Subgroups — C₃³×#362 #13146155

Order 1728.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃³×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃³×#108	C₂	1	1	yes	maximal, characteristic
4	C₃²×#362	C₃	1	13	yes	maximal, direct product factor
5	C₂×C₃³×Q₈	—	2	1	no	—
6	C₃³×Q₁₆	—	2	1	no	—
7	C₃³×Q₁₆	C₄	1	2	yes	—
8	C₃³×M₁₆	—	2	1	no	—
9	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
10	C₃³×C₁₆	—	2	1	no	—
11	C₂×C₃²×Q₁₆	C₂×C₃	1	13	yes	—
12	C₃²×M₃₂	C₂×C₃	1	13	yes	—
13	C₃²×#108	C₂×C₃	1	13	yes	—
14	C₃³×Q₈	—	2	2	no	—
15	C₃³×Q₈	—	4	1	no	—
16	C₂×C₃³×C₄	—	4	1	no	—
17	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
18	C₃³×C₈	—	4	1	no	—
19	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃³×C₈	D₄	1	1	yes	characteristic
21	C₃×#362	C₃²	1	13	yes	direct product factor
22	C₂×C₃²×Q₈	—	2	13	no	—
23	C₃²×Q₁₆	—	2	13	no	—
24	C₃²×Q₁₆	C₃×C₄	1	26	yes	—
25	C₃²×M₁₆	—	2	13	no	—
26	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
27	C₃²×C₁₆	—	2	13	no	—
28	C₂²×C₃³	SD₁₆	1	1	yes	characteristic
29	C₃³×C₄	—	4	2	no	—
30	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₃³×C₄	D₈	1	1	yes	characteristic
32	C₂×C₃×Q₁₆	C₂×C₃²	1	13	yes	—
33	C₃×M₃₂	C₂×C₃²	1	13	yes	—
34	C₃×#108	C₂×C₃²	1	13	yes	—
35	C₃²×Q₈	—	2	26	no	—
36	C₃²×Q₈	—	4	13	no	—
37	C₂×C₃²×C₄	—	4	13	no	—
38	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
39	C₃²×C₈	—	4	13	no	—
40	C₃²×C₈	C₂×C₃×C₄	1	13	yes	—
41	C₃²×C₈	C₃×D₄	1	13	yes	—
42	#362	C₃³	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃³	—	2	1	no	—
44	C₂×C₃³	#102	1	1	yes	characteristic
45	C₂×C₃×Q₈	—	2	13	no	—
46	C₃×Q₁₆	—	2	13	no	—
47	C₃×Q₁₆	C₃²×C₄	1	26	yes	—
48	C₃×M₁₆	—	2	13	no	—
49	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
50	C₃×C₁₆	—	2	13	no	—
51	C₂²×C₃²	C₃×SD₁₆	1	13	yes	—
52	C₃²×C₄	—	4	26	no	—
53	C₃²×C₄	C₃×C₂²⋊C₄	1	13	yes	—
54	C₃²×C₄	C₃×D₈	1	13	yes	—
55	C₂×Q₁₆	C₂×C₃³	1	1	yes	characteristic
56	M₃₂	C₂×C₃³	1	1	yes	characteristic
57	#108	C₂×C₃³	1	1	yes	characteristic
58	C₃³	#362	1	1	yes	characteristic, direct product factor, Sylow
59	C₃×Q₈	—	2	26	no	—
60	C₃×Q₈	—	4	13	no	—
61	C₂×C₃×C₄	—	4	13	no	—
62	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
63	C₃×C₈	—	4	13	no	—
64	C₃×C₈	C₂×C₃²×C₄	1	13	yes	—
65	C₃×C₈	C₃²×D₄	1	13	yes	—
66	C₂×C₃²	—	2	13	no	—
67	C₂×C₃²	C₃×#102	1	13	yes	—
68	C₂×Q₈	—	2	1	no	—
69	Q₁₆	—	2	1	no	—
70	Q₁₆	C₃³×C₄	1	2	yes	—
71	M₁₆	—	2	1	no	—
72	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₂²×C₃	C₃²×SD₁₆	1	13	yes	—
75	C₃×C₄	—	4	26	no	—
76	C₃×C₄	C₃²×C₂²⋊C₄	1	13	yes	—
77	C₃×C₄	C₃²×D₈	1	13	yes	—
78	C₃²	C₃×#362	1	13	yes	direct product factor
79	Q₈	—	2	2	no	—
80	Q₈	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
83	C₈	—	4	1	no	—
84	C₈	C₂×C₃³×C₄	1	1	yes	characteristic
85	C₈	C₃³×D₄	1	1	yes	characteristic
86	C₂×C₃	—	2	13	no	—
87	C₂×C₃	C₃²×#102	1	13	yes	—
88	C₂²	C₃³×SD₁₆	1	1	yes	characteristic
89	C₄	—	4	2	no	—
90	C₄	C₃³×C₂²⋊C₄	1	1	yes	characteristic
91	C₄	C₃³×D₈	1	1	yes	characteristic
92	C₃	C₃²×#362	1	13	yes	direct product factor
93	C₂	—	2	1	no	—
94	C₂	C₃³×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃³	#102
commutator	C₈	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₈	C₂²×C₃³
prime radical	C₂²×C₃³	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ C₂×C₃³×C₄ ◁ C₂×C₃³×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃²×#362	C₃	direct (N × H)
21. C₃×#362	C₃²	direct (N × H)
42. #362	C₃³	direct (N × H)
58. C₃³	#362	direct (N × H)
78. C₃²	C₃×#362	direct (N × H)
92. C₃	C₃²×#362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃³×Q₁₆	2
C₃³×M₃₂	2
C₃³×#108	2
C₃²×#362	3

Subgroups — C33×#362 #13146155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#362 #13146155