Subgroups — C₂³×#4178 #13048194

Order 1296.

718 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4178	C₂	1	14	yes	maximal, direct product factor
3	C₂³×#272	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃×C₉	—	3	1	no	maximal
5	C₂²×#732	C₂²	1	7	yes	complemented
6	C₂×#4178	C₂²	1	28	yes	direct product factor
7	C₂³×H(3,𝔽₃)	—	2	1	no	—
8	C₂³×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₃×C₉	D₃	1	1	yes	characteristic, complemented
10	C₂²×#272	C₂×C₃	1	14	yes	complemented
11	C₂²×D₃×C₉	—	3	14	no	—
12	C₂×#732	C₂³	1	7	yes	complemented
13	#4178	C₂³	1	8	yes	direct product factor
14	C₂³×C₃×D₃	—	3	1	no	—
15	C₂³×C₃×D₃	—	9	1	no	—
16	C₂⁴×C₉	—	9	1	no	—
17	C₂²×H(3,𝔽₃)	—	2	7	no	—
18	C₂²×H(3,𝔽₃)	C₂²×C₃	1	7	yes	—
19	C₂²×C₃×C₉	C₂×D₃	1	7	yes	complemented
20	C₂×#272	C₂²×C₃	1	28	yes	complemented
21	C₂×D₃×C₉	—	3	28	no	—
22	#732	C₂⁴	1	1	yes	characteristic, complemented, Sylow
23	C₂³×C₃²	—	3	1	no	—
24	C₂³×C₃²	—	6	1	no	—
25	C₂³×C₃²	C₃×D₃	1	1	yes	characteristic
26	C₂²×C₃×D₃	—	3	14	no	—
27	C₂²×C₃×D₃	—	9	14	no	—
28	C₂³×C₉	—	3	1	no	—
29	C₂³×C₉	—	9	14	no	—
30	C₂×H(3,𝔽₃)	—	2	7	no	—
31	C₂×H(3,𝔽₃)	C₂³×C₃	1	7	yes	—
32	C₂×C₃×C₉	C₂²×D₃	1	7	yes	complemented
33	#272	C₂³×C₃	1	8	yes	complemented
34	D₃×C₉	—	3	8	no	—
35	C₂⁴×C₃	—	9	1	no	—
36	C₂³×D₃	—	3	1	no	—
37	C₂³×D₃	—	9	1	no	—
38	C₂²×C₃²	—	3	7	no	—
39	C₂²×C₃²	—	6	7	no	—
40	C₂²×C₃²	C₂×C₃×D₃	1	7	yes	—
41	C₂×C₃×D₃	—	3	28	no	—
42	C₂×C₃×D₃	—	9	28	no	—
43	C₂²×C₉	—	3	7	no	—
44	C₂²×C₉	—	9	28	no	—
45	H(3,𝔽₃)	—	2	1	no	—
46	H(3,𝔽₃)	C₂⁴×C₃	1	1	yes	characteristic
47	C₃×C₉	C₂³×D₃	1	1	yes	characteristic, complemented
48	C₂³×C₃	—	3	1	no	—
49	C₂³×C₃	—	9	14	no	—
50	C₂³×C₃	—	9	1	no	—
51	C₂³×C₃	—	18	1	no	—
52	C₂³×C₃	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
53	C₂²×D₃	—	3	14	no	—
54	C₂²×D₃	—	9	14	no	—
55	C₂×C₃²	—	3	7	no	—
56	C₂×C₃²	—	6	7	no	—
57	C₂×C₃²	C₂²×C₃×D₃	1	7	yes	—
58	C₃×D₃	—	3	8	no	—
59	C₃×D₃	—	9	8	no	—
60	C₂×C₉	—	3	7	no	—
61	C₂×C₉	—	9	8	no	—
62	C₂⁴	—	9	1	no	Sylow
63	C₂²×C₃	—	3	7	no	—
64	C₂²×C₃	—	9	28	no	—
65	C₂²×C₃	—	9	7	no	—
66	C₂²×C₃	—	18	7	no	—
67	C₂²×C₃	C₂×(C₃⋊D₃)⋊C₃	1	7	yes	—
68	C₂×D₃	—	3	28	no	—
69	C₂×D₃	—	9	28	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	6	1	no	—
72	C₃²	C₂³×C₃×D₃	1	1	yes	characteristic
73	C₉	—	3	1	no	—
74	C₂³	—	9	14	no	—
75	C₂³	#4178	1	1	yes	characteristic, direct product factor
76	C₂×C₃	—	3	7	no	—
77	C₂×C₃	—	9	8	no	—
78	C₂×C₃	—	9	7	no	—
79	C₂×C₃	—	18	7	no	—
80	C₂×C₃	C₂²×(C₃⋊D₃)⋊C₃	1	7	yes	—
81	D₃	—	3	8	no	—
82	D₃	—	9	8	no	—
83	C₂²	—	9	28	no	—
84	C₂²	C₂×#4178	1	7	yes	direct product factor
85	C₃	—	3	1	no	—
86	C₃	—	9	1	no	—
87	C₃	—	18	1	no	—
88	C₃	C₂³×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
89	C₂	—	9	8	no	—
90	C₂	C₂²×#4178	1	7	yes	direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂³×C₃	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂⁴×C₃
Frattini	C₃²	C₂³×C₃×D₃
Fitting	C₂³×#732	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
1. C₂³×#732	C₂	semidirect
2. C₂²×#4178	C₂	direct (N × H)
3. C₂³×#272	C₃	semidirect
5. C₂²×#732	C₂²	semidirect
6. C₂×#4178	C₂²	direct (N × H)
9. C₂³×C₃×C₉	D₃	semidirect
10. C₂²×#272	C₂×C₃	semidirect
12. C₂×#732	C₂³	semidirect
13. #4178	C₂³	direct (N × H)
19. C₂²×C₃×C₉	C₂×D₃	semidirect
20. C₂×#272	C₂²×C₃	semidirect
22. #732	C₂⁴	semidirect
32. C₂×C₃×C₉	C₂²×D₃	semidirect
33. #272	C₂³×C₃	semidirect
47. C₃×C₉	C₂³×D₃	semidirect
75. C₂³	#4178	direct (N × H)
84. C₂²	C₂×#4178	direct (N × H)
90. C₂	C₂²×#4178	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#732	2
C₂²×#4178	2
C₂³×#272	3
C₂³×D₃×C₉	3

Subgroups — C23×#4178 #13048194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#4178 #13048194