Subgroups — #512707

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₆₄	C₂	1	2	yes	maximal
2	C₂×#2108	C₂	1	1	yes	maximal, characteristic
3	#13628	C₂	1	8	yes	maximal, complemented
4	#13627	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	C₄	1	1	yes	characteristic
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#405	C₄	1	1	yes	characteristic
8	C₂×M₆₄	—	2	8	no	—
9	C₂×M₆₄	C₂²	1	4	yes	—
10	C₂×M₆₄	C₂²	1	4	yes	—
11	C₂²×C₃₂	C₂²	1	4	yes	—
12	#2108	C₄	1	4	yes	—
13	#2108	C₂²	1	4	yes	—
14	C₂×#346	C₄	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	C₄⋊C₃₂	C₂²	1	16	yes	complemented
17	C₂²×M₁₆	C₈	1	1	yes	characteristic
18	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×#117	C₈	1	1	yes	characteristic
20	C₂×M₃₂	—	2	4	no	—
21	C₂×M₃₂	C₂×C₄	1	4	yes	—
22	C₂²×C₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	4	yes	—
26	C₂²×C₁₆	Q₈	1	4	yes	—
27	C₂²×C₁₆	C₂³	1	1	yes	characteristic
28	C₂²×C₁₆	C₂³	1	2	yes	—
29	#405	C₈	1	4	yes	—
30	#405	C₂×C₄	1	4	yes	—
31	C₂×#97	C₈	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	M₆₄	—	2	8	no	—
34	M₆₄	—	2	8	no	—
35	C₂×C₃₂	—	2	8	no	—
36	C₂×C₃₂	C₂³	1	8	yes	—
37	#346	C₂×C₄	1	8	yes	—
38	C₄×C₁₆	C₂×C₄	1	4	yes	—
39	C₄×C₁₆	C₂³	1	4	yes	—
40	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂×M₁₆	—	2	4	no	—
42	C₂×M₁₆	C₂×C₈	1	4	yes	—
43	C₂²×C₈	C₄⋊C₄	1	4	yes	—
44	C₂²×C₈	C₂×C₈	1	2	yes	—
45	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₈	C₂×C₈	1	2	yes	—
47	C₂²×C₈	M₁₆	1	4	yes	—
48	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	2	yes	—
50	#117	C₁₆	1	4	yes	—
51	#117	C₂×C₈	1	4	yes	—
52	C₂×C₄⋊C₄	C₁₆	1	2	yes	—
53	C₂×C₂²⋊C₄	C₁₆	1	2	yes	—
54	C₂×C₄²	C₂×C₈	1	2	yes	—
55	M₃₂	—	2	8	no	—
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
58	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
59	C₂×C₁₆	C₂²×C₄	1	4	yes	—
60	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×C₄	1	2	yes	—
62	C₂×C₁₆	C₂×D₄	1	4	yes	—
63	C₂×C₁₆	C₂×D₄	1	2	yes	—
64	C₂×C₁₆	C₂×Q₈	1	2	yes	—
65	C₂×C₁₆	C₂×Q₈	1	4	yes	—
66	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
67	#97	C₂×C₈	1	8	yes	—
68	C₄×C₈	C₂×C₈	1	4	yes	—
69	C₄×C₈	C₂²×C₄	1	4	yes	—
70	C₃₂	—	2	8	no	—
71	C₂⁴	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×C₄	C₄⋊C₈	1	4	yes	—
73	C₂²×C₄	C₂×C₁₆	1	4	yes	—
74	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₁₆	1	2	yes	—
76	C₂²×C₄	M₃₂	1	4	yes	—
77	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
78	C₂²×C₄	C₂²×C₈	1	2	yes	—
79	M₁₆	—	2	8	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₄⋊C₈	1	4	yes	—
82	C₂×C₈	C₄⋊C₈	1	8	yes	—
83	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₈	C₂²×C₈	1	4	yes	—
86	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₈	C₂²×C₈	1	2	yes	—
88	C₂×C₈	C₂×M₁₆	1	2	yes	—
89	C₂×C₈	C₂×M₁₆	1	4	yes	—
90	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
91	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
92	C₂²⋊C₄	C₂×C₁₆	1	8	yes	—
93	C₄²	C₂×C₁₆	1	4	yes	—
94	C₄²	C₂²×C₈	1	4	yes	—
95	C₁₆	—	2	4	no	—
96	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
97	C₁₆	C₂×C₄⋊C₄	1	4	yes	—
98	C₁₆	C₂²×D₄	1	1	yes	characteristic
99	C₁₆	C₂²×Q₈	1	1	yes	characteristic
100	C₂³	—	4	2	no	—
101	C₂³	C₄⋊C₁₆	1	4	yes	—
102	C₂³	C₂²×C₁₆	1	2	yes	—
103	C₂³	C₂²×C₁₆	1	1	yes	characteristic
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
106	C₂×C₄	C₄⋊C₁₆	1	8	yes	—
107	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
108	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
109	C₂×C₄	C₂²×C₁₆	1	8	yes	—
110	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
111	C₂×C₄	C₂²×C₁₆	1	2	yes	—
112	C₂×C₄	C₂×M₃₂	1	2	yes	—
113	C₂×C₄	C₂×M₃₂	1	4	yes	—
114	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	C₂×C₄⋊C₈	1	2	yes	—
117	C₈	C₂×C₄⋊C₈	1	4	yes	—
118	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₈	C₂²×M₁₆	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	—	2	4	no	—
122	C₂²	—	4	4	no	—
123	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
124	C₂²	C₂×C₄⋊C₁₆	1	4	yes	—
125	C₂²	C₂³×C₁₆	1	1	yes	characteristic
126	C₄	—	2	8	no	—
127	C₄	C₂×C₄⋊C₁₆	1	2	yes	—
128	C₄	C₂×C₄⋊C₁₆	1	4	yes	—
129	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
130	C₄	C₂²×M₃₂	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	2	4	no	—
133	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic
134	C₂	#33974	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂⁴	C₂×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
3. #13628	C₂	semidirect
4. #13627	C₂	semidirect
16. C₄⋊C₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₆₄	2
C₂×#2108	2
#13628	2
#13627	2