Finite Group Databasebeta

Subgroups — #140297

Order 512.

236 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 82 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	#13290	C₂	1	1	yes	maximal, characteristic
2	#12666	C₂	1	1	yes	maximal, characteristic
3	#11940	C₂	1	1	yes	maximal, characteristic
4	#9740	C₂	1	1	yes	maximal, characteristic
5	#9630	C₂	1	1	yes	maximal, characteristic
6	#7280	C₂	1	1	yes	maximal, characteristic
7	#7158	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	#2150	C₂²	1	1	yes	characteristic
10	C₂×#347	—	2	1	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	#1981	—	2	1	no	—
13	#1848	C₂²	1	1	yes	characteristic
14	#1770	C₂²	1	1	yes	characteristic
15	#1571	—	2	1	no	—
16	#1564	C₂²	1	1	yes	characteristic
17	C₂×#334	C₂²	1	1	yes	characteristic
18	C₂×#332	—	2	1	no	—
19	#1387	—	2	1	no	—
20	#1387	—	2	1	no	—
21	#1375	—	2	1	no	—
22	#1364	C₄	1	4	yes	complemented
23	#1303	—	2	1	no	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	D₄	1	1	yes	characteristic
27	C₂×M₃₂	D₄	1	1	yes	characteristic
28	#500	—	2	1	no	—
29	#500	—	2	1	no	—
30	#500	—	2	1	no	—
31	#500	C₂×C₄	1	2	yes	—
32	#431	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#108	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#107	—	2	1	no	—
37	#423	—	2	1	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
40	C₂×#103	—	4	2	no	—
41	C₂×#97	—	2	1	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	1	no	—
45	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
46	#347	—	4	2	no	—
47	#346	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	#336	—	2	1	no	—
50	#335	—	4	2	no	—
51	#335	C₂×C₄	1	2	yes	—
52	#334	—	2	1	no	—
53	#334	—	2	1	no	—
54	#334	—	2	1	no	—
55	#332	—	4	2	no	—
56	C₂²×Q₈	—	4	1	no	—
57	C₂×M₁₆	—	4	1	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	—	4	1	no	—
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	#41	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	4	1	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	1	no	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄⋊C₄	—	4	1	no	—
68	C₂×C₄⋊C₄	—	4	1	no	—
69	C₂×C₄²	—	2	1	no	—
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	M₃₂	—	2	1	no	—
72	M₃₂	—	2	1	no	—
73	M₃₂	—	2	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	Q₁₆	1	2	yes	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	1	no	—
80	#108	—	4	1	no	—
81	#108	—	4	1	no	—
82	#108	—	8	1	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#107	—	4	1	no	—
86	#107	—	4	1	no	—
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	4	1	no	—
94	#103	—	8	4	no	—
95	C₂²⋊C₈	—	4	1	no	—
96	#97	—	4	1	no	—
97	#97	—	4	1	no	—
98	C₄×C₈	—	4	1	no	—
99	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	#95	—	2	1	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	8	2	no	—
106	C₂×Q₈	—	8	2	no	—
107	C₂×Q₈	—	8	1	no	—
108	C₂²×C₄	—	4	1	no	—
109	C₂²×C₄	—	8	1	no	—
110	C₂²×C₄	#121	1	1	yes	characteristic
111	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂²×C₄	#137	1	1	yes	characteristic
113	M₁₆	—	8	1	no	—
114	M₁₆	—	8	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	8	2	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	#117	1	1	yes	characteristic
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	#123	1	1	yes	characteristic
127	C₂×C₈	#126	1	1	yes	characteristic
128	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₈	#135	1	1	yes	characteristic
130	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₈	#137	1	1	yes	characteristic
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	8	1	no	—
136	C₄⋊C₄	—	8	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	4	1	no	—
139	C₄²	C₄×D₄	1	1	yes	characteristic
140	C₄²	#124	1	1	yes	characteristic
141	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₁₆	—	2	2	no	—
144	C₁₆	—	4	2	no	—
145	C₁₆	—	4	1	no	—
146	C₂³	#468	1	1	yes	characteristic
147	Q₈	—	8	4	no	—
148	Q₈	—	16	1	no	—
149	Q₈	—	16	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	—	16	2	no	—
157	C₂×C₄	#388	1	1	yes	characteristic
158	C₂×C₄	#440	1	1	yes	characteristic
159	C₂×C₄	#442	1	1	yes	characteristic
160	C₂×C₄	#462	1	1	yes	characteristic
161	C₂×C₄	#463	1	1	yes	characteristic
162	C₂×C₄	#469	1	1	yes	characteristic
163	C₂×C₄	#471	1	1	yes	characteristic
164	C₂×C₄	#473	1	1	yes	characteristic
165	C₂×C₄	#489	1	1	yes	characteristic
166	C₂×C₄	#490	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	8	2	no	—
170	C₈	—	8	1	no	—
171	C₈	—	16	1	no	—
172	C₈	—	16	1	no	—
173	C₈	C₄×Q₁₆	1	1	yes	characteristic
174	C₈	#440	1	1	yes	characteristic
175	C₈	#484	1	1	yes	characteristic
176	C₈	#485	1	1	yes	characteristic
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#1692	1	1	yes	characteristic
180	C₂²	#1699	1	1	yes	characteristic
181	C₂²	#1936	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	16	1	no	—
186	C₄	—	16	2	no	—
187	C₄	#1694	1	1	yes	characteristic
188	C₄	#1701	1	1	yes	characteristic
189	C₄	#1928	1	1	yes	characteristic
190	C₄	#1930	1	1	yes	characteristic
191	C₄	AGL(1,ℤ/16)	1	2	yes	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#9574	1	1	yes	characteristic
195	C₂	#13155	1	1	yes	characteristic
196	C₂	#13157	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#468
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ 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
22. #1364	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13290	2
#12666	2
#11940	2
#9740	2
#9630	2
#7280	2
#7158	2