Subgroups — #11772391

Order 1152.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63770	C₂	1	2	yes	maximal
2	C₂×#63764	C₂	1	1	yes	maximal, characteristic
3	C₂×#3788	—	2	2	no	—
4	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#3708	C₄	1	2	yes	—
6	#63770	—	2	2	no	—
7	#63770	C₄	1	4	yes	—
8	#63764	—	2	2	no	—
9	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
10	C₂×#622	—	4	2	no	—
11	#3788	—	2	4	no	—
12	#3788	—	4	2	no	—
13	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
15	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
16	C₈×C₃⋊D₃	D₄	1	2	yes	—
17	#3708	—	2	2	no	—
18	#1381	—	9	1	no	maximal, Sylow
19	C₂×D₃×C₈	—	12	1	no	—
20	C₂²×C₃⋊D₃	Q₁₆	1	1	yes	characteristic
21	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	D₈	1	1	yes	characteristic
23	C₄×C₃⋊D₃	C₂²⋊C₄	1	2	yes	—
24	C₄×C₃⋊D₃	SD₁₆	1	2	yes	—
25	#622	—	4	4	no	—
26	C₃²×C₈	C₄⋊C₄	1	2	yes	—
27	C₃⋊(C₃⋊C₈)	C₄²	1	1	yes	characteristic
28	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₆	—	9	1	no	—
30	C₂×#108	—	9	2	no	—
31	C₂×C₄×D₃	—	12	1	no	—
32	C₂×C₃×C₈	—	4	1	no	—
33	C₂×C₃⋊C₈	—	12	1	no	—
34	D₃×C₈	—	12	2	no	—
35	D₃×C₈	—	12	2	no	—
36	C₂²×C₃²	#107	1	1	yes	characteristic
37	C₂×C₃⋊D₃	—	2	2	no	—
38	C₂×C₃⋊D₃	#103	1	2	yes	—
39	C₃²×C₄	#95	1	1	yes	characteristic
40	C₃²×C₄	#106	1	1	yes	characteristic
41	C₃⋊Dic₃	#102	1	2	yes	—
42	C₂×M₁₆	—	18	2	no	—
43	C₂²×C₈	—	9	1	no	—
44	C₂×C₁₆	—	9	2	no	—
45	C₂×C₁₆	—	18	2	no	—
46	#108	—	9	4	no	—
47	#108	—	18	2	no	—
48	C₂²×D₃	—	12	1	no	—
49	C₂×C₃×C₄	—	4	1	no	—
50	C₂×Dic₃	—	12	1	no	—
51	C₄×D₃	—	12	2	no	—
52	C₄×D₃	—	12	2	no	—
53	C₃×C₈	—	4	2	no	—
54	C₃⋊C₈	—	12	1	no	—
55	C₃⋊C₈	—	12	1	no	—
56	C₂×C₃²	#340	1	1	yes	characteristic
57	C₂×C₃²	#368	1	2	yes	—
58	C₃⋊D₃	—	2	2	no	—
59	C₂²×C₄	—	9	1	no	—
60	M₁₆	—	18	4	no	—
61	M₁₆	—	36	2	no	—
62	C₂×C₈	—	9	1	no	—
63	C₂×C₈	—	9	2	no	—
64	C₂×C₈	—	9	2	no	—
65	C₂×C₈	—	36	2	no	—
66	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
67	C₁₆	—	18	2	no	—
68	C₂²×C₃	—	4	1	no	—
69	C₂×D₃	—	12	2	no	—
70	C₂×D₃	—	12	4	no	—
71	C₃×C₄	—	4	1	no	—
72	C₃×C₄	—	4	1	no	—
73	Dic₃	—	12	2	no	—
74	C₃²	#1381	1	1	yes	characteristic, complemented, Sylow
75	C₂³	—	9	1	no	—
76	C₂×C₄	—	9	1	no	—
77	C₂×C₄	—	9	2	no	—
78	C₂×C₄	—	9	2	no	—
79	C₂×C₄	#3777	1	1	yes	characteristic
80	C₈	—	9	1	no	—
81	C₈	—	9	1	no	—
82	C₈	—	36	4	no	—
83	C₈	#3777	1	2	yes	—
84	C₂×C₃	—	4	2	no	—
85	C₂×C₃	—	4	1	no	—
86	D₃	—	12	4	no	—
87	C₂²	—	9	2	no	—
88	C₂²	—	18	2	no	—
89	C₂²	#63746	1	1	yes	characteristic
90	C₄	—	9	2	no	—
91	C₄	#63745	1	1	yes	characteristic
92	C₄	#63747	1	1	yes	characteristic
93	C₃	—	4	1	no	—
94	C₂	—	18	2	no	—
95	C₂	#10590087	1	2	yes	—
96	C₂	#10590093	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63746
commutator	C₃⋊(C₃⋊C₈)	C₄²
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂²×C₃²	#107
prime radical	C₂²×C₃⋊D₃	Q₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1381
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ 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
74. C₃²	#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63770	2
C₂×#63764	2
#1381	9