Subgroups — #64151

Order 288.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#3828	C₂	1	1	yes	maximal, characteristic
3	#3756	C₂	1	2	yes	maximal, complemented
4	#3755	C₂	1	2	yes	maximal, complemented
5	#3748	C₂	1	1	yes	maximal, characteristic, complemented
6	#942	—	3	4	no	maximal
7	C₃²×Q₈	—	2	1	no	—
8	C₃²×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄	—	2	1	no	—
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	#638	—	2	1	no	—
12	C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊D₃	—	2	1	no	—
14	#634	C₂²	1	1	yes	characteristic
15	C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₃×Q₈	D₃	1	4	yes	complemented
17	#235	—	3	4	no	—
18	#216	—	3	8	no	—
19	#215	—	3	8	no	—
20	#208	—	3	4	no	—
21	C₂²×C₃²	D₄	1	1	yes	characteristic
22	C₂×C₃⋊D₃	—	2	1	no	—
23	C₃²×C₄	—	2	2	no	—
24	C₃²×C₄	D₄	1	1	yes	characteristic
25	C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₃⋊Dic₃	—	2	1	no	—
27	#137	—	9	1	no	Sylow
28	C₃×Q₈	—	2	4	no	—
29	C₃×Q₈	C₂×D₃	1	8	yes	—
30	C₂×C₃×C₄	—	2	4	no	—
31	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
32	#67	—	6	4	no	—
33	D₁₂	—	3	4	no	—
34	C₄×D₃	—	6	4	no	—
35	Dic₆	—	3	4	no	—
36	C₃⋊C₈	—	3	8	no	—
37	C₂×C₃²	—	2	1	no	—
38	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
39	C₃⋊D₃	—	4	1	no	—
40	#41	—	9	1	no	—
41	C₂×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
42	Q₁₆	—	9	2	no	—
43	SD₁₆	—	9	2	no	—
44	M₁₆	—	9	1	no	—
45	C₂²×C₃	#67	1	4	yes	—
46	C₂×D₃	—	6	4	no	—
47	C₃×C₄	—	2	8	no	—
48	C₃×C₄	#67	1	4	yes	—
49	C₃×C₄	C₂²×D₃	1	4	yes	—
50	Dic₃	—	6	4	no	—
51	C₃²	#137	1	1	yes	characteristic, complemented, Sylow
52	Q₈	—	2	1	no	—
53	Q₈	—	9	1	no	—
54	Q₈	C₂×C₃⋊D₃	1	2	yes	—
55	D₄	—	9	1	no	—
56	D₄	—	18	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	18	1	no	—
59	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
60	C₈	—	9	2	no	—
61	C₂×C₃	—	2	4	no	—
62	C₂×C₃	C₂×#67	1	4	yes	—
63	D₃	—	12	4	no	—
64	C₂²	—	18	1	no	—
65	C₂²	#638	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	18	1	no	—
68	C₄	#638	1	1	yes	characteristic
69	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
70	C₃	#942	1	4	yes	complemented
71	C₂	—	2	1	no	—
72	C₂	—	36	1	no	—
73	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#638
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×Q₈	C₂
socle	C₂×C₃²	C₂×D₄
prime radical	#3828	C₂
hypercenter	C₂×Q₈	C₃⋊D₃
hypocenter	C₃²	#137
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×Q₈	C₂	semidirect
3. #3756	C₂	semidirect
4. #3755	C₂	semidirect
5. #3748	C₂	semidirect
16. C₂×C₃×Q₈	D₃	semidirect
41. C₂×Q₈	C₃⋊D₃	semidirect
51. C₃²	#137	semidirect
70. C₃	#942	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₈	2
#3828	2
#3756	2
#3755	2
#3748	2
#942	3