Finite Group Databasebeta

← #6568267

Subgroups — #6568267

Order 512.

1515 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#30168C212yesmaximal, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29259C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27584C212yesmaximal, complemented
9#27548C214yesmaximal, complemented
10#27120C214yesmaximal, complemented
11#26299C212yesmaximal, complemented
12#25886C212yesmaximal
13#23775C211yesmaximal, characteristic, complemented
14#23762C212yesmaximal, complemented
15#23221C212yesmaximal, complemented
16#23067C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C2×#487C2212yes—
20C2×#476C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563C2212yes—
25#2538—21no—
26#2497C2214yes—
27#2492—22no—
28#2439—22no—
29#2374—22no—
30#2369C2212yes—
31#2365C2211yescharacteristic
32#2363—21no—
33#2362C2212yes—
34#2355—21no—
35#2354—21no—
36#2354C2211yescharacteristic
37#2354C2211yescharacteristic
38#2345—21no—
39#2341C2212yes—
40#2335—22no—
41#2324—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2297C2212yes—
46#2293C2211yescharacteristic
47#2285C2212yes—
48#2274—21no—
49C42×D4C2211yescharacteristic
50#2100C2211yescharacteristic
51#2095C2212yes—
52#2094C2211yescharacteristic
53#2088C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2077C2214yes—
57#2075C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2065C2212yes—
63#2058C2211yescharacteristic
64#2035C2214yes—
65#2035C2214yes—
66#2024C2212yes—
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1987C2212yes—
70#1981C2212yes—
71#1954C2212yes—
72#1947C2212yes—
73#1936C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1925C2214yes—
77#1924C2214yes—
78#1919C2214yes—
79#1865C2214yes—
80#1863C2214yes—
81#1854C2214yes—
82#1831C2211yescharacteristic
83#1825C2214yes—
84#1817C2212yes—
85#1803C2214yes—
86#1788C2214yes—
87#1788C2214yes—
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96#547—24no—
97#547—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×#121—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C2×#117—21no—
116C2×#117—21no—
117C2×#117C2312yes—
118C22×C4⋊C4—21no—
119C22×C42C2312yes—
120#487—28no—
121#476—28no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2312yes—
125C2×#106C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#400—24no—
140#396—24no—
141#393—24no—
142#391—21no—
143#391—22no—
144#391—24no—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—21no—
149#390—22no—
150#390—22no—
151#390C2311yescharacteristic
152#390C2311yescharacteristic
153#390C2312yes—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389C2312yes—
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2314yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—24no—
168#388—21no—
169#388—21no—
170#388—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382—22no—
186#380—22no—
187#380—22no—
188#380—21no—
189#380—22no—
190#380C2314yes—
191#380C2314yes—
192#380C2312yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—42no—
254C4⋊Q8—82no—
255C4⋊Q8—82no—
256C4⋊Q8#4114yes—
257#123—216no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—84no—
261#121—28no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—42no—
270C4×D4#4114yes—
271#117—44no—
272#117—44no—
273#117—44no—
274#117—84no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—22no—
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—44no—
339#106—28no—
340#106—42no—
341#106—42no—
342C4⋊C8—24no—
343C4⋊C8—42no—
344#103—28no—
345#103—28no—
346#103—44no—
347#103—44no—
348#102—28no—
349#102—42no—
350#102—42no—
351C22⋊C8—44no—
352C22⋊C8—44no—
353#97—44no—
354C4×C8—24no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—21no—
374C24C22×D412yes—
375#41—88no—
376#41—88no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×Q8C2×#4114yes—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4C2×#4112yes—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C4⋊C4—28no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4C2×#4112yes—
473C22⋊C4—28no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—41no—
496C42—44no—
497C42—42no—
498C42—44no—
499C42—82no—
500C42—84no—
501C42—82no—
502C42—82no—
503C42#13714yes—
504C42C2×#4112yes—
505C42C2×#4112yes—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C23×D411yescharacteristic
521Q8—28no—
522Q8—44no—
523Q8—84no—
524Q8—164no—
525D4—24no—
526D4—44no—
527D4—42no—
528D4—88no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—84no—
578C2×C4—88no—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4C2×#13712yes—
584C2×C4#57512yes—
585C2×C4#57712yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4#58512yes—
589C2×C4#58512yes—
590C8—44no—
591C8—42no—
592C8—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C2×#52511yescharacteristic
608C22#348511yescharacteristic
609C22#350011yescharacteristic
610C22C22×#13711yescharacteristic
611C22C2×#57511yescharacteristic
612C22C2×#57711yescharacteristic
613C22#358511yescharacteristic
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—44no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4#311614yes—
631C4#351012yes—
632C4#352112yes—
633C2—82no—
634C2—82no—
635C2#3603311yescharacteristic
636C2C2×#311611yescharacteristic
637C2#6059311yescharacteristic
638C2#6171911yescharacteristic
639C2#6173111yescharacteristic
640C2#6254211yescharacteristic
641C2#6258611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #30168C2semidirect
4. #29868C2semidirect
5. #29259C2semidirect
6. #29113C2semidirect
7. #28717C2semidirect
8. #27584C2semidirect
9. #27548C2semidirect
10. #27120C2semidirect
11. #26299C2semidirect
13. #23775C2semidirect
14. #23762C2semidirect
15. #23221C2semidirect
16. #23067C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343162
#301682
#298682
#292592
#291132
#287172
#275842
#275482
#271202
#262992
#258862
#237752
#237622
#232212
#230672
#228392