Subgroups — #512784

Order 512.

324 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33474	C₂	1	1	yes	maximal, characteristic
2	#13660	C₂	1	4	yes	maximal, complemented
3	Q₈×C₃₂	C₂	1	4	yes	maximal, complemented
4	#13628	C₂	1	2	yes	maximal
5	#13627	C₂	1	1	yes	maximal, characteristic
6	#13549	C₂	1	2	yes	maximal
7	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
8	#2960	C₄	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	2	yes	—
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	#2183	C₄	1	4	yes	—
13	Q₈×C₁₆	C₂²	1	4	yes	—
14	#2151	C₄	1	2	yes	—
15	#2150	C₄	1	1	yes	characteristic
16	#2108	C₂²	1	2	yes	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₄⋊C₃₂	C₂²	1	4	yes	—
19	C₄⋊C₃₂	C₂²	1	8	yes	—
20	#1397	C₂²	1	4	yes	—
21	C₄×C₃₂	C₂²	1	2	yes	—
22	C₄×C₃₂	C₂²	1	2	yes	—
23	C₄×C₃₂	C₂²	1	4	yes	—
24	#533	C₈	1	1	yes	characteristic
25	C₂×M₃₂	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂³	1	1	yes	characteristic
28	#446	C₈	1	4	yes	—
29	C₈×Q₈	C₂×C₄	1	4	yes	—
30	#424	C₈	1	2	yes	—
31	#423	C₈	1	1	yes	characteristic
32	#405	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	M₆₄	—	2	4	no	—
35	C₂×C₃₂	—	2	2	no	—
36	C₂×C₃₂	Q₈	1	4	yes	complemented
37	C₂×C₃₂	C₂³	1	2	yes	—
38	C₂×C₃₂	C₂³	1	4	yes	—
39	C₂×C₃₂	C₂³	1	1	yes	characteristic
40	C₂×C₃₂	C₂³	1	1	yes	characteristic
41	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
42	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
43	#346	C₂×C₄	1	4	yes	—
44	C₄×C₁₆	C₂×C₄	1	2	yes	—
45	C₄×C₁₆	C₂³	1	2	yes	—
46	C₄×C₁₆	C₂³	1	4	yes	—
47	C₂×M₁₆	C₂×C₈	1	2	yes	—
48	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₄⋊Q₈	C₁₆	1	2	yes	—
51	#125	C₁₆	1	2	yes	—
52	C₂²⋊Q₈	C₁₆	1	4	yes	—
53	C₄×Q₈	C₂×C₈	1	4	yes	—
54	#117	C₂×C₈	1	2	yes	—
55	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
56	M₃₂	—	2	4	no	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	C₂²×C₄	1	2	yes	—
59	C₂×C₁₆	C₂²×C₄	1	4	yes	—
60	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₁₆	C₂×Q₈	1	2	yes	—
62	C₂×C₁₆	#41	1	2	yes	—
63	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₈	C₂×C₈	1	4	yes	—
65	C₄⋊C₈	C₂×C₈	1	8	yes	—
66	#97	C₂×C₈	1	4	yes	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	C₂²×C₄	1	2	yes	—
69	C₄×C₈	C₂²×C₄	1	4	yes	—
70	C₃₂	—	2	2	no	—
71	C₃₂	—	2	4	no	—
72	C₃₂	C₂×Q₈	1	2	yes	—
73	C₃₂	C₂×Q₈	1	2	yes	—
74	C₂×Q₈	C₂×C₁₆	1	4	yes	—
75	C₂²×C₄	C₂×C₁₆	1	2	yes	—
76	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	C₄×Q₈	1	2	yes	—
80	C₂×C₈	C₂²×C₈	1	2	yes	—
81	C₂×C₈	C₂²×C₈	1	4	yes	—
82	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₈	#131	1	2	yes	—
84	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
85	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
86	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
87	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
88	C₂²⋊C₄	C₂×C₁₆	1	4	yes	—
89	C₄²	C₂×C₁₆	1	1	yes	characteristic
90	C₄²	C₂×C₁₆	1	1	yes	characteristic
91	C₄²	C₂²×C₈	1	2	yes	—
92	C₄²	C₂²×C₈	1	4	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	4	no	—
95	C₁₆	C₄×Q₈	1	2	yes	—
96	C₁₆	C₂²×Q₈	1	1	yes	characteristic
97	C₁₆	C₂×#41	1	1	yes	characteristic
98	C₂³	C₂²×C₁₆	1	1	yes	characteristic
99	Q₈	—	2	8	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	C₈×Q₈	1	2	yes	—
102	C₂×C₄	C₂²×C₁₆	1	2	yes	—
103	C₂×C₄	C₂²×C₁₆	1	8	yes	—
104	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₁₆	1	2	yes	—
106	C₂×C₄	#504	1	2	yes	—
107	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	2	4	no	—
110	C₈	C₈×Q₈	1	2	yes	—
111	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
112	C₈	C₂×#131	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×C₁₆	1	2	yes	—
115	C₂²	C₂³×C₁₆	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	8	no	—
118	C₄	Q₈×C₁₆	1	2	yes	—
119	C₄	#2258	1	4	yes	—
120	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
121	C₄	C₂×#504	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	C₂×Q₈×C₁₆	1	1	yes	characteristic
124	C₂	C₂×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂	C₂³
commutator, socle	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13660	C₂	semidirect
3. Q₈×C₃₂	C₂	semidirect
36. C₂×C₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33474	2
#13660	2
Q₈×C₃₂	2
#13628	2
#13627	2
#13549	2
C₂×C₄×C₃₂	2