Subgroups — #64101

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#3752	C₂	1	2	yes	maximal
4	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
6	#892	—	3	4	no	maximal
7	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
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	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊D₃	C₄	1	4	yes	complemented
13	C₂×C₄×D₃	—	3	4	no	—
14	C₃×C₄⋊C₄	D₃	1	4	yes	complemented
15	(C₂×D₃)⋊C₄	—	3	8	no	—
16	C₄⋊Dic₃	—	3	4	no	—
17	C₄×Dic₃	—	3	8	no	—
18	C₂²×C₃²	C₂³	1	1	yes	characteristic
19	C₂×C₃⋊D₃	—	2	1	no	—
20	C₂×C₃⋊D₃	—	2	1	no	—
21	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
22	C₃²×C₄	—	2	2	no	—
23	C₃²×C₄	C₂×C₄	1	2	yes	—
24	C₃⋊Dic₃	—	2	2	no	—
25	C₃⋊Dic₃	C₂×C₄	1	2	yes	—
26	#117	—	9	1	no	Sylow
27	C₂²×D₃	—	3	4	no	—
28	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
29	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
30	C₂×Dic₃	—	3	4	no	—
31	C₂×Dic₃	—	3	8	no	—
32	C₄×D₃	—	3	16	no	—
33	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₃²	#41	1	1	yes	characteristic
35	C₂×C₃²	#41	1	1	yes	characteristic
36	C₃⋊D₃	—	2	2	no	—
37	C₂²×C₄	—	9	1	no	—
38	C₄⋊C₄	—	9	1	no	—
39	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
40	C₂²⋊C₄	—	9	2	no	—
41	C₄²	—	9	2	no	—
42	C₂²×C₃	C₂²×D₃	1	4	yes	—
43	C₂×D₃	—	3	8	no	—
44	C₂×D₃	—	6	4	no	—
45	C₂×D₃	—	6	4	no	—
46	C₃×C₄	—	2	8	no	—
47	C₃×C₄	C₄×D₃	1	8	yes	—
48	Dic₃	—	3	8	no	—
49	Dic₃	—	6	8	no	—
50	C₃²	#117	1	1	yes	characteristic, complemented, Sylow
51	C₂³	—	9	1	no	—
52	C₂×C₄	—	9	1	no	—
53	C₂×C₄	—	9	4	no	—
54	C₂×C₄	—	9	2	no	—
55	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
56	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
57	C₂×C₃	C₂×C₄×D₃	1	4	yes	—
58	C₂×C₃	#237	1	4	yes	—
59	C₂×C₃	#239	1	4	yes	—
60	D₃	—	6	8	no	—
61	C₂²	—	9	2	no	—
62	C₂²	—	18	1	no	—
63	C₂²	—	18	1	no	—
64	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	9	2	no	—
67	C₄	—	18	2	no	—
68	C₄	C₄×C₃⋊D₃	1	2	yes	—
69	C₃	#892	1	4	yes	complemented
70	C₂	—	18	2	no	—
71	C₂	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
72	C₂	#3830	1	1	yes	characteristic
73	C₂	#3832	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄⋊C₄	C₂	semidirect
4. C₄⋊(C₃⋊Dic₃)	C₂	semidirect
5. C₄×C₃⋊Dic₃	C₂	semidirect
12. C₄×C₃⋊D₃	C₄	semidirect
14. C₃×C₄⋊C₄	D₃	semidirect
39. C₄⋊C₄	C₃⋊D₃	semidirect
50. C₃²	#117	semidirect
69. C₃	#892	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊D₃	2
C₃²×C₄⋊C₄	2
#3752	2
C₄⋊(C₃⋊Dic₃)	2
C₄×C₃⋊Dic₃	2
#892	3