Subgroups — C₂×#33496 #7622695

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 144 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	#33496	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×C₁₆	C₂	1	8	yes	maximal, complemented
4	C₂×#2162	C₂	1	6	yes	maximal
5	C₂²×C₄×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₈×Q₈	C₄	1	4	yes	—
7	C₂×C₈×Q₈	C₄	1	8	yes	—
8	C₂×C₈×Q₈	C₂²	1	4	yes	—
9	C₂×C₈×Q₈	C₂²	1	8	yes	—
10	C₂²×C₄⋊C₈	C₄	1	3	yes	—
11	C₂²×C₄×C₈	C₂²	1	3	yes	—
12	Q₈×C₁₆	C₂²	1	64	yes	complemented
13	#2162	C₂²	1	48	yes	—
14	C₂×C₄⋊C₁₆	C₂²	1	12	yes	—
15	C₂×C₂²⋊C₁₆	C₂²	1	4	yes	—
16	C₂×C₄×C₁₆	C₂²	1	12	yes	—
17	C₂³×Q₈	C₈	1	1	yes	characteristic
18	C₂³×C₈	C₂×C₄	1	3	yes	—
19	C₂³×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄×Q₈	C₈	1	4	yes	—
21	C₂×C₄×Q₈	C₈	1	8	yes	—
22	C₂×C₄×Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₄×Q₈	C₂×C₄	1	8	yes	—
24	C₂²×C₄⋊C₄	C₈	1	3	yes	—
25	C₂²×C₄²	C₂×C₄	1	3	yes	—
26	C₂²×C₁₆	C₂³	1	8	yes	—
27	C₈×Q₈	C₂×C₄	1	16	yes	—
28	C₈×Q₈	C₂×C₄	1	32	yes	—
29	C₈×Q₈	C₂³	1	16	yes	—
30	C₂×C₄⋊C₈	C₂×C₄	1	12	yes	—
31	C₂×C₄⋊C₈	C₂×C₄	1	24	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	12	yes	—
34	C₂×C₄×C₈	C₂³	1	12	yes	—
35	C₂×C₄×C₈	C₂³	1	6	yes	—
36	C₄⋊C₁₆	C₂³	1	48	yes	—
37	C₂²⋊C₁₆	C₂³	1	16	yes	—
38	C₄×C₁₆	C₂³	1	48	yes	—
39	C₂²×Q₈	C₁₆	1	16	yes	complemented
40	C₂²×Q₈	C₂×C₈	1	4	yes	—
41	C₂²×Q₈	C₂×C₈	1	8	yes	—
42	C₂³×C₄	C₂×C₈	1	3	yes	—
43	C₂³×C₄	C₂×C₈	1	3	yes	—
44	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₈	—	2	4	no	—
46	C₂²×C₈	—	4	6	no	—
47	C₂²×C₈	C₂²×C₄	1	6	yes	—
48	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	2	yes	—
50	C₂²×C₈	C₂²×C₄	1	12	yes	—
51	C₂²×C₈	C₂⁴	1	2	yes	—
52	C₂²×C₈	C₂⁴	1	1	yes	characteristic
53	C₄×Q₈	C₂×C₈	1	32	yes	—
54	C₄×Q₈	C₂×C₈	1	16	yes	—
55	C₄×Q₈	C₂²×C₄	1	16	yes	—
56	C₂×C₄⋊C₄	C₂×C₈	1	12	yes	—
57	C₂×C₄⋊C₄	C₂×C₈	1	24	yes	—
58	C₂×C₄²	C₂×C₈	1	6	yes	—
59	C₂×C₄²	C₂×C₈	1	12	yes	—
60	C₂×C₄²	C₂²×C₄	1	6	yes	—
61	C₂×C₄²	C₂²×C₄	1	12	yes	—
62	C₂×C₁₆	—	2	16	no	—
63	C₂×C₁₆	C₂⁴	1	16	yes	—
64	C₄⋊C₈	C₂²×C₄	1	48	yes	—
65	C₄×C₈	C₂²×C₄	1	24	yes	—
66	C₄×C₈	C₂²×C₄	1	12	yes	—
67	C₄×C₈	C₂⁴	1	12	yes	—
68	C₂⁴	C₂²×C₈	1	1	yes	characteristic
69	C₂×Q₈	C₂×C₁₆	1	32	yes	—
70	C₂×Q₈	C₂×C₁₆	1	64	yes	complemented
71	C₂×Q₈	C₂²×C₈	1	16	yes	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	4	6	no	—
75	C₂²×C₄	C₂×C₁₆	1	24	yes	—
76	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₈	1	2	yes	—
78	C₂²×C₄	C₂²×C₈	1	6	yes	—
79	C₂²×C₄	C₂²×C₈	1	12	yes	—
80	C₂²×C₄	C₂²×C₈	1	6	yes	—
81	C₂²×C₄	C₂²×C₈	1	12	yes	—
82	C₂²×C₄	C₂³×C₄	1	2	yes	—
83	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	12	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	4	12	no	—
89	C₂×C₈	C₂³×C₄	1	2	yes	—
90	C₂×C₈	C₂³×C₄	1	12	yes	—
91	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₈	2₋¹⁺⁴	1	4	yes	—
93	C₂×C₈	C₂⁵	1	1	yes	characteristic
94	C₄⋊C₄	C₂²×C₈	1	48	yes	—
95	C₄²	C₂²×C₈	1	24	yes	—
96	C₄²	C₂²×C₈	1	12	yes	—
97	C₄²	C₂³×C₄	1	12	yes	—
98	C₁₆	—	2	16	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	C₂²×C₁₆	1	4	yes	—
102	C₂³	C₂³×C₈	1	2	yes	—
103	C₂³	C₂³×C₈	1	1	yes	characteristic
104	Q₈	C₂²×C₁₆	1	64	yes	—
105	C₂×C₄	—	2	12	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	12	no	—
111	C₂×C₄	C₂²×C₁₆	1	24	yes	—
112	C₂×C₄	C₂²×C₁₆	1	48	yes	—
113	C₂×C₄	#519	1	2	yes	—
114	C₂×C₄	C₂³×C₈	1	2	yes	—
115	C₂×C₄	C₂³×C₈	1	12	yes	—
116	C₂×C₄	C₂³×C₈	1	12	yes	—
117	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
118	C₂×C₄	#568	1	2	yes	—
119	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	2	12	no	—
122	C₈	#519	1	2	yes	—
123	C₈	C₂×2₋¹⁺⁴	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	8	no	—
127	C₂²	#2982	1	2	yes	—
128	C₂²	C₂³×C₁₆	1	4	yes	—
129	C₂²	C₂³×C₁₆	1	2	yes	—
130	C₂²	#3407	1	2	yes	—
131	C₂²	C₂⁴×C₈	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	12	no	—
134	C₄	#2982	1	2	yes	—
135	C₄	C₂³×C₁₆	1	24	yes	—
136	C₄	C₂×#519	1	1	yes	characteristic
137	C₄	C₂×#568	1	1	yes	characteristic
138	C₂	—	2	4	no	—
139	C₂	#33496	1	4	yes	direct product factor
140	C₂	C₂×#2982	1	1	yes	characteristic
141	C₂	C₂⁴×C₁₆	1	1	yes	characteristic
142	C₂	C₂×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂	C₂⁴×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. #33496	C₂	direct (N × H)
3. C₂×Q₈×C₁₆	C₂	semidirect
12. Q₈×C₁₆	C₂²	semidirect
39. C₂²×Q₈	C₁₆	semidirect
70. C₂×Q₈	C₂×C₁₆	semidirect
139. C₂	#33496	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
#33496	2
C₂×Q₈×C₁₆	2
C₂×#2162	2

Subgroups — C2×#33496 #7622695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#33496 #7622695