Finite Group Databasebeta

← #121010

Subgroups — #121010

Order 512.

413 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20823C211yesmaximal, characteristic
2#8106C211yesmaximal, characteristic
3#8094C211yesmaximal, characteristic
4#8088C211yesmaximal, characteristic
5#8088C211yesmaximal, characteristic
6#7976C211yesmaximal, characteristic
7#7953C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1847—21no—
16#1836—21no—
17#1808—21no—
18#1807—21no—
19#1776—22no—
20#1305—21no—
21#1305—21no—
22#1303—22no—
23#1302—21no—
24#1302—21no—
25#1302C414yescomplemented
26#1288—22no—
27#1285—22no—
28C23×C8—21no—
29C2×#117—21no—
30C22×C42C2311yescharacteristic
31#432—21no—
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432C2×C412yes—
38#432D412yes—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431C2×C412yes—
47C2×#107—22no—
48C2×#106—22no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423Q812yes—
53#407—21no—
54#407—21no—
55#407—21no—
56#407—21no—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16—21no—
60C4×M16—21no—
61C2×#97—21no—
62C2×#97—21no—
63C2×#97—21no—
64C2×#97C2×C412yes—
65C2×#97D411yescharacteristic
66C2×#97D411yescharacteristic
67C2×#97Q812yes—
68#384—21no—
69#383—21no—
70#382—21no—
71#382—21no—
72#381—21no—
73#380—21no—
74#380—21no—
75#380D411yescharacteristic
76#380D411yescharacteristic
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D412yes—
80C4×C22⋊C4—21no—
81#376—21no—
82#340—22no—
83#340—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C2×M16—22no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—44no—
96#117—82no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—41no—
105C2×C42—22no—
106C2×C42—41no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119#107—44no—
120#106—44no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#97—22no—
138#97—22no—
139#97—22no—
140#97—21no—
141#97—21no—
142#97—22no—
143#97—41no—
144#97—41no—
145#97C4⋊C412yes—
146#97C4⋊C412yes—
147C4×C8—22no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8—41no—
151#95—21no—
152#95—21no—
153#95—22no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158C24#12311yescharacteristic
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4#11711yescharacteristic
167C22×C4C4×D411yescharacteristic
168C22×C4C4×Q811yescharacteristic
169C22×C4#12011yescharacteristic
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174C22×C4C22⋊Q811yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4#12411yescharacteristic
177C22×C4#12411yescharacteristic
178C22×C4#12511yescharacteristic
179C22×C4#12511yescharacteristic
180C22×C4#12511yescharacteristic
181C22×C4#12611yescharacteristic
182C22×C4#12611yescharacteristic
183C22×C4#12611yescharacteristic
184M16—81no—
185M16—81no—
186M16—82no—
187M16—82no—
188M16—82no—
189C2×C8—24no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C42—82no—
209C42—82no—
210C42C2×C4⋊C411yescharacteristic
211C42#11711yescharacteristic
212C42C4×D411yescharacteristic
213C42C4×D411yescharacteristic
214C42C22⋊Q811yescharacteristic
215C42C22⋊Q811yescharacteristic
216C42C22⋊Q811yescharacteristic
217C42#12311yescharacteristic
218C42#12311yescharacteristic
219C42#12311yescharacteristic
220C42#12411yescharacteristic
221C42#12411yescharacteristic
222C42#12511yescharacteristic
223C42#12511yescharacteristic
224C42#12611yescharacteristic
225C42C4⋊Q811yescharacteristic
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23#38711yescharacteristic
231C23#39411yescharacteristic
232C23#40011yescharacteristic
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
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—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4#38211yescharacteristic
263C2×C4#38511yescharacteristic
264C2×C4#38711yescharacteristic
265C2×C4#38811yescharacteristic
266C2×C4#38811yescharacteristic
267C2×C4#38911yescharacteristic
268C2×C4#39311yescharacteristic
269C2×C4#39711yescharacteristic
270C2×C4#39811yescharacteristic
271C2×C4#39811yescharacteristic
272C2×C4#40011yescharacteristic
273C2×C4#40011yescharacteristic
274C8—24no—
275C8—42no—
276C8—82no—
277C8—82no—
278C8—82no—
279C8—82no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—42no—
289C22—42no—
290C22#144211yescharacteristic
291C22#204811yescharacteristic
292C22#204911yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—161no—
305C4—162no—
306C4—161no—
307C4#203112yes—
308C2—21no—
309C2—21no—
310C2—41no—
311C2—41no—
312C2#1126211yescharacteristic
313C2#1126511yescharacteristic
314C2#1127411yescharacteristic
subgroupG/H
center, socleC22#1442
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
25. #1302C4semidirect

Maximal subgroups

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

subgroupindex
#208232
#81062
#80942
#80882
#79762
#79532