Subgroups — #10596283

Order 576.

798 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	Q₈×C₃⋊Dic₃	C₂	1	8	yes	maximal, complemented
3	#64135	C₂	1	6	yes	maximal
4	#5812	—	3	4	no	maximal
5	C₂×C₃²×Q₈	C₄	1	8	yes	complemented
6	C₂×C₃²×Q₈	C₂²	1	4	yes	—
7	C₂²×C₃²×C₄	C₂²	1	3	yes	—
8	#3757	C₂²	1	4	yes	—
9	C₄⋊(C₃⋊Dic₃)	C₂²	1	12	yes	—
10	C₄×C₃⋊Dic₃	C₂²	1	12	yes	—
11	C₂²×C₃×Q₈	D₃	1	4	yes	—
12	Q₈×Dic₃	—	3	32	no	—
13	#926	—	3	24	no	—
14	C₂³×C₃²	C₂³	1	1	yes	characteristic
15	C₃²×Q₈	C₂×C₄	1	16	yes	—
16	C₂×C₃²×C₄	C₂×C₄	1	12	yes	—
17	C₂×C₃²×C₄	C₂³	1	6	yes	—
18	C₂×C₃⋊Dic₃	C₂³	1	8	yes	—
19	#519	—	9	1	no	Sylow
20	C₂×C₃×Q₈	Dic₃	1	32	yes	complemented
21	C₂×C₃×Q₈	C₂×D₃	1	16	yes	—
22	C₂²×C₃×C₄	C₂×D₃	1	12	yes	—
23	#217	—	3	16	no	—
24	C₄⋊Dic₃	—	3	48	no	—
25	C₄×Dic₃	—	3	48	no	—
26	C₂²×C₃²	—	2	2	no	—
27	C₂²×C₃²	C₂²×C₄	1	2	yes	—
28	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
29	C₃²×C₄	C₂²×C₄	1	12	yes	—
30	C₃⋊Dic₃	—	2	8	no	—
31	C₂²×Q₈	C₃⋊D₃	1	1	yes	characteristic
32	C₄×Q₈	—	9	8	no	—
33	#117	—	9	6	no	—
34	C₂³×C₃	C₂²×D₃	1	4	yes	—
35	C₃×Q₈	C₂×Dic₃	1	64	yes	—
36	C₂×C₃×C₄	C₂×Dic₃	1	48	yes	—
37	C₂×C₃×C₄	C₂²×D₃	1	24	yes	—
38	C₂×Dic₃	—	3	32	no	—
39	C₂×C₃²	—	2	2	no	—
40	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
42	C₂×Q₈	C₃⋊Dic₃	1	8	yes	complemented
43	C₂×Q₈	C₂×C₃⋊D₃	1	4	yes	—
44	C₂²×C₄	C₂×C₃⋊D₃	1	3	yes	—
45	C₄⋊C₄	—	9	12	no	—
46	C₂²⋊C₄	—	9	4	no	—
47	C₄²	—	9	12	no	—
48	C₂²×C₃	—	2	8	no	—
49	C₂²×C₃	C₂²×Dic₃	1	8	yes	—
50	C₂²×C₃	C₂³×D₃	1	4	yes	—
51	C₃×C₄	C₂²×Dic₃	1	48	yes	—
52	Dic₃	—	6	32	no	—
53	C₃²	#519	1	1	yes	characteristic, complemented, Sylow
54	C₂³	C₂²×C₃⋊D₃	1	1	yes	characteristic
55	Q₈	C₂×C₃⋊Dic₃	1	16	yes	—
56	C₂×C₄	—	9	8	no	—
57	C₂×C₄	C₂×C₃⋊Dic₃	1	12	yes	—
58	C₂×C₄	C₂²×C₃⋊D₃	1	6	yes	—
59	C₂×C₃	—	2	8	no	—
60	C₂×C₃	#1007	1	8	yes	—
61	C₂×C₃	C₂³×Dic₃	1	4	yes	—
62	C₂²	—	2	2	no	—
63	C₂²	C₂²×C₃⋊Dic₃	1	2	yes	—
64	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
65	C₄	—	18	8	no	—
66	C₄	C₂²×C₃⋊Dic₃	1	12	yes	—
67	C₃	#5812	1	4	yes	complemented
68	C₂	—	2	2	no	—
69	C₂	#64364	1	2	yes	—
70	C₂	C₂³×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₄
Fitting	C₂²×C₃²×Q₈	C₂
socle, cosocle	C₂²×C₃²	C₂⁴
prime radical	C₂³×C₃²	C₂³
hypercenter	C₂²×Q₈	C₃⋊D₃
hypocenter	C₃²	#519
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊Dic₃	C₂	semidirect
5. C₂×C₃²×Q₈	C₄	semidirect
20. C₂×C₃×Q₈	Dic₃	semidirect
42. C₂×Q₈	C₃⋊Dic₃	semidirect
53. C₃²	#519	semidirect
67. C₃	#5812	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
Q₈×C₃⋊Dic₃	2
#64135	2
#5812	3