9	1	no	—
189	C₂	—	9	1	no	—
190	C₂	#10593534	1	1	yes	characteristic
191	C₂	#10593536	1	1	yes	characteristic
192	C₂	#10593554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊PSU(3,𝔽₂)
commutator	C₄×C₃⋊D₃	C₂²×C₄
Frattini	C₂×C₄	C₂×PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂²×C₃²	C₄⋊Q₈
prime radical	C₂²×C₃⋊D₃	C₂×D₄
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	C₂×C₈	PSU(3,𝔽₂)
hypocenter	C₃²	#1942
LCS	G ▷ C₄×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
13. #63769	C₄	semidirect
57. C₃²×C₈	C₄⋊C₄	semidirect
58. C₃⋊(C₃⋊C₈)	C₄⋊C₄	semidirect
140. C₃²	#1942	semidirect
162. C₈	#3777	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63769	2
C₂×C₈×#157	2
#10593554	2
#10590093	2
#1942	9