Finite Group Databasebeta

← #6386226

Subgroups — #6386226

Order 512.

3048 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2#34371C212yesmaximal, complemented
3#22513C218yesmaximal, complemented
4Q8×#99C218yesmaximal, complemented
5#21828C214yesmaximal, complemented
6#21826C212yesmaximal, complemented
7#20725C214yesmaximal
8#20723C211yesmaximal, characteristic
9#20723C211yesmaximal, characteristic
10#3503—24no—
11#3503—24no—
12#3503—24no—
13#3503—24no—
14C2×Q82C412yescomplemented
15C2×#557C412yescomplemented
16C2×#554C412yescomplemented
17C2×#554C2214yes—
18C2×D4×Q8C412yescomplemented
19C2×D4×Q8C2214yes—
20C2×#544C414yescomplemented
21C2×#544C2212yes—
22C2×#533C412yescomplemented
23C2×#533C2211yescharacteristic
24C22×C4⋊Q8C412yescomplemented
25C22×C4⋊Q8C2211yescharacteristic
26C2×#520C2212yes—
27C2×C4×#41C2211yescharacteristic
28#2381—24no—
29#2365—24no—
30#2345C2218yes—
31Q8×C22⋊C4C2218yes—
32#2331—24no—
33#2327—24no—
34#2313C2218yes—
35#2283C2212yes—
36#2283C2212yes—
37#1996C2218yes—
38#1996C2218yes—
39#1996C22132yes—
40#1820C2218yes—
41#1820C2212yes—
42#1820C2212yes—
43#1799C2214yes—
44#1799C22116yes—
45#1799C2214yes—
46#1782C2212yes—
47#1782C2212yes—
48C4×#99C2214yes—
49C4×#99C22116yes—
50C4×#99C2214yes—
51C22×#41C2311yescharacteristic
52C23×Q8C2×C414yes—
53C23×Q8C2312yes—
54Q82—28no—
55Q82—28no—
56#557—28no—
57#557—28no—
58#554—28no—
59#554—28no—
60#554—216no—
61#554—216no—
62D4×Q8—216no—
63D4×Q8—216no—
64D4×Q8—28no—
65D4×Q8—28no—
66#544—216no—
67#544—216no—
68#544—28no—
69#544—28no—
70#533—24no—
71#533—24no—
72#533—24no—
73#533—24no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C4⋊Q8C2×C414yes—
84C2×C4⋊Q8C2×C414yes—
85C2×C4⋊Q8C2×C414yes—
86C2×C4⋊Q8C2×C411yescharacteristic
87C2×C4⋊Q8C2×C418yes—
88C2×C4⋊Q8C2×C414yes—
89C2×C4⋊Q8C2×C412yes—
90C2×C4⋊Q8C2×C412yes—
91C2×C4⋊Q8C2×C412yes—
92C2×C4⋊Q8C2×C412yes—
93C2×C4⋊Q8D414yes—
94C2×C4⋊Q8D414yes—
95C2×C4⋊Q8D414yes—
96C2×C4⋊Q8D414yes—
97C2×#125C2×C414yes—
98C2×#125C2×C412yes—
99C2×#125C2×C418yes—
100C2×#125C2×C414yes—
101C2×C22⋊Q8C2×C414yes—
102C2×C22⋊Q8C2×C414yes—
103C2×C22⋊Q8C2×C418yes—
104C2×C22⋊Q8C2×C418yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2318yes—
108#520—28no—
109#520—28no—
110C4×#41—22no—
111C4×#41—22no—
112C4×#41—22no—
113C4×#41—22no—
114C2×C4×Q8—28no—
115C2×C4×Q8C2×C411yescharacteristic
116C2×C4×Q8C2×C411yescharacteristic
117C2×C4×Q8C2×C414yes—
118C2×C4×Q8C2×C414yes—
119C2×C4×Q8C2×C414yes—
120C2×C4×Q8C2×C412yes—
121C2×C4×Q8C2×C412yes—
122C2×C4×D4C2×C411yescharacteristic
123C2×C4×D4C2×C411yescharacteristic
124C2×C4×D4C2×C414yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2318yes—
128C2×#117C2×C412yes—
129C2×#117C2×C414yes—
130C2×#117C2311yescharacteristic
131C2×#117C2312yes—
132C22×C4⋊C4C2×C414yes—
133C22×C4⋊C4C2×C414yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C42C2×C412yes—
137C22×C42C2311yescharacteristic
138C2×#99C2318yes—
139C2×#99C2318yes—
140#391—28no—
141#391—216no—
142#389—28no—
143#387—216no—
144#387—216no—
145#387—216no—
146#383—22no—
147#383—22no—
148#382—22no—
149#382—22no—
150#381C2314yes—
151#381C2314yes—
152#380C23116yes—
153#380C2314yes—
154#380C2314yes—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C22⋊C4C23116yes—
159C4×C22⋊C4C2314yes—
160C4×C22⋊C4C2314yes—
161#376—22no—
162#376—22no—
163#342C2318yes—
164#342C23132yes—
165#342C2318yes—
166C2×#41—28no—
167C2×#41—22no—
168C2×#41—22no—
169C22×Q8—28no—
170C22×Q8—28no—
171C22×Q8—84no—
172C22×Q8—88no—
173C22×Q8C22×C411yescharacteristic
174C22×Q8C22×C412yes—
175C22×Q8C22×C412yes—
176C22×Q8C22×C418yes—
177C22×Q8C22×C414yes—
178C22×Q8C22×C414yes—
179C22×Q8C2×D418yes—
180C22×Q8C2×D418yes—
181C22×D4C22×C411yescharacteristic
182C22×D4C2412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C22×C418yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2414yes—
191C4⋊Q8—28no—
192C4⋊Q8—28no—
193C4⋊Q8—232no—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—216no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8C22⋊C414yes—
208C4⋊Q8C22⋊C414yes—
209C4⋊Q8C22⋊C414yes—
210C4⋊Q8C22⋊C414yes—
211C4⋊Q8C22⋊C418yes—
212C4⋊Q8C22⋊C418yes—
213C4⋊Q8C22⋊C418yes—
214C4⋊Q8C22⋊C418yes—
215C4⋊Q8C22⋊C418yes—
216C4⋊Q8C22⋊C418yes—
217#125—28no—
218#125—28no—
219#125—232no—
220#125—216no—
221#125—24no—
222#125—24no—
223C22⋊Q8—28no—
224C22⋊Q8—28no—
225C22⋊Q8—232no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—232no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—232no—
234C4×Q8—28no—
235C4×Q8—28no—
236C4×Q8—216no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—816no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—232no—
253C4×D4—216no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258#117—22no—
259#117—22no—
260#117—24no—
261#117—24no—
262#117—28no—
263#117—28no—
264#117—24no—
265#117—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C418yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C418yes—
286C2×C4⋊C4C22×C414yes—
287C2×C4⋊C4C22×C414yes—
288C2×C4⋊C4C22×C414yes—
289C2×C4⋊C4C22×C414yes—
290C2×C4⋊C4C22×C414yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D418yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C412yes—
297C2×C22⋊C4C22×C412yes—
298C2×C22⋊C4C2418yes—
299C2×C22⋊C4C2418yes—
300C2×C22⋊C4C2414yes—
301C2×C22⋊C4C2412yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—216no—
309C2×C42C22×C412yes—
310C2×C42C22×C418yes—
311C2×C42C22×C414yes—
312C2×C42C22×C412yes—
313C2×C42C22×C412yes—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322#99—216no—
323#99—216no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—216no—
331#95—216no—
332#95—24no—
333#95—24no—
334C24C23×C412yes—
335C24C2511yescharacteristic
336#41—216no—
337#41—216no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—216no—
344C2×Q8—24no—
345C2×Q8—416no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—816no—
353C2×Q8—816no—
354C2×Q8C2×C22⋊C418yes—
355C2×Q8C2×C22⋊C418yes—
356C2×Q8C2×C22⋊C4116yes—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D42−1+418yes—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4—88no—
380C22×C4C23×C411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C23×C411yescharacteristic
383C22×C4C23×C411yescharacteristic
384C22×C4C23×C412yes—
385C22×C4C23×C414yes—
386C22×C4C23×C412yes—
387C22×C4C23×C412yes—
388C22×C4C23×C414yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C23×C418yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D4116yes—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—216no—
414C4⋊C4—28no—
415C4⋊C4—216no—
416C4⋊C4—28no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—816no—
420C4⋊C4C2×C22⋊C418yes—
421C4⋊C4C2×C22⋊C418yes—
422C4⋊C4C2×C22⋊C418yes—
423C4⋊C4C2×C22⋊C418yes—
424C4⋊C4C2×C22⋊C4116yes—
425C4⋊C4C2×C22⋊C4116yes—
426C22⋊C4—28no—
427C22⋊C4—24no—
428C22⋊C4—216no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—216no—
440C42—28no—
441C42—48no—
442C42—48no—
443C42—816no—
444C42C2×C22⋊C411yescharacteristic
445C42C2×C22⋊C411yescharacteristic
446C42C2×C22⋊C411yescharacteristic
447C42C2×C22⋊C411yescharacteristic
448C42C2×C22⋊C412yes—
449C42C2×C22⋊C412yes—
450C42C2×C22⋊C412yes—
451C42C2×C22⋊C412yes—
452C42C2×C22⋊C412yes—
453C42C2×C22⋊C412yes—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23#51914yes—
461C23C24×C411yescharacteristic
462C23C23×D412yes—
463C23C2×2−1+414yes—
464Q8—28no—
465Q8—216no—
466Q8—48no—
467Q8—416no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472Q8—816no—
473D4—28no—
474D4—216no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—416no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—816no—
512C2×C4C22×C22⋊C412yes—
513C2×C4C22×C22⋊C411yescharacteristic
514C2×C4C22×C22⋊C411yescharacteristic
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C22×C22⋊C4116yes—
518C2×C4C22×C22⋊C412yes—
519C2×C4C22×C22⋊C412yes—
520C2×C4C22×C22⋊C411yescharacteristic
521C2×C4C22×C22⋊C411yescharacteristic
522C2×C4#51914yes—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—24no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22C23×C22⋊C411yescharacteristic
534C22C2×#51912yes—
535C22#348614yes—
536C4—21no—
537C4—21no—
538C4—22no—
539C4—24no—
540C4—21no—
541C4—22no—
542C4—216no—
543C4—21no—
544C4—22no—
545C4—42no—
546C4—44no—
547C4—42no—
548C4—44no—
549C4—88no—
550C4—88no—
551C4#288312yes—
552C4#288412yes—
553C2—24no—
554C2—22no—
555C2—44no—
556C2#3417311yescharacteristic
557C2C2×#288311yescharacteristic
558C2C2×#288411yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34371C2semidirect
3. #22513C2semidirect
4. Q8×#99C2semidirect
5. #21828C2semidirect
6. #21826C2semidirect
14. C2×Q82C4semidirect
15. C2×#557C4semidirect
16. C2×#554C4semidirect
18. C2×D4×Q8C4semidirect
20. C2×#544C4semidirect
22. C2×#533C4semidirect
24. C22×C4⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
#343712
#225132
Q8×#992
#218282
#218262
#207252
#207232