Finite Group Databasebeta

← #108571

Subgroups — #108571

Order 512.

575 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 142 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#20528C211yesmaximal, characteristic
2C2×#1293C211yesmaximal, characteristic
3#7982C211yesmaximal, characteristic
4#7946C212yesmaximal
5#7925C212yesmaximal
6C2×#407C2212yes—
7C22×C22⋊C8C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9#2280C2212yes—
10#2280C2211yescharacteristic
11#1752—21no—
12#1752C412yescomplemented
13C8×C22⋊C4—21no—
14C8×C22⋊C4C412yescomplemented
15#1750—22no—
16#1293—21no—
17#1293—21no—
18#1289—22no—
19#1284—24no—
20#1282C418yescomplemented
21C24×C4C2311yescharacteristic
22C22×M16—22no—
23C23×C8—21no—
24C23×C8C2×C412yes—
25C2×#117—21no—
26C2×#117—22no—
27#407—22no—
28#407—22no—
29#407—22no—
30#407—22no—
31C2×C22⋊C8—21no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8C2×C414yes—
37C2×C22⋊C8C2×C414yes—
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8D412yes—
41C2×C22⋊C8D411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C2×#97—22no—
44C2×C4×C8—22no—
45#381—22no—
46#381D412yes—
47#380—21no—
48#380—22no—
49#380—22no—
50#380Q812yes—
51#379—21no—
52#379—22no—
53C4×C22⋊C4—21no—
54C4×C22⋊C4—22no—
55C4×C22⋊C4C2×C414yes—
56C4×C22⋊C4C2×C412yes—
57#336—22no—
58#336—22no—
59C25#4111yescharacteristic
60C23×C4—82no—
61C23×C4C22×C411yescharacteristic
62C23×C4C22×C412yes—
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×Q811yescharacteristic
67C23×C4#4112yes—
68C23×C4#4112yes—
69C23×C4#4111yescharacteristic
70C23×C4#4112yes—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—84no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—82no—
81C22×C8—82no—
82C22×C8C22⋊C412yes—
83C22×C8C22⋊C412yes—
84#117—82no—
85#117—84no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C42—22no—
95C2×C42—24no—
96C22⋊C8—21no—
97C22⋊C8—21no—
98C22⋊C8—24no—
99C22⋊C8—24no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102C22⋊C8—24no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8C4212yes—
107C22⋊C8C4212yes—
108#97—82no—
109C4×C8—82no—
110#95—22no—
111#95—24no—
112#95—42no—
113#95C4⋊C414yes—
114C24—42no—
115C24—42no—
116C24—81no—
117C24#11711yescharacteristic
118C24#11712yes—
119C24#12111yescharacteristic
120C24C22⋊Q811yescharacteristic
121C24#12612yes—
122C22×C4—24no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—82no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4—82no—
134C22×C4C2×C4211yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C2×C4⋊C411yescharacteristic
137C22×C4#11711yescharacteristic
138C22×C4#11711yescharacteristic
139C22×C4#11711yescharacteristic
140C22×C4#11712yes—
141C22×C4C4×D412yes—
142C22×C4C4×D412yes—
143C22×C4C4×D412yes—
144C22×C4C4×Q812yes—
145C22×C4#12111yescharacteristic
146C22×C4C22⋊Q811yescharacteristic
147C22×C4#12312yes—
148C22×C4#12312yes—
149C22×C4#12412yes—
150C22×C4#12512yes—
151C22×C4#12612yes—
152M16—84no—
153M16—84no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8—84no—
166C2×C8—84no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C22⋊C4—44no—
170C22⋊C4—44no—
171C22⋊C4—48no—
172C22⋊C4—82no—
173C22⋊C4—84no—
174C22⋊C4—84no—
175C42—42no—
176C42—42no—
177C42—84no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—24no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23—88no—
194C23#37611yescharacteristic
195C23#38011yescharacteristic
196C23#38311yescharacteristic
197C23#38712yes—
198C23#38812yes—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4C4×C22⋊C411yescharacteristic
224C2×C4C4×C4⋊C411yescharacteristic
225C2×C4#38111yescharacteristic
226C2×C4#38211yescharacteristic
227C2×C4#38712yes—
228C2×C4#38812yes—
229C8—42no—
230C8—42no—
231C8—42no—
232C8—42no—
233C8—84no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—21no—
243C22—22no—
244C22—22no—
245C22—21no—
246C22—22no—
247C22—44no—
248C22—44no—
249C22—82no—
250C22—84no—
251C22—84no—
252C22#143411yescharacteristic
253C22#174712yes—
254C22#181412yes—
255C4—21no—
256C4—22no—
257C4—21no—
258C4—21no—
259C4—22no—
260C4—84no—
261C4—164no—
262C4#182812yes—
263C2—22no—
264C2—22no—
265C2—42no—
266C2—41no—
267C2—42no—
268C2#1080011yescharacteristic
269C2#1082811yescharacteristic
270C2#1082911yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1752C4semidirect
14. C8×C22⋊C4C4semidirect
20. #1282C4semidirect

Maximal subgroups

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

subgroupindex
#205282
C2×#12932
#79822
#79462
#79252