Finite Group Databasebeta

← #319041

Subgroups — #319041

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#27972C211yesmaximal, characteristic, complemented
4#25903C211yesmaximal, characteristic
5#25856C211yesmaximal, characteristic, complemented
6C2×#1990C211yesmaximal, characteristic, complemented
7#10682C212yesmaximal, complemented
8#10311C212yesmaximal, complemented
9#9822C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C2×#486C2212yes—
15C2×#486C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857C2211yescharacteristic
18#2856C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2072—22no—
21#2071—22no—
22#2043—22no—
23#2023—22no—
24#1990—22no—
25#1990C2211yescharacteristic
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1957C2211yescharacteristic
30#1957C2212yes—
31#1956C2212yes—
32#1956C2212yes—
33#1934—22no—
34#1893—22no—
35#1874—22no—
36#1874—22no—
37#1872—22no—
38#1834C2211yescharacteristic
39#1834C2212yes—
40#1834C2212yes—
41#1787—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×C82C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C23×C8D412yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×#127C2311yescharacteristic
57C2×C22⋊Q8—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C42C2311yescharacteristic
67#487—24no—
68#487—42no—
69#486—24no—
70#486—42no—
71#486—42no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#102—24no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—24no—
79C2×C22⋊C8D412yes—
80C2×C22⋊C8D412yes—
81C2×C4×C8—42no—
82C2×C4×C8D412yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2312yes—
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87#391—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#389—21no—
92#389—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#384C2312yes—
99#380—21no—
100#380—21no—
101#340—24no—
102#340—24no—
103#340—22no—
104#340—24no—
105#340—22no—
106#340—24no—
107#340—24no—
108#336D412yes—
109#336D412yes—
110C82—42no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—44no—
114C22×Q8—41no—
115C22×Q8—44no—
116C22×Q8—82no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—41no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—24no—
127C22×C8—24no—
128C22×C8—42no—
129C22×C8C2×D412yes—
130C22×C8C2×D414yes—
131C22×C8C2×D414yes—
132C22×C8C2×D412yes—
133C4⋊Q8—24no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—44no—
140C4⋊Q8—44no—
141C4⋊Q8—42no—
142C4⋊Q8#4114yes—
143#127—22no—
144#127—22no—
145C22⋊Q8—44no—
146#121—44no—
147#121—44no—
148#121—44no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2411yescharacteristic
184#103—48no—
185#103—84no—
186#103—84no—
187#102—48no—
188#102—44no—
189#102—44no—
190#102—84no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—42no—
200#95—41no—
201#95—41no—
202C24—41no—
203C24#12711yescharacteristic
204C2×Q8—24no—
205C2×Q8—44no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—44no—
209C2×Q8—42no—
210C2×Q8—44no—
211C2×Q8—42no—
212C2×Q8—816no—
213C2×Q8—82no—
214C2×Q8—82no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—82no—
226C2×D4—82no—
227C2×D4—82no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4#12711yescharacteristic
247C22×C4#12711yescharacteristic
248C22×C4#12711yescharacteristic
249C22×C4#13514yes—
250C22×C4#13514yes—
251C22×C4#13514yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—48no—
267C2×C8#13714yes—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C42—21no—
295C42—21no—
296C42—22no—
297C42—22no—
298C42C2×#4112yes—
299C42H(5,𝔽2)12yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23#49512yes—
312C23#49512yes—
313C23#49512yes—
314C23C2×#12711yescharacteristic
315Q8—48no—
316Q8—88no—
317Q8—84no—
318Q8—84no—
319D4—82no—
320D4—82no—
321D4—84no—
322D4—84no—
323D4—164no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—44no—
335C2×C4—88no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—164no—
351C2×C4#49512yes—
352C2×C4#49512yes—
353C2×C4#49512yes—
354C2×C4#52511yescharacteristic
355C2×C4#53411yescharacteristic
356C2×C4#54612yes—
357C2×C4#54612yes—
358C2×C4C2×#13512yes—
359C2×C4C2×#13512yes—
360C2×C4C2×#13512yes—
361C2×C4C2×#13712yes—
362C2×C4#57612yes—
363C2×C4#57712yes—
364C8—24no—
365C8—42no—
366C8—44no—
367C8—44no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22—161no—
381C22#171114yes—
382C22#253911yescharacteristic
383C22C2×#49511yescharacteristic
384C22C2×#49511yescharacteristic
385C22C2×#49511yescharacteristic
386C22#315211yescharacteristic
387C22#315411yescharacteristic
388C22#315511yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—84no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4—162no—
397C4#310812yes—
398C4#319012yes—
399C4#329714yes—
400C4#330114yes—
401C2—24no—
402C2—321no—
403C2C2×#171111yescharacteristic
404C2#1962611yescharacteristic
405C2#1963811yescharacteristic
406C2#1965611yescharacteristic
407C2#2830611yescharacteristic
408C2#2833111yescharacteristic
409C2#2833411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27972C2semidirect
5. #25856C2semidirect
6. C2×#1990C2semidirect
7. #10682C2semidirect
8. #10311C2semidirect
9. #9822C2semidirect
10. #9821C2semidirect
11. C8×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#330422
#279732
#279722
#259032
#258562
C2×#19902
#106822
#103112
#98222
#98212
C8×C22⋊C82