Finite Group Databasebeta

← #6528508

Subgroups — #6528508

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#29111C212yesmaximal
4#29106C212yesmaximal
5#27966C212yesmaximal
6#27960C212yesmaximal
7#27832C214yesmaximal, complemented
8#27824C212yesmaximal, complemented
9#27815C212yesmaximal, complemented
10#27746C214yesmaximal, complemented
11#27738C212yesmaximal, complemented
12#27729C212yesmaximal, complemented
13#21840C211yesmaximal, characteristic
14#21833C211yesmaximal, characteristic
15#20740C211yesmaximal, characteristic
16#20735C211yesmaximal, characteristic
17#20535C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2666—22no—
25#2571C2211yescharacteristic
26#2570C2212yes—
27#2567—21no—
28#2567—21no—
29#2567—22no—
30#2567C2212yes—
31#2567C2212yes—
32#2567C2214yes—
33#2552C2211yescharacteristic
34#2552C2211yescharacteristic
35#2551C2211yescharacteristic
36#2550C2212yes—
37#2442—21no—
38#2440—21no—
39#2438—22no—
40C2×#395—22no—
41#2331—21no—
42#2330—21no—
43C2×#391C2212yes—
44C2×#389—21no—
45C2×#389—21no—
46#2285C2211yescharacteristic
47C2×#384C2212yes—
48#2283C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2214yes—
51#2100C2214yes—
52#2099C2214yes—
53#2099C2214yes—
54#2074C2214yes—
55#2072C2214yes—
56#2026C2214yes—
57#2024C2214yes—
58#1993C2212yes—
59#1993C2212yes—
60#1992C2214yes—
61#1990C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1988C2214yes—
65#1987C2212yes—
66#1987C2212yes—
67#1935C2214yes—
68#1935C2214yes—
69#1933C2214yes—
70#1933C2214yes—
71#1836C2214yes—
72#1834C2214yes—
73#1823C2212yes—
74#1823C2211yescharacteristic
75#1823C2212yes—
76#1823C2212yes—
77#1823C2211yescharacteristic
78#1823C2212yes—
79#1823C2212yes—
80#1805C2212yes—
81#1805C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1787C2212yes—
85#1787C2211yescharacteristic
86#1787C2211yescharacteristic
87#1771C2214yes—
88#1770C2212yes—
89C8×C4⋊C4C2212yes—
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—21no—
106C2×#125C2312yes—
107C2×C22⋊Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#432—24no—
127#431—24no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×C22⋊C8C2311yescharacteristic
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2312yes—
139C2×#97C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#395—22no—
150#395—24no—
151#395—24no—
152#395—24no—
153#395—24no—
154#395—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2314yes—
161#389—22no—
162#389—22no—
163#389—24no—
164#389—24no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—22no—
185#384—22no—
186#384C2314yes—
187#384C2314yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#383—21no—
192#382—21no—
193#382—21no—
194#382C2311yescharacteristic
195#382C2312yes—
196#382C2311yescharacteristic
197#381—21no—
198#380—22no—
199#380—21no—
200#380C2311yescharacteristic
201#380C2312yes—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214#376—21no—
215C2×#95—22no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#125—44no—
266#125—44no—
267C22⋊Q8—84no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—44no—
273C4×Q8—44no—
274#117—82no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C4⋊C8—24no—
343C4⋊C8—24no—
344#103—28no—
345#103—24no—
346#103—24no—
347#103—42no—
348#103—44no—
349#103—42no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—24no—
353#97—24no—
354#97—44no—
355C4×C8—24no—
356C4×C8—44no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—816no—
402C2×Q82−1+414yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4#12414yes—
434C22×C4#12414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2511yescharacteristic
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4#13714yes—
494C4⋊C4#13714yes—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C42—24no—
499C42—24no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—44no—
504C42—42no—
505C42—42no—
506C42—82no—
507C42#12414yes—
508C42#12414yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C2×#12412yes—
518C23#57612yes—
519C23#57712yes—
520C23C23×D411yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—48no—
524Q8—42no—
525Q8—42no—
526Q8—88no—
527Q8—88no—
528Q8—164no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4C2×#12412yes—
584C2×C4C2×#12412yes—
585C2×C4C2×#12412yes—
586C2×C4C2×#12412yes—
587C2×C4C2×#12412yes—
588C2×C4C2×#13714yes—
589C2×C4C2×#13714yes—
590C2×C4C2×#13712yes—
591C2×C4C2×#13712yes—
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×2−1+412yes—
595C2×C4#58512yes—
596C8—42no—
597C8—42no—
598C8—44no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22#313614yes—
606C22C22×#12411yescharacteristic
607C22#348511yescharacteristic
608C22#348611yescharacteristic
609C22C22×#13711yescharacteristic
610C22C22×#13711yescharacteristic
611C22C2×#57611yescharacteristic
612C22C2×#57711yescharacteristic
613C4—24no—
614C4—24no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4#313914yes—
631C4#349512yes—
632C4#349512yes—
633C2—24no—
634C2#3564411yescharacteristic
635C2C2×#313611yescharacteristic
636C2C2×#313911yescharacteristic
637C2#6172611yescharacteristic
638C2#6173511yescharacteristic
639C2#6183811yescharacteristic
640C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #27832C2semidirect
8. #27824C2semidirect
9. #27815C2semidirect
10. #27746C2semidirect
11. #27738C2semidirect
12. #27729C2semidirect

Maximal subgroups

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

subgroupindex
#352392
C2×#25672
#291112
#291062
#279662
#279602
#278322
#278242
#278152
#277462
#277382
#277292
#218402
#218332
#207402
#207352
#205352