Subgroups — #63581

Order 288.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#638	C₂	1	1	yes	maximal, characteristic
2	C₃×#217	C₂	1	2	yes	maximal, complemented
3	#806	—	3	2	no	maximal
4	C₂³×C₃²	C₂²	1	1	yes	characteristic
5	C₂²×C₃⋊D₃	C₄	1	1	yes	characteristic, complemented
6	#638	—	2	2	no	—
7	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃×Dic₃	—	2	2	no	—
9	C₂×#67	—	3	2	no	—
10	C₂×#67	—	6	1	no	—
11	C₃×C₂²⋊C₄	—	3	2	no	—
12	#217	D₃	1	2	yes	complemented
13	C₂²×C₃²	—	4	1	no	—
14	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃²	D₄	1	2	yes	—
16	C₂×C₃⋊D₃	—	2	1	no	—
17	C₂×C₃⋊D₃	—	4	1	no	—
18	C₃⋊Dic₃	—	2	1	no	—
19	C₃×Dic₃	—	4	2	no	—
20	#99	—	9	1	no	Sylow
21	C₂³×C₃	—	2	1	no	—
22	C₂³×C₃	C₂×D₃	1	2	yes	—
23	C₂²×D₃	—	3	2	no	—
24	C₂²×D₃	—	6	1	no	—
25	C₂×C₃×C₄	—	6	2	no	—
26	#67	—	6	4	no	—
27	#67	—	6	2	no	—
28	#67	—	6	2	no	—
29	C₂×Dic₃	—	2	2	no	—
30	C₂×Dic₃	—	3	2	no	—
31	C₂×Dic₃	—	6	1	no	—
32	C₂×C₃²	—	2	1	no	—
33	C₂×C₃²	—	2	2	no	—
34	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
35	C₃⋊D₃	—	4	1	no	—
36	C₂×D₄	—	9	1	no	—
37	C₂²⋊C₄	—	3	2	no	—
38	C₂²×C₃	—	2	1	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	—	4	2	no	—
41	C₂²×C₃	—	4	2	no	—
42	C₂²×C₃	C₄×D₃	1	2	yes	—
43	C₂²×C₃	D₁₂	1	2	yes	—
44	C₂²×C₃	#67	1	2	yes	—
45	C₂×D₃	—	6	2	no	—
46	C₂×D₃	—	6	2	no	—
47	C₂×D₃	—	12	2	no	—
48	C₂×D₃	—	12	2	no	—
49	C₃×C₄	—	12	2	no	—
50	Dic₃	—	4	2	no	—
51	Dic₃	—	6	2	no	—
52	Dic₃	—	6	2	no	—
53	C₃²	#99	1	1	yes	characteristic, complemented, Sylow
54	C₂³	—	9	1	no	—
55	C₂³	D₃²	1	1	yes	characteristic
56	D₄	—	18	2	no	—
57	C₂×C₄	—	6	2	no	—
58	C₂×C₄	—	9	1	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	—	2	2	no	—
61	C₂×C₃	—	2	2	no	—
62	C₂×C₃	—	2	2	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	—	4	2	no	—
65	C₂×C₃	(C₂×D₃)⋊C₄	1	2	yes	—
66	D₃	—	12	2	no	—
67	D₃	—	12	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	18	1	no	—
70	C₂²	—	36	1	no	—
71	C₂²	#624	1	1	yes	characteristic
72	C₂²	#626	1	2	yes	—
73	C₄	—	12	2	no	—
74	C₄	—	18	1	no	—
75	C₃	—	2	1	no	—
76	C₃	#806	1	2	yes	complemented
77	C₂	—	2	2	no	—
78	C₂	—	2	1	no	—
79	C₂	—	36	1	no	—
80	C₂	#3722	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3722
commutator	C₂²×C₃²	C₂×C₄
Frattini, hypercenter	C₂³	D₃²
Fitting, cosocle	C₂³×C₃²	C₂²
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₂×#638	C₂
hypocenter	C₃²	#99
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₃×#217	C₂	semidirect
5. C₂²×C₃⋊D₃	C₄	semidirect
7. C₂×C₃⋊Dic₃	C₄	semidirect
12. #217	D₃	semidirect
53. C₃²	#99	semidirect
76. C₃	#806	semidirect

Maximal subgroups

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

subgroup	index
C₂×#638	2
C₃×#217	2
#806	3