Finite Group Databasebeta

← #117285

Subgroups — #117285

Order 512.

667 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21811C211yesmaximal, characteristic
2C2×#1966C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17746C211yesmaximal, characteristic
5#8074C211yesmaximal, characteristic
6#7993C211yesmaximal, characteristic
7#7962C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×#101—22no—
14C2×C4×M16C2211yescharacteristic
15#2283C2211yescharacteristic
16#1981—21no—
17#1981—21no—
18#1966—22no—
19#1966—22no—
20#1960—22no—
21#1787—22no—
22#1573—22no—
23#1571—21no—
24#1570—21no—
25#1569—22no—
26#1558—22no—
27#1548—22no—
28#1548C414yescomplemented
29#1300—22no—
30#1291—22no—
31#1286—22no—
32C23×Q8D412yes—
33C22×M16—21no—
34C22×M16—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8Q812yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432D412yes—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#431D412yes—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423C2×C412yes—
67#423D412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D411yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×#103—24no—
73C2×#103—24no—
74C2×#101—28no—
75C2×#101—48no—
76#407—21no—
77#407—21no—
78C2×C22⋊C8—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16D412yes—
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88#383—21no—
89#382D412yes—
90#381—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4—21no—
94#376—21no—
95#335—24no—
96#334—24no—
97#333—28no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—44no—
104C22×Q8—44no—
105C22×Q8C4⋊C414yes—
106C23×C4—22no—
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—24no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—42no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—44no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—41no—
122C22×C8—41no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42—41no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—22no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8C22⋊C414yes—
178#103—84no—
179#103—84no—
180#101—416no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191#97—41no—
192#97—41no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197#95—22no—
198#95—22no—
199#95—41no—
200#95—41no—
201C24#12311yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—48no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—84no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—84no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—48no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12611yescharacteristic
242M16—24no—
243M16—44no—
244M16—42no—
245M16—44no—
246M16—82no—
247M16—82no—
248M16—82no—
249C2×C8—24no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—82no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—82no—
270C42—82no—
271C42C2×C22⋊C411yescharacteristic
272C42#11711yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42#12011yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42#12311yescharacteristic
285C42#12411yescharacteristic
286C42#12711yescharacteristic
287C23—22no—
288C23—22no—
289C23—41no—
290C23#38511yescharacteristic
291C23#39411yescharacteristic
292C23#39711yescharacteristic
293Q8—48no—
294Q8—84no—
295Q8—84no—
296Q8—82no—
297Q8—82no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4#38011yescharacteristic
325C2×C4#38511yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#39011yescharacteristic
330C2×C4#39211yescharacteristic
331C2×C4#39311yescharacteristic
332C2×C4#39411yescharacteristic
333C2×C4#39611yescharacteristic
334C2×C4#39611yescharacteristic
335C2×C4#39911yescharacteristic
336C2×C4#45312yes—
337C2×C4#45512yes—
338C8—44no—
339C8—44no—
340C8—82no—
341C8—81no—
342C8—82no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22#143911yescharacteristic
353C22#203712yes—
354C22#208212yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—84no—
365C4—162no—
366C4#185612yes—
367C4#190212yes—
368C4#200211yescharacteristic
369C4#200411yescharacteristic
370C2—22no—
371C2—22no—
372C2—41no—
373C2#1105911yescharacteristic
374C2#1113411yescharacteristic
375C2#1113511yescharacteristic
subgroupG/H
center, socleC22#1439
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
28. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#218112
C2×#19662
#178222
#177462
#80742
#79932
#79622