Finite Group Databasebeta

← #120551

Subgroups — #120551

Order 512.

449 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#21963C211yesmaximal, characteristic
3#8129C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#7957C211yesmaximal, characteristic
6#7953C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1950—22no—
16#1949—22no—
17#1948—21no—
18#1947—21no—
19#1855—22no—
20#1848C412yescomplemented
21#1837C412yescomplemented
22#1809—22no—
23#1309—22no—
24#1303—21no—
25#1303—21no—
26#1286—22no—
27#1285—22no—
28#1274—22no—
29C22×M16—21no—
30C22×M16D412yes—
31C2×#117—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431D412yes—
44#431Q812yes—
45C2×#108—24no—
46C2×#107—22no—
47C2×#106—22no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423C2×C411yescharacteristic
55#423C2×C411yescharacteristic
56#423C2×C412yes—
57#423D412yes—
58#423Q812yes—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64#407—21no—
65#407—21no—
66C2×C22⋊C8—21no—
67C4×M16—21no—
68C4×M16Q812yes—
69C2×C4×C8C2×C412yes—
70#384—21no—
71#383—21no—
72#382—21no—
73#382—21no—
74#381—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D412yes—
80C4×C4⋊C4D411yescharacteristic
81C4×C4⋊C4D411yescharacteristic
82C4×C22⋊C4—21no—
83#376—21no—
84#341—24no—
85#340—22no—
86#340—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—41no—
96C2×M16—82no—
97C2×M16—81no—
98C2×M16—81no—
99C2×M16C22⋊C414yes—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—41no—
103#117—81no—
104#117—81no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—22no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42C2×Q811yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127#108—84no—
128#107—82no—
129#106—82no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152#97—22no—
153#97—41no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—41no—
157C4×C8C4⋊C414yes—
158#95—22no—
159#95—21no—
160#95—21no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24#12411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4C4×Q811yescharacteristic
178C22×C4C4×Q811yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12511yescharacteristic
191C22×C4#12611yescharacteristic
192C22×C4#12611yescharacteristic
193C22×C4C4⋊Q811yescharacteristic
194M16—44no—
195M16—42no—
196M16—42no—
197M16—82no—
198M16—81no—
199M16—81no—
200M16—82no—
201M16—82no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—82no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—82no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—82no—
229C42C2×C4⋊C411yescharacteristic
230C42#11711yescharacteristic
231C42C4×D411yescharacteristic
232C42C4×D411yescharacteristic
233C42#12111yescharacteristic
234C42#12111yescharacteristic
235C42#12111yescharacteristic
236C42C22⋊Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42C22⋊Q811yescharacteristic
240C42C22⋊Q811yescharacteristic
241C42#12511yescharacteristic
242C42#12511yescharacteristic
243C42C4⋊Q811yescharacteristic
244C42C4⋊Q811yescharacteristic
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23#39111yescharacteristic
250C23#39411yescharacteristic
251C23#40011yescharacteristic
252C2×C4—21no—
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—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4#38011yescharacteristic
287C2×C4#38611yescharacteristic
288C2×C4#38711yescharacteristic
289C2×C4#38711yescharacteristic
290C2×C4#38911yescharacteristic
291C2×C4#38911yescharacteristic
292C2×C4#39511yescharacteristic
293C2×C4#39611yescharacteristic
294C2×C4#39611yescharacteristic
295C2×C4#39811yescharacteristic
296C2×C4#39811yescharacteristic
297C2×C4#39811yescharacteristic
298C2×C4#47111yescharacteristic
299C2×C4#47111yescharacteristic
300C2×C4#47211yescharacteristic
301C2×C4#47311yescharacteristic
302C8—44no—
303C8—44no—
304C8—82no—
305C8—82no—
306C8—81no—
307C8—81no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—81no—
317C22—81no—
318C22#144111yescharacteristic
319C22#205111yescharacteristic
320C22#205211yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—161no—
331C4—162no—
332C4—161no—
333C4#191111yescharacteristic
334C4#191211yescharacteristic
335C4#195111yescharacteristic
336C4#195211yescharacteristic
337C4#206311yescharacteristic
338C4#206411yescharacteristic
339C2—21no—
340C2—21no—
341C2—41no—
342C2—41no—
343C2#1122411yescharacteristic
344C2#1124711yescharacteristic
345C2#1124911yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
20. #1848C4semidirect
21. #1837C4semidirect

Maximal subgroups

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

subgroupindex
#219932
#219632
#81292
#80932
#79572
#79532
#78582