Subgroups — #64099

Order 288.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊Dic₃)	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	#3750	C₂	1	2	yes	maximal
5	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	#890	—	3	4	no	maximal
7	C₂×C₃²×C₄	C₂²	1	2	yes	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₃×C₄⋊C₄	D₃	1	4	yes	—
13	C₄⋊Dic₃	—	3	8	no	—
14	C₄⋊Dic₃	—	3	4	no	—
15	Dic₃⋊C₄	—	3	8	no	—
16	C₄×Dic₃	—	3	4	no	—
17	C₂²×C₃²	C₂³	1	1	yes	characteristic
18	C₃²×C₄	—	2	2	no	—
19	C₃²×C₄	Q₈	1	2	yes	—
20	C₃⋊Dic₃	—	2	1	no	—
21	C₃⋊Dic₃	—	2	1	no	—
22	C₃⋊Dic₃	—	2	2	no	—
23	#125	—	9	1	no	Sylow
24	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
25	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
26	C₂×Dic₃	—	3	4	no	—
27	C₂×Dic₃	—	3	4	no	—
28	C₂×Dic₃	—	3	8	no	—
29	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₃²	#41	1	1	yes	characteristic
31	C₂×C₃²	#41	1	1	yes	characteristic
32	C₄⋊C₄	—	9	2	no	—
33	C₄⋊C₄	—	9	2	no	—
34	C₄⋊C₄	—	9	1	no	—
35	C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
36	C₄²	—	9	1	no	—
37	C₂²×C₃	C₂²×D₃	1	4	yes	—
38	C₃×C₄	—	2	8	no	—
39	C₃×C₄	Dic₆	1	8	yes	—
40	Dic₃	—	6	4	no	—
41	Dic₃	—	6	4	no	—
42	Dic₃	—	6	8	no	—
43	C₃²	#125	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄	—	9	1	no	—
45	C₂×C₄	—	9	1	no	—
46	C₂×C₄	—	9	2	no	—
47	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
48	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
49	C₂×C₃	C₂×Dic₆	1	4	yes	—
50	C₂×C₃	#237	1	4	yes	—
51	C₂×C₃	#239	1	4	yes	—
52	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	18	1	no	—
55	C₄	—	18	1	no	—
56	C₄	—	18	2	no	—
57	C₄	#634	1	2	yes	—
58	C₃	#890	1	4	yes	complemented
59	C₂	C₂×#634	1	1	yes	characteristic
60	C₂	#3830	1	1	yes	characteristic
61	C₂	#3832	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃
commutator, socle, prime radical, cosocle	C₂²×C₃²	C₂³
Fitting	C₃²×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	C₃⋊D₃
hypocenter	C₃²	#125
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
43. C₃²	#125	semidirect
58. C₃	#890	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊C₄	2
C₄⋊(C₃⋊Dic₃)	2
#3750	2
C₄×C₃⋊Dic₃	2
#890	3