Subgroups — #512785

Order 512.

316 conjugacy classes of subgroups.

The subgroup lattice has 122 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	8	yes	maximal, complemented
3	#13628	C₂	1	2	yes	maximal
4	#13627	C₂	1	1	yes	maximal, characteristic
5	#13549	C₂	1	2	yes	maximal
6	C₂×#1397	C₂	1	1	yes	maximal, characteristic
7	#2960	C₄	1	1	yes	characteristic
8	C₂×M₆₄	C₂²	1	2	yes	—
9	C₂²×C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	#2183	C₄	1	4	yes	—
12	Q₈×C₁₆	C₂²	1	4	yes	—
13	#2151	C₄	1	2	yes	—
14	#2150	C₄	1	1	yes	characteristic
15	#2108	C₂²	1	2	yes	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₃₂	C₂²	1	4	yes	—
18	C₄⋊C₃₂	C₂²	1	8	yes	—
19	#1397	C₂²	1	4	yes	—
20	#1397	C₂²	1	4	yes	—
21	C₄×C₃₂	C₂²	1	4	yes	—
22	#533	C₈	1	1	yes	characteristic
23	C₂×M₃₂	C₂×C₄	1	2	yes	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂³	1	1	yes	characteristic
26	#446	C₈	1	4	yes	—
27	C₈×Q₈	C₂×C₄	1	4	yes	—
28	#424	C₈	1	2	yes	—
29	#423	C₈	1	1	yes	characteristic
30	#405	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	M₆₄	—	2	4	no	—
33	C₂×C₃₂	—	2	2	no	—
34	C₂×C₃₂	Q₈	1	4	yes	complemented
35	C₂×C₃₂	C₂³	1	4	yes	—
36	C₂×C₃₂	C₂³	1	2	yes	—
37	C₂×C₃₂	C₂³	1	2	yes	—
38	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
39	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
40	#346	C₂×C₄	1	4	yes	—
41	C₄×C₁₆	C₂×C₄	1	2	yes	—
42	C₄×C₁₆	C₂³	1	2	yes	—
43	C₄×C₁₆	C₂³	1	4	yes	—
44	C₂×M₁₆	C₂×C₈	1	2	yes	—
45	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊Q₈	C₁₆	1	2	yes	—
48	#125	C₁₆	1	2	yes	—
49	C₂²⋊Q₈	C₁₆	1	4	yes	—
50	C₄×Q₈	C₂×C₈	1	4	yes	—
51	#117	C₂×C₈	1	2	yes	—
52	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
53	M₃₂	—	2	4	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	C₂²×C₄	1	2	yes	—
56	C₂×C₁₆	C₂²×C₄	1	4	yes	—
57	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₁₆	C₂×Q₈	1	2	yes	—
59	C₂×C₁₆	#41	1	2	yes	—
60	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₈	C₂×C₈	1	4	yes	—
62	C₄⋊C₈	C₂×C₈	1	8	yes	—
63	#97	C₂×C₈	1	4	yes	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₄×C₈	C₂²×C₄	1	2	yes	—
66	C₄×C₈	C₂²×C₄	1	4	yes	—
67	C₃₂	—	2	2	no	—
68	C₃₂	—	2	4	no	—
69	C₃₂	C₂×Q₈	1	4	yes	—
70	C₂×Q₈	C₂×C₁₆	1	4	yes	—
71	C₂²×C₄	C₂×C₁₆	1	2	yes	—
72	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	C₄×Q₈	1	2	yes	—
76	C₂×C₈	C₂²×C₈	1	2	yes	—
77	C₂×C₈	C₂²×C₈	1	4	yes	—
78	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₈	#131	1	2	yes	—
80	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
81	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
82	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
83	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
84	C₂²⋊C₄	C₂×C₁₆	1	4	yes	—
85	C₄²	C₂×C₁₆	1	1	yes	characteristic
86	C₄²	C₂×C₁₆	1	1	yes	characteristic
87	C₄²	C₂²×C₈	1	2	yes	—
88	C₄²	C₂²×C₈	1	4	yes	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	2	4	no	—
91	C₁₆	C₄×Q₈	1	2	yes	—
92	C₁₆	C₂²×Q₈	1	1	yes	characteristic
93	C₁₆	C₂×#41	1	1	yes	characteristic
94	C₂³	C₂²×C₁₆	1	1	yes	characteristic
95	Q₈	—	4	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	C₈×Q₈	1	2	yes	—
98	C₂×C₄	C₂²×C₁₆	1	2	yes	—
99	C₂×C₄	C₂²×C₁₆	1	2	yes	—
100	C₂×C₄	C₂²×C₁₆	1	8	yes	—
101	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
102	C₂×C₄	#504	1	2	yes	—
103	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	2	4	no	—
106	C₈	C₈×Q₈	1	2	yes	—
107	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
108	C₈	C₂×#131	1	1	yes	characteristic
109	C₂²	—	2	2	no	—
110	C₂²	Q₈×C₁₆	1	2	yes	—
111	C₂²	C₂³×C₁₆	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	Q₈×C₁₆	1	2	yes	—
116	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
117	C₄	C₂×#504	1	1	yes	characteristic
118	C₂	—	2	2	no	—
119	C₂	C₂×Q₈×C₁₆	1	1	yes	characteristic
120	C₂	#33974	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ 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
34. 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
#13628	2
#13627	2
#13549	2
C₂×#1397	2