Finite Group Databasebeta

← #7533366

Subgroups — #7533366

Order 512.

1577 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36194C211yesmaximal, characteristic, complemented
3#31645C214yesmaximal, complemented
4#29141C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28186C212yesmaximal, complemented
7#27748C212yesmaximal, complemented
8#26693C214yesmaximal, complemented
9#26107C212yesmaximal, complemented
10#24995C212yesmaximal, complemented
11#24627C212yesmaximal, complemented
12#24186C212yesmaximal
13#23149C211yesmaximal, characteristic, complemented
14#22766C212yesmaximal, complemented
15#22543C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#559C2212yes—
19C2×#461C2211yescharacteristic
20C2×#459C2211yescharacteristic
21C2×C8×Q8C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2724—22no—
27#2582C2212yes—
28#2571C2212yes—
29#2563C2212yes—
30#2547—21no—
31#2547C2212yes—
32#2538C2212yes—
33#2497—22no—
34#2492—22no—
35#2486—22no—
36#2447—22no—
37#2447C2214yes—
38#2439—22no—
39#2369C2212yes—
40#2355C2212yes—
41#2354C2212yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47#2285—21no—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2088C2214yes—
51#2074C2214yes—
52#2055C2214yes—
53#2042C2214yes—
54#2024C2214yes—
55#2023C2214yes—
56#2012C2214yes—
57#1984C2212yes—
58#1983C2214yes—
59#1982C2211yescharacteristic
60#1981C2211yescharacteristic
61#1958C2214yes—
62#1957C2212yes—
63#1956C2212yes—
64#1948C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1934C2214yes—
68#1933C2214yes—
69#1928C2214yes—
70#1927C2212yes—
71#1925C2212yes—
72#1918C2214yes—
73#1916C2214yes—
74#1879C2214yes—
75#1879C2212yes—
76#1877C2212yes—
77#1864C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1804C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2212yes—
86#1802C2211yescharacteristic
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2214yes—
92C4×#102C2212yes—
93C22×SD16C2312yes—
94C22×D8C2312yes—
95#559—28no—
96#559—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#124—22no—
104C2×#124C2314yes—
105C2×#123—24no—
106C2×C22⋊Q8—22no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2314yes—
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C22×C4⋊C4—22no—
119C22×C42—21no—
120#461—24no—
121#459—24no—
122C8×Q8—44no—
123C4×SD16—44no—
124C4×D8—28no—
125C2×#107C2312yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2314yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#398—24no—
149#397—24no—
150#397—24no—
151#397—24no—
152#396—24no—
153#393—24no—
154#391—22no—
155#391—22no—
156#391—24no—
157#391D412yes—
158#391C2312yes—
159#391C2314yes—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2314yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389D412yes—
170#389D412yes—
171#389C2314yes—
172#389C2314yes—
173#388—24no—
174#387—24no—
175#385—24no—
176#385—24no—
177#384—21no—
178#384—21no—
179#384D411yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382C2314yes—
186#381—24no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C22×Q8—22no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C22×D4C2411yescharacteristic
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C2×SD16—24no—
233C2×SD16—42no—
234C2×SD16—42no—
235C2×D8—24no—
236C2×D8—24no—
237C2×D8—42no—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—82no—
248C4⋊Q8—82no—
249#127—22no—
250#127#4114yes—
251#124—28no—
252#124—44no—
253#124—48no—
254#123—416no—
255C22⋊Q8—84no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×D4—28no—
261C4×D4—24no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—24no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—44no—
271C4×D4—82no—
272C4×D4#4114yes—
273#117—84no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—84no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2414yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326#107—24no—
327#106—42no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—44no—
333#103—24no—
334#103—44no—
335#103—44no—
336#103—42no—
337#103—42no—
338#102—24no—
339#102—24no—
340#102—24no—
341#102—42no—
342#102—44no—
343#102—44no—
344#102—42no—
345C4×C8—24no—
346C4×C8—24no—
347C4×C8—44no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—24no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—21no—
363C24—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—81no—
391C2×D4—81no—
392C2×D4C2×#4112yes—
393C2×D4C2×#4114yes—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4C22×D414yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D414yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425SD16—48no—
426D8—48no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—81no—
465C4⋊C4—84no—
466C4⋊C4—81no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4C2×#4112yes—
471C22⋊C4—24no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—84no—
480C22⋊C4—82no—
481C22⋊C4—84no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—44no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—84no—
494C42#13514yes—
495C42C2×#4112yes—
496C42C2×#4112yes—
497C42H(5,𝔽2)12yes—
498C422−1+412yes—
499C23—24no—
500C23—22no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23C23×D411yescharacteristic
514Q8—48no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—164no—
520D4—22no—
521D4—24no—
522D4—22no—
523D4—24no—
524D4—42no—
525D4—41no—
526D4—41no—
527D4—42no—
528D4—48no—
529D4—84no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4—168no—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4#54812yes—
574C2×C4#54812yes—
575C2×C4C2×#13512yes—
576C2×C4C2×#13512yes—
577C2×C4C2×#13512yes—
578C2×C4#57612yes—
579C2×C4#57712yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4#58511yescharacteristic
585C2×C4#58511yescharacteristic
586C8—42no—
587C8—42no—
588C8—42no—
589C8—42no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—44no—
596C22—81no—
597C22—82no—
598C22—82no—
599C22—81no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22C2×#54611yescharacteristic
605C22C2×#54811yescharacteristic
606C22#348711yescharacteristic
607C22C22×#13511yescharacteristic
608C22C2×#57611yescharacteristic
609C22C2×#57711yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—41no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—81no—
622C4—81no—
623C4—84no—
624C4#330314yes—
625C4#336114yes—
626C4#347511yescharacteristic
627C4#350611yescharacteristic
628C4#351511yescharacteristic
629C4#354711yescharacteristic
630C2—84no—
631C2—82no—
632C2#5903411yescharacteristic
633C2C2×#330311yescharacteristic
634C2#6119911yescharacteristic
635C2C2×#336111yescharacteristic
636C2#6134611yescharacteristic
637C2#6188211yescharacteristic
638C2#6193711yescharacteristic
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. #36194C2semidirect
3. #31645C2semidirect
4. #29141C2semidirect
5. #29139C2semidirect
6. #28186C2semidirect
7. #27748C2semidirect
8. #26693C2semidirect
9. #26107C2semidirect
10. #24995C2semidirect
11. #24627C2semidirect
13. #23149C2semidirect
14. #22766C2semidirect
15. #22543C2semidirect
16. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#361942
#316452
#291412
#291392
#281862
#277482
#266932
#261072
#249952
#246272
#241862
#231492
#227662
#225432
C4×#4612
C4×#4592