Finite Group Databasebeta

← #6528534

Subgroups — #6528534

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic, complemented
2#35333C211yesmaximal, characteristic
3#29120C214yesmaximal
4#27971C214yesmaximal, complemented
5#27827C214yesmaximal, complemented
6#27825C214yesmaximal, complemented
7#27745C214yesmaximal, complemented
8#27743C214yesmaximal, complemented
9#21838C212yesmaximal, complemented
10#20743C212yesmaximal
11#20539C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#432C2212yes—
15#2851—22no—
16#2844—22no—
17#2629—21no—
18#2629—21no—
19#2628—24no—
20#2628C2214yes—
21#2579—24no—
22#2571—21no—
23#2571C2214yes—
24#2566C2214yes—
25#2565C2214yes—
26#2551—21no—
27#2551C2214yes—
28#2317C2212yes—
29#2311C2212yes—
30#2285C2212yes—
31#2285C2211yescharacteristic
32#2101C2214yes—
33#2101C2214yes—
34#2100C2214yes—
35#2099C2214yes—
36#2074C2214yes—
37#2073C2214yes—
38#2024C2214yes—
39#2023C2214yes—
40#1993C2214yes—
41#1992C2214yes—
42#1992C2214yes—
43#1990C2214yes—
44#1989C2214yes—
45#1989C2214yes—
46#1936C2214yes—
47#1935C2214yes—
48#1934C2214yes—
49#1933C2214yes—
50#1836C2214yes—
51#1835C2214yes—
52#1825C2214yes—
53#1824C2214yes—
54#1823C2214yes—
55#1803C2212yes—
56#1803C2212yes—
57#1802C2212yes—
58#1802C2212yes—
59#1788C2214yes—
60#1771C2214yes—
61#1771C2214yes—
62#1763C2214yes—
63#1762C2214yes—
64#1750C2214yes—
65C22×#41C2311yescharacteristic
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8C2314yes—
68C2×#125—21no—
69C2×#125—21no—
70C2×#125—28no—
71C2×#125C2314yes—
72C4×#41—28no—
73C2×C4×Q8C2312yes—
74C2×C4×D4—22no—
75C2×C4×D4C2312yes—
76C2×#117—22no—
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C2×#117C2312yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#432—28no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103C2314yes—
88C2×#103C2314yes—
89C2×#102C2314yes—
90C2×#102C2314yes—
91C2×C22⋊C8C2314yes—
92C2×#97C2314yes—
93C2×#97C2314yes—
94C2×C4×C8C2314yes—
95#400—24no—
96#400—24no—
97#400—24no—
98#399—24no—
99#398—24no—
100#398—24no—
101#396—24no—
102#396—24no—
103#395—24no—
104#395—24no—
105#395—24no—
106#393—24no—
107#391C2314yes—
108#390C2314yes—
109#389—24no—
110#389—24no—
111#389—22no—
112#389—22no—
113#389C2314yes—
114#389C2314yes—
115#387—24no—
116#387—24no—
117#386—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#384D411yescharacteristic
122#384D411yescharacteristic
123#384C2314yes—
124#384C2312yes—
125#384C2312yes—
126#382—24no—
127#382—22no—
128#382D412yes—
129#382C2314yes—
130#382C2314yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380D414yes—
135#380C2314yes—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139#340C2314yes—
140#340C2314yes—
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#340C2314yes—
146#340C2314yes—
147#340C2314yes—
148#340C2314yes—
149#340C2314yes—
150#340C2314yes—
151#336C2314yes—
152#336C2314yes—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—24no—
156C2×#41—42no—
157C22×Q8—24no—
158C22×Q8C2411yescharacteristic
159C22×D4—21no—
160C22×D4C2411yescharacteristic
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C2×D412yes—
165C23×C4C2411yescharacteristic
166C22×C8C2414yes—
167C22×C8C2414yes—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—88no—
171#125—28no—
172#125—28no—
173#125—44no—
174#125—44no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—48no—
180#117—24no—
181#117—42no—
182#117—42no—
183#117—44no—
184#117—84no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2412yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2414yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4C2×D414yes—
214C2×C42—21no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—44no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4118yes—
221C2×C42C2414yes—
222C2×C42C2412yes—
223C4⋊C8—24no—
224C4⋊C8—24no—
225#103—28no—
226#103—44no—
227#102—28no—
228#102—44no—
229C22⋊C8—28no—
230#97—28no—
231#97—44no—
232C4×C8—44no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95C2×D414yes—
244#95C2×D414yes—
245C24—21no—
246C24C22×D411yescharacteristic
247#41—48no—
248#41—48no—
249C2×Q8—21no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q82−1+412yes—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D42−1+412yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#12418yes—
283C22×C4C22×D414yes—
284C22×C4C22×D414yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C2×#4114yes—
289C22×C4C2511yescharacteristic
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C4⋊C4—28no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—24no—
327C42—24no—
328C42—48no—
329C42—42no—
330C42—44no—
331C42#12414yes—
332C42#12414yes—
333C42C2×#4114yes—
334C42C2×#4114yes—
335C23—22no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23C2×#12412yes—
345C23C23×D411yescharacteristic
346Q8—24no—
347Q8—42no—
348Q8—168no—
349D4—24no—
350D4—48no—
351D4—42no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—88no—
384C2×C4—88no—
385C2×C4C2×#12412yes—
386C2×C4C2×#12414yes—
387C2×C4C2×#12414yes—
388C2×C4C22×#4112yes—
389C2×C4C2×2−1+411yescharacteristic
390C2×C4C2×2−1+411yescharacteristic
391C2×C4#58512yes—
392C8—44no—
393C8—44no—
394C22—22no—
395C22—24no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—81no—
402C22—81no—
403C22—82no—
404C22C22×#12411yescharacteristic
405C22#348511yescharacteristic
406C22#348611yescharacteristic
407C22#358512yes—
408C22#358612yes—
409C4—22no—
410C4—24no—
411C4—24no—
412C4—42no—
413C4—44no—
414C4—44no—
415C4—84no—
416C4—84no—
417C4—84no—
418C4—82no—
419C4#349512yes—
420C4#349512yes—
421C2—42no—
422C2—82no—
423C2#3564411yescharacteristic
424C2#6062112yes—
425C2#6179711yescharacteristic
426C2#6179811yescharacteristic
427C2#6187812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37642C2semidirect
4. #27971C2semidirect
5. #27827C2semidirect
6. #27825C2semidirect
7. #27745C2semidirect
8. #27743C2semidirect
9. #21838C2semidirect
11. #20539C2semidirect

Maximal subgroups

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

subgroupindex
#376422
#353332
#291202
#279712
#278272
#278252
#277452
#277432
#218382
#207432
#205392