Subgroups — #96894

Order 512.

378 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#335	C₂	1	1	yes	maximal, characteristic
2	C₄×#334	C₂	1	1	yes	maximal, characteristic
3	#8261	C₂	1	2	yes	maximal
4	C₄×#322	C₂	1	1	yes	maximal, characteristic
5	#7015	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	C₄	1	2	yes	—
12	C₂×#334	C₄	1	2	yes	—
13	C₂×#322	—	2	2	no	—
14	C₂×#322	C₄	1	2	yes	—
15	C₂×#322	C₄	1	2	yes	—
16	#1277	C₄	1	2	yes	—
17	#1277	C₄	1	2	yes	—
18	#1273	—	2	2	no	—
19	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₈	—	2	2	no	—
22	C₂×C₄×C₈	—	2	2	no	—
23	C₂×C₄×C₈	—	2	1	no	—
24	C₂×C₄×C₈	—	2	1	no	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	Q₈	1	2	yes	—
35	C₄³	C₂³	1	1	yes	characteristic
36	#336	—	2	2	no	—
37	#336	—	2	2	no	—
38	#335	C₈	1	8	yes	complemented
39	#334	—	2	4	no	—
40	#322	—	2	4	no	—
41	#322	—	2	4	no	—
42	#322	—	4	4	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	—	2	4	no	—
45	C₂²×C₈	—	2	4	no	—
46	C₂²×C₈	C₄²	1	1	yes	characteristic
47	C₂²×C₈	C₄²	1	1	yes	characteristic
48	C₂²×C₈	C₄²	1	1	yes	characteristic
49	C₂²×C₈	C₄²	1	1	yes	characteristic
50	C₂²×C₈	C₄⋊C₄	1	4	yes	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄²	#41	1	1	yes	characteristic
57	C₂×C₄²	#41	1	1	yes	characteristic
58	C₄⋊C₈	—	2	4	no	—
59	C₄⋊C₈	—	2	4	no	—
60	C₄⋊C₈	—	2	4	no	—
61	C₄⋊C₈	C₂×C₈	1	8	yes	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	4	2	no	—
71	C₄×C₈	—	4	2	no	—
72	C₄×C₈	—	4	2	no	—
73	C₄×C₈	—	4	2	no	—
74	C₄×C₈	—	4	2	no	—
75	C₄×C₈	—	4	2	no	—
76	C₄×C₈	—	4	2	no	—
77	C₄×C₈	C₂×C₈	1	4	yes	—
78	C₄×C₈	M₁₆	1	4	yes	—
79	C₄×C₈	M₁₆	1	4	yes	—
80	C₄×C₈	M₁₆	1	4	yes	—
81	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
82	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	#117	1	1	yes	characteristic
84	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
87	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	8	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	—	4	4	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	C₄×C₈	1	2	yes	—
107	C₂×C₈	C₄×C₈	1	2	yes	—
108	C₂×C₈	#97	1	2	yes	—
109	C₂×C₈	#97	1	2	yes	—
110	C₂×C₈	#97	1	2	yes	—
111	C₂×C₈	#97	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	C₂²×C₈	1	2	yes	—
116	C₄²	C₂×M₁₆	1	2	yes	—
117	C₄²	C₂×M₁₆	1	2	yes	—
118	C₄²	C₂×M₁₆	1	2	yes	—
119	C₄²	#131	1	2	yes	—
120	C₄²	#131	1	2	yes	—
121	C₄²	AGL(1,ℤ/8)	1	2	yes	—
122	C₄²	#137	1	2	yes	—
123	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂×C₄	C₂×#97	1	1	yes	characteristic
132	C₂×C₄	C₂×#97	1	1	yes	characteristic
133	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
134	C₂×C₄	#405	1	1	yes	characteristic
135	C₂×C₄	#405	1	1	yes	characteristic
136	C₂×C₄	#423	1	2	yes	—
137	C₂×C₄	#428	1	2	yes	—
138	C₂×C₄	#430	1	2	yes	—
139	C₂×C₄	#432	1	2	yes	—
140	C₂×C₄	C₈×D₄	1	2	yes	—
141	C₂×C₄	#436	1	2	yes	—
142	C₂×C₄	#440	1	2	yes	—
143	C₂×C₄	#441	1	1	yes	characteristic
144	C₂×C₄	#442	1	1	yes	characteristic
145	C₂×C₄	#444	1	2	yes	—
146	C₂×C₄	#444	1	2	yes	—
147	C₂×C₄	C₈×Q₈	1	2	yes	—
148	C₂×C₄	#446	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	4	4	no	—
153	C₈	—	4	4	no	—
154	C₈	—	4	8	no	—
155	C₈	—	8	4	no	—
156	C₈	#322	1	4	yes	complemented
157	C₂²	#1767	1	1	yes	characteristic
158	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
159	C₂²	#1770	1	1	yes	characteristic
160	C₂²	#1773	1	1	yes	characteristic
161	C₂²	#1773	1	1	yes	characteristic
162	C₂²	#1776	1	1	yes	characteristic
163	C₂²	#1778	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	C₂×#322	1	1	yes	characteristic
168	C₄	C₈×M₁₆	1	1	yes	characteristic
169	C₄	#1451	1	1	yes	characteristic
170	C₄	#1452	1	1	yes	characteristic
171	C₄	#1569	1	4	yes	—
172	C₄	#1579	1	2	yes	—
173	C₄	#1580	1	2	yes	—
174	C₄	#1592	1	4	yes	—
175	C₂	#8261	1	1	yes	characteristic
176	C₂	#8269	1	1	yes	characteristic
177	C₂	#8276	1	1	yes	characteristic
178	C₂	#8284	1	1	yes	characteristic
179	C₂	#8286	1	1	yes	characteristic
180	C₂	#8294	1	1	yes	characteristic
181	C₂	#8295	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
38. #335	C₈	semidirect
156. C₈	#322	semidirect

Maximal subgroups

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

subgroup	index
C₄×#335	2
C₄×#334	2
#8261	2
C₄×#322	2
#7015	2