Subgroups — #64149

Order 288.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#638	C₂	1	4	yes	maximal, complemented
3	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#940	—	3	4	no	maximal
6	C₂³×C₃²	C₂²	1	2	yes	—
7	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
8	C₃²×D₄	—	2	2	no	—
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	#638	—	2	8	no	—
11	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₄⋊(C₃⋊D₃)	—	2	2	no	—
13	C₂×C₃×D₄	D₃	1	4	yes	complemented
14	C₂×#67	—	3	16	no	—
15	C₂×D₁₂	—	3	4	no	—
16	C₄×Dic₃	—	3	4	no	—
17	C₂²×C₃²	—	2	4	no	—
18	C₂²×C₃²	—	2	2	no	—
19	C₂²×C₃²	C₂³	1	1	yes	characteristic
20	C₂×C₃⋊D₃	—	2	2	no	—
21	C₂×C₃⋊D₃	—	2	4	no	—
22	C₃²×C₄	D₄	1	2	yes	complemented
23	C₃⋊Dic₃	D₄	1	4	yes	complemented
24	#127	—	9	1	no	Sylow
25	C₂³×C₃	C₂×D₃	1	8	yes	—
26	C₂²×D₃	—	3	8	no	—
27	C₃×D₄	—	2	8	no	—
28	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
29	#67	—	6	32	no	—
30	C₂×Dic₃	—	3	8	no	—
31	D₁₂	—	6	8	no	—
32	C₂×C₃²	—	4	2	no	—
33	C₂×C₃²	C₂×D₄	1	2	yes	—
34	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
35	C₃⋊D₃	—	4	2	no	—
36	C₂×D₄	—	9	4	no	—
37	C₂×D₄	—	9	1	no	—
38	C₂×D₄	C₃⋊D₃	1	1	yes	characteristic, complemented
39	C₄²	—	9	1	no	—
40	C₂²×C₃	—	2	16	no	—
41	C₂²×C₃	—	2	8	no	—
42	C₂²×C₃	C₂²×D₃	1	4	yes	—
43	C₂×D₃	—	6	16	no	—
44	C₂×D₃	—	6	8	no	—
45	C₃×C₄	#67	1	8	yes	complemented
46	Dic₃	—	3	16	no	—
47	C₃²	#127	1	1	yes	characteristic, complemented, Sylow
48	C₂³	—	9	2	no	—
49	C₂³	C₂×C₃⋊D₃	1	2	yes	—
50	D₄	—	2	2	no	—
51	D₄	—	18	8	no	—
52	D₄	—	18	2	no	—
53	C₂×C₄	—	9	2	no	—
54	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
55	C₂×C₃	—	4	8	no	—
56	C₂×C₃	D₃×D₄	1	8	yes	—
57	C₂×C₃	C₂×#67	1	4	yes	—
58	D₃	—	12	8	no	—
59	C₂²	—	2	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	—	18	4	no	—
62	C₂²	—	18	2	no	—
63	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
64	C₄	—	9	4	no	—
65	C₄	#638	1	2	yes	complemented
66	C₃	#940	1	4	yes	complemented
67	C₂	—	4	2	no	—
68	C₂	—	36	2	no	—
69	C₂	D₄×C₃⋊D₃	1	2	yes	—
70	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₂×C₃²×D₄	C₂
hypercenter	C₂×D₄	C₃⋊D₃
hypocenter	C₃²	#127
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₄	C₂	semidirect
2. C₂×#638	C₂	semidirect
3. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
4. C₄×C₃⋊Dic₃	C₂	semidirect
13. C₂×C₃×D₄	D₃	semidirect
22. C₃²×C₄	D₄	semidirect
23. C₃⋊Dic₃	D₄	semidirect
38. C₂×D₄	C₃⋊D₃	semidirect
45. C₃×C₄	#67	semidirect
47. C₃²	#127	semidirect
65. C₄	#638	semidirect
66. C₃	#940	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₄	2
C₂×#638	2
C₂×C₄⋊(C₃⋊D₃)	2
C₄×C₃⋊Dic₃	2
#940	3