Finite Group Databasebeta

← #6566754

Subgroups — #6566754

Order 512.

1615 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#30109C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic, complemented
5#28511C211yesmaximal, characteristic, complemented
6#28317C212yesmaximal, complemented
7#28161C211yesmaximal, characteristic, complemented
8#27584C212yesmaximal, complemented
9#27252C214yesmaximal, complemented
10#26810C214yesmaximal, complemented
11#25017C212yesmaximal, complemented
12#24241C212yesmaximal
13#23762C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23102C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22811C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×#485C2214yes—
20C2×#477C2214yes—
21C2×#462C2212yes—
22C2×#452C2214yes—
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2571—21no—
27#2563—21no—
28#2538C2212yes—
29#2497—22no—
30#2492C2214yes—
31#2439—22no—
32#2374—22no—
33#2369—21no—
34#2365C2211yescharacteristic
35#2363—21no—
36#2362C2212yes—
37#2355C2212yes—
38#2354—21no—
39#2354C2211yescharacteristic
40#2354C2211yescharacteristic
41#2345—21no—
42#2341C2212yes—
43#2335—22no—
44#2324—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2297C2212yes—
49#2293C2211yescharacteristic
50#2285C2212yes—
51#2274—21no—
52C42×D4C2211yescharacteristic
53#2100C2212yes—
54#2095C2212yes—
55#2095C2212yes—
56#2094C2212yes—
57#2084C2211yescharacteristic
58#2075C2214yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2066C2212yes—
62#2058C2212yes—
63#2055C2212yes—
64#2043C2211yescharacteristic
65#2036C2214yes—
66#2035C2214yes—
67#2023C2212yes—
68#2016C2212yes—
69#2013C2211yescharacteristic
70#2001C2214yes—
71#1973C2214yes—
72#1971C2214yes—
73#1956C2212yes—
74#1954C2212yes—
75#1948C2212yes—
76#1928C2214yes—
77#1927C2214yes—
78#1925C2214yes—
79#1924C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1854C2214yes—
83#1845C2212yes—
84#1831C2211yescharacteristic
85#1817C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1788C2214yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41C2312yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98#547—24no—
99#547—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×#121—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C42C2312yes—
122#485—48no—
123#477—48no—
124#462—28no—
125#452—48no—
126C4×SD16—28no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#396—24no—
148#393—24no—
149#391—24no—
150#391C2312yes—
151#391C2312yes—
152#391C2314yes—
153#391C2312yes—
154#390—22no—
155#390—21no—
156#390—22no—
157#390—22no—
158#390C2311yescharacteristic
159#390C2311yescharacteristic
160#390C2312yes—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#388—22no—
172#388—22no—
173#388—22no—
174#388—24no—
175#388—21no—
176#388—21no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#386—22no—
183#386—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234#336C2311yescharacteristic
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×Q16—28no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8#4114yes—
265#123—216no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270#121—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—42no—
280C4×D4#4114yes—
281#117—44no—
282#117—44no—
283#117—44no—
284#117—84no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—28no—
349#107—44no—
350#106—24no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—24no—
354#103—28no—
355#103—28no—
356#103—24no—
357#103—24no—
358#103—42no—
359#103—44no—
360#103—44no—
361#102—24no—
362#102—44no—
363C22⋊C8—28no—
364C22⋊C8—44no—
365#97—44no—
366C4×C8—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—21no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387#41—88no—
388#41—88no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8C2×#4114yes—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4C2×#4112yes—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446Q16—88no—
447SD16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8H(5,𝔽2)14yes—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4C2×#4112yes—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—81no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—42no—
510C42—42no—
511C42—41no—
512C42—44no—
513C42—44no—
514C42—42no—
515C42—82no—
516C42—84no—
517C42—82no—
518C42—82no—
519C42#13714yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—44no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—164no—
544D4—24no—
545D4—44no—
546D4—42no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4#52512yes—
599C2×C4#52512yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4#57512yes—
604C2×C4#57812yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4#58512yes—
609C8—24no—
610C8—42no—
611C8—44no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22C2×#52511yescharacteristic
627C22#348511yescharacteristic
628C22#349811yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57511yescharacteristic
631C22C2×#57811yescharacteristic
632C22#358511yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—41no—
643C4—81no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—84no—
649C4#311614yes—
650C4#349412yes—
651C4#351412yes—
652C2—82no—
653C2—82no—
654C2#3602211yescharacteristic
655C2C2×#311611yescharacteristic
656C2#6060011yescharacteristic
657C2#6171911yescharacteristic
658C2#6173011yescharacteristic
659C2#6242411yescharacteristic
660C2#6245111yescharacteristic
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. #30109C2semidirect
4. #29981C2semidirect
5. #28511C2semidirect
6. #28317C2semidirect
7. #28161C2semidirect
8. #27584C2semidirect
9. #27252C2semidirect
10. #26810C2semidirect
11. #25017C2semidirect
13. #23762C2semidirect
14. #23677C2semidirect
15. #23102C2semidirect
16. #22839C2semidirect
17. #22811C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343162
#301092
#299812
#285112
#283172
#281612
#275842
#272522
#268102
#250172
#242412
#237622
#236772
#231022
#228392
#228112