Finite Group Databasebeta

← #6495113

Subgroups — #6495113

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35336C211yesmaximal, characteristic, complemented
2#35317C211yesmaximal, characteristic
3#33237C211yesmaximal, characteristic, complemented
4#33234C211yesmaximal, characteristic, complemented
5#33232C211yesmaximal, characteristic, complemented
6#33232C211yesmaximal, characteristic, complemented
7#30223C214yesmaximal, complemented
8#28760C214yesmaximal, complemented
9#28757C212yesmaximal, complemented
10#28755C212yesmaximal, complemented
11#25868C214yesmaximal, complemented
12#25862C212yesmaximal
13#25861C212yesmaximal, complemented
14#21814C214yesmaximal, complemented
15#20465C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#536C2212yes—
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#491C2212yes—
21C2×#491C2212yes—
22C2×#491C2212yes—
23C2×#491C2214yes—
24C2×#490C2212yes—
25C2×#488C2214yes—
26C2×#433C2212yes—
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2851C2212yes—
34#2658C2212yes—
35#2658C2212yes—
36#2647C2212yes—
37#2629C2212yes—
38#2590—24no—
39#2587—24no—
40#2542—24no—
41#2501—24no—
42#2330C2212yes—
43#2314C2212yes—
44C4×#125C2212yes—
45C4×#117C2211yescharacteristic
46#2043C2218yes—
47#2043C2218yes—
48#2042C2218yes—
49#2042C2218yes—
50#1983C2218yes—
51#1981C2218yes—
52#1961C2212yes—
53#1960C2212yes—
54#1960C2212yes—
55#1960C2212yes—
56#1960C2214yes—
57#1959C2214yes—
58#1959C2214yes—
59#1958C2214yes—
60#1956C2214yes—
61#1956C2214yes—
62#1788C2218yes—
63#1787C2218yes—
64#1768C2212yes—
65#1767C2212yes—
66#1750C2212yes—
67#1749C2212yes—
68#1746C2212yes—
69C4×#97C2211yescharacteristic
70C4×#97C2211yescharacteristic
71C22×#41C2312yes—
72C23×Q8C2312yes—
73#536—216no—
74#536—48no—
75#532—44no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—44no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×#124C2312yes—
90C2×#124C2312yes—
91C2×#124C2312yes—
92C2×#124C2312yes—
93C2×#124C2312yes—
94C2×#123—24no—
95C2×#123—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×#121—22no—
99C2×C4×D4—22no—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C42C2311yescharacteristic
104#491—44no—
105#491—48no—
106#491—44no—
107#491—44no—
108#490—44no—
109#488—48no—
110#433—28no—
111C2×C4⋊C8C2314yes—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#103C2318yes—
115C2×#103C2318yes—
116C2×#103C2318yes—
117C2×#103C2318yes—
118C2×#102C2318yes—
119C2×#102C2318yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×#97C2312yes—
124C2×#97C2312yes—
125C2×C4×C8C2312yes—
126C2×C4×C8C2312yes—
127#400—24no—
128#399—28no—
129#399—28no—
130#398—24no—
131#398—24no—
132#396—24no—
133#390—24no—
134#390—24no—
135#389—24no—
136#389—24no—
137#389—24no—
138#389C2314yes—
139#389C2314yes—
140#388—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#385—24no—
146#384—24no—
147#384—24no—
148#384C2312yes—
149#384C2312yes—
150#382C2312yes—
151#382C2312yes—
152#380C2314yes—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D414yes—
155C4×C4⋊C4C2314yes—
156C4×C22⋊C4D414yes—
157#376C2311yescharacteristic
158#376C2311yescharacteristic
159C43C2311yescharacteristic
160C43C2311yescharacteristic
161#336C2314yes—
162#336C2314yes—
163C2×#41—28no—
164C2×#41—44no—
165C2×#41—44no—
166C22×Q8—22no—
167C22×Q8—28no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8C2412yes—
171C22×Q8C2412yes—
172C22×Q8C2412yes—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4C2412yes—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C2×D412yes—
179C23×C4C2411yescharacteristic
180C22×C8C2414yes—
181C22×C8C2414yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—28no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C4⋊Q8—48no—
191#127—24no—
192#125—28no—
193#125—24no—
194#125—24no—
195#124—28no—
196#124—28no—
197#124—28no—
198#124—44no—
199#124—44no—
200#123—416no—
201#123—416no—
202C22⋊Q8—416no—
203C22⋊Q8—416no—
204#121—48no—
205C4×D4—48no—
206#117—24no—
207#117#4118yes—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2414yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4C2×D412yes—
234C2×C22⋊C4C2×D412yes—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42C2×D412yes—
238C2×C42C2×D414yes—
239C2×C42C2×D414yes—
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245C2×C42C2411yescharacteristic
246C4⋊C8—28no—
247#103—48no—
248#103—48no—
249#103—48no—
250#103—48no—
251#103—48no—
252#103—48no—
253#102—48no—
254#102—48no—
255C22⋊C8—28no—
256#97—24no—
257#97—24no—
258#97—24no—
259#97—24no—
260C4×C8—24no—
261C4×C8—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95C2×D414yes—
269#95C2×D414yes—
270C24—22no—
271C24C22×D411yescharacteristic
272#41—816no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—816no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C22×D412yes—
314C22×C4C22×D412yes—
315C22×C4C22×D412yes—
316C22×C4C2×#4114yes—
317C22×C4C2511yescharacteristic
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4#12418yes—
343C22⋊C4—24no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C22⋊C4—48no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—84no—
350C22⋊C4#12418yes—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—44no—
355C42—44no—
356C42—84no—
357C42C2×#4114yes—
358C42C2×#4114yes—
359C42H(5,𝔽2)14yes—
360C42H(5,𝔽2)14yes—
361C23—24no—
362C23—21no—
363C23—21no—
364C23—48no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23C2×#12412yes—
369C23C23×D411yescharacteristic
370Q8—48no—
371Q8—48no—
372Q8—48no—
373Q8—88no—
374D4—48no—
375D4—88no—
376D4—88no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—88no—
405C2×C4—88no—
406C2×C4C2×#12414yes—
407C2×C4C2×#12414yes—
408C2×C4C2×#12412yes—
409C2×C4#53514yes—
410C2×C4#53514yes—
411C2×C4C22×#4112yes—
412C2×C4C2×H(5,𝔽2)12yes—
413C2×C4C2×H(5,𝔽2)12yes—
414C8—44no—
415C8—44no—
416C22—24no—
417C22—22no—
418C22—42no—
419C22—42no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22—82no—
424C22C22×#12411yescharacteristic
425C22C2×#53511yescharacteristic
426C22C2×#53511yescharacteristic
427C22#358614yes—
428C4—24no—
429C4—24no—
430C4—24no—
431C4—24no—
432C4—22no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—84no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4#350814yes—
441C2—42no—
442C2—162no—
443C2#3528511yescharacteristic
444C2#6062712yes—
445C2#6080712yes—
446C2#6083112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35336C2semidirect
3. #33237C2semidirect
4. #33234C2semidirect
5. #33232C2semidirect
6. #33232C2semidirect
7. #30223C2semidirect
8. #28760C2semidirect
9. #28757C2semidirect
10. #28755C2semidirect
11. #25868C2semidirect
13. #25861C2semidirect
14. #21814C2semidirect
15. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#353362
#353172
#332372
#332342
#332322
#302232
#287602
#287572
#287552
#258682
#258622
#258612
#218142
#204652