Finite Group Databasebeta

← #106151

Subgroups — #106151

Order 512.

1058 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 194 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
1C2×#2298C211yesmaximal, characteristic
2C2×#1822C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4C2×#1284C214yesmaximal
5C22×#117C2211yescharacteristic
6C2×#407C2212yes—
7#2298—28no—
8C2×#388—22no—
9C2×#387—22no—
10C2×#381C2212yes—
11C2×C4×C4⋊C4C412yescomplemented
12C2×C4×C22⋊C4C2212yes—
13C2×#376C411yescharacteristic, complemented
14C2×C43C411yescharacteristic, complemented
15#1822—22no—
16#1822C414yescomplemented
17#1820—22no—
18#1820C414yescomplemented
19C2×#343—22no—
20C2×#342—22no—
21#1284—28no—
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C2×#117—24no—
25C2×#117—24no—
26C2×#117C2×C412yes—
27C2×#117C2×C412yes—
28C2×#117D412yes—
29C2×#117Q812yes—
30C22×C4⋊C4—22no—
31C22×C4⋊C4D412yes—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—22no—
34C22×C22⋊C4D412yes—
35C22×C42—22no—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C412yes—
39C22×C42C2×C412yes—
40#407—24no—
41#407—22no—
42#407C2×C414yes—
43C2×C22⋊C8—24no—
44#388—216no—
45#387—216no—
46#381—24no—
47#381—22no—
48#381C2×C414yes—
49C4×C4⋊C4—24no—
50C4×C4⋊C4—24no—
51C4×C22⋊C4—24no—
52C4×C22⋊C4D418yes—
53#376—24no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—24no—
58C43—24no—
59#343—28no—
60#342—28no—
61C25C2×D411yescharacteristic
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C412yes—
69C23×C4C22×C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D414yes—
75C23×C4C2×Q811yescharacteristic
76C23×C4#4114yes—
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—44no—
80C2×M16—44no—
81C22×C8—42no—
82#117—28no—
83#117—28no—
84#117—28no—
85#117—28no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4C22⋊C414yes—
91C2×C4⋊C4C4⋊C414yes—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—28no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—48no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4C22⋊C414yes—
101C2×C22⋊C4C4⋊C414yes—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—28no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—48no—
114C2×C42C4212yes—
115C2×C42C4212yes—
116C2×C42C22⋊C414yes—
117C2×C42C22⋊C414yes—
118C2×C42C4⋊C412yes—
119C2×C42C4⋊C412yes—
120C22⋊C8—28no—
121C22⋊C8—44no—
122#95—28no—
123#95—28no—
124#95—28no—
125#95—28no—
126#95—44no—
127C24—22no—
128C24—22no—
129C24—44no—
130C24C2×C22⋊C411yescharacteristic
131C24C2×C22⋊C411yescharacteristic
132C24C2×C4⋊C411yescharacteristic
133C24#12012yes—
134C24#12312yes—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—28no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—88no—
167C22×C4#9514yes—
168C22×C4#9514yes—
169C22×C4C2×C4211yescharacteristic
170C22×C4C2×C22⋊C412yes—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C4⋊C411yescharacteristic
173C22×C4C2×C4⋊C411yescharacteristic
174C22×C4#11712yes—
175C22×C4C4×D414yes—
176C22×C4C4×D414yes—
177C22×C4#12012yes—
178C22×C4#12114yes—
179C22×C4C22⋊Q814yes—
180C22×C4#12312yes—
181M16—48no—
182M16—84no—
183C2×C8—44no—
184C2×C8—82no—
185C2×C8—82no—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—416no—
189C22⋊C4—28no—
190C22⋊C4—28no—
191C22⋊C4—416no—
192C22⋊C4—48no—
193C22⋊C4—84no—
194C42—24no—
195C42—24no—
196C42—24no—
197C42—24no—
198C42—44no—
199C42—44no—
200C42—44no—
201C42—44no—
202C42—416no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—28no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—44no—
211C23—42no—
212C23—42no—
213C23—84no—
214C23—84no—
215C23C2×#9511yescharacteristic
216C23#37911yescharacteristic
217C23#38111yescharacteristic
218C23#38512yes—
219C23#38612yes—
220C23#41012yes—
221C23#41012yes—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—48no—
244C2×C4—48no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4—88no—
257C2×C4—164no—
258C2×C4—164no—
259C2×C4C4×C22⋊C412yes—
260C2×C4#38012yes—
261C2×C4#38512yes—
262C2×C4#38612yes—
263C8—84no—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—44no—
273C22—44no—
274C22—84no—
275C22—82no—
276C22—82no—
277C22C22⋊#9511yescharacteristic
278C22#174012yes—
279C22#178212yes—
280C22#182011yescharacteristic
281C22#182011yescharacteristic
282C22#182714yes—
283C4—22no—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—48no—
288C4—44no—
289C4—44no—
290C4—84no—
291C4—164no—
292C2—22no—
293C2—22no—
294C2—42no—
295C2—42no—
296C2#874012yes—
297C2#874112yes—
298C2#1072611yescharacteristic
299C2#1075812yes—
subgroupG/H
center, socleC23#381
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C2×C4×C4⋊C4C4semidirect
13. C2×#376C4semidirect
14. C2×C43C4semidirect
16. #1822C4semidirect
18. #1820C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22982
C2×#18222
C2×#18202
C2×#12842