Subgroups — #151985

Order 512.

238 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1397	C₂	1	1	yes	maximal, characteristic
3	C₂×#1397	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#7465	C₂	1	1	yes	maximal, characteristic
7	#7465	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
17	C₄⋊C₃₂	C₄	1	4	yes	complemented
18	#1397	—	2	2	no	—
19	#1397	C₄	1	4	yes	complemented
20	C₄×C₃₂	—	2	2	no	—
21	#1374	C₄	1	1	yes	characteristic
22	#1374	C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄	C₈	1	1	yes	characteristic
32	C₂×C₃₂	—	2	1	no	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	—	2	1	no	—
35	C₂×C₃₂	—	2	1	no	—
36	C₂×C₃₂	—	2	1	no	—
37	C₂×C₃₂	—	2	1	no	—
38	C₂×C₃₂	C₂×C₄	1	2	yes	—
39	C₂×C₃₂	C₂×C₄	1	2	yes	—
40	C₂×C₃₂	C₂×C₄	1	2	yes	—
41	C₂×C₃₂	C₂×C₄	1	2	yes	—
42	C₂×C₃₂	D₄	1	2	yes	—
43	C₂×C₃₂	Q₈	1	2	yes	—
44	C₄⋊C₁₆	C₈	1	4	yes	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	C₂×C₄	1	2	yes	—
50	C₄×C₁₆	C₂×C₄	1	2	yes	—
51	#336	C₈	1	1	yes	characteristic
52	#336	C₈	1	1	yes	characteristic
53	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄	C₁₆	1	2	yes	—
58	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
61	C₂×C₁₆	—	2	1	no	—
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	2	1	no	—
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	C₄²	1	2	yes	—
68	C₂×C₁₆	C₂×C₈	1	2	yes	—
69	C₂×C₁₆	C₂×C₈	1	2	yes	—
70	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
73	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₁₆	#41	1	1	yes	characteristic
75	C₂×C₁₆	#41	1	1	yes	characteristic
76	C₄⋊C₈	C₁₆	1	4	yes	—
77	C₄×C₈	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₄²	1	2	yes	—
82	C₄×C₈	C₂×C₈	1	2	yes	—
83	#95	C₁₆	1	2	yes	—
84	C₃₂	—	2	2	no	—
85	C₃₂	—	2	2	no	—
86	C₃₂	—	2	2	no	—
87	C₃₂	C₄⋊C₄	1	4	yes	complemented
88	C₂²×C₄	C₂×C₁₆	1	2	yes	—
89	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
91	C₂²×C₄	C₂×C₁₆	1	2	yes	—
92	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	C₄×C₈	1	2	yes	—
100	C₂×C₈	C₂×C₁₆	1	2	yes	—
101	C₂×C₈	C₂×C₁₆	1	2	yes	—
102	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₈	C₄×D₄	1	1	yes	characteristic
104	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₈	#131	1	1	yes	characteristic
107	C₂×C₈	#131	1	1	yes	characteristic
108	C₄⋊C₄	—	2	4	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₄×C₈	1	2	yes	—
114	C₄²	C₂×C₁₆	1	2	yes	—
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	2	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₁₆	#117	1	1	yes	characteristic
120	C₁₆	C₄×D₄	1	1	yes	characteristic
121	C₁₆	C₄×Q₈	1	1	yes	characteristic
122	C₂³	C₂²×C₁₆	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	C₄×C₁₆	1	2	yes	—
131	C₂×C₄	C₄×C₁₆	1	2	yes	—
132	C₂×C₄	M₆₄	1	4	yes	—
133	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
134	C₂×C₄	C₈×D₄	1	1	yes	characteristic
135	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
137	C₂×C₄	#504	1	1	yes	characteristic
138	C₂×C₄	#504	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	2	no	—
142	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₈	#405	1	1	yes	characteristic
144	C₈	C₈×D₄	1	1	yes	characteristic
145	C₈	C₈×Q₈	1	1	yes	characteristic
146	C₂²	C₂×C₄×C₁₆	1	1	yes	characteristic
147	C₂²	D₄×C₁₆	1	1	yes	characteristic
148	C₂²	Q₈×C₁₆	1	1	yes	characteristic
149	C₂²	C₂×M₆₄	1	2	yes	—
150	C₂²	#2258	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	4	2	no	—
155	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
156	C₄	#2108	1	1	yes	characteristic
157	C₄	D₄×C₁₆	1	1	yes	characteristic
158	C₄	Q₈×C₁₆	1	1	yes	characteristic
159	C₂	C₁₆×C₄⋊C₄	1	1	yes	characteristic
160	C₂	C₄×M₆₄	1	1	yes	characteristic
161	C₂	#13549	1	1	yes	characteristic
162	C₂	#13645	1	2	yes	—
163	C₂	#13660	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
17. C₄⋊C₃₂	C₄	semidirect
19. #1397	C₄	semidirect
87. C₃₂	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₃₂	2
C₂×#1397	2
C₂×C₄×C₃₂	2
C₁₆×C₄⋊C₄	2
#7465	2