Subgroups — #14517261

Order 1944.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#6204	C₃	1	2	yes	maximal
3	C₃×#6204	C₃	1	1	yes	maximal, characteristic
4	#10619581	C₃	1	9	yes	maximal, complemented
5	C₂×#6912	—	4	1	no	maximal
6	C₃×C₄×3₋¹⁺²	—	3	1	no	—
7	#6912	—	4	1	no	Sylow
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	Q₈×3₋¹⁺²	—	3	2	no	—
10	Q₈×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃×Q₈×C₉	—	3	1	no	—
12	#6204	—	3	2	no	—
13	#6204	—	3	2	no	—
14	#6204	—	3	2	no	—
15	#6204	—	3	1	no	—
16	#6204	C₃²	1	6	yes	—
17	C₃×Q₈⋊C₉	—	3	2	no	—
18	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
19	C₂×C₃×3₋¹⁺²	—	4	1	no	—
20	C₂×C₃×3₋¹⁺²	—	4	2	no	—
21	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
22	C₂×#733	—	4	9	no	—
23	C₃³×C₄	—	3	1	no	—
24	C₄×3₋¹⁺²	—	3	6	no	—
25	C₄×3₋¹⁺²	—	3	3	no	—
26	C₃×C₄×C₉	—	3	3	no	—
27	C₃×3₋¹⁺²	—	4	1	no	—
28	C₃×3₋¹⁺²	—	4	2	no	—
29	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic
30	#733	—	4	9	no	—
31	C₃²×Q₈	—	9	1	no	—
32	C₃²×Q₈	H(3,𝔽₃)	1	2	yes	—
33	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²×Q₈	C₃³	1	1	yes	characteristic
35	Q₈×C₉	—	3	3	no	—
36	Q₈⋊C₉	—	3	6	no	—
37	Q₈⋊C₉	—	3	3	no	—
38	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
39	C₂×3₋¹⁺²	—	3	2	no	—
40	C₂×3₋¹⁺²	—	4	6	no	—
41	C₂×3₋¹⁺²	—	12	1	no	—
42	C₂×3₋¹⁺²	—	12	2	no	—
43	C₂×3₋¹⁺²	—	12	2	no	—
44	C₂×3₋¹⁺²	—	12	2	no	—
45	C₂×3₋¹⁺²	C₃×A₄	1	3	yes	—
46	C₂×C₃×C₉	—	3	1	no	—
47	C₂×C₃×C₉	—	4	3	no	—
48	C₂×C₃×C₉	—	12	2	no	—
49	C₃²×C₄	—	3	1	no	—
50	C₃²×C₄	—	3	2	no	—
51	C₃²×C₄	—	3	1	no	—
52	C₃²×C₄	—	9	3	no	—
53	C₄×C₉	—	3	9	no	—
54	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
55	3₋¹⁺²	—	3	2	no	—
56	3₋¹⁺²	—	4	6	no	—
57	3₋¹⁺²	—	12	1	no	—
58	3₋¹⁺²	—	12	2	no	—
59	3₋¹⁺²	—	12	2	no	—
60	3₋¹⁺²	—	12	2	no	—
61	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	3	yes	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	4	3	no	—
64	C₃×C₉	—	12	2	no	—
65	C₃×Q₈	—	3	1	no	—
66	C₃×Q₈	—	3	2	no	—
67	C₃×Q₈	—	3	1	no	—
68	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	C₂×C₃²	—	9	1	no	—
70	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
71	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
72	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
73	C₂×C₉	—	3	3	no	—
74	C₂×C₉	—	12	3	no	—
75	C₂×C₉	—	12	6	no	—
76	C₃×C₄	—	3	3	no	—
77	C₃×C₄	—	3	1	no	—
78	C₃×C₄	—	9	2	no	—
79	C₃×C₄	—	9	1	no	—
80	C₃²	—	9	1	no	—
81	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
82	C₃²	Q₈⋊H(3,𝔽₃)	1	2	yes	—
83	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
84	C₉	—	3	3	no	—
85	C₉	—	12	3	no	—
86	C₉	—	12	6	no	—
87	Q₈	#6912	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	3	1	no	—
89	C₂×C₃	—	3	2	no	—
90	C₂×C₃	—	3	1	no	—
91	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
92	C₄	—	3	1	no	—
93	C₃	—	3	1	no	—
94	C₃	—	3	2	no	—
95	C₃	—	3	1	no	—
96	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
97	C₂	#11757855	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₃²×Q₈	C₃³
Frattini	C₂×C₃²	C₃²×A₄
Fitting	C₃×Q₈×3₋¹⁺²	C₃
prime radical	C₂×C₃³	C₃×A₄
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6912
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ C₂×C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
4. #10619581	C₃	semidirect
87. Q₈	#6912	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×3₋¹⁺²	3
C₃×#6204	3
#10619581	3
C₂×#6912	4