Subgroups — #64135

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	#3757	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal, complemented
4	C₄×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
5	#926	—	3	4	no	maximal
6	C₂³×C₃²	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊Dic₃	C₂²	1	4	yes	—
11	C₂²×C₃×C₄	D₃	1	4	yes	—
12	#217	—	3	8	no	—
13	C₄⋊Dic₃	—	3	8	no	—
14	C₄×Dic₃	—	3	8	no	—
15	C₂²×C₃²	—	2	2	no	—
16	C₂²×C₃²	C₂×C₄	1	2	yes	—
17	C₂²×C₃²	C₂³	1	1	yes	characteristic
18	C₃²×C₄	C₂×C₄	1	2	yes	—
19	C₃²×C₄	C₂×C₄	1	2	yes	—
20	C₃⋊Dic₃	—	2	4	no	—
21	#117	—	9	1	no	Sylow
22	C₂³×C₃	C₂×D₃	1	4	yes	—
23	C₂×C₃×C₄	Dic₃	1	16	yes	complemented
24	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
25	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
26	C₂×Dic₃	—	3	16	no	—
27	C₂×C₃²	—	2	2	no	—
28	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₃²	#41	1	2	yes	—
30	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic
31	C₄⋊C₄	—	9	2	no	—
32	C₂²⋊C₄	—	9	2	no	—
33	C₄²	—	9	2	no	—
34	C₂²×C₃	—	2	8	no	—
35	C₂²×C₃	C₂×Dic₃	1	8	yes	—
36	C₂²×C₃	C₂²×D₃	1	4	yes	—
37	C₃×C₄	C₂×Dic₃	1	8	yes	—
38	C₃×C₄	C₂×Dic₃	1	8	yes	—
39	Dic₃	—	6	16	no	—
40	C₃²	#117	1	1	yes	characteristic, complemented, Sylow
41	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
42	C₂×C₄	—	9	4	no	—
43	C₂×C₄	C₃⋊Dic₃	1	4	yes	complemented
44	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₂×C₃	—	2	8	no	—
47	C₂×C₃	#235	1	8	yes	—
48	C₂×C₃	C₂²×Dic₃	1	4	yes	—
49	C₂²	—	2	2	no	—
50	C₂²	C₂×C₃⋊Dic₃	1	2	yes	—
51	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
52	C₄	—	18	4	no	—
53	C₄	C₂×C₃⋊Dic₃	1	2	yes	—
54	C₄	C₂×C₃⋊Dic₃	1	2	yes	—
55	C₃	#926	1	4	yes	complemented
56	C₂	—	2	2	no	—
57	C₂	#3828	1	2	yes	—
58	C₂	C₂²×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄
Frattini	C₂²	C₂²×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle, cosocle	C₂²×C₃²	C₂³
prime radical	C₂³×C₃²	C₂²
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	#117
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊Dic₃)	C₂	semidirect
4. C₄×C₃⋊Dic₃	C₂	semidirect
7. C₂×C₃²×C₄	C₄	semidirect
23. C₂×C₃×C₄	Dic₃	semidirect
40. C₃²	#117	semidirect
43. C₂×C₄	C₃⋊Dic₃	semidirect
55. C₃	#926	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
#3757	2
C₄⋊(C₃⋊Dic₃)	2
C₄×C₃⋊Dic₃	2
#926	3