Finite Group Databasebeta

← #6489420

Subgroups — #6489420

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33209C212yesmaximal, complemented
4#33042C212yesmaximal
5#30376C214yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29178C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28655C212yesmaximal, complemented
11#28508C211yesmaximal, characteristic, complemented
12#27856C214yesmaximal, complemented
13#24835C214yesmaximal, complemented
14#22110C214yesmaximal, complemented
15C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
16C2×#533—21no—
17C2×#529—21no—
18C2×#500C2212yes—
19C2×#486C2212yes—
20C2×#474C2214yes—
21C22×C4⋊C8C2212yes—
22C22×C4×C8C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862—21no—
26#2861—21no—
27#2860—21no—
28#2860C2212yes—
29#2859—21no—
30#2859C2211yescharacteristic
31#2859C2212yes—
32#2858C2211yescharacteristic
33#2857—21no—
34#2857—21no—
35#2857C2212yes—
36#2857C2212yes—
37#2857C2211yescharacteristic
38#2856C2211yescharacteristic
39#2854—21no—
40#2854C2212yes—
41#2850—21no—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#124—21no—
49C4×#121—21no—
50C4×#121C2212yes—
51#2293—21no—
52#2293C2212yes—
53#2285—21no—
54#2285C2212yes—
55C2×C43C2211yescharacteristic
56#2099C2214yes—
57#2093C2214yes—
58#2086C2218yes—
59#2083C2214yes—
60#2071C2214yes—
61#2067C2218yes—
62#1990C2212yes—
63#1990C2212yes—
64#1990C2212yes—
65#1988C2214yes—
66#1957C2212yes—
67#1947C2214yes—
68#1942C2218yes—
69#1934C2218yes—
70#1925C2218yes—
71#1918C2218yes—
72#1893C2218yes—
73#1845C2212yes—
74#1834C2212yes—
75#1808C2218yes—
76#1802C2214yes—
77C4×C4⋊C8C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79C4×C4⋊C8C2212yes—
80C4×C4⋊C8C2212yes—
81C4×C4⋊C8C2212yes—
82C2×#336C2212yes—
83C23×C8C2314yes—
84#533—48no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93#529—48no—
94C2×#126—22no—
95C2×#125—22no—
96C2×#125—21no—
97C2×#124—21no—
98C2×#124—21no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#500—44no—
123#486—44no—
124#474—216no—
125C2×#107C2318yes—
126C2×#107C2318yes—
127C2×C4⋊C8—28no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C2×C4×C8—44no—
134C2×C4×C8C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#391—22no—
138#391—22no—
139#391—21no—
140#391—22no—
141#391—22no—
142#391—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389C2314yes—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#386—22no—
178#386—22no—
179#384—21no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382—22no—
191#382—22no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2314yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2412yes—
228C23×C4—22no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8—28no—
237C22×C8—28no—
238C22×C8—44no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—28no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250#127—22no—
251#127—22no—
252#126—48no—
253#125—44no—
254#125—48no—
255#124—44no—
256#124—44no—
257#123—48no—
258C22⋊Q8—44no—
259C22⋊Q8—48no—
260C22⋊Q8—48no—
261#121—28no—
262#121—44no—
263#121—44no—
264C4×Q8—44no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×D4—28no—
268C4×D4—48no—
269C4×D4—44no—
270C4×D4—44no—
271#117—48no—
272#117—48no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42D818yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—216no—
350#107—48no—
351C4⋊C8—28no—
352C4⋊C8—24no—
353C4⋊C8#4118yes—
354#102—216no—
355#102—48no—
356C4×C8—24no—
357C4×C8—24no—
358C4×C8—44no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—24no—
368#95—22no—
369C24—22no—
370C24C22×D411yescharacteristic
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4C2×D814yes—
419C22×C4C2×D814yes—
420C22×C4C2×D814yes—
421C22×C4#13714yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C42−1+414yes—
436C22×C4C2511yescharacteristic
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—416no—
443C2×C8C2×#4118yes—
444C4⋊C4—28no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—41no—
505C42—41no—
506C42—44no—
507C42—48no—
508C42C2×D814yes—
509C42C2×D814yes—
510C42C2×D814yes—
511C42C2×D814yes—
512C42#13514yes—
513C42C2×#4114yes—
514C42H(5,𝔽2)12yes—
515C422−1+412yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23#53612yes—
528C23C22×D812yes—
529C23C2×#13712yes—
530C23C23×D411yescharacteristic
531Q8—88no—
532Q8—84no—
533Q8—88no—
534D4—44no—
535D4—44no—
536D4—88no—
537D4—84no—
538D4—84no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
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—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4#53212yes—
586C2×C4#53212yes—
587C2×C4#53612yes—
588C2×C4#53712yes—
589C2×C4#53712yes—
590C2×C4C22×D812yes—
591C2×C4C22×D812yes—
592C2×C4C22×D812yes—
593C2×C4C22×D812yes—
594C2×C4C22×D812yes—
595C2×C4C22×D812yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C2×#13512yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4#57512yes—
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+412yes—
605C2×C4C2×2−1+411yescharacteristic
606C8—28no—
607C8—44no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22#322614yes—
622C22C2×#53211yescharacteristic
623C22C2×#53611yescharacteristic
624C22C2×#53711yescharacteristic
625C22C23×D811yescharacteristic
626C22C22×#13511yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—82no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—84no—
644C4#316914yes—
645C4#324514yes—
646C4#324814yes—
647C4#352012yes—
648C4#352312yes—
649C2—24no—
650C2—162no—
651C2#3524111yescharacteristic
652C2C2×#316911yescharacteristic
653C2#6070111yescharacteristic
654C2C2×#322611yescharacteristic
655C2#6084611yescharacteristic
656C2C2×#324511yescharacteristic
657C2C2×#324811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33209C2semidirect
5. #30376C2semidirect
6. #29219C2semidirect
7. #29178C2semidirect
8. #28716C2semidirect
9. #28689C2semidirect
10. #28655C2semidirect
11. #28508C2semidirect
12. #27856C2semidirect
13. #24835C2semidirect
14. #22110C2semidirect
15. C2×C4×C4⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366312
#332092
#330422
#303762
#292192
#291782
#287162
#286892
#286552
#285082
#278562
#248352
#221102
C2×C4×C4⋊C82