Subgroups — #64156

Order 288.

228 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	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
3	#3752	C₂	1	4	yes	maximal, complemented
4	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#947	—	3	4	no	maximal
6	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
7	C₃²×Q₈	—	2	2	no	—
8	C₂×C₃²×C₄	C₂²	1	2	yes	—
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
11	C₄⋊(C₃⋊D₃)	—	2	2	no	—
12	C₂×C₃×Q₈	D₃	1	4	yes	complemented
13	C₂×D₁₂	—	3	4	no	—
14	(C₂×D₃)⋊C₄	—	3	16	no	—
15	C₄×Dic₃	—	3	4	no	—
16	C₂²×C₃²	C₂³	1	1	yes	characteristic
17	C₂×C₃⋊D₃	—	2	2	no	—
18	C₂×C₃⋊D₃	—	2	4	no	—
19	C₃²×C₄	—	2	2	no	—
20	C₃²×C₄	D₄	1	2	yes	—
21	C₃⋊Dic₃	—	2	2	no	—
22	#124	—	9	1	no	Sylow
23	C₂²×D₃	—	3	8	no	—
24	C₃×Q₈	—	2	8	no	—
25	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
26	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
27	C₂×Dic₃	—	3	8	no	—
28	D₁₂	—	6	8	no	—
29	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
30	C₂×C₃²	#41	1	2	yes	—
31	C₃⋊D₃	—	4	2	no	—
32	C₂×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
33	C₂×D₄	—	9	1	no	—
34	C₂²⋊C₄	—	9	4	no	—
35	C₄²	—	9	1	no	—
36	C₂²×C₃	C₂²×D₃	1	4	yes	—
37	C₂×D₃	—	6	16	no	—
38	C₂×D₃	—	6	8	no	—
39	C₃×C₄	—	2	8	no	—
40	C₃×C₄	#67	1	8	yes	—
41	Dic₃	—	6	8	no	—
42	C₃²	#124	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	9	2	no	—
44	Q₈	—	2	2	no	—
45	D₄	—	18	2	no	—
46	C₂×C₄	—	9	2	no	—
47	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
48	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
49	C₂×C₃	#239	1	8	yes	—
50	C₂×C₃	C₂×#67	1	4	yes	—
51	D₃	—	12	8	no	—
52	C₂²	—	18	4	no	—
53	C₂²	—	18	2	no	—
54	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	18	2	no	—
57	C₄	#638	1	2	yes	—
58	C₃	#947	1	4	yes	complemented
59	C₂	—	36	2	no	—
60	C₂	#3832	1	2	yes	—
61	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×Q₈	C₂
prime radical	C₂×C₄⋊(C₃⋊D₃)	C₂
hypercenter	C₂×Q₈	C₃⋊D₃
hypocenter	C₃²	#124
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ 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. #3752	C₂	semidirect
4. C₄×C₃⋊Dic₃	C₂	semidirect
12. C₂×C₃×Q₈	D₃	semidirect
32. C₂×Q₈	C₃⋊D₃	semidirect
42. C₃²	#124	semidirect
58. C₃	#947	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
C₂×C₄⋊(C₃⋊D₃)	2
#3752	2
C₄×C₃⋊Dic₃	2
#947	3