Subgroups — C₃²×D₄×C₁₆ #11769216

Order 1152.

522 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₄×C₁₆	C₃	1	4	yes	maximal, direct product factor
7	C₂²×C₃²×C₈	C₂²	1	2	yes	—
8	C₃²×C₄×D₄	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₁₆	—	2	4	no	—
10	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
12	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
13	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₁₆	C₂×C₃	1	8	yes	complemented
17	C₃×C₈×D₄	C₂×C₃	1	4	yes	—
18	C₃×C₄⋊C₁₆	C₂×C₃	1	4	yes	complemented
19	C₃×C₂²⋊C₁₆	C₂×C₃	1	8	yes	complemented
20	C₃×C₄×C₁₆	C₂×C₃	1	4	yes	complemented
21	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
22	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
28	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₃²×C₂²⋊C₄	C₈	1	2	yes	—
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₁₆	—	2	2	no	—
32	C₃²×C₁₆	—	2	1	no	—
33	C₃²×C₁₆	D₄	1	2	yes	direct product factor
34	D₄×C₁₆	C₃²	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
36	C₃×C₄×D₄	C₃×C₄	1	4	yes	—
37	C₂×C₃×C₁₆	—	2	16	no	—
38	C₂×C₃×C₁₆	C₂²×C₃	1	8	yes	—
39	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
40	C₂×C₃×C₁₆	C₂²×C₃	1	4	yes	—
41	C₃×C₄⋊C₈	C₃×C₄	1	4	yes	—
42	C₃×C₂²⋊C₈	C₃×C₄	1	8	yes	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₂³×C₃²	C₂×C₈	1	2	yes	—
45	C₃²×D₄	C₁₆	1	4	yes	direct product factor
46	C₂×C₃²×C₄	—	2	2	no	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
50	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	—	2	1	no	—
54	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
55	C₃²×C₈	#41	1	1	yes	characteristic
56	C₂²×C₁₆	C₂×C₃²	1	2	yes	complemented
57	C₈×D₄	C₂×C₃²	1	1	yes	characteristic
58	C₄⋊C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
59	C₂²⋊C₁₆	C₂×C₃²	1	2	yes	complemented
60	C₄×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
61	C₂×C₃×D₄	C₃×C₈	1	4	yes	—
62	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
63	C₂×C₃×C₈	—	2	8	no	—
64	C₂×C₃×C₈	—	2	8	no	—
65	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
67	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
68	C₃×C₄⋊C₄	C₃×C₈	1	4	yes	—
69	C₃×C₂²⋊C₄	C₃×C₈	1	8	yes	—
70	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
71	C₃×C₁₆	—	2	8	no	—
72	C₃×C₁₆	—	2	4	no	—
73	C₃×C₁₆	C₃×D₄	1	8	yes	direct product factor
74	C₂²×C₃²	—	2	2	no	—
75	C₂²×C₃²	—	2	2	no	—
76	C₂²×C₃²	C₂×C₁₆	1	4	yes	complemented
77	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
78	C₃²×C₄	—	2	2	no	—
79	C₃²×C₄	—	2	1	no	—
80	C₃²×C₄	C₂×C₁₆	1	2	yes	complemented
81	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
82	C₃²×C₄	#131	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₃²	1	2	yes	—
84	C₄×D₄	C₃²×C₄	1	1	yes	characteristic
85	C₂×C₁₆	—	2	4	no	—
86	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
87	C₂×C₁₆	C₂²×C₃²	1	2	yes	—
88	C₂×C₁₆	C₂²×C₃²	1	1	yes	characteristic
89	C₄⋊C₈	C₃²×C₄	1	1	yes	characteristic
90	C₂²⋊C₈	C₃²×C₄	1	2	yes	—
91	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
92	C₂³×C₃	C₂×C₃×C₈	1	8	yes	—
93	C₃×D₄	C₃×C₁₆	1	16	yes	direct product factor
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	8	no	—
96	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
97	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
98	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
99	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
100	C₃×C₈	—	2	8	no	—
101	C₃×C₈	—	2	4	no	—
102	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
103	C₃×C₈	C₃×#41	1	4	yes	—
104	C₂×C₃²	—	2	4	no	—
105	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
106	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
107	C₂×C₃²	#504	1	1	yes	characteristic
108	C₂×D₄	C₃²×C₈	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₃²×C₄	1	2	yes	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
113	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
114	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
115	C₄⋊C₄	C₃²×C₈	1	1	yes	characteristic
116	C₂²⋊C₄	C₃²×C₈	1	2	yes	—
117	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
118	C₁₆	—	2	2	no	—
119	C₁₆	—	2	1	no	—
120	C₁₆	C₃²×D₄	1	2	yes	direct product factor
121	C₂²×C₃	—	2	8	no	—
122	C₂²×C₃	—	2	8	no	—
123	C₂²×C₃	C₂×C₃×C₁₆	1	16	yes	complemented
124	C₂²×C₃	C₂²×C₃×C₈	1	4	yes	—
125	C₃×C₄	—	2	4	no	—
126	C₃×C₄	—	2	8	no	—
127	C₃×C₄	C₂×C₃×C₁₆	1	8	yes	complemented
128	C₃×C₄	C₃×C₄×D₄	1	4	yes	—
129	C₃×C₄	C₃×#131	1	4	yes	—
130	C₃²	D₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³	C₂×C₃²×C₈	1	2	yes	—
132	D₄	C₃²×C₁₆	1	4	yes	direct product factor
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
136	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
137	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	C₂×C₃²×D₄	1	1	yes	characteristic
142	C₈	C₃²×#41	1	1	yes	characteristic
143	C₂×C₃	—	2	16	no	—
144	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
145	C₂×C₃	C₂²×C₃×C₁₆	1	4	yes	—
146	C₂×C₃	C₃×#504	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂×C₃²×C₁₆	1	4	yes	complemented
150	C₂²	C₂²×C₃²×C₈	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	2	no	—
153	C₄	C₂×C₃²×C₁₆	1	2	yes	complemented
154	C₄	C₃²×C₄×D₄	1	1	yes	characteristic
155	C₄	C₃²×#131	1	1	yes	characteristic
156	C₃	C₃×D₄×C₁₆	1	4	yes	direct product factor
157	C₂	—	2	4	no	—
158	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
159	C₂	C₂²×C₃²×C₁₆	1	1	yes	characteristic
160	C₂	C₃²×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₁₆	C₂²
commutator	C₂	C₂²×C₃²×C₁₆
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃²×D₄	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₁₆	C₂	semidirect
3. C₃²×C₄⋊C₁₆	C₂	semidirect
4. C₃²×C₂²⋊C₁₆	C₂	semidirect
5. C₃²×C₄×C₁₆	C₂	semidirect
6. C₃×D₄×C₁₆	C₃	direct (N × H)
16. C₂²×C₃×C₁₆	C₂×C₃	semidirect
18. C₃×C₄⋊C₁₆	C₂×C₃	semidirect
19. C₃×C₂²⋊C₁₆	C₂×C₃	semidirect
20. C₃×C₄×C₁₆	C₂×C₃	semidirect
33. C₃²×C₁₆	D₄	direct (N × H)
34. D₄×C₁₆	C₃²	direct (N × H)
45. C₃²×D₄	C₁₆	direct (N × H)
56. C₂²×C₁₆	C₂×C₃²	semidirect
58. C₄⋊C₁₆	C₂×C₃²	semidirect
59. C₂²⋊C₁₆	C₂×C₃²	semidirect
60. C₄×C₁₆	C₂×C₃²	semidirect
73. C₃×C₁₆	C₃×D₄	direct (N × H)
76. C₂²×C₃²	C₂×C₁₆	semidirect
80. C₃²×C₄	C₂×C₁₆	semidirect
93. C₃×D₄	C₃×C₁₆	direct (N × H)
120. C₁₆	C₃²×D₄	direct (N × H)
123. C₂²×C₃	C₂×C₃×C₁₆	semidirect
127. C₃×C₄	C₂×C₃×C₁₆	semidirect
130. C₃²	D₄×C₁₆	direct (N × H)
132. D₄	C₃²×C₁₆	direct (N × H)
149. C₂²	C₂×C₃²×C₁₆	semidirect
153. C₄	C₂×C₃²×C₁₆	semidirect
156. C₃	C₃×D₄×C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₁₆	2
C₃²×C₈×D₄	2
C₃²×C₄⋊C₁₆	2
C₃²×C₂²⋊C₁₆	2
C₃²×C₄×C₁₆	2
C₃×D₄×C₁₆	3

Subgroups — C32×D4×C16 #11769216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₄×C₁₆ #11769216