Subgroups — C₃³×#356 #13146149

Order 1728.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃³×#101	C₂	1	2	yes	maximal
3	C₃²×#356	C₃	1	13	yes	maximal, direct product factor
4	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
5	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
6	C₃³×M₁₆	—	2	2	no	—
7	C₃³×C₄⋊C₄	—	2	2	no	—
8	C₃³×C₄²	C₄	1	1	yes	characteristic
9	C₃²×C₄⋊Q₈	C₂×C₃	1	13	yes	—
10	C₃²×#101	C₂×C₃	1	26	yes	—
11	C₃³×Q₈	—	4	1	no	—
12	C₃³×Q₈	—	4	1	no	—
13	C₂×C₃³×C₄	—	2	1	no	—
14	C₂×C₃³×C₄	—	2	1	no	—
15	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃³×C₄	D₄	1	2	yes	—
17	C₃³×C₈	—	4	2	no	—
18	C₃×#356	C₃²	1	13	yes	direct product factor
19	C₂×C₃²×Q₈	C₃×C₄	1	13	yes	—
20	C₂×C₃²×Q₈	C₂²×C₃	1	13	yes	—
21	C₃²×M₁₆	—	2	26	no	—
22	C₃²×C₄⋊C₄	—	2	26	no	—
23	C₃²×C₄²	C₃×C₄	1	13	yes	—
24	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
25	C₃³×C₄	—	2	2	no	—
26	C₃³×C₄	—	2	2	no	—
27	C₃³×C₄	—	2	1	no	—
28	C₃³×C₄	—	4	1	no	—
29	C₃×C₄⋊Q₈	C₂×C₃²	1	13	yes	—
30	C₃×#101	C₂×C₃²	1	26	yes	—
31	C₃²×Q₈	—	4	13	no	—
32	C₃²×Q₈	—	4	13	no	—
33	C₂×C₃²×C₄	—	2	13	no	—
34	C₂×C₃²×C₄	—	2	13	no	—
35	C₂×C₃²×C₄	C₂×C₃×C₄	1	13	yes	—
36	C₂×C₃²×C₄	C₃×D₄	1	26	yes	—
37	C₃²×C₈	—	4	26	no	—
38	#356	C₃³	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃³	—	2	1	no	—
40	C₂×C₃³	#99	1	1	yes	characteristic
41	C₂×C₃×Q₈	C₃²×C₄	1	13	yes	—
42	C₂×C₃×Q₈	C₂²×C₃²	1	13	yes	—
43	C₃×M₁₆	—	2	26	no	—
44	C₃×C₄⋊C₄	—	2	26	no	—
45	C₃×C₄²	C₃²×C₄	1	13	yes	—
46	C₂²×C₃²	C₃×C₂²⋊C₄	1	13	yes	—
47	C₃²×C₄	—	2	26	no	—
48	C₃²×C₄	—	2	26	no	—
49	C₃²×C₄	—	2	13	no	—
50	C₃²×C₄	—	4	13	no	—
51	C₄⋊Q₈	C₂×C₃³	1	1	yes	characteristic
52	#101	C₂×C₃³	1	2	yes	—
53	C₃³	#356	1	1	yes	characteristic, direct product factor, Sylow
54	C₃×Q₈	—	4	13	no	—
55	C₃×Q₈	—	4	13	no	—
56	C₂×C₃×C₄	—	2	13	no	—
57	C₂×C₃×C₄	—	2	13	no	—
58	C₂×C₃×C₄	C₂×C₃²×C₄	1	13	yes	—
59	C₂×C₃×C₄	C₃²×D₄	1	26	yes	—
60	C₃×C₈	—	4	26	no	—
61	C₂×C₃²	—	2	13	no	—
62	C₂×C₃²	C₃×#99	1	13	yes	—
63	C₂×Q₈	C₃³×C₄	1	1	yes	characteristic
64	C₂×Q₈	C₂²×C₃³	1	1	yes	characteristic
65	M₁₆	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄²	C₃³×C₄	1	1	yes	characteristic
68	C₂²×C₃	C₃²×C₂²⋊C₄	1	13	yes	—
69	C₃×C₄	—	2	26	no	—
70	C₃×C₄	—	2	26	no	—
71	C₃×C₄	—	2	13	no	—
72	C₃×C₄	—	4	13	no	—
73	C₃²	C₃×#356	1	13	yes	direct product factor
74	Q₈	—	4	1	no	—
75	Q₈	—	4	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	C₂×C₃³×C₄	1	1	yes	characteristic
79	C₂×C₄	C₃³×D₄	1	2	yes	—
80	C₈	—	4	2	no	—
81	C₂×C₃	—	2	13	no	—
82	C₂×C₃	C₃²×#99	1	13	yes	—
83	C₂²	C₃³×C₂²⋊C₄	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₃	C₃²×#356	1	13	yes	direct product factor
89	C₂	—	2	1	no	—
90	C₂	C₃³×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃³	#99
commutator	C₂×C₄	C₂×C₃³×C₄
Frattini, cosocle	C₂×Q₈	C₂²×C₃³
prime radical	C₂²×C₃³	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ C₂²×C₃³ ◁ C₂×C₃³×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#356	C₃	direct (N × H)
18. C₃×#356	C₃²	direct (N × H)
38. #356	C₃³	direct (N × H)
53. C₃³	#356	direct (N × H)
73. C₃²	C₃×#356	direct (N × H)
88. C₃	C₃²×#356	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₃³×#101	2
C₃²×#356	3

Subgroups — C33×#356 #13146149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#356 #13146149