Subgroups — #10592929

Order 576.

546 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#3750	C₂	1	3	yes	maximal
3	#63658	C₂	1	3	yes	maximal
4	#5230	—	3	4	no	maximal
5	C₂×C₃²×Q₈	—	2	6	no	—
6	C₂²×C₃²×C₄	C₂²	1	3	yes	—
7	C₂²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₃	C₂²	1	3	yes	—
9	#3750	—	2	6	no	—
10	C₂²×C₃×Q₈	D₃	1	4	yes	—
11	C₂×Dic₃⋊C₄	—	3	12	no	—
12	C₃⋊#95	—	3	12	no	—
13	C₂³×C₃²	C₂³	1	1	yes	characteristic
14	C₃²×Q₈	—	2	8	no	—
15	C₂×C₃²×C₄	—	2	6	no	—
16	C₂×C₃²×C₄	D₄	1	6	yes	—
17	C₂×C₃⋊Dic₃	—	2	3	no	—
18	C₂×C₃⋊Dic₃	—	2	3	no	—
19	C₂×C₃⋊Dic₃	—	2	3	no	—
20	C₂×C₃⋊Dic₃	Q₈	1	6	yes	complemented
21	#395	—	9	1	no	Sylow
22	C₂×C₃×Q₈	—	2	24	no	—
23	C₂²×C₃×C₄	C₂×D₃	1	12	yes	—
24	C₂²×Dic₃	—	3	12	no	—
25	C₂²×Dic₃	—	3	4	no	—
26	Dic₃⋊C₄	—	6	24	no	—
27	C₂²×C₃²	C₂×D₄	1	3	yes	—
28	C₂²×C₃²	C₂×Q₈	1	3	yes	—
29	C₂²×C₃²	#41	1	1	yes	characteristic
30	C₃²×C₄	—	2	6	no	—
31	C₃⋊Dic₃	—	2	6	no	—
32	C₃⋊Dic₃	—	4	1	no	—
33	C₂²×Q₈	C₃⋊D₃	1	1	yes	characteristic
34	C₂×C₄⋊C₄	—	9	3	no	—
35	#95	—	9	3	no	—
36	C₂³×C₃	C₂²×D₃	1	4	yes	—
37	C₃×Q₈	—	2	32	no	—
38	C₂×C₃×C₄	—	2	24	no	—
39	C₂×C₃×C₄	#67	1	24	yes	—
40	C₂×Dic₃	—	3	24	no	—
41	C₂×Dic₃	—	6	12	no	—
42	C₂×Dic₃	—	6	12	no	—
43	C₂×Dic₃	—	6	12	no	—
44	C₂×C₃²	#120	1	1	yes	characteristic
45	C₂×C₃²	C₂²⋊Q₈	1	3	yes	—
46	C₂×C₃²	C₄⋊Q₈	1	3	yes	—
47	C₂×Q₈	—	2	6	no	—
48	C₂²×C₄	—	9	3	no	—
49	C₂²×C₄	—	9	1	no	—
50	C₂²×C₄	C₂×C₃⋊D₃	1	3	yes	—
51	C₄⋊C₄	—	18	6	no	—
52	C₂²×C₃	D₃×Q₈	1	12	yes	—
53	C₂²×C₃	#239	1	4	yes	—
54	C₂²×C₃	C₂×#67	1	12	yes	—
55	C₃×C₄	—	2	24	no	—
56	Dic₃	—	6	24	no	—
57	Dic₃	—	12	4	no	—
58	C₃²	#395	1	1	yes	characteristic, complemented, Sylow
59	C₂³	C₂²×C₃⋊D₃	1	1	yes	characteristic
60	Q₈	—	2	8	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	—	9	6	no	—
63	C₂×C₄	—	18	3	no	—
64	C₂×C₄	—	18	3	no	—
65	C₂×C₄	—	18	3	no	—
66	C₂×C₄	#638	1	6	yes	—
67	C₂×C₃	#944	1	12	yes	—
68	C₂×C₃	#946	1	12	yes	—
69	C₂×C₃	#953	1	4	yes	—
70	C₂²	Q₈×C₃⋊D₃	1	3	yes	—
71	C₂²	#3832	1	1	yes	characteristic
72	C₂²	C₂×#638	1	3	yes	—
73	C₄	—	2	6	no	—
74	C₄	—	18	6	no	—
75	C₄	—	36	1	no	—
76	C₃	#5230	1	4	yes	complemented
77	C₂	#64153	1	3	yes	—
78	C₂	#64155	1	3	yes	—
79	C₂	#64162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₃⋊D₃
commutator, socle, prime radical, cosocle	C₂³×C₃²	C₂³
Fitting	C₂²×C₃²×Q₈	C₂
hypercenter	C₂²×Q₈	C₃⋊D₃
hypocenter	C₃²	#395
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
20. C₂×C₃⋊Dic₃	Q₈	semidirect
58. C₃²	#395	semidirect
76. C₃	#5230	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×Q₈	2
C₂×#3750	2
#63658	2
#5230	3