Subgroups — #10589414

Order 576.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#63663	C₂	1	2	yes	maximal
3	#4548	—	3	4	no	maximal
4	C₂×C₃²×Q₈	C₄	1	1	yes	characteristic
5	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
6	C₃²×C₄⋊C₄	—	2	2	no	—
7	C₃²×C₄²	C₄	1	1	yes	characteristic
8	#3748	—	2	2	no	—
9	C₃×C₄⋊Q₈	D₃	1	4	yes	—
10	#836	—	3	8	no	—
11	C₃²×Q₈	—	4	1	no	—
12	C₃²×Q₈	—	4	1	no	—
13	C₂×C₃²×C₄	—	2	1	no	—
14	C₂×C₃²×C₄	—	2	1	no	—
15	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃²×C₄	D₄	1	2	yes	—
17	C₃⋊(C₃⋊C₈)	—	4	2	no	—
18	#356	—	9	1	no	Sylow
19	C₂×C₃×Q₈	Dic₃	1	4	yes	—
20	C₂×C₃×Q₈	C₂×D₃	1	4	yes	—
21	C₃×C₄⋊C₄	—	2	8	no	—
22	C₃×C₄²	Dic₃	1	4	yes	—
23	#208	—	6	8	no	—
24	C₂²×C₃²	C₂²⋊C₄	1	1	yes	characteristic
25	C₃²×C₄	—	2	2	no	—
26	C₃²×C₄	—	2	2	no	—
27	C₃²×C₄	—	2	1	no	—
28	C₃²×C₄	—	4	1	no	—
29	C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
30	#101	—	9	2	no	—
31	C₃×Q₈	—	4	4	no	—
32	C₃×Q₈	—	4	4	no	—
33	C₂×C₃×C₄	—	2	4	no	—
34	C₂×C₃×C₄	—	2	4	no	—
35	C₂×C₃×C₄	C₂×Dic₃	1	4	yes	—
36	C₂×C₃×C₄	#67	1	8	yes	—
37	C₃⋊C₈	—	12	8	no	—
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	#99	1	1	yes	characteristic
40	C₂×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
41	C₂×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
42	M₁₆	—	18	2	no	—
43	C₄⋊C₄	—	2	2	no	—
44	C₄²	C₃⋊Dic₃	1	1	yes	characteristic
45	C₂²×C₃	#217	1	4	yes	—
46	C₃×C₄	—	2	8	no	—
47	C₃×C₄	—	2	8	no	—
48	C₃×C₄	—	2	4	no	—
49	C₃×C₄	—	4	4	no	—
50	C₃²	#356	1	1	yes	characteristic, complemented, Sylow
51	Q₈	—	4	1	no	—
52	Q₈	—	4	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
56	C₂×C₄	#638	1	2	yes	—
57	C₈	—	36	2	no	—
58	C₂×C₃	—	2	4	no	—
59	C₂×C₃	#834	1	4	yes	—
60	C₂²	#3757	1	1	yes	characteristic
61	C₄	—	2	2	no	—
62	C₄	—	2	2	no	—
63	C₄	—	2	1	no	—
64	C₄	—	4	1	no	—
65	C₃	#4548	1	4	yes	complemented
66	C₂	—	2	1	no	—
67	C₂	#63661	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#63661
commutator	C₂×C₃²×C₄	C₂×C₄
Frattini	C₂×Q₈	C₂×C₃⋊D₃
Fitting	C₃²×C₄⋊Q₈	C₂
socle	C₂×C₃²	#99
prime radical	C₂²×C₃²	C₂²⋊C₄
cosocle	C₂×C₃²×Q₈	C₂²
hypercenter	C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	#356
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
50. C₃²	#356	semidirect
65. C₃	#4548	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊Q₈	2
#63663	2
#4548	3