Subgroups — #14517180

Order 1944.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic
2	Q₈×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
3	#10619584	C₃	1	2	yes	maximal
4	C₉⋊(Q₈⋊C₉)	C₃	1	3	yes	maximal, complemented
5	(Q₈⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₂×#6895	—	4	1	no	maximal
7	C₄×C₃²⋊C₉	—	3	1	no	—
8	#6895	—	4	1	no	Sylow
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	C₃×Q₈×C₉	C₃²	1	3	yes	—
11	C₃×Q₈⋊C₉	—	3	3	no	—
12	C₃×Q₈⋊C₉	C₃²	1	2	yes	—
13	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×Q₈⋊C₉	C₃²	1	6	yes	—
15	C₂×C₃²×C₉	—	4	1	no	—
16	C₂×C₉⋊C₉	—	4	6	no	—
17	C₂×C₉⋊C₉	—	4	3	no	—
18	C₂×C₃²⋊C₉	—	4	2	no	—
19	C₂×C₃²⋊C₉	A₄	1	1	yes	characteristic
20	C₃³×C₄	—	3	1	no	—
21	C₃×C₄×C₉	—	3	3	no	—
22	C₃²×C₉	—	4	1	no	—
23	C₉⋊C₉	—	4	6	no	—
24	C₉⋊C₉	—	4	3	no	—
25	C₃²⋊C₉	—	4	2	no	—
26	C₃²⋊C₉	SL(2,𝔽₃)	1	1	yes	characteristic
27	C₃²×Q₈	—	3	2	no	—
28	C₃²×Q₈	—	3	1	no	—
29	C₃²×Q₈	3₋¹⁺²	1	3	yes	—
30	C₃²×Q₈	C₃³	1	1	yes	characteristic
31	Q₈×C₉	—	3	3	no	—
32	Q₈⋊C₉	—	3	2	no	—
33	Q₈⋊C₉	—	3	1	no	—
34	Q₈⋊C₉	—	3	6	no	—
35	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
36	C₂×C₃×C₉	—	4	6	no	—
37	C₂×C₃×C₉	—	4	2	no	—
38	C₂×C₃×C₉	—	4	1	no	—
39	C₂×C₃×C₉	—	12	3	no	—
40	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
41	C₃²×C₄	—	3	1	no	—
42	C₃²×C₄	—	3	3	no	—
43	C₃²×C₄	—	9	1	no	—
44	C₃²×C₄	—	9	2	no	—
45	C₄×C₉	—	9	3	no	—
46	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
47	C₃×C₉	—	4	6	no	—
48	C₃×C₉	—	4	2	no	—
49	C₃×C₉	—	4	1	no	—
50	C₃×C₉	—	12	3	no	—
51	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	—
52	C₃×Q₈	—	3	3	no	—
53	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
54	C₃×Q₈	#737	1	2	yes	—
55	C₃×Q₈	#737	1	1	yes	characteristic
56	C₂×C₃²	—	3	1	no	—
57	C₂×C₃²	—	3	2	no	—
58	C₂×C₃²	C₉⋊A₄	1	3	yes	—
59	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
60	C₂×C₉	—	3	3	no	—
61	C₂×C₉	—	12	2	no	—
62	C₂×C₉	—	12	1	no	—
63	C₂×C₉	—	12	6	no	—
64	C₃×C₄	—	3	1	no	—
65	C₃×C₄	—	3	1	no	—
66	C₃×C₄	—	3	2	no	—
67	C₃×C₄	—	9	3	no	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	2	no	—
70	C₃²	C₉⋊SL(2,𝔽₃)	1	3	yes	—
71	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
72	C₉	—	3	3	no	—
73	C₉	—	12	2	no	—
74	C₉	—	12	1	no	—
75	C₉	—	12	6	no	—
76	Q₈	#6895	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₃	—	3	3	no	—
78	C₂×C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
79	C₂×C₃	#66437	1	1	yes	characteristic
80	C₂×C₃	#66440	1	2	yes	—
81	C₄	—	3	1	no	—
82	C₃	—	3	3	no	—
83	C₃	C₃×C₉⋊SL(2,𝔽₃)	1	1	yes	characteristic
84	C₃	#10619913	1	1	yes	characteristic
85	C₃	#10619916	1	2	yes	—
86	C₂	#11757771	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	Q₈×C₃²⋊C₉	C₃
prime radical	C₂×C₃³	C₃×A₄
hypercenter	C₂×C₃²⋊C₉	A₄
hypocenter	Q₈	#6895
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
4. C₉⋊(Q₈⋊C₉)	C₃	semidirect
5. (Q₈⋊C₉)⋊C₉	C₃	semidirect
76. Q₈	#6895	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Q₈⋊C₉	3
Q₈×C₃²⋊C₉	3
#10619584	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₂×#6895	4