Finite Group Databasebeta

← #6532037

Subgroups — #6532037

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic, complemented
3#27623C214yesmaximal, complemented
4#27593C214yesmaximal
5#25071C214yesmaximal, complemented
6#25069C214yesmaximal, complemented
7#25004C214yesmaximal, complemented
8#25002C214yesmaximal, complemented
9#21986C212yesmaximal, complemented
10#21823C211yesmaximal, characteristic, complemented
11#20743C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#432C2212yes—
15#2851—21no—
16#2851—21no—
17#2628—22no—
18#2581C2214yes—
19#2580C2214yes—
20#2564—24no—
21#2382—21no—
22#2382—21no—
23#2365—22no—
24#2355C2214yes—
25Q8×C4⋊C4C2214yes—
26#2323—24no—
27#2316C2212yes—
28#2314C2214yes—
29#2311C2212yes—
30#2285C2212yes—
31C2×#382C2211yescharacteristic
32#2100C2214yes—
33#2099C2214yes—
34#2097C2214yes—
35#2092C2214yes—
36#2085C2214yes—
37#2085C2214yes—
38#2072C2214yes—
39#2071C2214yes—
40#1985C2214yes—
41#1985C2214yes—
42#1949C2214yes—
43#1949C2214yes—
44#1928C2214yes—
45#1927C2214yes—
46#1926C2214yes—
47#1925C2214yes—
48#1921C2214yes—
49#1920C2214yes—
50#1919C2214yes—
51#1918C2214yes—
52#1853C2214yes—
53#1852C2214yes—
54#1847C2214yes—
55#1836C2214yes—
56#1825C2214yes—
57#1818C2214yes—
58#1803C2212yes—
59#1803C2212yes—
60#1802C2212yes—
61#1802C2212yes—
62#1788C2214yes—
63#1763C2214yes—
64#1762C2214yes—
65C22×#41C2311yescharacteristic
66#533—28no—
67C2×C4⋊Q8D412yes—
68C2×#125D411yescharacteristic
69C2×#125D411yescharacteristic
70C2×C22⋊Q8D414yes—
71C4×#41—28no—
72C2×C4×Q8—24no—
73C2×C4×Q8C2312yes—
74C2×C4×Q8C2314yes—
75C2×C4×D4—22no—
76C2×C4×D4C2312yes—
77C2×#117—22no—
78C2×#117—21no—
79C2×#117C2311yescharacteristic
80C2×#117C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#432—28no—
85C2×#107C2314yes—
86C2×#107C2314yes—
87C2×#106C2314yes—
88C2×#106C2314yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103C2314yes—
92C2×#103C2314yes—
93C2×#102C2314yes—
94C2×#102C2314yes—
95C2×C22⋊C8C2314yes—
96C2×#97C2314yes—
97#400—24no—
98#399—24no—
99#395—24no—
100#393—24no—
101#391—24no—
102#391C2314yes—
103#390C2314yes—
104#389—22no—
105#389—22no—
106#389—24no—
107#389—24no—
108#389—24no—
109#389C2314yes—
110#389C2314yes—
111#388—24no—
112#388—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#386—24no—
120#385—24no—
121#385—24no—
122#384C2312yes—
123#384C2312yes—
124#384C2314yes—
125#382—21no—
126#382—21no—
127#382—22no—
128#382C2314yes—
129#382C2314yes—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—24no—
134#380C2314yes—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C2×#95—22no—
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#340C2314yes—
146#340C2314yes—
147#340C2314yes—
148#340C2314yes—
149#336C2314yes—
150#336C2314yes—
151#336C2314yes—
152#336C2314yes—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—24no—
156C2×#41—42no—
157C22×Q8C2×D414yes—
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—28no—
169#125—24no—
170#125—24no—
171C22⋊Q8—216no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—88no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—48no—
180#117—28no—
181#117—24no—
182#117—44no—
183#117—84no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4C2×D414yes—
211C2×C42—21no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—44no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4118yes—
220C2×C42C2414yes—
221C2×C42C2412yes—
222#107—44no—
223#107—44no—
224#106—44no—
225#106—44no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228#103—28no—
229#103—44no—
230#102—28no—
231#102—44no—
232C22⋊C8—28no—
233#97—28no—
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#95—24no—
244#95—24no—
245#95—24no—
246C24—21no—
247C24C22×D411yescharacteristic
248#41—48no—
249#41—48no—
250C2×Q8—28no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q82−1+412yes—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D42−1+412yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4#12318yes—
284C22×C4C22×D414yes—
285C22×C4C22×D414yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C2×#4114yes—
290C22×C4C2511yescharacteristic
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C22⋊C4—28no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—28no—
327C42—48no—
328C42—42no—
329C42—44no—
330C42—84no—
331C42—84no—
332C42#12314yes—
333C42#12314yes—
334C42C2×#4114yes—
335C42C2×#4114yes—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23C2×#12312yes—
346C23C23×D411yescharacteristic
347Q8—24no—
348Q8—416no—
349Q8—42no—
350D4—24no—
351D4—48no—
352D4—42no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—88no—
385C2×C4—164no—
386C2×C4C2×#12312yes—
387C2×C4C2×#12314yes—
388C2×C4C2×#12314yes—
389C2×C4C22×#4112yes—
390C2×C4C2×2−1+411yescharacteristic
391C2×C4C2×2−1+411yescharacteristic
392C2×C4#58512yes—
393C8—44no—
394C8—44no—
395C22—22no—
396C22—24no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22C22×#12311yescharacteristic
406C22#348611yescharacteristic
407C22#348711yescharacteristic
408C22#358512yes—
409C22#358612yes—
410C4—22no—
411C4—24no—
412C4—24no—
413C4—42no—
414C4—44no—
415C4—44no—
416C4—82no—
417C4—84no—
418C4—84no—
419C4—84no—
420C4#349512yes—
421C4#349512yes—
422C2—42no—
423C2—82no—
424C2#3567211yescharacteristic
425C2#6051712yes—
426C2#6187611yescharacteristic
427C2#6187711yescharacteristic
428C2#6194412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34374C2semidirect
3. #27623C2semidirect
5. #25071C2semidirect
6. #25069C2semidirect
7. #25004C2semidirect
8. #25002C2semidirect
9. #21986C2semidirect
10. #21823C2semidirect

Maximal subgroups

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

subgroupindex
#353342
#343742
#276232
#275932
#250712
#250692
#250042
#250022
#219862
#218232
#207432