Finite Group Databasebeta

← #120943

Subgroups — #120943

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#10855C211yesmaximal, characteristic
3#8094C211yesmaximal, characteristic
4#8084C211yesmaximal, characteristic
5#8065C211yesmaximal, characteristic
6#7993C211yesmaximal, characteristic
7#7900C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1950—21no—
16#1950—21no—
17#1855—22no—
18#1849—21no—
19#1838—21no—
20C2×#344—22no—
21#1303—21no—
22#1303—21no—
23#1302—22no—
24#1299—22no—
25#1291C414yescomplemented
26#1279—21no—
27#1279—21no—
28C22×M16—21no—
29C22×M16D412yes—
30C2×#117—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432C2×C412yes—
38#432C2×C412yes—
39#432Q812yes—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431D412yes—
48C2×#108—22no—
49C2×#108—22no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423D411yescharacteristic
56#423D411yescharacteristic
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62#407—21no—
63#407—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16C2×C412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76#384—21no—
77#382—21no—
78#380—21no—
79#380Q812yes—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82#344—44no—
83#341—22no—
84#341—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—22no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—41no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8—41no—
106#117—42no—
107#117—42no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C22⋊C4—22no—
112C2×C42—41no—
113C2×C42C22×C411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×Q811yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125#108—82no—
126#108—82no—
127C4⋊C8—22no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—22no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8C4⋊C412yes—
142C4⋊C8C4⋊C412yes—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#97—22no—
152#97—22no—
153#97—22no—
154#97—22no—
155#97—41no—
156#97—41no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—41no—
163#95—22no—
164#95—41no—
165C24C22⋊Q811yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4#11711yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4C4×Q811yescharacteristic
176C22×C4#12011yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4#12311yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12411yescharacteristic
184C22×C4#12411yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12511yescharacteristic
187C22×C4#12511yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12611yescharacteristic
190C22×C4C4⋊Q811yescharacteristic
191M16—24no—
192M16—42no—
193M16—42no—
194M16—42no—
195M16—41no—
196M16—42no—
197M16—41no—
198M16—41no—
199M16—42no—
200M16—41no—
201M16—82no—
202M16—82no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—44no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—82no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C22⋊C4—44no—
223C42—81no—
224C42—81no—
225C42C2×C4⋊C411yescharacteristic
226C42#11711yescharacteristic
227C42C4×D411yescharacteristic
228C42C4×D411yescharacteristic
229C42#12111yescharacteristic
230C42#12111yescharacteristic
231C42C22⋊Q811yescharacteristic
232C42#12311yescharacteristic
233C42#12311yescharacteristic
234C42#12311yescharacteristic
235C42#12411yescharacteristic
236C42#12511yescharacteristic
237C42#12511yescharacteristic
238C42#12611yescharacteristic
239C42#12611yescharacteristic
240C42#12611yescharacteristic
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#38711yescharacteristic
246C23#39311yescharacteristic
247C23#39811yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4#38211yescharacteristic
274C2×C4#38511yescharacteristic
275C2×C4#38711yescharacteristic
276C2×C4#38811yescharacteristic
277C2×C4#38811yescharacteristic
278C2×C4#38911yescharacteristic
279C2×C4#39411yescharacteristic
280C2×C4#39711yescharacteristic
281C2×C4#39811yescharacteristic
282C2×C4#40011yescharacteristic
283C2×C4#40011yescharacteristic
284C2×C4#40011yescharacteristic
285C2×C4#45311yescharacteristic
286C2×C4#45411yescharacteristic
287C2×C4#45511yescharacteristic
288C2×C4#45611yescharacteristic
289C8—44no—
290C8—42no—
291C8—42no—
292C8—82no—
293C8—82no—
294C8—82no—
295C8—81no—
296C8—81no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—81no—
306C22—81no—
307C22#144211yescharacteristic
308C22#202711yescharacteristic
309C22#202811yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—162no—
320C4#185911yescharacteristic
321C4#186011yescharacteristic
322C4#203711yescharacteristic
323C4#203811yescharacteristic
324C4#210312yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2#1127311yescharacteristic
330C2#1127611yescharacteristic
331C2#1129511yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1442
prime radicalC24C22⋊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
25. #1291C4semidirect

Maximal subgroups

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

subgroupindex
#219942
#108552
#80942
#80842
#80652
#79932
#79002