Subgroups — #10593543

Order 576.

141 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64286	C₂	1	2	yes	maximal
2	C₂×#3787	C₂	1	1	yes	maximal, characteristic
3	#63746	C₂	1	2	yes	maximal, complemented
4	#63745	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₂²⋊#157	—	2	2	no	—
7	C₂×#622	C₄	1	2	yes	complemented
8	C₄⋊#157	C₂²	1	4	yes	—
9	C₄×#157	—	2	2	no	—
10	#3787	C₄	1	2	yes	complemented
11	#3787	C₂²	1	1	yes	characteristic
12	#3787	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊D₃	D₄	1	1	yes	characteristic
14	C₂×#157	—	2	4	no	—
15	C₂×C₃²×C₄	Q₈	1	1	yes	characteristic
16	C₂×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
17	C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
19	C₄×C₃⋊D₃	Q₈	1	1	yes	characteristic
20	C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
21	#622	C₂×C₄	1	2	yes	—
22	#622	C₂×C₄	1	2	yes	—
23	#427	—	9	1	no	maximal, Sylow
24	C₂×C₄×D₃	—	12	1	no	—
25	C₂²×C₃²	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₃⋊D₃	—	2	2	no	—
27	C₂×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
29	#157	—	4	4	no	—
30	C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
31	C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
32	C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
33	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₈	—	9	1	no	—
35	#117	—	9	2	no	—
36	#107	—	9	2	no	—
37	#106	—	9	2	no	—
38	C₂²×D₃	—	12	1	no	—
39	C₂×C₃×C₄	—	4	1	no	—
40	C₂×Dic₃	—	12	1	no	—
41	C₄×D₃	—	12	1	no	—
42	C₄×D₃	—	12	1	no	—
43	C₄×D₃	—	12	1	no	—
44	C₄×D₃	—	12	1	no	—
45	C₂×C₃²	—	2	1	no	—
46	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₃⋊D₃	—	2	1	no	—
48	C₃⋊D₃	#135	1	2	yes	—
49	C₂²×C₄	—	9	1	no	—
50	C₂×C₈	—	9	1	no	—
51	C₂×C₈	—	9	1	no	—
52	C₂×C₈	—	9	2	no	—
53	C₂×C₈	—	9	2	no	—
54	C₄⋊C₄	—	9	4	no	—
55	C₂²⋊C₄	—	18	2	no	—
56	C₄²	—	18	2	no	—
57	C₂²×C₃	—	4	1	no	—
58	C₂×D₃	—	12	1	no	—
59	C₂×D₃	—	12	1	no	—
60	C₂×D₃	—	12	4	no	—
61	C₃×C₄	—	4	1	no	—
62	C₃×C₄	—	4	1	no	—
63	Dic₃	—	12	1	no	—
64	Dic₃	—	12	1	no	—
65	C₃²	#427	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	9	1	no	—
67	C₂×C₄	—	9	1	no	—
68	C₂×C₄	—	9	1	no	—
69	C₂×C₄	—	9	1	no	—
70	C₂×C₄	—	9	1	no	—
71	C₂×C₄	—	9	1	no	—
72	C₂×C₄	—	18	4	no	—
73	C₂×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
74	C₈	—	9	2	no	—
75	C₈	—	9	2	no	—
76	C₂×C₃	—	4	2	no	—
77	C₂×C₃	—	4	1	no	—
78	D₃	—	12	2	no	—
79	D₃	—	12	2	no	—
80	C₂²	—	9	1	no	—
81	C₂²	—	9	1	no	—
82	C₂²	—	18	2	no	—
83	C₂²	#3777	1	1	yes	characteristic
84	C₄	—	9	1	no	—
85	C₄	—	9	1	no	—
86	C₄	—	36	4	no	—
87	C₄	#3777	1	1	yes	characteristic
88	C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
89	C₃	—	4	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	9	2	no	—
92	C₂	—	18	1	no	—
93	C₂	C₂×#3777	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#3777
commutator	C₃⋊Dic₃	C₂²×C₄
Frattini	C₄	C₂×PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₄	Q₈
socle	C₂×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	D₄
cosocle	C₄×C₃⋊D₃	C₂³
hypercenter	C₂×C₄	PSU(3,𝔽₂)
hypocenter	C₃²	#427
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. #63746	C₂	semidirect
4. #63745	C₂	semidirect
7. C₂×#622	C₄	semidirect
10. #3787	C₄	semidirect
65. C₃²	#427	semidirect

Maximal subgroups

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

subgroup	index
#64286	2
C₂×#3787	2
#63746	2
#63745	2
#427	9