Subgroups — #512786

Order 512.

320 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	#33474	C₂	1	1	yes	maximal, characteristic
2	#13660	C₂	1	2	yes	maximal, complemented
3	#13660	C₂	1	4	yes	maximal, complemented
4	Q₈×C₃₂	C₂	1	2	yes	maximal, complemented
5	#13628	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
7	#13549	C₂	1	2	yes	maximal
8	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic
9	#2960	C₄	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×M₆₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃₂	C₂²	1	2	yes	—
13	#2183	C₄	1	2	yes	—
14	#2183	C₄	1	2	yes	—
15	Q₈×C₁₆	C₂²	1	2	yes	—
16	Q₈×C₁₆	C₂²	1	2	yes	—
17	#2151	C₄	1	2	yes	—
18	#2150	C₄	1	1	yes	characteristic
19	#2108	C₂²	1	2	yes	—
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
22	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
23	C₄⋊C₃₂	C₂²	1	2	yes	—
24	C₄⋊C₃₂	C₂²	1	4	yes	—
25	C₄⋊C₃₂	C₂²	1	4	yes	—
26	#1397	C₂²	1	2	yes	—
27	#1397	C₂²	1	4	yes	—
28	C₄×C₃₂	C₂²	1	2	yes	—
29	C₄×C₃₂	C₂²	1	4	yes	—
30	#533	C₈	1	1	yes	characteristic
31	C₂×M₃₂	C₂×C₄	1	2	yes	—
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	#446	C₈	1	2	yes	—
35	#446	C₈	1	2	yes	—
36	C₈×Q₈	C₂×C₄	1	2	yes	—
37	C₈×Q₈	C₂×C₄	1	2	yes	—
38	#424	C₈	1	2	yes	—
39	#423	C₈	1	1	yes	characteristic
40	#405	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	M₆₄	—	2	2	no	—
43	M₆₄	Q₈	1	4	yes	complemented
44	C₂×C₃₂	—	2	4	no	—
45	C₂×C₃₂	C₂³	1	2	yes	—
46	C₂×C₃₂	C₂³	1	2	yes	—
47	C₂×C₃₂	C₂³	1	2	yes	—
48	C₂×C₃₂	C₂³	1	2	yes	—
49	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
50	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
52	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
53	#346	C₂×C₄	1	4	yes	—
54	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₄×C₁₆	C₂³	1	1	yes	characteristic
57	C₄×C₁₆	C₂³	1	1	yes	characteristic
58	C₄×C₁₆	C₂³	1	4	yes	—
59	C₂×M₁₆	C₂×C₈	1	2	yes	—
60	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₄⋊Q₈	C₁₆	1	1	yes	characteristic
63	C₄⋊Q₈	C₁₆	1	1	yes	characteristic
64	#125	C₁₆	1	2	yes	—
65	C₂²⋊Q₈	C₁₆	1	2	yes	—
66	C₂²⋊Q₈	C₁₆	1	2	yes	—
67	C₄×Q₈	C₂×C₈	1	2	yes	—
68	C₄×Q₈	C₂×C₈	1	2	yes	—
69	#117	C₂×C₈	1	2	yes	—
70	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
71	M₃₂	—	2	4	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×C₄	1	4	yes	—
76	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₁₆	C₂×Q₈	1	2	yes	—
78	C₂×C₁₆	#41	1	2	yes	—
79	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₈	C₂×C₈	1	2	yes	—
81	C₄⋊C₈	C₂×C₈	1	2	yes	—
82	C₄⋊C₈	C₂×C₈	1	4	yes	—
83	C₄⋊C₈	C₂×C₈	1	4	yes	—
84	#97	C₂×C₈	1	4	yes	—
85	C₄×C₈	C₂×C₈	1	1	yes	characteristic
86	C₄×C₈	C₂×C₈	1	1	yes	characteristic
87	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₄×C₈	C₂²×C₄	1	4	yes	—
90	C₃₂	—	2	2	no	—
91	C₃₂	—	2	2	no	—
92	C₃₂	—	2	2	no	—
93	C₃₂	C₂×Q₈	1	4	yes	—
94	C₂×Q₈	C₂×C₁₆	1	2	yes	—
95	C₂×Q₈	C₂×C₁₆	1	2	yes	—
96	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₄×Q₈	1	2	yes	—
102	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₈	1	4	yes	—
105	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₈	#131	1	2	yes	—
107	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
108	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
109	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
110	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
111	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
112	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
113	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
114	C₂²⋊C₄	C₂×C₁₆	1	4	yes	—
115	C₄²	C₂×C₁₆	1	1	yes	characteristic
116	C₄²	C₂×C₁₆	1	1	yes	characteristic
117	C₄²	C₂²×C₈	1	1	yes	characteristic
118	C₄²	C₂²×C₈	1	1	yes	characteristic
119	C₄²	C₂²×C₈	1	4	yes	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	4	no	—
123	C₁₆	C₄×Q₈	1	2	yes	—
124	C₁₆	C₂²×Q₈	1	1	yes	characteristic
125	C₁₆	C₂×#41	1	1	yes	characteristic
126	C₂³	C₂²×C₁₆	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	C₈×Q₈	1	2	yes	—
132	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
133	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₁₆	1	4	yes	—
136	C₂×C₄	C₂²×C₁₆	1	4	yes	—
137	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
139	C₂×C₄	#504	1	2	yes	—
140	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	4	no	—
144	C₈	C₈×Q₈	1	2	yes	—
145	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₈	C₂×#131	1	1	yes	characteristic
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	Q₈×C₁₆	1	2	yes	—
150	C₂²	C₂³×C₁₆	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	4	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	4	no	—
156	C₄	Q₈×C₁₆	1	2	yes	—
157	C₄	#2258	1	2	yes	—
158	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
159	C₄	C₂×#504	1	1	yes	characteristic
160	C₂	—	2	2	no	—
161	C₂	C₂×Q₈×C₁₆	1	1	yes	characteristic
162	C₂	C₂×#2258	1	1	yes	characteristic
163	C₂	#33974	1	1	yes	characteristic

	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
3. #13660	C₂	semidirect
4. Q₈×C₃₂	C₂	semidirect
43. M₆₄	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
C₂×C₄⋊C₃₂	2
#13549	2
C₄×M₆₄	2