Finite Group Databasebeta

← #6855854

Subgroups — #6855854

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#32235C214yesmaximal, complemented
5#30810C214yesmaximal, complemented
6#29760C212yesmaximal
7#28790C212yesmaximal, complemented
8#27972C211yesmaximal, characteristic, complemented
9#27768C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic
11#26461C212yesmaximal, complemented
12#26021C212yesmaximal, complemented
13#25907C212yesmaximal
14#24307C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22078C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic
18C22×#125C2212yes—
19C2×#488C2214yes—
20C2×#483C2212yes—
21C2×#480C2211yescharacteristic
22C2×#479C2214yes—
23#2857C2211yescharacteristic
24#2850C2212yes—
25#2596—22no—
26#2571C2211yescharacteristic
27#2552—22no—
28#2552—21no—
29#2552—21no—
30#2552C2214yes—
31#2552C2212yes—
32#2552C2212yes—
33#2547C2211yescharacteristic
34#2514C2212yes—
35#2510—22no—
36#2494—21no—
37#2487—21no—
38#2448C2212yes—
39C2×#398—22no—
40#2365C2211yescharacteristic
41#2354C2211yescharacteristic
42#2352—21no—
43#2347—21no—
44C4×C22⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384—21no—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yes—
53#2094C2212yes—
54#2085C2214yes—
55#2072C2214yes—
56#2068C2214yes—
57#2065C2214yes—
58#2059C2214yes—
59#2042C2214yes—
60#2026C2214yes—
61#2026C2214yes—
62#2025C2214yes—
63#1991C2214yes—
64#1989C2214yes—
65#1981C2212yes—
66#1956C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1941C2214yes—
70#1939C2212yes—
71#1933C2214yes—
72#1927C2214yes—
73#1925C2212yes—
74#1919C2214yes—
75#1918C2212yes—
76#1916C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2214yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1834C2212yes—
86#1823C2211yescharacteristic
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—22no—
97C2×#125—22no—
98C2×#125—48no—
99C2×#125C2314yes—
100C2×#125C2314yes—
101C2×#124C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×#121C2311yescharacteristic
104C2×C4×Q8C2312yes—
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⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#488—48no—
122#483—28no—
123#480—24no—
124#479—48no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#401—22no—
144#400—22no—
145#400—22no—
146#399—22no—
147#398—24no—
148#398—24no—
149#398—24no—
150#398—24no—
151#397—22no—
152#397—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390C2312yes—
157#390C2311yescharacteristic
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#389C2311yescharacteristic
171#388—22no—
172#388—22no—
173#388—22no—
174#388—21no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#386—21no—
183#385—22no—
184#384—22no—
185#384—22no—
186#384—24no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#383—22no—
193#382—21no—
194#382C2312yes—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2412yes—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
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#124—44no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258#121—22no—
259#121—22no—
260C4×Q8—44no—
261C4×D4—24no—
262C4×D4—44no—
263#117—44no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
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—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—84no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4#4114yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—24no—
337#106—28no—
338#106—44no—
339C4⋊C8—24no—
340C4⋊C8—44no—
341#103—28no—
342#103—28no—
343#103—44no—
344#103—44no—
345#102—24no—
346#102—24no—
347#102—44no—
348C22⋊C8—22no—
349C22⋊C8—24no—
350C22⋊C8—22no—
351C4×C8—24no—
352C4×C8—44no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C24C22×D411yescharacteristic
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4#13514yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4112yes—
431C22×C42−1+412yes—
432C22×C4C2511yescharacteristic
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4114yes—
480C4⋊C4H(5,𝔽2)14yes—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—44no—
500C42—44no—
501C42—42no—
502C42—82no—
503C422−1+414yes—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23#54812yes—
516C23C2×#13512yes—
517C23#57712yes—
518C23C23×D411yescharacteristic
519Q8—44no—
520Q8—44no—
521D4—42no—
522D4—42no—
523D4—84no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4#53612yes—
580C2×C4#53612yes—
581C2×C4#54812yes—
582C2×C4C2×#13512yes—
583C2×C4C2×#13512yes—
584C2×C4#57512yes—
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4C2×2−1+412yes—
588C2×C4C2×2−1+411yescharacteristic
589C8—42no—
590C8—44no—
591C8—42no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22#336014yes—
606C22C2×#53611yescharacteristic
607C22C2×#54811yescharacteristic
608C22#348411yescharacteristic
609C22C22×#13511yescharacteristic
610C22C2×#57511yescharacteristic
611C22C2×#57711yescharacteristic
612C22#358611yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—84no—
628C4—81no—
629C4#323214yes—
630C4#351112yes—
631C4#355212yes—
632C2—24no—
633C2—161no—
634C2#3972011yescharacteristic
635C2C2×#323211yescharacteristic
636C2#6087211yescharacteristic
637C2C2×#336011yescharacteristic
638C2#6132011yescharacteristic
639C2#6156511yescharacteristic
640C2#6159611yescharacteristic
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. #33108C2semidirect
4. #32235C2semidirect
5. #30810C2semidirect
7. #28790C2semidirect
8. #27972C2semidirect
9. #27768C2semidirect
11. #26461C2semidirect
12. #26021C2semidirect
14. #24307C2semidirect
15. #22625C2semidirect
16. #22078C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25522
#331082
#322352
#308102
#297602
#287902
#279722
#277682
#276782
#264612
#260212
#259072
#243072
#226252
#220782
C4×#4802