Subgroups — #10593566

Order 576.

253 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#157	C₂	1	3	yes	maximal
2	C₂×#3777	C₂	1	3	yes	maximal, complemented
3	#63747	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₂²×#157	C₂²	1	3	yes	—
6	C₂²×#157	C₂²	1	3	yes	—
7	C₂²⋊#157	—	2	6	no	—
8	#3777	—	2	6	no	—
9	C₂³×C₃²	Q₈	1	1	yes	characteristic
10	C₂²×C₃⋊D₃	—	2	3	no	—
11	C₂²×C₃⋊D₃	—	2	3	no	—
12	C₂²×C₃⋊D₃	Q₈	1	1	yes	characteristic
13	C₂²×C₃⋊D₃	C₂³	1	1	yes	characteristic
14	C₂×#157	—	2	6	no	—
15	C₂×#157	—	2	6	no	—
16	C₂×#157	—	2	3	no	—
17	C₂×#157	D₄	1	6	yes	—
18	#399	—	9	1	no	maximal, Sylow
19	C₂³×D₃	—	12	1	no	—
20	C₂²×C₃²	—	2	3	no	—
21	C₂²×C₃²	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₃⋊D₃	—	2	3	no	—
23	C₂×C₃⋊D₃	—	4	3	no	—
24	C₂×C₃⋊D₃	—	4	1	no	—
25	C₂×C₃⋊D₃	C₂×D₄	1	3	yes	—
26	C₂×C₃⋊D₃	#41	1	3	yes	—
27	#157	—	2	6	no	—
28	#157	—	4	3	no	—
29	C₂×C₄⋊C₄	—	9	3	no	—
30	C₂×C₂²⋊C₄	—	9	3	no	—
31	#95	—	9	1	no	—
32	C₂³×C₃	—	4	1	no	—
33	C₂²×D₃	—	12	1	no	—
34	C₂²×D₃	—	12	1	no	—
35	C₂²×D₃	—	12	6	no	—
36	C₂²×D₃	—	12	6	no	—
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₃²	C₂²⋊Q₈	1	3	yes	—
39	C₃⋊D₃	—	4	1	no	—
40	C₃⋊D₃	#123	1	3	yes	—
41	C₃⋊D₃	#127	1	1	yes	characteristic
42	C₂⁴	—	9	1	no	—
43	C₂²×C₄	—	9	3	no	—
44	C₂²×C₄	—	9	3	no	—
45	C₄⋊C₄	—	18	6	no	—
46	C₂²⋊C₄	—	18	6	no	—
47	C₂²×C₃	—	4	6	no	—
48	C₂²×C₃	—	4	1	no	—
49	C₂×D₃	—	12	3	no	—
50	C₂×D₃	—	12	3	no	—
51	C₂×D₃	—	12	6	no	—
52	C₂×D₃	—	12	12	no	—
53	C₂×D₃	—	12	4	no	—
54	C₃²	#399	1	1	yes	characteristic, complemented, Sylow
55	C₂³	—	9	1	no	—
56	C₂³	—	9	1	no	—
57	C₂³	—	18	3	no	—
58	C₂³	—	18	3	no	—
59	C₂³	PSU(3,𝔽₂)	1	1	yes	characteristic
60	C₂×C₄	—	9	6	no	—
61	C₂×C₄	—	18	6	no	—
62	C₂×C₄	—	18	3	no	—
63	C₂×C₄	—	18	6	no	—
64	C₂×C₃	—	4	4	no	—
65	C₂×C₃	—	4	3	no	—
66	D₃	—	12	3	no	—
67	D₃	—	12	1	no	—
68	D₃	—	12	4	no	—
69	C₂²	—	2	3	no	—
70	C₂²	—	9	3	no	—
71	C₂²	—	9	3	no	—
72	C₂²	—	18	3	no	—
73	C₂²	—	36	3	no	—
74	C₂²	—	36	1	no	—
75	C₂²	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
76	C₄	—	18	6	no	—
77	C₄	—	36	3	no	—
78	C₃	—	4	1	no	—
79	C₂	—	4	1	no	—
80	C₂	—	9	3	no	—
81	C₂	—	9	1	no	—
82	C₂	—	36	1	no	—
83	C₂	C₂²⋊PSU(3,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂×PSU(3,𝔽₂)
commutator, cosocle	C₂²×C₃⋊D₃	C₂³
Fitting	C₂³×C₃²	Q₈
socle	C₂²×C₃²	C₂×Q₈
prime radical	C₂³×C₃⋊D₃	C₂²
hypercenter	C₂³	PSU(3,𝔽₂)
hypocenter	C₃²	#399
LCS	G ▷ C₂²×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×#3777	C₂	semidirect
3. #63747	C₂	semidirect
54. C₃²	#399	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#157	2
C₂×#3777	2
#63747	2
#399	9