Subgroups — C₃×#63700 #11722575

Order 864.

254 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41⋊C₉	C₂	1	3	yes	maximal, complemented
2	C₃²×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic
3	#63700	C₃	1	3	yes	maximal, direct product factor
4	C₃×Q₈×C₉	—	4	1	no	maximal
5	C₃×Q₈⋊C₉	C₂²	1	1	yes	characteristic, complemented
6	C₃²×#41	—	3	1	no	—
7	C₃²×#41	C₂×C₃	1	3	yes	—
8	C₂×C₃²×D₄	—	3	3	no	—
9	#41⋊C₉	C₂×C₃	1	9	yes	complemented
10	C₃×C₄×C₉	—	4	3	no	—
11	C₃×H(5,𝔽₂)	C₉	1	3	yes	complemented
12	C₃×H(5,𝔽₂)	C₃²	1	1	yes	characteristic
13	C₂³×C₃²	—	3	2	no	—
14	C₃²×Q₈	A₄	1	1	yes	characteristic
15	C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
16	C₃²×D₄	—	3	3	no	—
17	C₃²×D₄	—	3	3	no	—
18	C₂×C₃²×C₄	—	3	3	no	—
19	Q₈×C₉	—	4	3	no	—
20	Q₈⋊C₉	C₂²×C₃	1	3	yes	complemented
21	C₂×C₃×C₉	—	4	1	no	—
22	C₃×#41	—	3	3	no	—
23	C₃×#41	—	3	1	no	—
24	C₃×#41	C₂×C₉	1	9	yes	—
25	C₃×#41	C₂×C₃²	1	3	yes	—
26	C₂×C₃×D₄	—	3	9	no	—
27	C₂×C₃×D₄	—	3	3	no	—
28	C₂²×C₃²	—	3	3	no	—
29	C₂²×C₃²	—	12	2	no	—
30	C₃²×C₄	—	3	1	no	—
31	C₃²×C₄	C₂×A₄	1	3	yes	—
32	C₄×C₉	—	4	9	no	—
33	H(5,𝔽₂)	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
34	C₃×C₉	—	4	1	no	Sylow
35	C₂³×C₃	—	3	2	no	—
36	C₂³×C₃	—	3	6	no	—
37	C₃×Q₈	C₂²⋊C₉	1	3	yes	—
38	C₃×Q₈	C₂²×C₉	1	3	yes	—
39	C₃×Q₈	C₃×A₄	1	1	yes	characteristic
40	C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
41	C₃×D₄	—	3	9	no	—
42	C₃×D₄	—	3	3	no	—
43	C₃×D₄	—	3	9	no	—
44	C₃×D₄	—	3	3	no	—
45	C₂×C₃×C₄	—	3	9	no	—
46	C₂×C₃×C₄	—	3	3	no	—
47	C₂×C₃²	—	6	3	no	—
48	C₂×C₃²	C₂²×A₄	1	1	yes	characteristic
49	C₂×C₉	—	4	3	no	—
50	#41	—	3	1	no	—
51	#41	C₂×C₃×C₉	1	3	yes	—
52	C₂×D₄	—	3	3	no	—
53	C₂²×C₃	—	3	3	no	—
54	C₂²×C₃	—	3	9	no	—
55	C₂²×C₃	—	12	2	no	—
56	C₂²×C₃	—	12	6	no	—
57	C₃×C₄	—	3	1	no	—
58	C₃×C₄	—	3	3	no	—
59	C₃×C₄	C₂×C₂²⋊C₉	1	9	yes	—
60	C₃×C₄	C₂×C₃×A₄	1	3	yes	—
61	C₃²	#994	1	1	yes	characteristic
62	C₉	—	4	3	no	—
63	C₂³	—	3	2	no	—
64	Q₈	C₃×C₂²⋊C₉	1	1	yes	characteristic
65	Q₈	C₂²×C₃×C₉	1	1	yes	characteristic
66	D₄	—	3	3	no	—
67	D₄	—	3	3	no	—
68	C₂×C₄	—	3	3	no	—
69	C₂×C₃	—	6	3	no	—
70	C₂×C₃	—	6	9	no	—
71	C₂×C₃	C₂²×C₂²⋊C₉	1	3	yes	—
72	C₂×C₃	C₂²×C₃×A₄	1	1	yes	characteristic
73	C₂²	—	3	3	no	—
74	C₂²	—	12	2	no	—
75	C₄	—	3	1	no	—
76	C₄	C₂×C₃×C₂²⋊C₉	1	3	yes	—
77	C₃	#63700	1	3	yes	direct product factor
78	C₃	C₃×#994	1	1	yes	characteristic
79	C₂	—	6	3	no	—
80	C₂	C₂²×C₃×C₂²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂²×A₄
commutator, hypocenter	Q₈	C₂²×C₃×C₉
Frattini	C₂×C₃	C₂²×C₃×A₄
Fitting, prime radical	C₃²×H(5,𝔽₂)	C₃
cosocle	C₃×Q₈	C₂²×C₃²
hypercenter	C₃²×Q₈	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×#41⋊C₉	C₂	semidirect
3. #63700	C₃	direct (N × H)
5. C₃×Q₈⋊C₉	C₂²	semidirect
9. #41⋊C₉	C₂×C₃	semidirect
11. C₃×H(5,𝔽₂)	C₉	semidirect
20. Q₈⋊C₉	C₂²×C₃	semidirect
33. H(5,𝔽₂)	C₃×C₉	semidirect
77. C₃	#63700	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#41⋊C₉	2
C₃²×H(5,𝔽₂)	3
#63700	3
C₃×Q₈×C₉	4

Subgroups — C3×#63700 #11722575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#63700 #11722575