Subgroups — #64153

Order 288.

204 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
2	C₂×#634	C₂	1	1	yes	maximal, characteristic
3	#3750	C₂	1	4	yes	maximal
4	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
5	#944	—	3	4	no	maximal
6	C₃²×Q₈	—	2	2	no	—
7	C₂×C₃²×C₄	C₂²	1	2	yes	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
11	#634	—	2	2	no	—
12	C₂×C₃×Q₈	D₃	1	4	yes	—
13	C₂×Dic₆	—	3	4	no	—
14	Dic₃⋊C₄	—	3	16	no	—
15	C₄×Dic₃	—	3	4	no	—
16	C₂²×C₃²	C₂³	1	1	yes	characteristic
17	C₃²×C₄	—	2	2	no	—
18	C₃²×C₄	D₄	1	2	yes	—
19	C₃⋊Dic₃	—	2	2	no	—
20	C₃⋊Dic₃	Q₈	1	4	yes	complemented
21	C₄⋊Q₈	—	9	1	no	Sylow
22	C₃×Q₈	—	2	8	no	—
23	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
24	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
25	C₂×Dic₃	—	3	8	no	—
26	C₂×Dic₃	—	3	8	no	—
27	Dic₆	—	6	8	no	—
28	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
29	C₂×C₃²	C₂×Q₈	1	2	yes	—
30	C₂×Q₈	—	9	1	no	—
31	C₂×Q₈	C₃⋊D₃	1	1	yes	characteristic
32	C₄⋊C₄	—	9	4	no	—
33	C₄²	—	9	1	no	—
34	C₂²×C₃	C₂²×D₃	1	4	yes	—
35	C₃×C₄	—	2	8	no	—
36	C₃×C₄	#67	1	8	yes	—
37	Dic₃	—	3	16	no	—
38	Dic₃	—	6	8	no	—
39	C₃²	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
40	Q₈	—	2	2	no	—
41	Q₈	—	18	2	no	—
42	C₂×C₄	—	9	2	no	—
43	C₂×C₄	—	9	2	no	—
44	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
45	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₂×C₃	D₃×Q₈	1	8	yes	—
47	C₂×C₃	C₂×#67	1	4	yes	—
48	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	9	4	no	—
51	C₄	—	18	2	no	—
52	C₄	#638	1	2	yes	—
53	C₃	#944	1	4	yes	complemented
54	C₂	Q₈×C₃⋊D₃	1	2	yes	—
55	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, prime radical, cosocle	C₂²×C₃²	C₂³
Fitting	C₂×C₃²×Q₈	C₂
hypercenter	C₂×Q₈	C₃⋊D₃
hypocenter	C₃²	C₄⋊Q₈
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
20. C₃⋊Dic₃	Q₈	semidirect
39. C₃²	C₄⋊Q₈	semidirect
53. C₃	#944	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×#634	2
#3750	2
C₄×C₃⋊Dic₃	2
#944	3