Finite Group Databasebeta

← #121784

Subgroups — #121784

Order 512.

792 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 170 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#13966C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C2×#1294C212yesmaximal
4C2×#1289C212yesmaximal
5#7907C211yesmaximal, characteristic
6C22×C22⋊C8C2212yes—
7C2×#380C2212yes—
8C2×#380C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C2211yescharacteristic
11C2×#342—22no—
12#1442—22no—
13#1434C414yescomplemented
14C22⋊#95—22no—
15#1294—24no—
16#1289—24no—
17#1280—22no—
18C24×C4—21no—
19C24×C4C2311yescharacteristic
20C23×C8—22no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4—21no—
23C22×C22⋊C4—22no—
24C22×C22⋊C4—21no—
25C22×C22⋊C4D412yes—
26C22×C42C2×C412yes—
27C2×C22⋊C8—22no—
28C2×C22⋊C8—22no—
29C2×C22⋊C8—22no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8D414yes—
33#380—22no—
34#380—22no—
35#380—22no—
36#380—22no—
37#380—22no—
38#380Q814yes—
39C4×C22⋊C4—22no—
40C4×C22⋊C4—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—24no—
45C2×#95—21no—
46C2×#95—24no—
47C2×#95C2×C412yes—
48C2×#95C2×C412yes—
49C2×#95D412yes—
50#342—44no—
51C25C2×D411yescharacteristic
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—44no—
55C23×C4—41no—
56C23×C4—41no—
57C23×C4—42no—
58C23×C4—44no—
59C23×C4—82no—
60C23×C4—82no—
61C23×C4C22⋊C414yes—
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D412yes—
65C23×C4C2×Q812yes—
66C23×C4#4112yes—
67C23×C4#4112yes—
68C23×C4#4112yes—
69C23×C4#4112yes—
70C22×C8—24no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8—84no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—82no—
78C2×C4⋊C4—84no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—44no—
85C2×C22⋊C4—44no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—82no—
94C22⋊C8—24no—
95C22⋊C8—24no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—24no—
102#95—48no—
103#95—42no—
104#95—42no—
105#95—84no—
106C24—42no—
107C24—42no—
108C24C2×C22⋊C411yescharacteristic
109C24C22⋊Q812yes—
110C24#12312yes—
111C24#12411yescharacteristic
112C24#12711yescharacteristic
113C24#13514yes—
114C24AGL(1,ℤ/8)12yes—
115C24#13712yes—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—816no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4—82no—
145C22×C4—82no—
146C22×C4—84no—
147C22×C4—84no—
148C22×C4—84no—
149C22×C4#11711yescharacteristic
150C22×C4#11711yescharacteristic
151C22×C4C4×D412yes—
152C22×C4C4×Q812yes—
153C22×C4#12112yes—
154C22×C4C22⋊Q812yes—
155C22×C4C22⋊Q812yes—
156C22×C4#12312yes—
157C22×C4#12312yes—
158C22×C4#12312yes—
159C22×C4#12411yescharacteristic
160C22×C4#12512yes—
161C22×C4#12512yes—
162C22×C4#12612yes—
163C22×C4#12612yes—
164C22×C4C4⋊Q811yescharacteristic
165C2×C8—44no—
166C2×C8—84no—
167C2×C8—84no—
168C2×C8—84no—
169C4⋊C4—84no—
170C4⋊C4—84no—
171C4⋊C4—84no—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C22⋊C4—48no—
175C22⋊C4—82no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C42—82no—
181C42—82no—
182C23—22no—
183C23—24no—
184C23—22no—
185C23—24no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—42no—
191C23—48no—
192C23—42no—
193C23—44no—
194C23—44no—
195C23#38111yescharacteristic
196C23#39011yescharacteristic
197C23#39111yescharacteristic
198C23#39912yes—
199C23#40012yes—
200C23#41612yes—
201C23#41712yes—
202C23#47814yes—
203C23#47914yes—
204C23#48214yes—
205C23#48514yes—
206C23#48812yes—
207C23#48912yes—
208C23#49512yes—
209C23#49611yescharacteristic
210C23#49711yescharacteristic
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—84no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4—88no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—84no—
239C2×C4—82no—
240C2×C4—84no—
241C2×C4—88no—
242C2×C4—82no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—82no—
246C2×C4—168no—
247C2×C4#38712yes—
248C2×C4#38712yes—
249C2×C4#39712yes—
250C2×C4#39812yes—
251C8—84no—
252C22—24no—
253C22—24no—
254C22—28no—
255C22—28no—
256C22—21no—
257C22—22no—
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—42no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22#144511yescharacteristic
272C22#182512yes—
273C22#182712yes—
274C22#182812yes—
275C22#195811yescharacteristic
276C22#195911yescharacteristic
277C22#198912yes—
278C22#209212yes—
279C22#209712yes—
280C22#210114yes—
281C4—22no—
282C4—22no—
283C4—22no—
284C4—22no—
285C4—84no—
286C4—84no—
287C4—162no—
288C4—164no—
289C2—21no—
290C2—22no—
291C2—21no—
292C2—22no—
293C2—24no—
294C2—22no—
295C2#876314yes—
296C2#1134811yescharacteristic
297C2#1135111yescharacteristic
298C2#1137211yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×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
13. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#139662
#108462
C2×#12942
C2×#12892
#79072