Finite Group Databasebeta

← #127379

Subgroups — #127379

Order 512.

637 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 136 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#21847C211yesmaximal, characteristic
2#20723C212yesmaximal
3#7237C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C22×C4⋊C8C411yescharacteristic, complemented
7#2283—22no—
8#1984C412yescomplemented
9#1954C2212yes—
10#1869—22no—
11#1820C414yescomplemented
12#1799—22no—
13#1799C2214yes—
14#1797—22no—
15#1782—22no—
16C4×#99—22no—
17C4×#99C414yescomplemented
18C22×#41D411yescharacteristic
19C23×C8—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#431—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—42no—
36C2×C4⋊C8C2×C412yes—
37C2×#99—24no—
38C2×#99—24no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—24no—
41C2×C4×C8C2×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—24no—
53C2×#41—42no—
54C2×#41—42no—
55C22×Q8C4⋊C411yescharacteristic
56C22×D4C22⋊C412yes—
57C22×D4C2×D411yescharacteristic
58C23×C4C4211yescharacteristic
59C23×C4C22⋊C412yes—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C412yes—
62C23×C4C4⋊C411yescharacteristic
63C23×C4C2×D412yes—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—42no—
67C22×C8—82no—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×Q8—42no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—42no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—41no—
77C4×D4—41no—
78#117—42no—
79#117—42no—
80#117—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4C4212yes—
83C2×C4⋊C4C4⋊C412yes—
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C2×Q811yescharacteristic
86C2×C22⋊C4—28no—
87C2×C22⋊C4C22⋊C412yes—
88C2×C22⋊C4C4⋊C412yes—
89C2×C22⋊C4C22×C412yes—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—44no—
97C2×C42C4211yescharacteristic
98C2×C42C22⋊C412yes—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C4⋊C412yes—
101C2×C42C4⋊C412yes—
102C2×C42C4⋊C411yescharacteristic
103C2×C42C22×C411yescharacteristic
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107#99—48no—
108#99—48no—
109C22⋊C8—42no—
110C22⋊C8—84no—
111C4×C8—42no—
112#95—44no—
113#95—44no—
114#95—44no—
115#95—44no—
116C24#9511yescharacteristic
117C24C2×C22⋊C412yes—
118#41—416no—
119C2×Q8—22no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—42no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4#13514yes—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—48no—
147C22×C4—82no—
148C22×C4—81no—
149C22×C4#9512yes—
150C22×C4#9511yescharacteristic
151C22×C4#9511yescharacteristic
152C22×C4#9512yes—
153C22×C4#9511yescharacteristic
154C22×C4C2×C4211yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4C2×C4⋊C412yes—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—84no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—48no—
174C22⋊C4—48no—
175C42—22no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42—21no—
182C42—21no—
183C42—22no—
184C42—22no—
185C42—88no—
186C42#9912yes—
187C42#9912yes—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23H(3,ℤ/4)12yes—
197C23C2×#9511yescharacteristic
198C23#41614yes—
199Q8—44no—
200D4—44no—
201D4—48no—
202C2×C4—28no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—88no—
233C2×C4—88no—
234C2×C4—88no—
235C2×C4#34212yes—
236C2×C4#34212yes—
237C2×C4C2×#9911yescharacteristic
238C2×C4C2×#9911yescharacteristic
239C2×C4#42712yes—
240C8—82no—
241C8—82no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—24no—
246C22—42no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—82no—
254C22C2×H(3,ℤ/4)11yescharacteristic
255C22#173511yescharacteristic
256C22C2×#34211yescharacteristic
257C4—22no—
258C4—21no—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—41no—
263C4—41no—
264C4—42no—
265C4—42no—
266C4—168no—
267C4#212811yescharacteristic
268C4#212911yescharacteristic
269C2—21no—
270C2—21no—
271C2—44no—
272C2—42no—
273C2#798211yescharacteristic
274C2#1257711yescharacteristic
275C2#1258311yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7237C2semidirect
5. C2×#431C4semidirect
6. C22×C4⋊C8C4semidirect
8. #1984C4semidirect
11. #1820C4semidirect
17. C4×#99C4semidirect

Maximal subgroups

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

subgroupindex
#218472
#207232
#72372