Finite Group Databasebeta

← #95191

Subgroups — #95191

Order 512.

1100 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20714C211yesmaximal, characteristic
2#13753C212yesmaximal
3#7035C214yesmaximal
4C23×C42C2211yescharacteristic
5C2×#431C412yescomplemented
6C2×C4×C22⋊C4—22no—
7C2×#376—22no—
8#1817C412yescomplemented
9#1797—22no—
10#1779C2212yes—
11C4×C22⋊C8—22no—
12#1434—22no—
13#1434C414yescomplemented
14C22⋊#95—22no—
15C22⋊#95C2214yes—
16C4×#95C414yescomplemented
17C24×C4D411yescharacteristic
18C24×C4D411yescharacteristic
19C24×C4C2311yescharacteristic
20C22×C22⋊C4—24no—
21C22×C42—24no—
22C22×C42—44no—
23C22×C42C2×C411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C22×C42C2×C412yes—
26C22×C42C2×C412yes—
27C22×C42D412yes—
28C22×C42D412yes—
29C22×C42Q811yescharacteristic
30C22×C42Q811yescharacteristic
31#431—44no—
32C2×C4⋊C8—22no—
33C2×C22⋊C8—24no—
34C2×C22⋊C8C2×C414yes—
35C2×C4×C8—22no—
36C4×C22⋊C4—216no—
37#376—48no—
38C2×#95—24no—
39C2×#95—24no—
40C2×#95—24no—
41C2×#95—24no—
42C2×#95C2×C418yes—
43#336—24no—
44C25C2×D411yescharacteristic
45C23×C4—24no—
46C23×C4—28no—
47C23×C4—28no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4—42no—
52C23×C4—44no—
53C23×C4C22⋊C412yes—
54C23×C4C22⋊C412yes—
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C412yes—
57C23×C4C4⋊C411yescharacteristic
58C23×C4C4⋊C411yescharacteristic
59C23×C4C4⋊C411yescharacteristic
60C23×C4C4⋊C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C22×C412yes—
63C23×C4C2×D412yes—
64C23×C4C2×Q811yescharacteristic
65C22×C8—24no—
66C2×C22⋊C4—216no—
67C2×C22⋊C4—48no—
68C2×C22⋊C4—48no—
69C2×C42—28no—
70C2×C42—24no—
71C2×C42—216no—
72C2×C42—44no—
73C2×C42—44no—
74C2×C42—42no—
75C2×C42—44no—
76C2×C42—42no—
77C2×C42—42no—
78C2×C42—48no—
79C2×C42—48no—
80C2×C42C4211yescharacteristic
81C2×C42C4211yescharacteristic
82C2×C42C4212yes—
83C2×C42C22⋊C412yes—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C4⋊C412yes—
87C2×C42C4⋊C412yes—
88C2×C42C4⋊C412yes—
89C2×C42C4⋊C412yes—
90C4⋊C8—44no—
91C22⋊C8—48no—
92C22⋊C8—44no—
93C4×C8—44no—
94#95—216no—
95#95—48no—
96#95—48no—
97#95—48no—
98#95—48no—
99C24—24no—
100C24—24no—
101C24—41no—
102C24—41no—
103C24C2×C22⋊C412yes—
104C24C2×C4⋊C411yescharacteristic
105C24#13514yes—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—216no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—416no—
139C22×C4—416no—
140C22×C4—416no—
141C22×C4—48no—
142C22×C4—48no—
143C22×C4—48no—
144C22×C4—48no—
145C22×C4—48no—
146C22×C4—48no—
147C22×C4#9512yes—
148C22×C4#9511yescharacteristic
149C22×C4#9511yescharacteristic
150C22×C4#9512yes—
151C22×C4#9511yescharacteristic
152C22×C4#9511yescharacteristic
153C22×C4#10418yes—
154C22×C4C2×C4211yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×C4⋊C412yes—
157C2×C8—48no—
158C2×C8—44no—
159C22⋊C4—432no—
160C42—22no—
161C42—22no—
162C42—24no—
163C42—24no—
164C42—22no—
165C42—24no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—432no—
170C42#9914yes—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—21no—
176C23—22no—
177C23—21no—
178C23—22no—
179C23—216no—
180C23—44no—
181C23—44no—
182C23—44no—
183C23—48no—
184C23C2×#9511yescharacteristic
185C23#41014yes—
186C23#41614yes—
187C23C2×#10414yes—
188C23#42712yes—
189C2×C4—28no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—216no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—44no—
209C2×C4—48no—
210C2×C4—48no—
211C2×C4—48no—
212C2×C4—48no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—416no—
219C2×C4—416no—
220C2×C4—416no—
221C2×C4—88no—
222C2×C4—88no—
223C2×C4—88no—
224C2×C4—88no—
225C2×C4H(3,ℤ/4)12yes—
226C2×C4#33914yes—
227C2×C4#33914yes—
228C2×C4#34212yes—
229C2×C4#34212yes—
230C2×C4C2×#9912yes—
231C8—84no—
232C22—24no—
233C22—24no—
234C22—24no—
235C22—24no—
236C22—24no—
237C22—24no—
238C22—22no—
239C22—24no—
240C22—416no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22#150018yes—
245C22C2×H(3,ℤ/4)11yescharacteristic
246C22C2×#33912yes—
247C22#173511yescharacteristic
248C22C2×#34211yescharacteristic
249C22#174012yes—
250C4—22no—
251C4—22no—
252C4—24no—
253C4—22no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—42no—
258C4—816no—
259C4#129714yes—
260C2—28no—
261C2—42no—
262C2#794112yes—
263C2#798211yescharacteristic
264C2#801312yes—
265C2C2×#129711yescharacteristic
266C2#810011yescharacteristic
subgroupG/H
centerC22×C4#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC23C2×#95
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#431C4semidirect
8. #1817C4semidirect
13. #1434C4semidirect
16. C4×#95C4semidirect

Maximal subgroups

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

subgroupindex
#207142
#137532
#70352