Subgroups — #512809

Order 512.

291 conjugacy classes of subgroups.

The subgroup lattice has 156 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	#13641	C₂	1	4	yes	maximal
6	#13640	C₂	1	1	yes	maximal, characteristic
7	#13639	C₂	1	1	yes	maximal, characteristic
8	#2960	C₄	1	1	yes	characteristic
9	#2183	C₄	1	2	yes	—
10	#2183	C₄	1	2	yes	—
11	Q₈×C₁₆	C₂²	1	2	yes	—
12	Q₈×C₁₆	C₂²	1	2	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	2	yes	—
18	C₄⋊C₃₂	C₂²	1	2	yes	—
19	C₄⋊C₃₂	C₂²	1	4	yes	—
20	C₄⋊C₃₂	C₂²	1	4	yes	—
21	C₂²⋊C₃₂	C₂²	1	2	yes	—
22	C₂²⋊C₃₂	C₂²	1	2	yes	—
23	#1397	C₂²	1	2	yes	—
24	#1397	C₂²	1	4	yes	—
25	C₄×C₃₂	C₂²	1	1	yes	characteristic
26	C₄×C₃₂	C₂²	1	1	yes	characteristic
27	C₄×C₃₂	C₂²	1	4	yes	—
28	#533	C₈	1	1	yes	characteristic
29	C₂×M₃₂	C₂×C₄	1	2	yes	—
30	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	#446	C₈	1	2	yes	—
33	#446	C₈	1	2	yes	—
34	C₈×Q₈	C₂×C₄	1	2	yes	—
35	C₈×Q₈	C₂×C₄	1	2	yes	—
36	#424	C₈	1	2	yes	—
37	#423	C₈	1	1	yes	characteristic
38	#405	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃₂	C₂³	1	2	yes	—
41	C₂×C₃₂	C₂³	1	2	yes	—
42	C₂×C₃₂	C₂³	1	4	yes	—
43	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
44	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
45	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
46	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
47	#346	C₂×C₄	1	4	yes	—
48	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
49	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₄×C₁₆	C₂³	1	1	yes	characteristic
51	C₄×C₁₆	C₂³	1	1	yes	characteristic
52	C₄×C₁₆	C₂³	1	4	yes	—
53	C₂×M₁₆	C₂×C₈	1	2	yes	—
54	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₄⋊Q₈	C₁₆	1	1	yes	characteristic
57	C₄⋊Q₈	C₁₆	1	1	yes	characteristic
58	#125	C₁₆	1	2	yes	—
59	C₂²⋊Q₈	C₁₆	1	2	yes	—
60	C₂²⋊Q₈	C₁₆	1	2	yes	—
61	C₄×Q₈	C₂×C₈	1	2	yes	—
62	C₄×Q₈	C₂×C₈	1	2	yes	—
63	#117	C₂×C₈	1	2	yes	—
64	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
65	M₃₂	—	4	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	4	1	no	—
68	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₁₆	C₂²×C₄	1	4	yes	—
72	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₈	C₂×C₈	1	2	yes	—
74	C₄⋊C₈	C₂×C₈	1	2	yes	—
75	C₄⋊C₈	C₂×C₈	1	4	yes	—
76	C₄⋊C₈	C₂×C₈	1	4	yes	—
77	#97	C₂×C₈	1	4	yes	—
78	C₄×C₈	C₂×C₈	1	1	yes	characteristic
79	C₄×C₈	C₂×C₈	1	1	yes	characteristic
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₄×C₈	C₂²×C₄	1	4	yes	—
83	C₃₂	—	2	2	no	—
84	C₃₂	—	2	2	no	—
85	C₃₂	—	2	4	no	—
86	C₂×Q₈	C₂×C₁₆	1	2	yes	—
87	C₂×Q₈	C₂×C₁₆	1	2	yes	—
88	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
89	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
91	M₁₆	—	4	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₈	C₂²×C₈	1	4	yes	—
99	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
100	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
101	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
102	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
103	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
104	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
105	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
106	C₂²⋊C₄	C₂×C₁₆	1	4	yes	—
107	C₄²	C₂×C₁₆	1	1	yes	characteristic
108	C₄²	C₂×C₁₆	1	1	yes	characteristic
109	C₄²	C₂²×C₈	1	1	yes	characteristic
110	C₄²	C₂²×C₈	1	1	yes	characteristic
111	C₄²	C₂²×C₈	1	4	yes	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	2	4	no	—
116	C₁₆	2₋¹⁺⁴	1	2	yes	—
117	C₂³	C₂²×C₁₆	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
125	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
127	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₁₆	1	4	yes	—
129	C₂×C₄	C₂²×C₁₆	1	4	yes	—
130	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
131	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	4	no	—
135	C₈	—	2	1	no	—
136	C₈	#519	1	1	yes	characteristic
137	C₈	#568	1	1	yes	characteristic
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂³×C₁₆	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	4	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	4	no	—
147	C₄	—	2	1	no	—
148	C₄	#2258	1	2	yes	—
149	C₄	#2982	1	1	yes	characteristic
150	C₄	#3407	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#33496	1	1	yes	characteristic
153	C₂	C₂×#2258	1	1	yes	characteristic
154	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

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
#13641	2
#13640	2
#13639	2