Subgroups — #7622644

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 123 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#60191	C₂	1	1	yes	maximal, characteristic
2	#33475	C₂	1	12	yes	maximal, complemented
3	#33474	C₂	1	12	yes	maximal, complemented
4	#33397	C₂	1	3	yes	maximal
5	C₂×#2108	C₂	1	3	yes	maximal
6	#3458	C₄	1	1	yes	characteristic
7	C₂²×M₃₂	C₂²	1	3	yes	—
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	#2961	C₄	1	12	yes	—
10	#2960	C₂²	1	12	yes	—
11	#2905	C₄	1	3	yes	—
12	C₂×#405	C₂²	1	3	yes	—
13	#2183	C₂²	1	48	yes	complemented
14	Q₈×C₁₆	C₂²	1	16	yes	complemented
15	#2151	C₂²	1	24	yes	—
16	#2150	C₂²	1	12	yes	—
17	#2108	C₂²	1	12	yes	—
18	#2108	C₂²	1	12	yes	—
19	C₂×#346	C₂²	1	6	yes	—
20	C₂×C₄×C₁₆	C₂²	1	6	yes	—
21	C₂²×M₁₆	C₂×C₄	1	3	yes	—
22	C₂³×C₈	C₂³	1	1	yes	characteristic
23	#544	C₈	1	12	yes	—
24	#543	C₈	1	4	yes	—
25	#533	C₂×C₄	1	12	yes	—
26	C₂×#117	C₂×C₄	1	3	yes	—
27	C₂×M₃₂	—	2	12	no	—
28	C₂×M₃₂	C₂³	1	12	yes	—
29	C₂²×C₁₆	Q₈	1	8	yes	complemented
30	C₂²×C₁₆	C₂³	1	6	yes	—
31	C₂²×C₁₆	C₂³	1	3	yes	—
32	C₂²×C₁₆	C₂³	1	3	yes	—
33	#446	C₂×C₄	1	48	yes	—
34	C₈×Q₈	C₂³	1	16	yes	—
35	#424	C₂×C₄	1	24	yes	—
36	#423	C₂×C₄	1	12	yes	—
37	#405	C₂×C₄	1	12	yes	—
38	#405	C₂³	1	12	yes	—
39	C₂×#97	C₂×C₄	1	6	yes	—
40	C₂×C₄×C₈	C₂³	1	6	yes	—
41	C₄⋊C₁₆	C₂³	1	48	yes	—
42	#346	C₂³	1	24	yes	—
43	C₄×C₁₆	C₂³	1	12	yes	—
44	C₄×C₁₆	C₂³	1	12	yes	—
45	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×M₁₆	—	2	12	no	—
47	C₂×M₁₆	C₂²×C₄	1	12	yes	—
48	C₂²×C₈	C₂²×C₄	1	6	yes	—
49	C₂²×C₈	C₂²×C₄	1	3	yes	—
50	C₂²×C₈	C₂×Q₈	1	4	yes	—
51	C₂²×C₈	#41	1	4	yes	—
52	C₂²×C₈	C₂⁴	1	3	yes	—
53	C₄⋊Q₈	C₂×C₈	1	24	yes	—
54	#125	C₂×C₈	1	24	yes	—
55	C₂²⋊Q₈	C₂×C₈	1	48	yes	—
56	C₄×Q₈	C₂²×C₄	1	16	yes	—
57	#117	C₂×C₈	1	12	yes	—
58	#117	C₂²×C₄	1	12	yes	—
59	C₂×C₄⋊C₄	C₂×C₈	1	6	yes	—
60	C₂×C₂²⋊C₄	C₂×C₈	1	6	yes	—
61	C₂×C₄²	C₂²×C₄	1	6	yes	—
62	M₃₂	—	2	24	no	—
63	C₂×C₁₆	—	2	12	no	—
64	C₂×C₁₆	C₂×Q₈	1	12	yes	—
65	C₂×C₁₆	C₂×Q₈	1	12	yes	—
66	C₂×C₁₆	C₂⁴	1	12	yes	—
67	C₂×C₁₆	C₂⁴	1	3	yes	—
68	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₈	C₂²×C₄	1	48	yes	—
70	#97	C₂²×C₄	1	24	yes	—
71	C₄×C₈	C₂²×C₄	1	12	yes	—
72	C₄×C₈	C₂⁴	1	12	yes	—
73	C₂⁴	C₂²×C₈	1	1	yes	characteristic
74	C₂×Q₈	C₂²×C₈	1	16	yes	—
75	C₂²×C₄	C₄×Q₈	1	4	yes	—
76	C₂²×C₄	C₂²×C₈	1	12	yes	—
77	C₂²×C₄	C₂²×C₈	1	3	yes	—
78	C₂²×C₄	#131	1	4	yes	—
79	C₂²×C₄	C₂³×C₄	1	3	yes	—
80	M₁₆	—	2	24	no	—
81	C₂×C₈	—	2	12	no	—
82	C₂×C₈	C₄×Q₈	1	12	yes	—
83	C₂×C₈	C₂³×C₄	1	12	yes	—
84	C₂×C₈	C₂³×C₄	1	3	yes	—
85	C₂×C₈	C₂²×Q₈	1	6	yes	—
86	C₂×C₈	C₂×#41	1	6	yes	—
87	C₂×C₈	C₂⁵	1	1	yes	characteristic
88	C₄⋊C₄	C₂²×C₈	1	24	yes	—
89	C₄⋊C₄	C₂²×C₈	1	48	yes	—
90	C₂²⋊C₄	C₂²×C₈	1	24	yes	—
91	C₄²	C₂²×C₈	1	12	yes	—
92	C₄²	C₂³×C₄	1	12	yes	—
93	C₁₆	—	2	12	no	—
94	C₁₆	C₂²×Q₈	1	6	yes	—
95	C₁₆	C₂²×Q₈	1	2	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₈×Q₈	1	4	yes	—
98	C₂³	C₂³×C₈	1	3	yes	—
99	Q₈	—	4	16	no	—
100	C₂×C₄	—	4	12	no	—
101	C₂×C₄	C₈×Q₈	1	12	yes	—
102	C₂×C₄	C₂×C₄×Q₈	1	6	yes	—
103	C₂×C₄	C₂³×C₈	1	24	yes	—
104	C₂×C₄	C₂³×C₈	1	3	yes	—
105	C₂×C₄	C₂×#131	1	6	yes	—
106	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
107	C₈	—	2	12	no	—
108	C₈	C₂×C₄×Q₈	1	6	yes	—
109	C₈	C₂³×Q₈	1	1	yes	characteristic
110	C₈	C₂²×#41	1	1	yes	characteristic
111	C₂²	—	2	6	no	—
112	C₂²	—	4	4	no	—
113	C₂²	C₂×C₈×Q₈	1	6	yes	—
114	C₂²	C₂⁴×C₈	1	1	yes	characteristic
115	C₄	—	2	24	no	—
116	C₄	C₂×C₈×Q₈	1	6	yes	—
117	C₄	C₂²×C₄×Q₈	1	1	yes	characteristic
118	C₄	C₂²×#131	1	1	yes	characteristic
119	C₂	—	2	6	no	—
120	C₂	C₂²×C₈×Q₈	1	1	yes	characteristic
121	C₂	#62624	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ 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. #33475	C₂	semidirect
3. #33474	C₂	semidirect
13. #2183	C₂²	semidirect
14. Q₈×C₁₆	C₂²	semidirect
29. C₂²×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#60191	2
#33475	2
#33474	2
#33397	2
C₂×#2108	2