Subgroups — C₃²×#2164 #11769213

Order 1152.

414 conjugacy classes of subgroups.

The subgroup lattice has 153 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#405	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₃²×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2164	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₃²×#117	C₄	1	1	yes	characteristic
10	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
11	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
12	C₃²×#97	C₄	1	1	yes	characteristic
13	C₃²×#97	C₄	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×#405	C₂×C₃	1	4	yes	—
17	C₃×C₄⋊C₁₆	C₂×C₃	1	8	yes	complemented
18	C₃×C₂²⋊C₁₆	C₂×C₃	1	8	yes	—
19	C₃×#346	C₂×C₃	1	4	yes	complemented
20	C₃×C₄×C₁₆	C₂×C₃	1	4	yes	complemented
21	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₃²×M₁₆	—	2	2	no	—
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	C₈	1	2	yes	—
29	C₃²×C₂²⋊C₄	C₈	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃²×C₁₆	—	2	2	no	—
33	C₃²×C₁₆	—	2	2	no	—
34	#2164	C₃²	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×M₁₆	C₃×C₄	1	4	yes	—
36	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
37	C₃×#117	C₃×C₄	1	4	yes	—
38	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
39	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
40	C₃×#97	C₃×C₄	1	4	yes	—
41	C₃×#97	C₃×C₄	1	4	yes	—
42	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	—	2	1	no	—
46	C₂×C₃²×C₄	—	2	1	no	—
47	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
48	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
49	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₃²×C₈	—	2	1	no	—
52	C₃²×C₈	—	2	1	no	—
53	C₃²×C₈	M₁₆	1	2	yes	—
54	C₃²×C₈	#41	1	2	yes	—
55	#405	C₂×C₃²	1	1	yes	characteristic
56	C₄⋊C₁₆	C₂×C₃²	1	2	yes	complemented
57	C₂²⋊C₁₆	C₂×C₃²	1	2	yes	—
58	#346	C₂×C₃²	1	1	yes	characteristic, complemented
59	C₄×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
60	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
61	C₃×M₁₆	—	2	8	no	—
62	C₂×C₃×C₈	—	2	8	no	—
63	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
64	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
65	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
67	C₃×C₄⋊C₄	C₃×C₈	1	8	yes	—
68	C₃×C₂²⋊C₄	C₃×C₈	1	8	yes	—
69	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
70	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
71	C₃×C₁₆	—	2	8	no	—
72	C₃×C₁₆	—	2	8	no	—
73	C₂²×C₃²	—	2	1	no	—
74	C₂²×C₃²	—	2	2	no	—
75	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
76	C₃²×C₄	—	2	1	no	—
77	C₃²×C₄	—	2	2	no	—
78	C₃²×C₄	—	2	2	no	—
79	C₃²×C₄	#117	1	1	yes	characteristic
80	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
81	C₂×M₁₆	C₃²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
83	#117	C₃²×C₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
85	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
86	#97	C₃²×C₄	1	1	yes	characteristic
87	#97	C₃²×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
89	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
90	C₂³×C₃	C₂×C₃×C₈	1	4	yes	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	2	4	no	—
93	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
94	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
95	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
96	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
97	C₃×C₈	—	2	4	no	—
98	C₃×C₈	—	2	4	no	—
99	C₃×C₈	C₃×M₁₆	1	8	yes	—
100	C₃×C₈	C₃×#41	1	8	yes	—
101	C₂×C₃²	—	4	1	no	—
102	C₂×C₃²	#431	1	1	yes	characteristic
103	C₂×C₃²	#504	1	2	yes	—
104	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
108	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
109	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
110	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
111	C₄⋊C₄	C₃²×C₈	1	2	yes	—
112	C₂²⋊C₄	C₃²×C₈	1	2	yes	—
113	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
114	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	2	no	—
117	C₂²×C₃	—	2	8	no	—
118	C₂²×C₃	—	2	4	no	—
119	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
120	C₃×C₄	—	2	8	no	—
121	C₃×C₄	—	2	8	no	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	C₃×#117	1	4	yes	—
124	C₃×C₄	C₂×C₃×M₁₆	1	4	yes	—
125	C₃²	#2164	1	1	yes	characteristic, direct product factor, Sylow
126	C₂³	C₂×C₃²×C₈	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
130	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
131	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	C₃²×M₁₆	1	2	yes	—
136	C₈	C₃²×#41	1	2	yes	—
137	C₂×C₃	—	4	4	no	—
138	C₂×C₃	C₃×#431	1	4	yes	—
139	C₂×C₃	C₃×#504	1	8	yes	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂²×C₃²×C₈	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	1	no	—
146	C₄	C₃²×#117	1	1	yes	characteristic
147	C₄	C₂×C₃²×M₁₆	1	1	yes	characteristic
148	C₃	C₃×#2164	1	4	yes	direct product factor
149	C₂	—	4	1	no	—
150	C₂	C₃²×#431	1	1	yes	characteristic
151	C₂	C₃²×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₈	C₂³
commutator	C₂²	C₂²×C₃²×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂³×C₃²	C₂×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
2. C₃²×C₄⋊C₁₆	C₂	semidirect
4. C₃²×#346	C₂	semidirect
5. C₃²×C₄×C₁₆	C₂	semidirect
6. C₃×#2164	C₃	direct (N × H)
17. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
19. C₃×#346	C₂×C₃	semidirect
20. C₃×C₄×C₁₆	C₂×C₃	semidirect
34. #2164	C₃²	direct (N × H)
56. C₄⋊C₁₆	C₂×C₃²	semidirect
58. #346	C₂×C₃²	semidirect
59. C₄×C₁₆	C₂×C₃²	semidirect
125. C₃²	#2164	direct (N × H)
148. C₃	C₃×#2164	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×#405	2
C₃²×C₄⋊C₁₆	2
C₃²×C₂²⋊C₁₆	2
C₃²×#346	2
C₃²×C₄×C₁₆	2
C₃×#2164	3

Subgroups — C32×#2164 #11769213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#2164 #11769213