Finite Group Databasebeta

Subgroups — #93461

Order 512.

230 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12900	C₂	1	1	yes	maximal, characteristic
2	#8266	C₂	1	1	yes	maximal, characteristic
3	#8260	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	1	no	—
5	C₂×#364	C₄	1	2	yes	—
6	#2150	—	2	1	no	—
7	C₄×C₄⋊C₈	—	2	1	no	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	#1564	C₄	1	1	yes	characteristic
10	#1564	C₄	1	1	yes	characteristic
11	#1450	—	2	1	no	—
12	C₂×#322	—	2	1	no	—
13	#1276	—	2	1	no	—
14	#1273	C₄	1	2	yes	—
15	C₂×M₃₂	—	2	2	no	—
16	#431	—	2	1	no	—
17	#431	—	2	1	no	—
18	C₂×C₄⋊C₈	—	2	2	no	—
19	C₂×C₄×C₈	—	2	2	no	—
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
26	C₄³	D₄	1	1	yes	characteristic
27	#364	—	4	2	no	—
28	#364	—	4	1	no	—
29	#364	—	4	1	no	—
30	C₄⋊C₁₆	—	4	1	no	—
31	C₄⋊C₁₆	—	4	1	no	—
32	#336	—	2	2	no	—
33	#335	—	2	1	no	—
34	#335	C₈	1	2	yes	complemented
35	#334	—	2	1	no	—
36	#334	C₈	1	2	yes	complemented
37	#322	—	4	1	no	—
38	#322	—	4	1	no	—
39	#322	—	4	1	no	—
40	C₈²	—	4	1	no	—
41	C₂²×C₈	—	2	4	no	—
42	C₂²×C₈	C₄²	1	1	yes	characteristic
43	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄²	—	2	1	no	—
47	C₂×C₄²	—	2	1	no	—
48	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
51	M₃₂	—	4	2	no	—
52	M₃₂	—	4	2	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	4	4	no	—
57	C₂²⋊C₈	—	4	1	no	—
58	C₂²⋊C₈	—	4	1	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	—	4	2	no	—
65	C₄×C₈	—	4	1	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	—	4	2	no	—
68	C₄×C₈	C₂×C₈	1	1	yes	characteristic
69	C₄×C₈	C₂×C₈	1	1	yes	characteristic
70	C₄×C₈	M₁₆	1	1	yes	characteristic
71	C₄×C₈	M₁₆	1	1	yes	characteristic
72	C₄×C₈	D₈	1	1	yes	characteristic
73	C₄×C₈	D₈	1	1	yes	characteristic
74	C₄×C₈	SD₁₆	1	1	yes	characteristic
75	C₄×C₈	SD₁₆	1	1	yes	characteristic
76	C₄×C₈	SD₁₆	1	1	yes	characteristic
77	C₄×C₈	SD₁₆	1	1	yes	characteristic
78	C₄×C₈	Q₁₆	1	1	yes	characteristic
79	C₄×C₈	Q₁₆	1	1	yes	characteristic
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	#95	1	1	yes	characteristic
83	C₂²×C₄	#104	1	1	yes	characteristic
84	C₂²×C₄	#104	1	1	yes	characteristic
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	4	4	no	—
90	C₂×C₈	—	4	4	no	—
91	C₂×C₈	—	4	4	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	C₄×C₈	1	1	yes	characteristic
95	C₂×C₈	#97	1	1	yes	characteristic
96	C₂×C₈	#100	1	1	yes	characteristic
97	C₂×C₈	#101	1	1	yes	characteristic
98	C₂×C₈	#102	1	1	yes	characteristic
99	C₂×C₈	#102	1	1	yes	characteristic
100	C₂×C₈	#103	1	1	yes	characteristic
101	C₂×C₈	#103	1	1	yes	characteristic
102	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₈	#106	1	1	yes	characteristic
105	C₂×C₈	#106	1	1	yes	characteristic
106	C₂×C₈	#107	1	1	yes	characteristic
107	C₂×C₈	#107	1	1	yes	characteristic
108	C₂×C₈	#108	1	1	yes	characteristic
109	C₂×C₈	#108	1	1	yes	characteristic
110	C₄²	—	2	4	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	4	2	no	—
115	C₄²	C₂²⋊C₈	1	1	yes	characteristic
116	C₄²	C₂²⋊C₈	1	1	yes	characteristic
117	C₄²	#102	1	1	yes	characteristic
118	C₄²	#102	1	1	yes	characteristic
119	C₄²	#103	1	1	yes	characteristic
120	C₄²	#103	1	1	yes	characteristic
121	C₁₆	—	8	2	no	—
122	C₂³	#339	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
133	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
134	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
135	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
136	C₂×C₄	#336	1	1	yes	characteristic
137	C₂×C₄	#340	1	1	yes	characteristic
138	C₂×C₄	#340	1	1	yes	characteristic
139	C₂×C₄	#341	1	1	yes	characteristic
140	C₂×C₄	#343	1	1	yes	characteristic
141	C₂×C₄	#344	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	4	no	—
148	C₈	—	8	2	no	—
149	C₈	#331	1	1	yes	characteristic
150	C₈	#332	1	1	yes	characteristic
151	C₈	#332	1	1	yes	characteristic
152	C₈	#333	1	1	yes	characteristic
153	C₈	#334	1	1	yes	characteristic
154	C₈	#334	1	1	yes	characteristic
155	C₈	#335	1	1	yes	characteristic
156	C₈	#335	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#1279	1	1	yes	characteristic
160	C₂²	#1299	1	1	yes	characteristic
161	C₂²	#1303	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	4	2	no	—
168	C₄	#1277	1	1	yes	characteristic
169	C₄	#1283	1	1	yes	characteristic
170	C₄	#1296	1	1	yes	characteristic
171	C₄	#1301	1	1	yes	characteristic
172	C₄	#1336	1	2	yes	—
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#7031	1	1	yes	characteristic
176	C₂	#7180	1	1	yes	characteristic
177	C₂	#7182	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1279
prime radical	C₂³	#339
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
34. #335	C₈	semidirect
36. #334	C₈	semidirect

Maximal subgroups

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

subgroup	index
#12900	2
#8266	2
#8260	2