Finite Group Databasebeta

← #6798437

Subgroups — #6798437

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#36114C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31381C214yesmaximal, complemented
5#31377C214yesmaximal, complemented
6#29884C212yesmaximal, complemented
7#29148C211yesmaximal, characteristic, complemented
8#28693C211yesmaximal, characteristic, complemented
9#27967C212yesmaximal, complemented
10#26109C212yesmaximal, complemented
11#26107C212yesmaximal, complemented
12#25679C212yesmaximal, complemented
13#24972C212yesmaximal, complemented
14#24968C212yesmaximal
15#21868C212yesmaximal, complemented
16C4×#489C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#501C2212yes—
20C2×#489C2211yescharacteristic
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2571C2212yes—
27#2563—21no—
28#2538—21no—
29#2538C2211yescharacteristic
30#2537C2211yescharacteristic
31#2497—22no—
32#2496—21no—
33#2492—22no—
34#2470—22no—
35#2466—22no—
36#2439C2214yes—
37#2422—21no—
38#2421C2212yes—
39#2420C2212yes—
40#2369—21no—
41#2361—21no—
42#2355—21no—
43#2354C2212yes—
44Q8×C4⋊C4C2211yescharacteristic
45D4×C4⋊C4C2211yescharacteristic
46#2344—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#124C2211yescharacteristic
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2088C2214yes—
54#2086C2214yes—
55#2085C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2035C2214yes—
59#2033C2214yes—
60#2024C2212yes—
61#2023C2212yes—
62#1993C2214yes—
63#1993C2211yescharacteristic
64#1992C2212yes—
65#1992C2212yes—
66#1992C2212yes—
67#1959C2212yes—
68#1959C2211yescharacteristic
69#1954C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1939C2212yes—
73#1936C2212yes—
74#1935C2212yes—
75#1924C2214yes—
76#1875C2214yes—
77#1873C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1836C2211yescharacteristic
81#1836C2212yes—
82#1834C2212yes—
83#1834C2212yes—
84#1825C2214yes—
85#1825C2214yes—
86#1805C2214yes—
87#1804C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99#529—28no—
100#529—44no—
101C2×#126—22no—
102C2×#125C2312yes—
103C2×#124C2311yescharacteristic
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117C2314yes—
120C22×C4⋊C4—21no—
121C22×C42C2312yes—
122#501—44no—
123#489—24no—
124#432—44no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2311yescharacteristic
138C2×#97C2312yes—
139C2×#97C2311yescharacteristic
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#400—22no—
144#397—22no—
145#397—22no—
146#396—24no—
147#396—22no—
148#394—22no—
149#394—22no—
150#393—24no—
151#391—21no—
152#391—22no—
153#391—24no—
154#391C2311yescharacteristic
155#390—22no—
156#390—21no—
157#390C2311yescharacteristic
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#388—24no—
172#388—22no—
173#388—21no—
174#388—21no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—24no—
179#386—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—22no—
192#382C2314yes—
193#382C2311yescharacteristic
194#382C2312yes—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×D4—22no—
237C22×D4C2×D411yescharacteristic
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—82no—
252#126—48no—
253#125—24no—
254#125—24no—
255#124—22no—
256#124#4114yes—
257#123—48no—
258C22⋊Q8—24no—
259C22⋊Q8—42no—
260C22⋊Q8—84no—
261#121—24no—
262#121—42no—
263#121—44no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—48no—
275#117—48no—
276#117—44no—
277#117—44no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—44no—
342#106—44no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345#103—24no—
346#103—42no—
347#103—44no—
348#102—24no—
349#102—42no—
350#102—44no—
351C22⋊C8—44no—
352C22⋊C8—44no—
353#97—22no—
354#97—22no—
355#97—24no—
356#97—42no—
357C4×C8—24no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
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#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D412yes—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8C2×#4112yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—82no—
398C2×D4C2×#4112yes—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—82no—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—82no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—41no—
501C42—41no—
502C42—44no—
503C42—42no—
504C42—44no—
505C42—84no—
506C42C2×#4112yes—
507C42H(5,𝔽2)12yes—
508C422−1+414yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C23×D411yescharacteristic
524Q8—24no—
525Q8—42no—
526Q8—84no—
527Q8—164no—
528D4—24no—
529D4—44no—
530D4—42no—
531D4—88no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—88no—
582C2×C4—84no—
583C2×C4#53612yes—
584C2×C4#53612yes—
585C2×C4#54612yes—
586C2×C4#54612yes—
587C2×C4#57512yes—
588C2×C4#57512yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×2−1+412yes—
592C2×C4#58512yes—
593C8—42no—
594C8—42no—
595C8—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22C2×#53611yescharacteristic
611C22C2×#54611yescharacteristic
612C22#350011yescharacteristic
613C22C2×#57511yescharacteristic
614C22C2×#57511yescharacteristic
615C22#358511yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—84no—
627C4—82no—
628C4—81no—
629C4—81no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#324014yes—
634C4#351512yes—
635C4#354712yes—
636C2—82no—
637C2—82no—
638C2#3895511yescharacteristic
639C2C2×#324011yescharacteristic
640C2#6088711yescharacteristic
641C2#6116511yescharacteristic
642C2#6116611yescharacteristic
643C2#6258011yescharacteristic
644C2#6259111yescharacteristic
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. #33297C2semidirect
4. #31381C2semidirect
5. #31377C2semidirect
6. #29884C2semidirect
7. #29148C2semidirect
8. #28693C2semidirect
9. #27967C2semidirect
10. #26109C2semidirect
11. #26107C2semidirect
12. #25679C2semidirect
13. #24972C2semidirect
15. #21868C2semidirect
16. C4×#489C2semidirect
17. #21270C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361142
#332972
#313812
#313772
#298842
#291482
#286932
#279672
#261092
#261072
#256792
#249722
#249682
#218682
C4×#4892
#212702