Subgroups — #10590090

Order 576.

145 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#157	C₂	1	1	yes	maximal, characteristic
2	C₂×#3787	C₂	1	2	yes	maximal
3	C₂²×#157	C₄	1	2	yes	—
4	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂×#622	—	2	2	no	—
6	C₄×#157	—	2	2	no	—
7	#3787	—	2	2	no	—
8	#3787	C₄	1	4	yes	complemented
9	C₂²×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
10	C₂×#157	—	2	2	no	—
11	C₂×#157	—	2	2	no	—
12	C₂×#157	C₈	1	4	yes	complemented
13	C₂×C₃²×C₄	Q₈	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
15	C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
16	C₄×C₃⋊D₃	D₄	1	2	yes	—
17	#622	—	2	4	no	—
18	#336	—	9	1	no	maximal, Sylow
19	C₂×C₄×D₃	—	12	1	no	—
20	C₂²×C₃²	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₃⋊D₃	C₄²	1	1	yes	characteristic
22	C₂×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₃⋊D₃	C₂×C₈	1	2	yes	—
24	C₂×C₃⋊D₃	M₁₆	1	2	yes	—
25	#157	—	2	4	no	—
26	C₃²×C₄	C₄⋊C₄	1	2	yes	—
27	C₃⋊Dic₃	C₂²⋊C₄	1	2	yes	—
28	C₂²×C₈	—	9	2	no	—
29	C₂×C₄²	—	9	1	no	—
30	C₂²×D₃	—	12	1	no	—
31	C₂×C₃×C₄	—	4	1	no	—
32	C₂×Dic₃	—	12	1	no	—
33	C₄×D₃	—	12	2	no	—
34	C₄×D₃	—	12	2	no	—
35	C₂×C₃²	#95	1	1	yes	characteristic
36	C₂×C₃²	C₄⋊C₈	1	2	yes	—
37	C₃⋊D₃	C₄×C₈	1	1	yes	characteristic
38	C₃⋊D₃	#97	1	1	yes	characteristic
39	C₃⋊D₃	C₂²⋊C₈	1	2	yes	—
40	C₂²×C₄	—	9	1	no	—
41	C₂²×C₄	—	9	2	no	—
42	C₂×C₈	—	9	4	no	—
43	C₂×C₈	—	18	2	no	—
44	C₂×C₈	—	18	2	no	—
45	C₄²	—	18	2	no	—
46	C₂²×C₃	—	4	1	no	—
47	C₂×D₃	—	12	1	no	—
48	C₂×D₃	—	12	1	no	—
49	C₂×D₃	—	12	2	no	—
50	C₂×D₃	—	12	2	no	—
51	C₃×C₄	—	4	2	no	—
52	Dic₃	—	12	2	no	—
53	C₃²	#336	1	1	yes	characteristic, complemented, Sylow
54	C₂³	—	9	1	no	—
55	C₂×C₄	—	9	1	no	—
56	C₂×C₄	—	9	2	no	—
57	C₂×C₄	—	9	2	no	—
58	C₂×C₄	—	9	4	no	—
59	C₂×C₄	—	18	2	no	—
60	C₂×C₄	—	18	2	no	—
61	C₂×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
62	C₈	—	18	4	no	—
63	C₂×C₃	—	4	2	no	—
64	C₂×C₃	—	4	1	no	—
65	D₃	—	12	1	no	—
66	D₃	—	12	1	no	—
67	D₃	—	12	2	no	—
68	C₂²	—	9	2	no	—
69	C₂²	—	9	2	no	—
70	C₂²	—	9	1	no	—
71	C₂²	—	9	1	no	—
72	C₂²	#3777	1	1	yes	characteristic
73	C₄	—	9	2	no	—
74	C₄	—	18	4	no	—
75	C₄	#3777	1	2	yes	—
76	C₃	—	4	1	no	—
77	C₂	—	9	2	no	—
78	C₂	—	9	1	no	—
79	C₂	—	9	1	no	—
80	C₂	#63744	1	2	yes	—
81	C₂	#63747	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	PSU(3,𝔽₂)
commutator	C₃⋊D₃	C₄×C₈
Fitting	C₂×C₃²×C₄	Q₈
socle	C₂²×C₃²	C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	C₂×C₄
cosocle	C₂×C₄×C₃⋊D₃	C₂²
hypocenter	C₃²	#336
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. #3787	C₄	semidirect
12. C₂×#157	C₈	semidirect
53. C₃²	#336	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#157	2
C₂×#3787	2
#336	9