Finite Group Databasebeta

← #120346

Subgroups — #120346

Order 512.

433 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21965C211yesmaximal, characteristic
2#17796C211yesmaximal, characteristic
3#8106C211yesmaximal, characteristic
4#8003C211yesmaximal, characteristic
5#7973C211yesmaximal, characteristic
6#7972C211yesmaximal, characteristic
7#7951C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—22no—
16#1949—22no—
17#1809—22no—
18#1569—22no—
19#1567C412yescomplemented
20#1566C412yescomplemented
21#1565—22no—
22#1305—22no—
23#1292—22no—
24#1288—21no—
25#1288—21no—
26#1288—21no—
27#1288—21no—
28#1285—22no—
29C22×M16—21no—
30C22×M16—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432Q812yes—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44C2×#107—22no—
45C2×#106—22no—
46#423—21no—
47#423—21no—
48#423—21no—
49#423C2×C411yescharacteristic
50#423C2×C411yescharacteristic
51#423C2×C412yes—
52#423D412yes—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8D412yes—
56#407—21no—
57#407—21no—
58C2×C22⋊C8—21no—
59C4×M16—21no—
60C4×M16—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16D411yescharacteristic
64C4×M16D411yescharacteristic
65C4×M16Q812yes—
66C2×#97—21no—
67C2×#97Q812yes—
68C2×C4×C8C2×C412yes—
69#383—21no—
70#383—21no—
71#382—21no—
72#382—21no—
73#381—21no—
74#381—21no—
75#380—21no—
76#380D412yes—
77C4×C4⋊C4—21no—
78C4×C4⋊C4D412yes—
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81#376—21no—
82#376—21no—
83#340—24no—
84#335—24no—
85#334—24no—
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—44no—
97C2×M16—41no—
98C2×M16—81no—
99C2×M16—81no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—41no—
107C2×C42—22no—
108C2×C42—41no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×Q811yescharacteristic
115C2×C42C2×Q811yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121#107—82no—
122#106—82no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8C22⋊C414yes—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#97—21no—
142#97—21no—
143#97—22no—
144#97—22no—
145#97—22no—
146#97—22no—
147#97—41no—
148C4×C8—22no—
149C4×C8—41no—
150C4×C8—41no—
151C4×C8C4⋊C412yes—
152C4×C8C4⋊C412yes—
153#95—22no—
154#95—22no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161C24#12311yescharacteristic
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4#11711yescharacteristic
169C22×C4C4×D411yescharacteristic
170C22×C4C4×Q811yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4#12311yescharacteristic
180C22×C4#12411yescharacteristic
181C22×C4#12511yescharacteristic
182C22×C4#12511yescharacteristic
183C22×C4#12511yescharacteristic
184C22×C4#12611yescharacteristic
185C22×C4C4⋊Q811yescharacteristic
186M16—24no—
187M16—44no—
188M16—42no—
189M16—82no—
190M16—82no—
191M16—81no—
192M16—81no—
193M16—81no—
194M16—81no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—24no—
198C2×C8—44no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—82no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—82no—
213C42—82no—
214C42C2×C22⋊C411yescharacteristic
215C42C2×C4⋊C411yescharacteristic
216C42C4×D411yescharacteristic
217C42C4×Q811yescharacteristic
218C42#12111yescharacteristic
219C42#12111yescharacteristic
220C42C22⋊Q811yescharacteristic
221C42C22⋊Q811yescharacteristic
222C42C22⋊Q811yescharacteristic
223C42C22⋊Q811yescharacteristic
224C42C22⋊Q811yescharacteristic
225C42#12311yescharacteristic
226C42#12511yescharacteristic
227C42#12611yescharacteristic
228C42C4⋊Q811yescharacteristic
229C42C4⋊Q811yescharacteristic
230C23—22no—
231C23—41no—
232C23—41no—
233C23#38711yescharacteristic
234C23#39411yescharacteristic
235C23#39811yescharacteristic
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#38011yescharacteristic
264C2×C4#38611yescharacteristic
265C2×C4#38711yescharacteristic
266C2×C4#38911yescharacteristic
267C2×C4#38911yescharacteristic
268C2×C4#39111yescharacteristic
269C2×C4#39511yescharacteristic
270C2×C4#39611yescharacteristic
271C2×C4#39611yescharacteristic
272C2×C4#39811yescharacteristic
273C2×C4#39811yescharacteristic
274C2×C4#40011yescharacteristic
275C8—24no—
276C8—42no—
277C8—44no—
278C8—82no—
279C8—82no—
280C8—81no—
281C8—81no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—82no—
291C22#144111yescharacteristic
292C22#205012yes—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—162no—
305C4—162no—
306C4#190411yescharacteristic
307C4#190511yescharacteristic
308C2—22no—
309C2—41no—
310C2—41no—
311C2#1121311yescharacteristic
312C2#1124311yescharacteristic
313C2#1124711yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1567C4semidirect
20. #1566C4semidirect

Maximal subgroups

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

subgroupindex
#219652
#177962
#81062
#80032
#79732
#79722
#79512