Finite Group Databasebeta

← #6986813

Subgroups — #6986813

Order 512.

1573 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#36505C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#32789C212yesmaximal, complemented
4#32265C214yesmaximal, complemented
5#32008C214yesmaximal, complemented
6#30294C212yesmaximal, complemented
7#30175C212yesmaximal, complemented
8#29936C211yesmaximal, characteristic, complemented
9#29159C212yesmaximal, complemented
10#28706C212yesmaximal, complemented
11#26279C212yesmaximal, complemented
12#25836C212yesmaximal, complemented
13#25811C211yesmaximal, characteristic, complemented
14#25340C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic, complemented
16#23756C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#498C2212yes—
22C2×#477C2212yes—
23C2×C8×D4C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2628—22no—
29#2568—21no—
30#2567C2212yes—
31#2563C2212yes—
32#2562C2212yes—
33#2547C2211yescharacteristic
34#2542—21no—
35#2538C2211yescharacteristic
36#2536C2212yes—
37#2487C2214yes—
38#2475—22no—
39#2460—22no—
40#2446—21no—
41#2432—21no—
42#2409—22no—
43#2367—21no—
44#2362—21no—
45#2355C2211yescharacteristic
46#2317—21no—
47#2311—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2096C2211yescharacteristic
55#2085C2214yes—
56#2084C2211yescharacteristic
57#2083C2212yes—
58#2077C2214yes—
59#2077C2212yes—
60#2077C2214yes—
61#2076C2214yes—
62#2075C2214yes—
63#2072C2211yescharacteristic
64#2059C2214yes—
65#2059C2214yes—
66#2058C2212yes—
67#2058C2212yes—
68#2057C2212yes—
69#2036C2214yes—
70#2035C2212yes—
71#2024C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1987C2212yes—
75#1984C2212yes—
76#1947C2212yes—
77#1919C2214yes—
78#1918C2214yes—
79#1892C2214yes—
80#1872C2212yes—
81#1869C2214yes—
82#1865C2214yes—
83#1865C2214yes—
84#1853C2214yes—
85#1852C2212yes—
86#1817C2212yes—
87#1808C2214yes—
88#1807C2214yes—
89#1807C2214yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#557—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125C2312yes—
108C2×#124—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C22⋊Q8C2312yes—
114C4×#41—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117D411yescharacteristic
126C2×#117D412yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#498—44no—
132#477—44no—
133C8×D4—44no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—24no—
150#399—24no—
151#398—22no—
152#397—22no—
153#396—22no—
154#395—24no—
155#394—22no—
156#394—22no—
157#393—24no—
158#391—22no—
159#391—21no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391—21no—
164#391C2311yescharacteristic
165#390—22no—
166#390—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389C2314yes—
172#389C2314yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—21no—
184#386—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—21no—
189#385—24no—
190#384—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382—21no—
196#382C2312yes—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380C2311yescharacteristic
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C22×C8—28no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267#125—44no—
268#125—48no—
269#124—44no—
270C22⋊Q8—22no—
271C22⋊Q8—24no—
272C22⋊Q8—41no—
273C22⋊Q8—42no—
274C22⋊Q8—44no—
275C22⋊Q8—84no—
276C22⋊Q8—84no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—42no—
284C4×D4—44no—
285C4×D4—44no—
286#117—44no—
287#117—42no—
288#117—44no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—44no—
351#107—44no—
352#106—24no—
353#106—42no—
354#106—44no—
355#106—44no—
356C4⋊C8—24no—
357#103—24no—
358#103—24no—
359#103—44no—
360C22⋊C8—24no—
361C22⋊C8—42no—
362C22⋊C8—44no—
363C4×C8—24no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24C22×D411yescharacteristic
384C24C22×D412yes—
385#41—416no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—816no—
402C2×Q82−1+412yes—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4#13714yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4#13514yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—88no—
459C2×C82−1+414yes—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—81no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—42no—
515C42—42no—
516C42—44no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42—82no—
521C42H(5,𝔽2)14yes—
522C23—22no—
523C23—21no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—21no—
534C23C2×#13512yes—
535C23C2×#13714yes—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—44no—
539Q8—42no—
540Q8—88no—
541Q8—88no—
542Q8—84no—
543D4—28no—
544D4—48no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4#53412yes—
600C2×C4#53412yes—
601C2×C4C2×#13512yes—
602C2×C4C2×#13512yes—
603C2×C4C2×#13712yes—
604C2×C4#57712yes—
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4C2×2−1+411yescharacteristic
607C2×C4C2×2−1+412yes—
608C2×C4#58511yescharacteristic
609C8—24no—
610C8—42no—
611C8—44no—
612C22—22no—
613C22—24no—
614C22—22no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—41no—
626C22C2×#53411yescharacteristic
627C22#348911yescharacteristic
628C22#349911yescharacteristic
629C22C22×#13511yescharacteristic
630C22C22×#13711yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358511yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4#319014yes—
650C4#355212yes—
651C4#356112yes—
652C2—42no—
653C2—44no—
654C2#4291211yescharacteristic
655C2C2×#319011yescharacteristic
656C2#6075711yescharacteristic
657C2#6210211yescharacteristic
658C2#6213111yescharacteristic
659C2#6247011yescharacteristic
660C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32789C2semidirect
4. #32265C2semidirect
5. #32008C2semidirect
6. #30294C2semidirect
7. #30175C2semidirect
8. #29936C2semidirect
9. #29159C2semidirect
10. #28706C2semidirect
11. #26279C2semidirect
12. #25836C2semidirect
13. #25811C2semidirect
14. #25340C2semidirect
15. #24302C2semidirect
16. #23756C2semidirect

Maximal subgroups

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

subgroupindex
#365052
#352792
#327892
#322652
#320082
#302942
#301752
#299362
#291592
#287062
#262792
#258362
#258112
#253402
#243022
#237562
#224692