Finite Group Databasebeta

← #108677

Subgroups — #108677

Order 512.

581 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 140 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#20527C211yesmaximal, characteristic
2#8102C212yesmaximal
3#7982C211yesmaximal, characteristic
4#7982C211yesmaximal, characteristic
5#7946C212yesmaximal
6C2×#407C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2280C2212yes—
9#2280C2212yes—
10#2280C2211yescharacteristic
11#1753—22no—
12#1752C412yescomplemented
13C8×C22⋊C4C412yescomplemented
14#1750—22no—
15#1304—24no—
16#1289—22no—
17#1289—22no—
18#1284C418yescomplemented
19C24×C4C2311yescharacteristic
20C22×M16—21no—
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C2×#117—21no—
24C2×#117—22no—
25C2×#117—22no—
26#407—22no—
27#407C2×C414yes—
28C2×C22⋊C8—21no—
29C2×C22⋊C8—22no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8C2×C412yes—
32C2×C22⋊C8D412yes—
33C2×C22⋊C8D412yes—
34C2×#97—22no—
35C2×C4×C8—22no—
36#381—21no—
37#381—22no—
38#381C2×C414yes—
39#380—21no—
40#380—22no—
41#380—22no—
42#380—22no—
43#380—22no—
44#380Q812yes—
45#379—22no—
46#379—22no—
47#379D412yes—
48C4×C22⋊C4—21no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4C2×C414yes—
53C4×C22⋊C4C2×C412yes—
54#336—22no—
55#336—22no—
56C25C2×D411yescharacteristic
57C23×C4—81no—
58C23×C4—81no—
59C23×C4C22×C411yescharacteristic
60C23×C4C22×C412yes—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4C2×Q811yescharacteristic
64C23×C4#4112yes—
65C23×C4#4112yes—
66C23×C4#4111yescharacteristic
67C23×C4#4111yescharacteristic
68C23×C4#4112yes—
69C2×M16—24no—
70C2×M16—81no—
71C2×M16—81no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—81no—
78C22×C8—81no—
79C22×C8C22⋊C412yes—
80C22×C8C22⋊C412yes—
81#117—81no—
82#117—81no—
83#117—82no—
84#117—82no—
85#117—82no—
86#117—82no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4C4⋊C414yes—
99C2×C42—22no—
100C2×C42—24no—
101C2×C42—42no—
102C22⋊C8—22no—
103C22⋊C8—24no—
104C22⋊C8—24no—
105C22⋊C8—24no—
106C22⋊C8C4214yes—
107#97—82no—
108C4×C8—82no—
109#95—22no—
110#95—22no—
111#95—24no—
112#95—24no—
113#95—24no—
114#95—42no—
115C24—42no—
116C24—81no—
117C24—81no—
118C24C2×C4⋊C411yescharacteristic
119C24C4×D412yes—
120C24#12111yescharacteristic
121C24#12111yescharacteristic
122C24#12412yes—
123C22×C4—24no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—82no—
137C22×C4—81no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4—82no—
141C22×C4C2×C4211yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4#11711yescharacteristic
144C22×C4#11711yescharacteristic
145C22×C4#11711yescharacteristic
146C22×C4#11712yes—
147C22×C4#11712yes—
148C22×C4#11711yescharacteristic
149C22×C4C4×D412yes—
150C22×C4C4×D412yes—
151C22×C4C4×Q812yes—
152C22×C4C22⋊Q811yescharacteristic
153C22×C4C22⋊Q811yescharacteristic
154C22×C4#12312yes—
155C22×C4#12312yes—
156C22×C4#12512yes—
157C22×C4#12612yes—
158C22×C4#12612yes—
159M16—84no—
160C2×C8—24no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—81no—
170C2×C8—81no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C22⋊C4—44no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C22⋊C4—84no—
180C22⋊C4—84no—
181C22⋊C4—84no—
182C22⋊C4—84no—
183C22⋊C4—84no—
184C22⋊C4—84no—
185C42—42no—
186C42—42no—
187C42—84no—
188C42—84no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23—41no—
196C23—42no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—88no—
205C23C4×C4⋊C411yescharacteristic
206C23#38011yescharacteristic
207C23#38211yescharacteristic
208C23#38812yes—
209C23#38812yes—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4—84no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—84no—
242C2×C4#37611yescharacteristic
243C2×C4C4×C22⋊C411yescharacteristic
244C2×C4#38111yescharacteristic
245C2×C4#38311yescharacteristic
246C2×C4#38712yes—
247C2×C4#38712yes—
248C8—44no—
249C8—42no—
250C8—42no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—22no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—22no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22—82no—
267C22—82no—
268C22—84no—
269C22—84no—
270C22#143411yescharacteristic
271C22#181212yes—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—84no—
278C4—164no—
279C4—164no—
280C4#182811yescharacteristic
281C4#182811yescharacteristic
282C2—22no—
283C2—42no—
284C2—42no—
285C2—41no—
286C2—41no—
287C2#1081111yescharacteristic
288C2#1081411yescharacteristic
289C2#1082911yescharacteristic
subgroupG/H
center, socleC22#1434
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
13. C8×C22⋊C4C4semidirect
18. #1284C4semidirect

Maximal subgroups

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

subgroupindex
#205272
#81022
#79822
#79462