Finite Group Databasebeta

← #7534840

Subgroups — #7534840

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#31586C214yesmaximal, complemented
4#28717C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic, complemented
6#28414C212yesmaximal, complemented
7#28272C212yesmaximal, complemented
8#27206C214yesmaximal, complemented
9#25275C212yesmaximal, complemented
10#25062C212yesmaximal, complemented
11#24811C212yesmaximal, complemented
12#24186C212yesmaximal
13#23153C211yesmaximal, characteristic, complemented
14#22968C212yesmaximal, complemented
15#22546C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#559C2212yes—
19C2×#533—21no—
20C2×#487C2212yes—
21C2×#474C2212yes—
22C2×#474C2212yes—
23C2×#461C2211yescharacteristic
24C2×#459C2211yescharacteristic
25C2×#446C2212yes—
26C2×C4×SD16C2212yes—
27C2×C4×D8C2212yes—
28#2868C2211yescharacteristic
29#2857—21no—
30#2724—22no—
31#2582C2212yes—
32#2552C2212yes—
33#2547—21no—
34#2547C2212yes—
35#2542C2212yes—
36#2538C2212yes—
37#2486—22no—
38#2447—22no—
39#2447C2214yes—
40#2441—22no—
41#2429—22no—
42#2423—22no—
43#2365C2212yes—
44#2354C2212yes—
45Q8×C4⋊C4C2212yes—
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2095C2214yes—
53#2093C2214yes—
54#2083C2214yes—
55#2072C2214yes—
56#2068C2214yes—
57#2067C2214yes—
58#2042C2214yes—
59#2012C2214yes—
60#1990C2212yes—
61#1984C2212yes—
62#1983C2214yes—
63#1982C2211yescharacteristic
64#1981C2211yescharacteristic
65#1973C2214yes—
66#1959C2214yes—
67#1957C2212yes—
68#1948C2211yescharacteristic
69#1947C2211yescharacteristic
70#1943C2214yes—
71#1939C2212yes—
72#1928C2214yes—
73#1927C2212yes—
74#1925C2212yes—
75#1921C2214yes—
76#1917C2214yes—
77#1879C2212yes—
78#1879C2214yes—
79#1877C2212yes—
80#1864C2214yes—
81#1845C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1802C2214yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93C22×SD16C2312yes—
94C22×D8C2312yes—
95#559—28no—
96#559—44no—
97#533—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125—22no—
105C2×#124—22no—
106C2×#124C2314yes—
107C2×#123—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2314yes—
123C2×#117—22no—
124C22×C4⋊C4—22no—
125C22×C42—21no—
126#487—44no—
127#474—28no—
128#474—44no—
129#461—24no—
130#459—24no—
131#446—44no—
132C4×SD16—44no—
133C4×D8—44no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158#399—24no—
159#398—24no—
160#398—24no—
161#397—24no—
162#397—24no—
163#397—24no—
164#396—24no—
165#391—22no—
166#391—22no—
167#391D412yes—
168#391C2312yes—
169#391C2314yes—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389—21no—
179#389—21no—
180#389D412yes—
181#389D412yes—
182#389C2314yes—
183#389C2314yes—
184#388—24no—
185#387—24no—
186#385—24no—
187#385—24no—
188#384—21no—
189#384—21no—
190#384D411yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2314yes—
194#384C2312yes—
195#384C2312yes—
196#381—24no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C43C2311yescharacteristic
214#340C2312yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#336C2312yes—
221#336C2314yes—
222#336C2312yes—
223#336C2314yes—
224C22×Q8—22no—
225C22×Q8C2×D412yes—
226C22×Q8C2412yes—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C22×D4C2411yescharacteristic
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C2×SD16—24no—
237C2×SD16—24no—
238C2×SD16—42no—
239C2×D8—24no—
240C2×D8—24no—
241C2×D8—42no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253#127—22no—
254#127#4114yes—
255#125—48no—
256#124—28no—
257#124—44no—
258#124—48no—
259#123—416no—
260C22⋊Q8—48no—
261C22⋊Q8—48no—
262C22⋊Q8—84no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—48no—
268C4×Q8—48no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—28no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—44no—
281C4×D4—82no—
282#117—48no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—84no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—44no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2414yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—28no—
338#107—42no—
339#107—44no—
340#106—42no—
341C4⋊C8—22no—
342C4⋊C8—22no—
343C4⋊C8—22no—
344C4⋊C8—44no—
345C4⋊C8#4114yes—
346#103—24no—
347#103—24no—
348#103—42no—
349#103—44no—
350#103—44no—
351#102—24no—
352#102—28no—
353#102—24no—
354#102—42no—
355#102—42no—
356#102—42no—
357#102—44no—
358#97—44no—
359C4×C8—24no—
360C4×C8—24no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—21no—
374C24—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4C2×#4114yes—
404C2×D42−1+412yes—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436SD16—48no—
437D8—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8C2×#4114yes—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—84no—
479C4⋊C4—81no—
480C4⋊C4—84no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C42−1+412yes—
484C22⋊C4—24no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C42—22no—
496C42—22no—
497C42—24no—
498C42—24no—
499C42—21no—
500C42—21no—
501C42—41no—
502C42—48no—
503C42—44no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42AGL(1,ℤ/8)14yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C42H(5,𝔽2)12yes—
511C42H(5,𝔽2)12yes—
512C23—24no—
513C23—22no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23C23×D411yescharacteristic
527Q8—48no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—88no—
533D4—24no—
534D4—22no—
535D4—24no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—42no—
540D4—42no—
541D4—41no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4—168no—
584C2×C4#54612yes—
585C2×C4#54612yes—
586C2×C4#55012yes—
587C2×C4#55012yes—
588C2×C4C2×AGL(1,ℤ/8)12yes—
589C2×C4C2×AGL(1,ℤ/8)12yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4#57612yes—
592C2×C4#57812yes—
593C2×C4C22×#4111yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—24no—
600C8—42no—
601C8—42no—
602C8—42no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—81no—
610C22—82no—
611C22—82no—
612C22—81no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—82no—
617C22C2×#54611yescharacteristic
618C22C2×#55011yescharacteristic
619C22#348611yescharacteristic
620C22C22×AGL(1,ℤ/8)11yescharacteristic
621C22C2×#57611yescharacteristic
622C22C2×#57811yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—42no—
629C4—41no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4—84no—
636C4—81no—
637C4#333014yes—
638C4#339914yes—
639C4#348211yescharacteristic
640C4#352011yescharacteristic
641C4#352011yescharacteristic
642C4#354811yescharacteristic
643C2—84no—
644C2—82no—
645C2#5904011yescharacteristic
646C2C2×#333011yescharacteristic
647C2#6119911yescharacteristic
648C2C2×#339911yescharacteristic
649C2#6142211yescharacteristic
650C2#6182711yescharacteristic
651C2#6187011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
2. #36134C2semidirect
3. #31586C2semidirect
4. #28717C2semidirect
5. #28508C2semidirect
6. #28414C2semidirect
7. #28272C2semidirect
8. #27206C2semidirect
9. #25275C2semidirect
10. #25062C2semidirect
11. #24811C2semidirect
13. #23153C2semidirect
14. #22968C2semidirect
15. #22546C2semidirect
16. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#361342
#315862
#287172
#285082
#284142
#282722
#272062
#252752
#250622
#248112
#241862
#231532
#229682
#225462
C4×#4612
C4×#4592