Subgroups — C₃²×#2183 #11769232

Order 1152.

378 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₁₆	C₂	1	2	yes	maximal
4	C₃²×#346	C₂	1	2	yes	maximal
5	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₃×#2183	C₃	1	4	yes	maximal, direct product factor
7	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
10	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊C₈	C₄	1	2	yes	—
12	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	2	yes	—
15	C₃×C₈×Q₈	C₂×C₃	1	4	yes	—
16	C₃×C₄⋊C₁₆	C₂×C₃	1	4	yes	—
17	C₃×C₄⋊C₁₆	C₂×C₃	1	8	yes	—
18	C₃×#346	C₂×C₃	1	8	yes	—
19	C₃×C₄×C₁₆	C₂×C₃	1	4	yes	—
20	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
24	C₃²×C₄⋊C₄	C₈	1	2	yes	—
25	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
26	C₃²×C₄²	C₂×C₄	1	2	yes	—
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₁₆	—	2	1	no	—
29	C₃²×C₁₆	—	2	2	no	—
30	C₃²×C₁₆	Q₈	1	2	yes	complemented
31	#2183	C₃²	1	1	yes	characteristic, direct product factor, Sylow
32	C₃×C₄×Q₈	C₃×C₄	1	4	yes	—
33	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
34	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
35	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
36	C₃×C₄⋊C₈	C₃×C₄	1	8	yes	—
37	C₃×C₄⋊C₈	C₃×C₄	1	4	yes	—
38	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
39	C₃×C₄×C₈	C₂²×C₃	1	8	yes	—
40	C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
42	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
43	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
44	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₃²×C₈	—	2	1	no	—
47	C₃²×C₈	—	2	2	no	—
48	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₃²×C₈	#41	1	1	yes	characteristic
50	C₈×Q₈	C₂×C₃²	1	1	yes	characteristic
51	C₄⋊C₁₆	C₂×C₃²	1	1	yes	characteristic
52	C₄⋊C₁₆	C₂×C₃²	1	2	yes	—
53	#346	C₂×C₃²	1	2	yes	—
54	C₄×C₁₆	C₂×C₃²	1	1	yes	characteristic
55	C₂×C₃×Q₈	C₃×C₈	1	4	yes	—
56	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
57	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
58	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
59	C₃×C₄⋊C₄	C₃×C₈	1	8	yes	—
60	C₃×C₄⋊C₄	C₃×C₈	1	4	yes	—
61	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
62	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
63	C₃×C₁₆	—	2	4	no	—
64	C₃×C₁₆	—	2	8	no	—
65	C₃×C₁₆	C₃×Q₈	1	8	yes	complemented
66	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
67	C₃²×C₄	—	2	2	no	—
68	C₃²×C₄	—	2	2	no	—
69	C₃²×C₄	—	2	1	no	—
70	C₃²×C₄	M₃₂	1	2	yes	—
71	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
72	C₃²×C₄	#131	1	1	yes	characteristic
73	C₄×Q₈	C₃²×C₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
76	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
77	C₄⋊C₈	C₃²×C₄	1	2	yes	—
78	C₄⋊C₈	C₃²×C₄	1	1	yes	characteristic
79	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
80	C₄×C₈	C₂²×C₃²	1	2	yes	—
81	C₃×Q₈	—	2	8	no	—
82	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
83	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
84	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
85	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
86	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
87	C₃×C₈	—	2	4	no	—
88	C₃×C₈	—	2	8	no	—
89	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
90	C₃×C₈	C₃×#41	1	4	yes	—
91	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
92	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
93	C₂×C₃²	#504	1	1	yes	characteristic
94	C₂×Q₈	C₃²×C₈	1	1	yes	characteristic
95	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
97	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
98	C₄⋊C₄	C₃²×C₈	1	2	yes	—
99	C₄⋊C₄	C₃²×C₈	1	1	yes	characteristic
100	C₄²	C₂×C₃²×C₄	1	2	yes	—
101	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
102	C₁₆	—	2	1	no	—
103	C₁₆	—	2	2	no	—
104	C₁₆	C₃²×Q₈	1	2	yes	complemented
105	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
106	C₃×C₄	—	2	8	no	—
107	C₃×C₄	—	2	8	no	—
108	C₃×C₄	—	2	4	no	—
109	C₃×C₄	C₃×M₃₂	1	8	yes	—
110	C₃×C₄	C₃×C₄×Q₈	1	4	yes	—
111	C₃×C₄	C₃×#131	1	4	yes	—
112	C₃²	#2183	1	1	yes	characteristic, direct product factor, Sylow
113	Q₈	—	2	2	no	—
114	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
115	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
116	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
117	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	2	no	—
121	C₈	C₂×C₃²×Q₈	1	1	yes	characteristic
122	C₈	C₃²×#41	1	1	yes	characteristic
123	C₂×C₃	C₃×C₈×Q₈	1	4	yes	—
124	C₂×C₃	C₂×C₃×M₃₂	1	4	yes	—
125	C₂×C₃	C₃×#504	1	4	yes	—
126	C₂²	C₂²×C₃²×C₈	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	C₃²×M₃₂	1	2	yes	—
131	C₄	C₃²×C₄×Q₈	1	1	yes	characteristic
132	C₄	C₃²×#131	1	1	yes	characteristic
133	C₃	C₃×#2183	1	4	yes	direct product factor
134	C₂	C₃²×C₈×Q₈	1	1	yes	characteristic
135	C₂	C₂×C₃²×M₃₂	1	1	yes	characteristic
136	C₂	C₃²×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₈	C₂³
commutator	C₂²	C₂²×C₃²×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃×#2183	C₃	direct (N × H)
30. C₃²×C₁₆	Q₈	semidirect
31. #2183	C₃²	direct (N × H)
65. C₃×C₁₆	C₃×Q₈	semidirect
104. C₁₆	C₃²×Q₈	semidirect
112. C₃²	#2183	direct (N × H)
133. C₃	C₃×#2183	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×C₈×Q₈	2
C₃²×C₄⋊C₁₆	2
C₃²×#346	2
C₃²×C₄×C₁₆	2
C₃×#2183	3

Subgroups — C32×#2183 #11769232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2183 #11769232