Finite Group Databasebeta

← #241292

Subgroups — #241292

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#10582C214yesmaximal, complemented
6#10581C214yesmaximal, complemented
7#8040C211yesmaximal, characteristic
8#7904C211yesmaximal, characteristic
9#7904C211yesmaximal, characteristic
10C22×#125—21no—
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2552—22no—
15#2552—22no—
16#2552—22no—
17#2552C2214yes—
18C2×#398—22no—
19C2×#389—21no—
20C2×#389—21no—
21#2285C2211yescharacteristic
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1948C2212yes—
27#1948C2212yes—
28#1947C2212yes—
29#1947C2212yes—
30#1853—21no—
31#1853—21no—
32#1852—21no—
33#1852—21no—
34#1836—22no—
35#1834C2212yes—
36#1834C2212yes—
37#1776—22no—
38#1776—22no—
39#1736—22no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44C2×#125—24no—
45C2×#125—24no—
46C2×#125—48no—
47C2×#117—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432—22no—
63#432Q814yes—
64#432Q814yes—
65C2×#107—22no—
66C2×#107—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C4×M16—22no—
74C4×M16Q814yes—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×C4×C8C2312yes—
79#398—28no—
80#398—28no—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#384—24no—
86#384—24no—
87#384—21no—
88#384—21no—
89#384D412yes—
90#384C2311yescharacteristic
91#384C2311yescharacteristic
92#384C2312yes—
93#382—21no—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2312yes—
105C4×C4⋊C4C2312yes—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—24no—
124C2×M16—42no—
125C2×M16—42no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129#125—416no—
130#125—48no—
131#125—48no—
132#117—44no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D414yes—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—44no—
172C2×C42—44no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×Q812yes—
178C2×C42C2×Q812yes—
179C2×C42C2×Q812yes—
180C2×C42#4112yes—
181C2×C42C2411yescharacteristic
182#107—44no—
183#107—44no—
184#106—44no—
185#106—44no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8C2×Q814yes—
189C4⋊C8C2×Q814yes—
190C22⋊C8—24no—
191C22⋊C8—24no—
192#97—24no—
193#97—24no—
194#97—24no—
195C4×C8—22no—
196C4×C8C2×Q814yes—
197#95—28no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—41no—
203#95—41no—
204C24#12011yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4C22⋊Q812yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4C4⋊Q812yes—
240C22×C4C4⋊Q812yes—
241C22×C4C4⋊Q812yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245M16—48no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—416no—
269C4⋊C4—416no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—82no—
279C4⋊C4—88no—
280C4⋊C4AGL(1,ℤ/8)14yes—
281C4⋊C4#13714yes—
282C22⋊C4—82no—
283C42—44no—
284C42—44no—
285C42—44no—
286C42—44no—
287C42—82no—
288C42C22⋊Q812yes—
289C42C22⋊Q812yes—
290C42C22⋊Q812yes—
291C42C4⋊Q812yes—
292C42C4⋊Q812yes—
293C42C4⋊Q812yes—
294C42C22×Q811yescharacteristic
295C42C22×Q811yescharacteristic
296C42C22×Q811yescharacteristic
297C42C2×#4111yescharacteristic
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#39512yes—
303C23#45412yes—
304C23#45512yes—
305C23C2×#12011yescharacteristic
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—88no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4#39512yes—
347C2×C4#39512yes—
348C2×C4#39512yes—
349C2×C4C2×C22⋊Q811yescharacteristic
350C2×C4C2×C22⋊Q811yescharacteristic
351C2×C4C2×C22⋊Q811yescharacteristic
352C2×C4C2×C4⋊Q811yescharacteristic
353C2×C4C2×C4⋊Q811yescharacteristic
354C2×C4C2×C4⋊Q811yescharacteristic
355C2×C4C2×AGL(1,ℤ/8)12yes—
356C2×C4C2×#13712yes—
357C2×C4#57512yes—
358C2×C4#57512yes—
359C8—44no—
360C8—44no—
361C8—44no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22#206114yes—
369C22C2×#39511yescharacteristic
370C22#300311yescharacteristic
371C22#300411yescharacteristic
372C22#300511yescharacteristic
373C22#300511yescharacteristic
374C22C2×#45511yescharacteristic
375C22C2×#45411yescharacteristic
376C4—22no—
377C4—22no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—44no—
382C4—44no—
383C4—82no—
384C4—82no—
385C4—84no—
386C4—161no—
387C4—161no—
388C4#307912yes—
389C4#308012yes—
390C4#316512yes—
391C4#316612yes—
392C2—24no—
393C2#1853011yescharacteristic
394C2#1853311yescharacteristic
395C2#1853511yescharacteristic
396C2#1855011yescharacteristic
397C2#2355211yescharacteristic
398C2#2355311yescharacteristic
399C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
5. #10582C2semidirect
6. #10581C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#213652
#207642
#105822
#105812
#80402
#79042