Subgroups — #64104

Order 288.

282 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
2	C₂×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#3752	C₂	1	2	yes	maximal, complemented
6	#895	—	3	4	no	maximal
7	C₂²×C₃⋊D₃	C₂²	1	2	yes	—
8	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄	C₂²	1	2	yes	—
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊D₃)	—	2	4	no	—
13	C₄⋊(C₃⋊D₃)	—	2	2	no	—
14	C₄×C₃⋊D₃	—	2	2	no	—
15	C₂×D₁₂	—	3	8	no	—
16	C₂×D₁₂	—	3	4	no	—
17	C₂×C₄×D₃	—	3	4	no	—
18	C₃×C₄⋊C₄	D₃	1	4	yes	complemented
19	(C₂×D₃)⋊C₄	—	3	8	no	—
20	C₂²×C₃²	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊D₃	—	2	1	no	—
22	C₂×C₃⋊D₃	—	2	1	no	—
23	C₂×C₃⋊D₃	—	2	2	no	—
24	C₂×C₃⋊D₃	—	2	2	no	—
25	C₂×C₃⋊D₃	—	2	2	no	—
26	C₂×C₃⋊D₃	D₄	1	2	yes	complemented
27	C₃²×C₄	—	2	2	no	—
28	C₃²×C₄	D₄	1	2	yes	complemented
29	C₃⋊Dic₃	—	2	1	no	—
30	#121	—	9	1	no	Sylow
31	C₂²×D₃	—	3	4	no	—
32	C₂²×D₃	—	3	8	no	—
33	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
34	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
35	C₂×Dic₃	—	3	4	no	—
36	D₁₂	—	6	16	no	—
37	D₁₂	—	6	8	no	—
38	C₄×D₃	—	6	8	no	—
39	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃²	#41	1	1	yes	characteristic
42	C₃⋊D₃	—	2	2	no	—
43	C₃⋊D₃	—	4	2	no	—
44	C₂×D₄	—	9	2	no	—
45	C₂×D₄	—	9	1	no	—
46	C₂²×C₄	—	9	1	no	—
47	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	9	2	no	—
49	C₂²×C₃	C₂²×D₃	1	4	yes	—
50	C₂×D₃	—	3	8	no	—
51	C₂×D₃	—	6	4	no	—
52	C₂×D₃	—	6	4	no	—
53	C₂×D₃	—	6	8	no	—
54	C₂×D₃	—	6	8	no	—
55	C₂×D₃	—	6	8	no	—
56	C₃×C₄	—	2	8	no	—
57	C₃×C₄	D₁₂	1	8	yes	complemented
58	Dic₃	—	6	4	no	—
59	C₃²	#121	1	1	yes	characteristic, complemented, Sylow
60	C₂³	—	9	1	no	—
61	C₂³	—	9	2	no	—
62	D₄	—	18	4	no	—
63	D₄	—	18	2	no	—
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	18	2	no	—
66	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
67	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
68	C₂×C₃	C₂×D₁₂	1	4	yes	—
69	C₂×C₃	D₃×D₄	1	4	yes	—
70	C₂×C₃	#239	1	4	yes	—
71	D₃	—	6	8	no	—
72	D₃	—	12	8	no	—
73	C₂²	—	9	2	no	—
74	C₂²	—	18	1	no	—
75	C₂²	—	18	1	no	—
76	C₂²	—	18	2	no	—
77	C₂²	—	18	2	no	—
78	C₂²	—	18	2	no	—
79	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	18	1	no	—
82	C₄	C₄⋊(C₃⋊D₃)	1	2	yes	complemented
83	C₃	#895	1	4	yes	complemented
84	C₂	—	18	2	no	—
85	C₂	—	36	2	no	—
86	C₂	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₂	D₄×C₃⋊D₃	1	1	yes	characteristic
88	C₂	#3832	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₃²×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#121
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
2. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
3. C₂×C₄×C₃⋊D₃	C₂	semidirect
4. C₃²×C₄⋊C₄	C₂	semidirect
5. #3752	C₂	semidirect
18. C₃×C₄⋊C₄	D₃	semidirect
26. C₂×C₃⋊D₃	D₄	semidirect
28. C₃²×C₄	D₄	semidirect
47. C₄⋊C₄	C₃⋊D₃	semidirect
57. C₃×C₄	D₁₂	semidirect
59. C₃²	#121	semidirect
82. C₄	C₄⋊(C₃⋊D₃)	semidirect
83. C₃	#895	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
C₂×C₄×C₃⋊D₃	2
C₃²×C₄⋊C₄	2
#3752	2
#895	3