Subgroups — #64133

Order 288.

222 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	C₂×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	#3757	C₂	1	2	yes	maximal
4	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	#3750	C₂	1	2	yes	maximal, complemented
6	#924	—	3	4	no	maximal
7	C₂³×C₃²	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄	—	2	2	no	—
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
13	#634	—	2	2	no	—
14	C₂²×C₃×C₄	D₃	1	4	yes	—
15	C₂×Dic₆	—	3	4	no	—
16	#217	—	3	8	no	—
17	C₄⋊Dic₃	—	3	4	no	—
18	Dic₃⋊C₄	—	3	8	no	—
19	C₂²×C₃²	—	2	1	no	—
20	C₂²×C₃²	—	2	1	no	—
21	C₂²×C₃²	Q₈	1	2	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	D₄	1	2	yes	—
25	C₃⋊Dic₃	—	2	2	no	—
26	C₃⋊Dic₃	—	2	2	no	—
27	C₂²⋊Q₈	—	9	1	no	Sylow
28	C₂³×C₃	C₂×D₃	1	4	yes	—
29	C₂×C₃×C₄	—	2	8	no	—
30	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
31	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
32	C₂×Dic₃	—	3	8	no	—
33	C₂×Dic₃	—	3	8	no	—
34	Dic₆	—	6	8	no	—
35	C₂×C₃²	—	2	2	no	—
36	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
37	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₃²	#41	1	1	yes	characteristic
39	C₂×Q₈	—	9	1	no	—
40	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic
41	C₄⋊C₄	—	9	2	no	—
42	C₄⋊C₄	—	9	1	no	—
43	C₂²⋊C₄	—	9	2	no	—
44	C₂²×C₃	—	2	4	no	—
45	C₂²×C₃	—	2	4	no	—
46	C₂²×C₃	Dic₆	1	8	yes	complemented
47	C₂²×C₃	C₂²×D₃	1	4	yes	—
48	C₃×C₄	—	2	4	no	—
49	C₃×C₄	#67	1	8	yes	—
50	Dic₃	—	6	8	no	—
51	Dic₃	—	6	8	no	—
52	C₃²	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
53	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
54	Q₈	—	18	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	9	2	no	—
57	C₂×C₄	—	9	2	no	—
58	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
59	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
60	C₂×C₃	—	2	8	no	—
61	C₂×C₃	C₂×Dic₆	1	4	yes	—
62	C₂×C₃	#235	1	4	yes	—
63	C₂×C₃	C₂×#67	1	4	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#634	1	2	yes	complemented
67	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	18	2	no	—
70	C₄	—	18	2	no	—
71	C₄	#638	1	2	yes	—
72	C₃	#924	1	4	yes	complemented
73	C₂	—	2	2	no	—
74	C₂	C₂×#634	1	1	yes	characteristic
75	C₂	#3828	1	1	yes	characteristic
76	C₂	C₂×#638	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, cosocle	C₂²×C₃²	C₂³
Fitting	C₂²×C₃²×C₄	C₂
prime radical	C₂³×C₃²	C₂²
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	C₂²⋊Q₈
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#634	C₂	semidirect
4. C₄⋊(C₃⋊Dic₃)	C₂	semidirect
5. #3750	C₂	semidirect
21. C₂²×C₃²	Q₈	semidirect
46. C₂²×C₃	Dic₆	semidirect
52. C₃²	C₂²⋊Q₈	semidirect
66. C₂²	#634	semidirect
72. C₃	#924	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
C₂×#634	2
#3757	2
C₄⋊(C₃⋊Dic₃)	2
#3750	2
#924	3