Subgroups — C₂²×#2164 #7622544

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 181 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#2164	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₁₆	C₂	1	2	yes	maximal
5	C₂²×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×M₁₆	C₄	1	1	yes	characteristic
8	C₂⁴×C₈	C₂²	1	1	yes	characteristic
9	C₂²×#117	C₄	1	1	yes	characteristic
10	C₂³×C₁₆	C₂²	1	2	yes	—
11	C₂³×C₁₆	C₂²	1	2	yes	—
12	C₂×#405	C₄	1	12	yes	—
13	C₂×#405	C₂²	1	12	yes	—
14	C₂²×#97	C₄	1	1	yes	characteristic
15	C₂²×#97	C₄	1	1	yes	characteristic
16	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
18	#2164	C₂²	1	64	yes	direct product factor
19	C₂×C₄⋊C₁₆	C₂²	1	24	yes	complemented
20	C₂×C₂²⋊C₁₆	C₂²	1	24	yes	—
21	C₂×#346	C₂²	1	12	yes	complemented
22	C₂×C₄×C₁₆	C₂²	1	12	yes	complemented
23	C₂⁴×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×M₁₆	—	2	8	no	—
25	C₂²×M₁₆	C₂×C₄	1	12	yes	—
26	C₂³×C₈	—	2	8	no	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	C₂×C₄	1	6	yes	—
30	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	6	yes	—
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	C₈	1	12	yes	—
34	C₂×#117	C₂×C₄	1	12	yes	—
35	C₂²×C₄⋊C₄	C₈	1	2	yes	—
36	C₂²×C₂²⋊C₄	C₈	1	2	yes	—
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₆	—	2	8	no	—
40	C₂²×C₁₆	—	2	8	no	—
41	C₂²×C₁₆	C₂³	1	12	yes	—
42	C₂²×C₁₆	C₂³	1	12	yes	—
43	#405	C₂×C₄	1	48	yes	—
44	#405	C₂³	1	16	yes	—
45	C₂×#97	C₂×C₄	1	12	yes	—
46	C₂×#97	C₂×C₄	1	12	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
49	C₂×C₄×C₈	C₂³	1	6	yes	—
50	C₂×C₄×C₈	C₂³	1	6	yes	—
51	C₄⋊C₁₆	C₂³	1	32	yes	complemented
52	C₂²⋊C₁₆	C₂³	1	32	yes	—
53	#346	C₂³	1	16	yes	complemented
54	C₄×C₁₆	C₂³	1	16	yes	complemented
55	C₂⁵	C₂×C₈	1	1	yes	characteristic
56	C₂³×C₄	—	2	4	no	—
57	C₂³×C₄	—	2	4	no	—
58	C₂³×C₄	C₂×C₈	1	2	yes	—
59	C₂³×C₄	C₂×C₈	1	2	yes	—
60	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
61	C₂³×C₄	C₂×C₈	1	6	yes	—
62	C₂³×C₄	C₂²×C₄	1	6	yes	—
63	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×M₁₆	—	2	48	no	—
65	C₂×M₁₆	C₂²×C₄	1	16	yes	—
66	C₂²×C₈	—	2	4	no	—
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	—	2	24	no	—
69	C₂²×C₈	—	2	24	no	—
70	C₂²×C₈	M₁₆	1	8	yes	—
71	C₂²×C₈	C₂²×C₄	1	6	yes	—
72	C₂²×C₈	C₂²×C₄	1	12	yes	—
73	C₂²×C₈	C₂²×C₄	1	6	yes	—
74	C₂²×C₈	C₂²×C₄	1	3	yes	—
75	C₂²×C₈	C₂²×C₄	1	6	yes	—
76	C₂²×C₈	#41	1	8	yes	—
77	C₂²×C₈	C₂⁴	1	4	yes	—
78	C₂²×C₈	C₂⁴	1	3	yes	—
79	#117	C₂×C₈	1	48	yes	—
80	#117	C₂²×C₄	1	16	yes	—
81	C₂×C₄⋊C₄	C₂×C₈	1	24	yes	—
82	C₂×C₂²⋊C₄	C₂×C₈	1	24	yes	—
83	C₂×C₄²	C₂×C₈	1	6	yes	—
84	C₂×C₄²	C₂×C₈	1	6	yes	—
85	C₂×C₄²	C₂²×C₄	1	6	yes	—
86	C₂×C₄²	C₂²×C₄	1	6	yes	—
87	C₂×C₁₆	—	2	24	no	—
88	C₂×C₁₆	—	2	24	no	—
89	C₂×C₁₆	C₂⁴	1	8	yes	—
90	C₂×C₁₆	C₂⁴	1	8	yes	—
91	#97	C₂²×C₄	1	16	yes	—
92	#97	C₂²×C₄	1	16	yes	—
93	C₄×C₈	C₂²×C₄	1	12	yes	—
94	C₄×C₈	C₂²×C₄	1	12	yes	—
95	C₄×C₈	C₂⁴	1	4	yes	—
96	C₄×C₈	C₂⁴	1	4	yes	—
97	C₂⁴	—	2	4	no	—
98	C₂⁴	—	2	8	no	—
99	C₂⁴	C₂²×C₈	1	6	yes	—
100	C₂⁴	C₂²×C₈	1	1	yes	characteristic
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	8	no	—
103	C₂²×C₄	—	2	8	no	—
104	C₂²×C₄	—	2	12	no	—
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	—	2	24	no	—
107	C₂²×C₄	#117	1	4	yes	—
108	C₂²×C₄	C₂²×C₈	1	12	yes	—
109	C₂²×C₄	C₂²×C₈	1	12	yes	—
110	C₂²×C₄	C₂²×C₈	1	6	yes	—
111	C₂²×C₄	C₂²×C₈	1	12	yes	—
112	C₂²×C₄	C₂²×C₈	1	3	yes	—
113	C₂²×C₄	C₂×M₁₆	1	4	yes	—
114	C₂²×C₄	C₂³×C₄	1	4	yes	—
115	C₂²×C₄	C₂³×C₄	1	3	yes	—
116	M₁₆	—	2	32	no	—
117	C₂×C₈	—	2	12	no	—
118	C₂×C₈	—	2	12	no	—
119	C₂×C₈	—	2	24	no	—
120	C₂×C₈	—	2	8	no	—
121	C₂×C₈	#117	1	12	yes	—
122	C₂×C₈	C₂×M₁₆	1	24	yes	—
123	C₂×C₈	C₂³×C₄	1	4	yes	—
124	C₂×C₈	C₂³×C₄	1	4	yes	—
125	C₂×C₈	C₂³×C₄	1	3	yes	—
126	C₂×C₈	C₂³×C₄	1	4	yes	—
127	C₂×C₈	C₂×#41	1	12	yes	—
128	C₂×C₈	C₂⁵	1	1	yes	characteristic
129	C₄⋊C₄	C₂²×C₈	1	32	yes	—
130	C₂²⋊C₄	C₂²×C₈	1	32	yes	—
131	C₄²	C₂²×C₈	1	12	yes	—
132	C₄²	C₂²×C₈	1	12	yes	—
133	C₄²	C₂³×C₄	1	4	yes	—
134	C₄²	C₂³×C₄	1	4	yes	—
135	C₁₆	—	2	8	no	—
136	C₁₆	—	2	8	no	—
137	C₂³	—	2	24	no	—
138	C₂³	—	2	12	no	—
139	C₂³	—	4	16	no	—
140	C₂³	#431	1	4	yes	—
141	C₂³	#504	1	8	yes	—
142	C₂³	C₂³×C₈	1	4	yes	—
143	C₂³	C₂³×C₈	1	3	yes	—
144	C₂×C₄	—	2	24	no	—
145	C₂×C₄	—	2	24	no	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	24	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	#431	1	12	yes	—
151	C₂×C₄	C₂×#117	1	6	yes	—
152	C₂×C₄	C₂³×C₈	1	8	yes	—
153	C₂×C₄	C₂³×C₈	1	8	yes	—
154	C₂×C₄	C₂³×C₈	1	4	yes	—
155	C₂×C₄	C₂³×C₈	1	3	yes	—
156	C₂×C₄	C₂²×M₁₆	1	6	yes	—
157	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	2	4	no	—
160	C₈	C₂×#117	1	6	yes	—
161	C₈	C₂²×M₁₆	1	8	yes	—
162	C₈	C₂²×#41	1	2	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	4	24	no	—
166	C₂²	#2164	1	16	yes	direct product factor
167	C₂²	C₂×#431	1	6	yes	—
168	C₂²	C₂×#504	1	12	yes	—
169	C₂²	C₂⁴×C₈	1	1	yes	characteristic
170	C₄	—	2	8	no	—
171	C₄	—	2	8	no	—
172	C₄	—	2	4	no	—
173	C₄	C₂×#431	1	6	yes	—
174	C₄	C₂²×#117	1	1	yes	characteristic
175	C₄	C₂³×M₁₆	1	1	yes	characteristic
176	C₂	—	4	4	no	—
177	C₂	C₂×#2164	1	12	yes	direct product factor
178	C₂	C₂²×#431	1	1	yes	characteristic
179	C₂	C₂²×#504	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂⁴	C₂²×C₈
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#2164	C₂	direct (N × H)
3. C₂²×C₄⋊C₁₆	C₂	semidirect
5. C₂²×#346	C₂	semidirect
6. C₂²×C₄×C₁₆	C₂	semidirect
18. #2164	C₂²	direct (N × H)
19. C₂×C₄⋊C₁₆	C₂²	semidirect
21. C₂×#346	C₂²	semidirect
22. C₂×C₄×C₁₆	C₂²	semidirect
51. C₄⋊C₁₆	C₂³	semidirect
53. #346	C₂³	semidirect
54. C₄×C₁₆	C₂³	semidirect
166. C₂²	#2164	direct (N × H)
177. C₂	C₂×#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
C₂²×#405	2
C₂×#2164	2
C₂²×C₄⋊C₁₆	2
C₂²×C₂²⋊C₁₆	2
C₂²×#346	2
C₂²×C₄×C₁₆	2

Subgroups — C22×#2164 #7622544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#2164 #7622544