Finite Group Databasebeta

Subgroups — #142110

Order 512.

362 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 126 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	#12900	C₂	1	1	yes	maximal, characteristic
3	C₄×#364	C₂	1	1	yes	maximal, characteristic
4	#7269	C₂	1	4	yes	maximal
5	C₂×#500	—	2	1	no	—
6	C₂×#500	—	2	1	no	—
7	#2868	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2150	—	2	1	no	—
12	C₄×M₃₂	—	2	1	no	—
13	#1845	—	2	1	no	—
14	#1845	—	2	1	no	—
15	#1845	C₄	1	2	yes	—
16	#1845	C₂²	1	2	yes	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1376	—	2	1	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂×C₄⋊Q₈	—	4	2	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	#500	—	4	1	no	—
24	#500	—	4	1	no	—
25	#500	—	4	1	no	—
26	#500	—	4	1	no	—
27	#500	—	4	1	no	—
28	#500	—	4	1	no	—
29	#500	—	4	1	no	—
30	#500	—	4	1	no	—
31	C₂×#107	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	#384	—	2	2	no	—
42	#384	—	4	2	no	—
43	#384	C₂×C₄	1	4	yes	—
44	C₄³	C₂×C₄	1	1	yes	characteristic
45	#364	—	2	2	no	—
46	#364	—	2	4	no	—
47	#364	—	4	2	no	—
48	#364	Q₈	1	4	yes	complemented
49	C₄⋊C₁₆	—	4	1	no	—
50	C₄⋊C₁₆	—	4	1	no	—
51	#346	—	4	1	no	—
52	C₄×C₁₆	—	4	1	no	—
53	C₂²×Q₈	—	4	2	no	—
54	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Q₈	—	4	2	no	—
59	C₄⋊Q₈	—	4	2	no	—
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	—	4	2	no	—
62	C₄⋊Q₈	—	8	8	no	—
63	C₂×C₄⋊C₄	—	2	4	no	—
64	C₂×C₄⋊C₄	—	4	4	no	—
65	C₂×C₄⋊C₄	—	4	4	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄²	—	2	1	no	—
68	C₂×C₄²	—	2	1	no	—
69	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	M₃₂	—	2	4	no	—
73	M₃₂	—	4	2	no	—
74	M₃₂	—	4	2	no	—
75	M₃₂	—	4	2	no	—
76	C₂×C₁₆	—	4	2	no	—
77	C₂×C₁₆	—	4	2	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	#107	—	4	2	no	—
81	#107	—	4	2	no	—
82	#107	—	4	2	no	—
83	#107	—	4	2	no	—
84	#107	—	4	2	no	—
85	#107	—	4	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	D₈	1	1	yes	characteristic
89	C₄×C₈	D₈	1	1	yes	characteristic
90	C₄×C₈	D₈	1	1	yes	characteristic
91	C₄×C₈	D₈	1	1	yes	characteristic
92	C₄×C₈	SD₁₆	1	1	yes	characteristic
93	C₄×C₈	SD₁₆	1	1	yes	characteristic
94	C₄×C₈	SD₁₆	1	1	yes	characteristic
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₄×C₈	SD₁₆	1	2	yes	—
97	C₄×C₈	Q₁₆	1	2	yes	—
98	C₄×C₈	C₂×Q₈	1	2	yes	—
99	C₄×C₈	#41	1	2	yes	—
100	C₂×Q₈	—	8	8	no	—
101	C₂×Q₈	—	8	2	no	—
102	C₂×Q₈	—	8	2	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	4	4	no	—
106	C₂²×C₄	#100	1	1	yes	characteristic
107	C₂²×C₄	#100	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	#101	1	2	yes	—
112	C₂×C₈	#102	1	1	yes	characteristic
113	C₂×C₈	#102	1	1	yes	characteristic
114	C₂×C₈	#102	1	1	yes	characteristic
115	C₂×C₈	#102	1	1	yes	characteristic
116	C₂×C₈	#103	1	2	yes	—
117	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₈	#124	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₈	C₂×D₈	1	1	yes	characteristic
121	C₂×C₈	C₂×D₈	1	1	yes	characteristic
122	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₈	#137	1	2	yes	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄⋊C₄	—	4	4	no	—
129	C₄⋊C₄	—	8	8	no	—
130	C₄⋊C₄	—	8	8	no	—
131	C₄⋊C₄	—	8	2	no	—
132	C₄⋊C₄	—	8	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	#102	1	1	yes	characteristic
138	C₄²	#102	1	1	yes	characteristic
139	C₄²	#102	1	1	yes	characteristic
140	C₄²	#102	1	1	yes	characteristic
141	C₄²	#103	1	2	yes	—
142	C₄²	C₄×Q₈	1	2	yes	—
143	C₁₆	—	4	4	no	—
144	C₁₆	—	8	2	no	—
145	C₂³	C₂×#100	1	1	yes	characteristic
146	Q₈	—	16	4	no	—
147	Q₈	—	16	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	8	8	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	#331	1	1	yes	characteristic
159	C₂×C₄	#331	1	1	yes	characteristic
160	C₂×C₄	#331	1	1	yes	characteristic
161	C₂×C₄	#331	1	1	yes	characteristic
162	C₂×C₄	#391	1	1	yes	characteristic
163	C₂×C₄	C₂×#101	1	1	yes	characteristic
164	C₂×C₄	C₂×#102	1	1	yes	characteristic
165	C₂×C₄	C₂×#102	1	1	yes	characteristic
166	C₂×C₄	C₂×#103	1	1	yes	characteristic
167	C₂×C₄	#419	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	#332	1	2	yes	—
171	C₈	#332	1	2	yes	—
172	C₈	#474	1	2	yes	—
173	C₈	#475	1	2	yes	—
174	C₈	#486	1	1	yes	characteristic
175	C₈	#487	1	1	yes	characteristic
176	C₈	#498	1	1	yes	characteristic
177	C₈	#500	1	1	yes	characteristic
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	C₂×#331	1	1	yes	characteristic
181	C₂²	#1548	1	1	yes	characteristic
182	C₂²	#1997	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	4	no	—
188	C₄	#1357	1	2	yes	—
189	C₄	#1358	1	2	yes	—
190	C₄	C₂×#332	1	1	yes	characteristic
191	C₄	#1546	1	1	yes	characteristic
192	C₄	#1988	1	1	yes	characteristic
193	C₄	#1990	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#9898	1	1	yes	characteristic
197	C₂	C₂×#1357	1	1	yes	characteristic
198	C₂	C₂×#1358	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
48. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33312	2
#12900	2
C₄×#364	2
#7269	2