Finite Group Databasebeta

Subgroups — #141008

Order 512.

408 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 166 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	#33312	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1365	C₂	1	4	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	1	no	—
6	C₂×#500	C₂²	1	2	yes	—
7	#2868	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	—	2	1	no	—
10	#1845	—	2	1	no	—
11	#1845	—	2	1	no	—
12	#1845	C₄	1	2	yes	—
13	#1845	C₄	1	2	yes	—
14	#1842	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#335	—	2	1	no	—
17	C₂×#335	C₂²	1	2	yes	—
18	#1374	—	2	1	no	—
19	#1365	—	2	8	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	4	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#500	—	2	4	no	—
24	#500	—	2	4	no	—
25	#500	—	4	2	no	—
26	#500	—	4	2	no	—
27	C₂×#107	—	2	2	no	—
28	C₂×#107	—	2	2	no	—
29	C₂×#107	—	2	2	no	—
30	C₂×#107	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	—	4	2	no	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#384	—	2	2	no	—
40	#384	—	2	2	no	—
41	#384	—	4	2	no	—
42	C₄³	D₄	1	1	yes	characteristic
43	C₄⋊C₁₆	—	2	2	no	—
44	C₄⋊C₁₆	D₄	1	4	yes	—
45	C₄×C₁₆	—	4	2	no	—
46	#335	—	2	4	no	—
47	#335	—	4	2	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂²×C₈	—	4	2	no	—
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	8	8	no	—
59	C₂×C₄⋊C₄	—	2	4	no	—
60	C₂×C₄⋊C₄	—	4	4	no	—
61	C₂×C₄⋊C₄	—	4	4	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	D₈	1	2	yes	—
66	C₂×C₄²	SD₁₆	1	2	yes	—
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₁₆	—	2	4	no	—
69	C₂×C₁₆	—	4	2	no	—
70	C₂×C₁₆	—	4	2	no	—
71	#107	—	2	8	no	—
72	#107	—	4	4	no	—
73	#107	—	4	4	no	—
74	#107	—	4	4	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	8	4	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	D₈	1	2	yes	—
82	C₄×C₈	SD₁₆	1	2	yes	—
83	C₄×C₈	SD₁₆	1	2	yes	—
84	C₄×C₈	Q₁₆	1	2	yes	—
85	C₄×C₈	C₂×D₄	1	2	yes	—
86	C₄×C₈	#41	1	2	yes	—
87	C₂×Q₈	—	8	8	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×Q₈	—	8	2	no	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	#102	1	2	yes	—
92	C₂²×C₄	#102	1	2	yes	—
93	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	8	2	no	—
101	C₂×C₈	—	8	4	no	—
102	C₂×C₈	#101	1	2	yes	—
103	C₂×C₈	#102	1	2	yes	—
104	C₂×C₈	#102	1	2	yes	—
105	C₂×C₈	#103	1	2	yes	—
106	C₂×C₈	C₄×D₄	1	2	yes	—
107	C₂×C₈	#121	1	2	yes	—
108	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₈	#137	1	2	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	8	8	no	—
114	C₄⋊C₄	—	8	8	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	#101	1	2	yes	—
121	C₄²	#103	1	2	yes	—
122	C₄²	SD₃₂	1	4	yes	—
123	C₄²	Q₃₂	1	4	yes	—
124	C₄²	#124	1	1	yes	characteristic
125	C₄²	#127	1	1	yes	characteristic
126	C₄²	C₂×D₈	1	1	yes	characteristic
127	C₄²	C₂×SD₁₆	1	1	yes	characteristic
128	C₁₆	—	4	4	no	—
129	C₂³	C₂×#102	1	1	yes	characteristic
130	Q₈	—	16	4	no	—
131	Q₈	—	16	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	8	8	no	—
136	C₂×C₄	—	8	4	no	—
137	C₂×C₄	#332	1	4	yes	—
138	C₂×C₄	#358	1	4	yes	—
139	C₂×C₄	#358	1	4	yes	—
140	C₂×C₄	#361	1	4	yes	—
141	C₂×C₄	#390	1	1	yes	characteristic
142	C₂×C₄	C₂×#101	1	1	yes	characteristic
143	C₂×C₄	C₂×#101	1	1	yes	characteristic
144	C₂×C₄	C₂×#102	1	1	yes	characteristic
145	C₂×C₄	C₂×#103	1	1	yes	characteristic
146	C₂×C₄	#419	1	1	yes	characteristic
147	C₂×C₄	#486	1	1	yes	characteristic
148	C₂×C₄	#486	1	1	yes	characteristic
149	C₂×C₄	#492	1	1	yes	characteristic
150	C₂×C₄	#493	1	1	yes	characteristic
151	C₂×C₄	C₂×SD₃₂	1	2	yes	—
152	C₂×C₄	C₂×Q₃₂	1	2	yes	—
153	C₈	—	2	2	no	—
154	C₈	—	2	2	no	—
155	C₈	—	16	2	no	—
156	C₈	#332	1	4	yes	—
157	C₈	#461	1	2	yes	—
158	C₈	#462	1	2	yes	—
159	C₂²	C₂×#332	1	1	yes	characteristic
160	C₂²	#1546	1	1	yes	characteristic
161	C₂²	#1957	1	1	yes	characteristic
162	C₂²	C₂×#358	1	2	yes	—
163	C₂²	C₂×#361	1	2	yes	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	4	no	—
167	C₄	—	16	4	no	—
168	C₄	#1365	1	8	yes	complemented
169	C₄	C₂×#332	1	1	yes	characteristic
170	C₄	#1546	1	1	yes	characteristic
171	C₄	#1966	1	1	yes	characteristic
172	C₄	#1981	1	1	yes	characteristic
173	C₄	#2241	1	2	yes	—
174	C₄	#2247	1	1	yes	characteristic
175	C₄	#2248	1	1	yes	characteristic
176	C₂	#9692	1	1	yes	characteristic
177	C₂	C₂×#1365	1	2	yes	—
178	C₂	#11972	1	2	yes	—
179	C₂	#13203	1	1	yes	characteristic
180	C₂	#13213	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
168. C₄	#1365	semidirect

Maximal subgroups

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

subgroup	index
#33312	2
C₄×C₄⋊C₁₆	2
C₂×#1365	2
#8806	2