Finite Group Databasebeta

← #6528552

Subgroups — #6528552

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic, complemented
2#35279C211yesmaximal, characteristic
3#29120C214yesmaximal
4#27971C214yesmaximal, complemented
5#27823C212yesmaximal, complemented
6#27821C212yesmaximal, complemented
7#27815C212yesmaximal, complemented
8#27813C212yesmaximal, complemented
9#27738C212yesmaximal, complemented
10#27736C212yesmaximal, complemented
11#27730C212yesmaximal, complemented
12#27728C212yesmaximal, complemented
13#21834C211yesmaximal, characteristic, complemented
14#21833C211yesmaximal, characteristic, complemented
15#20737C211yesmaximal, characteristic
16#20737C211yesmaximal, characteristic
17#20538C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857—21no—
23#2857—21no—
24#2844—22no—
25#2628—22no—
26#2628—22no—
27#2628—22no—
28#2628C2214yes—
29#2571—21no—
30#2571C2212yes—
31#2571C2212yes—
32#2568C2212yes—
33#2567C2212yes—
34#2563C2212yes—
35#2551—21no—
36#2551C2212yes—
37#2551C2212yes—
38#2542C2212yes—
39#2487—22no—
40#2409—22no—
41#2317C2212yes—
42#2311C2212yes—
43#2285C2211yescharacteristic
44#2285C2211yescharacteristic
45#2285C2211yescharacteristic
46#2101C2214yes—
47#2101C2214yes—
48#2100C2214yes—
49#2099C2214yes—
50#2074C2214yes—
51#2073C2214yes—
52#2024C2214yes—
53#2023C2214yes—
54#1993C2214yes—
55#1990C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1987C2212yes—
63#1987C2212yes—
64#1936C2214yes—
65#1935C2214yes—
66#1934C2214yes—
67#1933C2214yes—
68#1836C2214yes—
69#1835C2214yes—
70#1825C2212yes—
71#1825C2212yes—
72#1824C2214yes—
73#1823C2214yes—
74#1803C2211yescharacteristic
75#1803C2211yescharacteristic
76#1803C2211yescharacteristic
77#1803C2211yescharacteristic
78#1802C2211yescharacteristic
79#1802C2211yescharacteristic
80#1802C2211yescharacteristic
81#1802C2211yescharacteristic
82#1788C2212yes—
83#1788C2212yes—
84#1770C2212yes—
85#1770C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92#1750C2214yes—
93C22×#41C2311yescharacteristic
94C23×Q8—21no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2314yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—21no—
116C2×#117—21no—
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123#431—24no—
124#431—24no—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—24no—
144#400—24no—
145#400—24no—
146#400—24no—
147#399—24no—
148#399—24no—
149#395—24no—
150#395—24no—
151#395—24no—
152#395—24no—
153#393—24no—
154#393—24no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#386—24no—
170#385—24no—
171#384D411yescharacteristic
172#384D411yescharacteristic
173#384C2314yes—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382—22no—
179#382—22no—
180#382—21no—
181#382—21no—
182#382D412yes—
183#382C2314yes—
184#382C2314yes—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380D412yes—
192#380D412yes—
193#380C2312yes—
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201#340C2312yes—
202#340C2312yes—
203#340C2312yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—41no—
224C2×#41—41no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—24no—
228C22×Q8—44no—
229C22×Q8—82no—
230C22×Q8—82no—
231C22×Q8C2411yescharacteristic
232C22×D4—21no—
233C22×D4C2411yescharacteristic
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—84no—
254C4⋊Q8—84no—
255C22⋊Q8—84no—
256C22⋊Q8—84no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—44no—
266C4×D4—44no—
267#117—22no—
268#117—22no—
269#117—42no—
270#117—42no—
271#117—44no—
272#117—82no—
273#117—82no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4C2×D414yes—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—44no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4114yes—
325C2×C42#4114yes—
326C2×C42C2414yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333#103—24no—
334#103—24no—
335#103—42no—
336#103—42no—
337#102—24no—
338#102—24no—
339#102—42no—
340#102—42no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343#97—44no—
344C4×C8—24no—
345C4×C8—24no—
346C4×C8—44no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95C2×D414yes—
360#95C2×D414yes—
361C24—21no—
362C24C22×D411yescharacteristic
363#41—44no—
364#41—44no—
365#41—44no—
366#41—44no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×Q8—84no—
381C2×Q8—816no—
382C2×Q82−1+412yes—
383C2×D4—24no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D42−1+412yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4#12414yes—
421C22×C4#12414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2511yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—24no—
491C42—24no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—44no—
496C42—44no—
497C42#12414yes—
498C42#12414yes—
499C42C2×#4112yes—
500C42C2×#4112yes—
501C42C2×#4112yes—
502C42C2×#4112yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×#12412yes—
517C23C23×D411yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—88no—
523Q8—88no—
524Q8—168no—
525D4—22no—
526D4—22no—
527D4—48no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—88no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4C2×#12412yes—
577C2×C4C2×#12412yes—
578C2×C4C2×#12412yes—
579C2×C4C2×#12412yes—
580C2×C4C2×#12412yes—
581C2×C4#57612yes—
582C2×C4#57712yes—
583C2×C4#57712yes—
584C2×C4#57812yes—
585C2×C4C22×#4111yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C2×C4C2×2−1+411yescharacteristic
589C2×C4#58511yescharacteristic
590C2×C4#58511yescharacteristic
591C8—42no—
592C8—42no—
593C8—42no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C22×#12411yescharacteristic
610C22#348511yescharacteristic
611C22#348611yescharacteristic
612C22C2×#57611yescharacteristic
613C22C2×#57711yescharacteristic
614C22C2×#57711yescharacteristic
615C22C2×#57811yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—44no—
621C4—44no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—81no—
627C4—84no—
628C4—81no—
629C4#313614yes—
630C4#313714yes—
631C4#349511yescharacteristic
632C4#349511yescharacteristic
633C4#349511yescharacteristic
634C4#349511yescharacteristic
635C2—42no—
636C2—82no—
637C2#3564411yescharacteristic
638C2C2×#313611yescharacteristic
639C2C2×#313711yescharacteristic
640C2#6178811yescharacteristic
641C2#6178911yescharacteristic
642C2#6187011yescharacteristic
643C2#6187111yescharacteristic
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. #27823C2semidirect
6. #27821C2semidirect
7. #27815C2semidirect
8. #27813C2semidirect
9. #27738C2semidirect
10. #27736C2semidirect
11. #27730C2semidirect
12. #27728C2semidirect
13. #21834C2semidirect
14. #21833C2semidirect
17. #20538C2semidirect

Maximal subgroups

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

subgroupindex
#376422
#352792
#291202
#279712
#278232
#278212
#278152
#278132
#277382
#277362
#277302
#277282
#218342
#218332
#207372
#205382