Subgroups — C₃²×#7463 #72254864

Order 2304.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#1397	C₂	1	1	yes	maximal, characteristic
2	C₃²×#1372	C₂	1	2	yes	maximal
3	C₃×#7463	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₃₂	C₄	1	1	yes	characteristic
6	C₃²×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₃²×#335	—	2	2	no	—
8	C₃×#1397	C₂×C₃	1	4	yes	—
9	C₃×#1372	C₂×C₃	1	8	yes	—
10	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	Q₈	1	1	yes	characteristic
12	C₃²×C₄⋊C₈	—	4	2	no	—
13	C₃²×C₄×C₈	D₄	1	1	yes	characteristic
14	C₃²×C₃₂	—	2	1	no	—
15	C₃²×C₃₂	C₈	1	2	yes	complemented
16	#7463	C₃²	1	1	yes	characteristic, direct product factor, Sylow
17	C₂×C₃×C₃₂	C₃×C₄	1	4	yes	—
18	C₂×C₃×C₃₂	C₃×C₄	1	4	yes	—
19	C₃×C₄×C₁₆	C₂²×C₃	1	4	yes	—
20	C₃×#335	—	2	8	no	—
21	C₂×C₃²×C₈	—	8	2	no	—
22	C₂×C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	Q₁₆	1	1	yes	characteristic
24	C₃²×C₄²	D₈	1	1	yes	characteristic
25	C₃²×C₁₆	—	2	1	no	—
26	C₃²×C₁₆	C₂×C₈	1	1	yes	characteristic
27	C₃²×C₁₆	M₁₆	1	1	yes	characteristic
28	#1397	C₂×C₃²	1	1	yes	characteristic
29	#1372	C₂×C₃²	1	2	yes	—
30	C₂×C₃×C₁₆	C₂×C₃×C₄	1	4	yes	—
31	C₂×C₃×C₁₆	C₃×Q₈	1	4	yes	—
32	C₃×C₄⋊C₈	—	4	8	no	—
33	C₃×C₄×C₈	C₃×D₄	1	4	yes	—
34	C₃×C₃₂	—	2	4	no	—
35	C₃×C₃₂	C₃×C₈	1	8	yes	complemented
36	C₂×C₃²×C₄	#107	1	1	yes	characteristic
37	C₂×C₃²×C₄	SD₃₂	1	2	yes	—
38	C₃²×C₈	—	2	1	no	—
39	C₃²×C₈	—	16	2	no	—
40	C₃²×C₈	C₄⋊C₈	1	1	yes	characteristic
41	C₃²×C₈	#108	1	1	yes	characteristic
42	C₂×C₃₂	C₃²×C₄	1	1	yes	characteristic
43	C₂×C₃₂	C₃²×C₄	1	1	yes	characteristic
44	C₄×C₁₆	C₂²×C₃²	1	1	yes	characteristic
45	#335	—	2	2	no	—
46	C₂×C₃×C₈	—	8	8	no	—
47	C₂×C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
48	C₂×C₃×C₈	C₃×Q₁₆	1	4	yes	—
49	C₃×C₄²	C₃×D₈	1	4	yes	—
50	C₃×C₁₆	—	2	4	no	—
51	C₃×C₁₆	C₂×C₃×C₈	1	4	yes	—
52	C₃×C₁₆	C₃×M₁₆	1	4	yes	—
53	C₂²×C₃²	#367	1	1	yes	characteristic
54	C₃²×C₄	—	2	2	no	—
55	C₃²×C₄	#335	1	1	yes	characteristic
56	C₃²×C₄	#368	1	1	yes	characteristic
57	C₂×C₁₆	C₂×C₃²×C₄	1	1	yes	characteristic
58	C₂×C₁₆	C₃²×Q₈	1	1	yes	characteristic
59	C₄⋊C₈	—	4	2	no	—
60	C₄×C₈	C₃²×D₄	1	1	yes	characteristic
61	C₃₂	—	2	1	no	—
62	C₃₂	C₃²×C₈	1	2	yes	complemented
63	C₂×C₃×C₄	C₃×#107	1	4	yes	—
64	C₂×C₃×C₄	C₃×SD₃₂	1	8	yes	—
65	C₃×C₈	—	2	4	no	—
66	C₃×C₈	—	16	8	no	—
67	C₃×C₈	C₃×C₄⋊C₈	1	4	yes	—
68	C₃×C₈	C₃×#108	1	4	yes	—
69	C₂×C₃²	#1372	1	1	yes	characteristic
70	C₂×C₃²	#1426	1	2	yes	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈	C₃²×Q₁₆	1	1	yes	characteristic
74	C₄²	C₃²×D₈	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₂×C₃²×C₈	1	1	yes	characteristic
77	C₁₆	C₃²×M₁₆	1	1	yes	characteristic
78	C₂²×C₃	C₃×#367	1	4	yes	—
79	C₃×C₄	—	2	8	no	—
80	C₃×C₄	C₃×#335	1	4	yes	—
81	C₃×C₄	C₃×#368	1	4	yes	—
82	C₃²	#7463	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₄	C₃²×#107	1	1	yes	characteristic
84	C₂×C₄	C₃²×SD₃₂	1	2	yes	—
85	C₈	—	2	1	no	—
86	C₈	—	16	2	no	—
87	C₈	C₃²×C₄⋊C₈	1	1	yes	characteristic
88	C₈	C₃²×#108	1	1	yes	characteristic
89	C₂×C₃	C₃×#1372	1	4	yes	—
90	C₂×C₃	C₃×#1426	1	8	yes	—
91	C₂²	C₃²×#367	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	C₃²×#335	1	1	yes	characteristic
94	C₄	C₃²×#368	1	1	yes	characteristic
95	C₃	C₃×#7463	1	4	yes	direct product factor
96	C₂	C₃²×#1372	1	1	yes	characteristic
97	C₂	C₃²×#1426	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#367
commutator	C₁₆	C₂×C₃²×C₈
Frattini, cosocle	C₄×C₁₆	C₂²×C₃²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ C₃²×C₄×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#7463	C₃	direct (N × H)
15. C₃²×C₃₂	C₈	semidirect
16. #7463	C₃²	direct (N × H)
35. C₃×C₃₂	C₃×C₈	semidirect
62. C₃₂	C₃²×C₈	semidirect
82. C₃²	#7463	direct (N × H)
95. C₃	C₃×#7463	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#1397	2
C₃²×#1372	2
C₃×#7463	3

Subgroups — C32×#7463 #72254864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#7463 #72254864