Finite Group Databasebeta

← #6795683

Subgroups — #6795683

Order 512.

1419 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#31697C214yesmaximal, complemented
5#31082C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#29206C211yesmaximal, characteristic, complemented
9#28830C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#27646C212yesmaximal, complemented
12#26299C212yesmaximal, complemented
13#25013C212yesmaximal
14#24701C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21985C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#482C2214yes—
19C2×#476C2211yescharacteristic
20C2×#474C2212yes—
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2572—22no—
25#2571C2212yes—
26#2571C2211yescharacteristic
27#2563—21no—
28#2563C2211yescharacteristic
29#2563C2211yescharacteristic
30#2538—21no—
31#2537C2211yescharacteristic
32#2497—22no—
33#2495—21no—
34#2495C2212yes—
35#2492—22no—
36#2439C2214yes—
37#2420—21no—
38#2420C2212yes—
39#2398—22no—
40#2369—21no—
41#2355—21no—
42#2354C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121—21no—
46C4×#121C2212yes—
47#2285C2212yes—
48#2283—21no—
49#2283C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2101C2214yes—
53#2087C2214yes—
54#2085C2214yes—
55#2085C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2075C2214yes—
59#2073C2214yes—
60#2058C2214yes—
61#2034C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1987C2211yescharacteristic
66#1987C2212yes—
67#1954C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1941C2214yes—
76#1934C2214yes—
77#1923C2214yes—
78#1920C2212yes—
79#1864C2212yes—
80#1854C2214yes—
81#1854C2214yes—
82#1846C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1835C2214yes—
86#1825C2214yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×D4—22no—
106C2×C4×D4D411yescharacteristic
107C2×C4×D4D411yescharacteristic
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×#117C2314yes—
113C22×C42—21no—
114C22×C42C2312yes—
115#482—48no—
116#476—24no—
117#474—44no—
118#431—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106C2312yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#106C2312yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×#102C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2314yes—
136C2×C4×C8C2312yes—
137#400—22no—
138#400—24no—
139#400—22no—
140#397—22no—
141#397—22no—
142#396—24no—
143#394—22no—
144#394—22no—
145#393—24no—
146#393—22no—
147#393—22no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—24no—
152#390—22no—
153#390—21no—
154#390C2312yes—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#388—22no—
165#388—24no—
166#388—22no—
167#387—24no—
168#386—22no—
169#386—22no—
170#384—21no—
171#384—21no—
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2312yes—
175#384C2312yes—
176#383—22no—
177#383—22no—
178#382—22no—
179#382—21no—
180#382—21no—
181#382C2312yes—
182#382C2314yes—
183#382C2312yes—
184#381—22no—
185#381—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×D4—22no—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C22⋊Q8—84no—
256#121—24no—
257#121—42no—
258#121—44no—
259C4×Q8—44no—
260C4×Q8—82no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—42no—
270C4×D4—41no—
271C4×D4—84no—
272#117—44no—
273#117—44no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—44no—
334#107—44no—
335#106—24no—
336#106—42no—
337#106—44no—
338#106—44no—
339C4⋊C8—24no—
340C4⋊C8—24no—
341C4⋊C8—22no—
342C4⋊C8—42no—
343C4⋊C8#4114yes—
344#102—28no—
345#102—24no—
346#102—24no—
347C22⋊C8—28no—
348C22⋊C8—44no—
349C4×C8—28no—
350C4×C8—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—21no—
374C24C22×D412yes—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—81no—
381C2×Q8—81no—
382C2×Q8—82no—
383C2×Q8—82no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—82no—
395C2×D4H(5,𝔽2)12yes—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2511yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8C2×#4114yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—84no—
490C22⋊C4—82no—
491C22⋊C4—84no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—22no—
499C42—21no—
500C42—41no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—44no—
505C42—44no—
506C42—84no—
507C42#13514yes—
508C42C2×#4112yes—
509C422−1+412yes—
510C422−1+412yes—
511C422−1+412yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C23×D411yescharacteristic
527Q8—84no—
528Q8—84no—
529Q8—164no—
530D4—22no—
531D4—22no—
532D4—44no—
533D4—41no—
534D4—41no—
535D4—88no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4#53612yes—
588C2×C4#53612yes—
589C2×C4#54712yes—
590C2×C4#54712yes—
591C2×C4C2×#13512yes—
592C2×C4C2×#13512yes—
593C2×C4C2×#13512yes—
594C2×C4#57512yes—
595C2×C4#57712yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C2×C4#58511yescharacteristic
602C8—24no—
603C8—44no—
604C8—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#53611yescharacteristic
620C22C2×#54711yescharacteristic
621C22#348811yescharacteristic
622C22C22×#13511yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57711yescharacteristic
625C22#358511yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—84no—
641C4#323314yes—
642C4#334014yes—
643C4#351511yescharacteristic
644C4#352211yescharacteristic
645C4#354411yescharacteristic
646C4#355511yescharacteristic
647C2—82no—
648C2—82no—
649C2#3892111yescharacteristic
650C2C2×#323311yescharacteristic
651C2#6089211yescharacteristic
652C2C2×#334011yescharacteristic
653C2#6125611yescharacteristic
654C2#6194911yescharacteristic
655C2#6199611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #33195C2semidirect
4. #31697C2semidirect
5. #31082C2semidirect
6. #29942C2semidirect
7. #29290C2semidirect
8. #29206C2semidirect
9. #28830C2semidirect
10. #27728C2semidirect
11. #27646C2semidirect
12. #26299C2semidirect
14. #24701C2semidirect
15. #22795C2semidirect
16. #21985C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349432
#331952
#316972
#310822
#299422
#292902
#292062
#288302
#277282
#276462
#262992
#250132
#247012
#227952
#219852
C4×#4762