Finite Group Databasebeta

← #121543

Subgroups — #121543

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#17817C211yesmaximal, characteristic
3#17795C211yesmaximal, characteristic
4#8090C211yesmaximal, characteristic
5#8058C211yesmaximal, characteristic
6#7918C211yesmaximal, characteristic
7#7876C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17#1848—21no—
18#1837—21no—
19#1574—22no—
20#1573—22no—
21#1569—21no—
22#1569—21no—
23#1569—21no—
24#1569—21no—
25#1567—21no—
26#1566—21no—
27#1564—22no—
28#1302—22no—
29#1298—22no—
30#1281C412yes—
31#1281C412yes—
32#1277—22no—
33C22×M16—21no—
34C22×M16Q812yes—
35C22×C42C2311yescharacteristic
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432—21no—
43#432Q812yes—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431—21no—
51#431—21no—
52#431C2×C412yes—
53#431C2×C412yes—
54#431D412yes—
55#431Q812yes—
56C2×#108—24no—
57#423—21no—
58#423—21no—
59#423—21no—
60#423D411yescharacteristic
61#423D411yescharacteristic
62#423D411yescharacteristic
63#423D411yescharacteristic
64#423D412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D412yes—
74C2×C22⋊C8Q812yes—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C2×#97—21no—
82C2×#97C2×C411yescharacteristic
83C2×#97C2×C411yescharacteristic
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8D412yes—
87#341—24no—
88#335—22no—
89#335—22no—
90#335—24no—
91#334—22no—
92#334—22no—
93#334—24no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×Q811yescharacteristic
96C23×C4C2×Q811yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—24no—
100C2×M16—22no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—21no—
105C2×M16—42no—
106C2×M16—41no—
107C2×M16—82no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8C4⋊C412yes—
113C22×C8C4⋊C412yes—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42C2×Q811yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126#108—84no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8C4⋊C414yes—
154C22⋊C8C4⋊C414yes—
155#97—21no—
156#97—21no—
157#97—21no—
158#97—21no—
159#97—22no—
160#97—41no—
161#97—41no—
162#97—41no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—41no—
170C4×C8—41no—
171C24C4⋊Q811yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—81no—
177C22×C4C2×C4⋊C411yescharacteristic
178C22×C4C2×C4⋊C411yescharacteristic
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12711yescharacteristic
191C22×C4C4⋊Q811yescharacteristic
192C22×C4C4⋊Q811yescharacteristic
193C22×C4C4⋊Q811yescharacteristic
194C22×C4C4⋊Q811yescharacteristic
195M16—44no—
196M16—44no—
197M16—42no—
198M16—44no—
199M16—42no—
200M16—82no—
201M16—81no—
202M16—81no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—24no—
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—82no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—82no—
222C2×C8#10814yescomplemented
223C42C4×D411yescharacteristic
224C42C4×D411yescharacteristic
225C42C4×D411yescharacteristic
226C42C4×Q811yescharacteristic
227C42#12011yescharacteristic
228C42#12011yescharacteristic
229C42#12111yescharacteristic
230C42#12111yescharacteristic
231C42C22⋊Q811yescharacteristic
232C42C22⋊Q811yescharacteristic
233C42C22⋊Q811yescharacteristic
234C42C22⋊Q811yescharacteristic
235C42#12311yescharacteristic
236C42#12311yescharacteristic
237C42#12511yescharacteristic
238C42C4⋊Q811yescharacteristic
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23#38411yescharacteristic
244C23#39511yescharacteristic
245C23#39811yescharacteristic
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4#38511yescharacteristic
268C2×C4#38511yescharacteristic
269C2×C4#38511yescharacteristic
270C2×C4#38911yescharacteristic
271C2×C4#38911yescharacteristic
272C2×C4#38911yescharacteristic
273C2×C4#39211yescharacteristic
274C2×C4#39511yescharacteristic
275C2×C4#39511yescharacteristic
276C2×C4#39811yescharacteristic
277C2×C4#39811yescharacteristic
278C2×C4#39911yescharacteristic
279C2×C4C2×#10812yes—
280C2×C4#43012yes—
281C8—24no—
282C8—44no—
283C8—44no—
284C8—42no—
285C8—82no—
286C8—82no—
287C8—81no—
288C8—81no—
289C8—81no—
290C8—81no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—81no—
300C22—81no—
301C22#144411yescharacteristic
302C22#184811yescharacteristic
303C22#184911yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4#185512yes—
314C4#195012yes—
315C4#200811yescharacteristic
316C4#200911yescharacteristic
317C2—21no—
318C2—21no—
319C2—41no—
320C2—41no—
321C2#1132611yescharacteristic
322C2#1133311yescharacteristic
323C2#1133411yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1444
prime radicalC24C4⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
222. C2×C8#108semidirect

Maximal subgroups

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

subgroupindex
#219932
#178172
#177952
#80902
#80582
#79182
#78762