Finite Group Databasebeta

← #6848160

Subgroups — #6848160

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32241C214yesmaximal, complemented
5#32145C214yesmaximal, complemented
6#29640C212yesmaximal
7#28792C212yesmaximal, complemented
8#27960C211yesmaximal, characteristic, complemented
9#27814C212yesmaximal, complemented
10#27644C211yesmaximal, characteristic
11#27616C212yesmaximal, complemented
12#26445C212yesmaximal, complemented
13#26025C212yesmaximal, complemented
14#24600C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21836C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C22×#125C2212yes—
19C2×#481C2211yescharacteristic
20C2×#480C2212yes—
21C2×#431C2212yes—
22#2857C2211yescharacteristic
23#2854C2212yes—
24#2571C2211yescharacteristic
25#2563C2211yescharacteristic
26#2552—21no—
27#2552—21no—
28#2552—22no—
29#2552C2212yes—
30#2552C2212yes—
31#2552C2214yes—
32#2532—22no—
33#2513C2212yes—
34#2508C2212yes—
35#2495—21no—
36#2488—22no—
37#2486—21no—
38C2×#398—22no—
39#2369C2211yescharacteristic
40#2354C2211yescharacteristic
41#2352—21no—
42#2346—21no—
43C4×#121C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384—21no—
47C2×#384C2211yescharacteristic
48#2283C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2099C2214yes—
53#2086C2214yes—
54#2085C2214yes—
55#2083C2212yes—
56#2071C2212yes—
57#2067C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2025C2214yes—
61#2025C2214yes—
62#1991C2214yes—
63#1991C2214yes—
64#1990C2212yes—
65#1989C2214yes—
66#1989C2214yes—
67#1987C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1942C2214yes—
71#1939C2212yes—
72#1934C2214yes—
73#1926C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1919C2212yes—
78#1916C2214yes—
79#1874C2211yescharacteristic
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1834C2212yes—
83#1824C2211yescharacteristic
84#1824C2212yes—
85#1823C2212yes—
86#1823C2212yes—
87#1823C2212yes—
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—28no—
97C2×#125—22no—
98C2×#125—22no—
99C2×#125—44no—
100C2×#125C2314yes—
101C2×#125C2314yes—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×#121C2311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4C2311yescharacteristic
107C2×#117—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#481—24no—
122#480—44no—
123#431—44no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2312yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2311yescharacteristic
135C2×C22⋊C8C2311yescharacteristic
136C2×C4×C8C2312yes—
137C2×C4×C8C2314yes—
138#400—22no—
139#400—22no—
140#398—22no—
141#398—24no—
142#398—24no—
143#398—24no—
144#398—24no—
145#397—22no—
146#397—22no—
147#393—22no—
148#391—21no—
149#391—22no—
150#390C2312yes—
151#390C2312yes—
152#390C2311yescharacteristic
153#389—21no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389—24no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#389C2311yescharacteristic
165#389C2314yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—21no—
171#388—22no—
172#388—22no—
173#387—22no—
174#386—22no—
175#386—21no—
176#384—22no—
177#384—22no—
178#384—24no—
179#384—21no—
180#384—21no—
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2312yes—
184#383—22no—
185#382—21no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195#380C2311yescharacteristic
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212#376—22no—
213C2×#95—21no—
214C2×#95—22no—
215C2×#95D412yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—24no—
249#125—28no—
250#125—28no—
251#125—44no—
252#125—44no—
253#125—48no—
254#125—48no—
255#123—84no—
256C22⋊Q8—82no—
257#121—22no—
258#121—22no—
259C4×D4—24no—
260C4×D4—44no—
261#117—44no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—84no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—28no—
336#107—44no—
337#106—24no—
338#106—24no—
339C4⋊C8—24no—
340C4⋊C8—44no—
341#102—24no—
342#102—24no—
343#102—44no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346C22⋊C8—24no—
347C4×C8—24no—
348C4×C8—44no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—21no—
371C24C22×D411yescharacteristic
372C2×Q8—42no—
373C2×Q8—81no—
374C2×Q8—81no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4#13514yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4C2511yescharacteristic
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—42no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4C2×#4114yes—
474C4⋊C42−1+414yes—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—44no—
495C42—44no—
496C42—42no—
497C42—44no—
498C42—82no—
499C422−1+414yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23#54612yes—
512C23C2×#13512yes—
513C23#57612yes—
514C23C23×D411yescharacteristic
515Q8—162no—
516D4—42no—
517D4—42no—
518D4—84no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—88no—
573C2×C4—84no—
574C2×C4#53612yes—
575C2×C4#53612yes—
576C2×C4#54612yes—
577C2×C4C2×#13512yes—
578C2×C4C2×#13512yes—
579C2×C4#57512yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×2−1+412yes—
583C2×C4C2×2−1+412yes—
584C8—42no—
585C8—42no—
586C8—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22#330514yes—
601C22C2×#53611yescharacteristic
602C22C2×#54611yescharacteristic
603C22#348611yescharacteristic
604C22C22×#13511yescharacteristic
605C22C2×#57511yescharacteristic
606C22C2×#57611yescharacteristic
607C22#358611yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—81no—
619C4—82no—
620C4—81no—
621C4—84no—
622C4—81no—
623C4—82no—
624C4#323214yes—
625C4#350912yes—
626C4#354412yes—
627C2—24no—
628C2—161no—
629C2#3946511yescharacteristic
630C2C2×#323211yescharacteristic
631C2#6087111yescharacteristic
632C2C2×#330511yescharacteristic
633C2#6115911yescharacteristic
634C2#6182211yescharacteristic
635C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #33149C2semidirect
4. #32241C2semidirect
5. #32145C2semidirect
7. #28792C2semidirect
8. #27960C2semidirect
9. #27814C2semidirect
11. #27616C2semidirect
12. #26445C2semidirect
13. #26025C2semidirect
15. #22599C2semidirect
16. #21836C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25522
#331492
#322412
#321452
#296402
#287922
#279602
#278142
#276442
#276162
#264452
#260252
#246002
#225992
#218362
C4×#4812