Finite Group Databasebeta

← #6568270

Subgroups — #6568270

Order 512.

1571 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34330C211yesmaximal, characteristic
3#30165C212yesmaximal, complemented
4#29869C211yesmaximal, characteristic, complemented
5#29258C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#27584C212yesmaximal, complemented
9#27544C214yesmaximal, complemented
10#27124C214yesmaximal, complemented
11#26297C212yesmaximal, complemented
12#25883C212yesmaximal
13#23777C211yesmaximal, characteristic, complemented
14#23760C212yesmaximal, complemented
15#23223C212yesmaximal, complemented
16#23069C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×#486C2212yes—
20C2×#477C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563C2212yes—
25#2538—21no—
26#2497C2214yes—
27#2492—22no—
28#2439—22no—
29#2369C2212yes—
30#2366—21no—
31#2366C2211yescharacteristic
32#2364—22no—
33#2363C2212yes—
34#2357—21no—
35#2355—21no—
36#2355C2211yescharacteristic
37#2354—21no—
38#2354C2211yescharacteristic
39#2339C2212yes—
40#2333—22no—
41#2328—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#121C2212yes—
45#2297C2212yes—
46#2295C2211yescharacteristic
47#2285C2212yes—
48#2275—21no—
49C42×D4C2211yescharacteristic
50#2100C2211yescharacteristic
51#2095C2211yescharacteristic
52#2094C2212yes—
53#2087C2212yes—
54#2083C2212yes—
55#2083C2212yes—
56#2077C2214yes—
57#2075C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2068C2212yes—
63#2057C2211yescharacteristic
64#2033C2214yes—
65#2033C2214yes—
66#2024C2211yescharacteristic
67#2023C2212yes—
68#1990C2212yes—
69#1987C2212yes—
70#1982C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1925C2214yes—
77#1924C2214yes—
78#1918C2214yes—
79#1864C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1831C2211yescharacteristic
83#1825C2214yes—
84#1817C2212yes—
85#1802C2214yes—
86#1788C2214yes—
87#1788C2214yes—
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96#550—24no—
97#550—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127—21no—
103C2×#127C2311yescharacteristic
104C2×C22⋊Q8—22no—
105C2×#121—22no—
106C2×#121C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117C2312yes—
121C22×C42C2312yes—
122#486—28no—
123#477—28no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#400—24no—
142#396—24no—
143#393—24no—
144#391—21no—
145#391—22no—
146#391—24no—
147#391C2311yescharacteristic
148#391C2311yescharacteristic
149#391C2312yes—
150#390—22no—
151#390—21no—
152#390—21no—
153#390—22no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—21no—
160#389—22no—
161#389C2312yes—
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#389C2314yes—
165#388—22no—
166#388—24no—
167#388—22no—
168#388—22no—
169#387—21no—
170#387—21no—
171#387—22no—
172#387—24no—
173#386—22no—
174#386—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382—21no—
188#382—22no—
189#381—22no—
190#380—22no—
191#380—22no—
192#380C2314yes—
193#380C2314yes—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376—21no—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—42no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#127—28no—
260#127—22no—
261#127#4114yes—
262C22⋊Q8—84no—
263#121—216no—
264#121—24no—
265#121—24no—
266C4×Q8—22no—
267C4×Q8—44no—
268C4×Q8#4114yes—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—42no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—82no—
279C4×D4—84no—
280#117—44no—
281#117—84no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—44no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—28no—
339#107—42no—
340#107—42no—
341#106—44no—
342C4⋊C8—24no—
343C4⋊C8—42no—
344#103—28no—
345#103—42no—
346#103—42no—
347#102—28no—
348#102—28no—
349#102—44no—
350#102—44no—
351C22⋊C8—44no—
352C22⋊C8—44no—
353#97—44no—
354C4×C8—24no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—22no—
367#95—24no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—22no—
376C24C22×D412yes—
377#41—88no—
378#41—88no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8C2×#4112yes—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C2×#4114yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4C2×#4112yes—
483C22⋊C4—28no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42—42no—
505C42—41no—
506C42—42no—
507C42—44no—
508C42—82no—
509C42—84no—
510C42—82no—
511C42#13714yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C23×D411yescharacteristic
529Q8—24no—
530Q8—42no—
531Q8—84no—
532Q8—164no—
533D4—28no—
534D4—44no—
535D4—48no—
536D4—44no—
537D4—84no—
538D4—88no—
539D4—88no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—164no—
590C2×C4—164no—
591C2×C4#52512yes—
592C2×C4#52512yes—
593C2×C4C2×#13712yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4#57512yes—
597C2×C4#57612yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4#58512yes—
601C2×C4#58512yes—
602C8—44no—
603C8—42no—
604C8—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#52511yescharacteristic
620C22#348511yescharacteristic
621C22#350011yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57611yescharacteristic
625C22#358511yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—81no—
641C4—84no—
642C4#311614yes—
643C4#351012yes—
644C4#352112yes—
645C2—82no—
646C2—84no—
647C2#3603311yescharacteristic
648C2C2×#311611yescharacteristic
649C2#6059011yescharacteristic
650C2#6171911yescharacteristic
651C2#6174011yescharacteristic
652C2#6253711yescharacteristic
653C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #30165C2semidirect
4. #29869C2semidirect
5. #29258C2semidirect
6. #29113C2semidirect
7. #28716C2semidirect
8. #27584C2semidirect
9. #27544C2semidirect
10. #27124C2semidirect
11. #26297C2semidirect
13. #23777C2semidirect
14. #23760C2semidirect
15. #23223C2semidirect
16. #23069C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343302
#301652
#298692
#292582
#291132
#287162
#275842
#275442
#271242
#262972
#258832
#237772
#237602
#232232
#230692
#228392