Finite Group Databasebeta

← #127810

Subgroups — #127810

Order 512.

618 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 128 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#21853C211yesmaximal, characteristic
2#20723C212yesmaximal
3#7283C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5C2×#432C411yescharacteristic, complemented
6C2×#423C411yescharacteristic, complemented
7#2283—22no—
8#1985C412yescomplemented
9#1955C2212yes—
10#1870—22no—
11#1820C414yescomplemented
12#1799—22no—
13#1799C414yescomplemented
14#1797—22no—
15#1782—22no—
16C4×#99—22no—
17C4×#99C2214yes—
18C22×#41D411yescharacteristic
19C22×M16—21no—
20C4×#41—42no—
21C4×#41—42no—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×Q8Q811yescharacteristic
24C2×C4×D4C2×C412yes—
25C2×C4×D4D412yes—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4C2311yescharacteristic
28C2×#117C2×C412yes—
29C2×#117Q811yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#432—42no—
33#423—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C412yes—
37C2×#99—24no—
38C2×#99—24no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—24no—
41C2×#97C2×C412yes—
42#383—42no—
43#382—42no—
44#381—24no—
45#380—24no—
46C4×C4⋊C4—42no—
47C4×C22⋊C4—24no—
48#376—42no—
49#342—24no—
50#342C2×C418yes—
51#336—24no—
52C2×#41—42no—
53C2×#41—44no—
54C22×Q8C4⋊C411yescharacteristic
55C22×D4C22⋊C412yes—
56C22×D4C2×D411yescharacteristic
57C23×C4C4211yescharacteristic
58C23×C4C22⋊C412yes—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C4⋊C411yescharacteristic
62C23×C4C2×D412yes—
63C2×M16—42no—
64C2×M16—82no—
65C22×C8—22no—
66C22×C8—22no—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×Q8—42no—
70C4×D4—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—42no—
74C4×D4—42no—
75C4×D4—41no—
76C4×D4—41no—
77#117—42no—
78#117—42no—
79#117—42no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4C4212yes—
82C2×C4⋊C4C4⋊C412yes—
83C2×C4⋊C4C4⋊C411yescharacteristic
84C2×C4⋊C4C2×Q811yescharacteristic
85C2×C22⋊C4—28no—
86C2×C22⋊C4C22⋊C412yes—
87C2×C22⋊C4C4⋊C412yes—
88C2×C22⋊C4C22×C412yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—44no—
96C2×C42C4211yescharacteristic
97C2×C42C22⋊C412yes—
98C2×C42C22⋊C411yescharacteristic
99C2×C42C4⋊C412yes—
100C2×C42C4⋊C412yes—
101C2×C42C4⋊C411yescharacteristic
102C2×C42C22×C411yescharacteristic
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106#99—48no—
107#99—48no—
108C22⋊C8—42no—
109C22⋊C8—84no—
110#97—42no—
111#95—44no—
112#95—44no—
113#95—44no—
114#95—44no—
115C24#9511yescharacteristic
116C24C2×C22⋊C412yes—
117#41—416no—
118C2×Q8—41no—
119C2×Q8—42no—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—42no—
123C2×D4—41no—
124C2×D4—42no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—48no—
143C22×C4—82no—
144C22×C4—81no—
145C22×C4#9512yes—
146C22×C4#9511yescharacteristic
147C22×C4#9511yescharacteristic
148C22×C4#9512yes—
149C22×C4#9511yescharacteristic
150C22×C4C2×C4211yescharacteristic
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×C4⋊C411yescharacteristic
153C22×C4C2×C4⋊C412yes—
154M16—84no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—88no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—48no—
169C22⋊C4—48no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—22no—
176C42—21no—
177C42—22no—
178C42—21no—
179C42—22no—
180C42—84no—
181C42—84no—
182C42#9912yes—
183C42#9912yes—
184C23—22no—
185C23—22no—
186C23—21no—
187C23—22no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—41no—
192C23C2×#9511yescharacteristic
193C23#42114yes—
194Q8—44no—
195D4—44no—
196D4—48no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—88no—
227C2×C4—88no—
228C2×C4—88no—
229C2×C4#34212yes—
230C2×C4#34212yes—
231C2×C4C2×#9911yescharacteristic
232C2×C4C2×#9911yescharacteristic
233C8—82no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—24no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—82no—
245C22—82no—
246C22#173312yes—
247C22C2×#34211yescharacteristic
248C4—22no—
249C4—21no—
250C4—22no—
251C4—42no—
252C4—41no—
253C4—41no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—168no—
258C4#212811yescharacteristic
259C4#212911yescharacteristic
260C2—22no—
261C2—44no—
262C2—81no—
263C2#804911yescharacteristic
264C2#1257711yescharacteristic
265C2#1258311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7283C2semidirect
5. C2×#432C4semidirect
6. C2×#423C4semidirect
8. #1985C4semidirect
11. #1820C4semidirect
13. #1799C4semidirect

Maximal subgroups

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

subgroupindex
#218532
#207232
#72832