Subgroups — C₂²×#2183 #7622639

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#2183	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊C₁₆	C₂	1	2	yes	maximal
5	C₂²×#346	C₂	1	2	yes	maximal
6	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×Q₈	C₄	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	2	yes	—
10	C₂³×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₈×Q₈	C₄	1	12	yes	—
12	C₂×C₈×Q₈	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₈	C₄	1	2	yes	—
14	C₂²×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₈	C₂²	1	2	yes	—
17	#2183	C₂²	1	64	yes	direct product factor
18	C₂×C₄⋊C₁₆	C₂²	1	12	yes	—
19	C₂×C₄⋊C₁₆	C₂²	1	24	yes	—
20	C₂×#346	C₂²	1	24	yes	—
21	C₂×C₄×C₁₆	C₂²	1	12	yes	—
22	C₂³×Q₈	C₈	1	1	yes	characteristic
23	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₈	C₂×C₄	1	2	yes	—
25	C₂³×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄×Q₈	C₈	1	12	yes	—
27	C₂×C₄×Q₈	C₂×C₄	1	12	yes	—
28	C₂²×C₄⋊C₄	C₈	1	2	yes	—
29	C₂²×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₂²×C₄²	C₂×C₄	1	2	yes	—
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	4	no	—
33	C₂²×C₁₆	—	2	8	no	—
34	C₂²×C₁₆	Q₈	1	8	yes	complemented
35	C₂²×C₁₆	C₂³	1	6	yes	—
36	C₂²×C₁₆	C₂³	1	12	yes	—
37	C₂²×C₁₆	C₂³	1	6	yes	—
38	C₈×Q₈	C₂×C₄	1	48	yes	—
39	C₈×Q₈	C₂³	1	16	yes	—
40	C₂×C₄⋊C₈	C₂×C₄	1	24	yes	—
41	C₂×C₄⋊C₈	C₂×C₄	1	12	yes	—
42	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
44	C₂×C₄×C₈	C₂³	1	6	yes	—
45	C₂×C₄×C₈	C₂³	1	12	yes	—
46	C₄⋊C₁₆	C₂³	1	16	yes	—
47	C₄⋊C₁₆	C₂³	1	32	yes	—
48	#346	C₂³	1	32	yes	—
49	C₄×C₁₆	C₂³	1	16	yes	—
50	C₂²×Q₈	—	2	8	no	—
51	C₂²×Q₈	C₂×C₈	1	12	yes	—
52	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
53	C₂³×C₄	C₂×C₈	1	2	yes	—
54	C₂³×C₄	C₂×C₈	1	2	yes	—
55	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	—	2	4	no	—
58	C₂²×C₈	—	2	8	no	—
59	C₂²×C₈	C₂²×C₄	1	6	yes	—
60	C₂²×C₈	C₂²×C₄	1	12	yes	—
61	C₂²×C₈	C₂²×C₄	1	3	yes	—
62	C₂²×C₈	C₂×Q₈	1	4	yes	—
63	C₂²×C₈	#41	1	4	yes	—
64	C₂²×C₈	C₂⁴	1	3	yes	—
65	C₄×Q₈	C₂×C₈	1	48	yes	—
66	C₄×Q₈	C₂²×C₄	1	16	yes	—
67	C₂×C₄⋊C₄	C₂×C₈	1	24	yes	—
68	C₂×C₄⋊C₄	C₂×C₈	1	12	yes	—
69	C₂×C₄²	C₂×C₈	1	12	yes	—
70	C₂×C₄²	C₂×C₈	1	6	yes	—
71	C₂×C₄²	C₂²×C₄	1	12	yes	—
72	C₂×C₄²	C₂²×C₄	1	6	yes	—
73	C₂×C₁₆	—	2	12	no	—
74	C₂×C₁₆	—	2	24	no	—
75	C₂×C₁₆	C₂×Q₈	1	24	yes	complemented
76	C₂×C₁₆	C₂⁴	1	4	yes	—
77	C₂×C₁₆	C₂⁴	1	8	yes	—
78	C₂×C₁₆	C₂⁴	1	4	yes	—
79	C₄⋊C₈	C₂²×C₄	1	32	yes	—
80	C₄⋊C₈	C₂²×C₄	1	16	yes	—
81	C₄×C₈	C₂²×C₄	1	12	yes	—
82	C₄×C₈	C₂²×C₄	1	24	yes	—
83	C₄×C₈	C₂⁴	1	4	yes	—
84	C₄×C₈	C₂⁴	1	8	yes	—
85	C₂⁴	C₂²×C₈	1	1	yes	characteristic
86	C₂×Q₈	—	2	48	no	—
87	C₂×Q₈	C₂²×C₈	1	16	yes	—
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	—	2	8	no	—
91	C₂²×C₄	M₃₂	1	8	yes	—
92	C₂²×C₄	C₄×Q₈	1	4	yes	—
93	C₂²×C₄	C₂²×C₈	1	12	yes	—
94	C₂²×C₄	C₂²×C₈	1	6	yes	—
95	C₂²×C₄	C₂²×C₈	1	12	yes	—
96	C₂²×C₄	C₂²×C₈	1	3	yes	—
97	C₂²×C₄	C₂²×C₈	1	6	yes	—
98	C₂²×C₄	#131	1	4	yes	—
99	C₂²×C₄	C₂³×C₄	1	3	yes	—
100	C₂×C₈	—	2	12	no	—
101	C₂×C₈	—	2	24	no	—
102	C₂×C₈	C₄×Q₈	1	12	yes	—
103	C₂×C₈	C₂³×C₄	1	4	yes	—
104	C₂×C₈	C₂³×C₄	1	8	yes	—
105	C₂×C₈	C₂³×C₄	1	3	yes	—
106	C₂×C₈	C₂²×Q₈	1	6	yes	—
107	C₂×C₈	C₂×#41	1	6	yes	—
108	C₂×C₈	C₂⁵	1	1	yes	characteristic
109	C₄⋊C₄	C₂²×C₈	1	32	yes	—
110	C₄⋊C₄	C₂²×C₈	1	16	yes	—
111	C₄²	C₂²×C₈	1	24	yes	—
112	C₄²	C₂²×C₈	1	12	yes	—
113	C₄²	C₂³×C₄	1	8	yes	—
114	C₄²	C₂³×C₄	1	4	yes	—
115	C₁₆	—	2	4	no	—
116	C₁₆	—	2	8	no	—
117	C₁₆	C₂²×Q₈	1	8	yes	complemented
118	C₂³	C₈×Q₈	1	4	yes	—
119	C₂³	C₂×M₃₂	1	4	yes	—
120	C₂³	#504	1	4	yes	—
121	C₂³	C₂³×C₈	1	3	yes	—
122	Q₈	—	2	32	no	—
123	C₂×C₄	—	2	24	no	—
124	C₂×C₄	—	2	24	no	—
125	C₂×C₄	—	2	12	no	—
126	C₂×C₄	C₈×Q₈	1	12	yes	—
127	C₂×C₄	C₂×M₃₂	1	24	yes	—
128	C₂×C₄	C₂×C₄×Q₈	1	6	yes	—
129	C₂×C₄	C₂³×C₈	1	8	yes	—
130	C₂×C₄	C₂³×C₈	1	4	yes	—
131	C₂×C₄	C₂³×C₈	1	8	yes	—
132	C₂×C₄	C₂³×C₈	1	4	yes	—
133	C₂×C₄	C₂³×C₈	1	3	yes	—
134	C₂×C₄	C₂×#131	1	6	yes	—
135	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₈	—	2	8	no	—
138	C₈	C₂×C₄×Q₈	1	6	yes	—
139	C₈	C₂³×Q₈	1	1	yes	characteristic
140	C₈	C₂²×#41	1	1	yes	characteristic
141	C₂²	#2183	1	16	yes	direct product factor
142	C₂²	C₂×C₈×Q₈	1	6	yes	—
143	C₂²	C₂²×M₃₂	1	6	yes	—
144	C₂²	C₂×#504	1	6	yes	—
145	C₂²	C₂⁴×C₈	1	1	yes	characteristic
146	C₄	—	2	8	no	—
147	C₄	—	2	8	no	—
148	C₄	—	2	4	no	—
149	C₄	C₂×C₈×Q₈	1	6	yes	—
150	C₄	C₂²×M₃₂	1	8	yes	—
151	C₄	C₂²×C₄×Q₈	1	1	yes	characteristic
152	C₄	C₂²×#131	1	1	yes	characteristic
153	C₂	C₂×#2183	1	12	yes	direct product factor
154	C₂	C₂²×C₈×Q₈	1	1	yes	characteristic
155	C₂	C₂³×M₃₂	1	1	yes	characteristic
156	C₂	C₂²×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂³
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle, prime radical	C₂⁴	C₂²×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×#2183	C₂	direct (N × H)
17. #2183	C₂²	direct (N × H)
34. C₂²×C₁₆	Q₈	semidirect
75. C₂×C₁₆	C₂×Q₈	semidirect
117. C₁₆	C₂²×Q₈	semidirect
141. C₂²	#2183	direct (N × H)
153. C₂	C₂×#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
C₂²×C₈×Q₈	2
C₂×#2183	2
C₂²×C₄⋊C₁₆	2
C₂²×#346	2
C₂²×C₄×C₁₆	2

Subgroups — C22×#2183 #7622639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#2183 #7622639