Finite Group Databasebeta

← #6624353

Subgroups — #6624353

Order 512.

1545 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33150C212yesmaximal
4#32954C211yesmaximal, characteristic
5#29112C212yesmaximal
6#28121C212yesmaximal
7#27961C211yesmaximal, characteristic
8#27648C211yesmaximal, characteristic
9#27444C214yesmaximal, complemented
10#27378C214yesmaximal, complemented
11#26023C212yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#23068C212yesmaximal, complemented
14#22763C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#21581C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#475C2212yes—
21C2×#451C2211yescharacteristic
22C2×C4×Q16C2212yes—
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2590—22no—
26#2568C2211yescharacteristic
27#2552C2211yescharacteristic
28#2511—21no—
29#2508C2212yes—
30#2505—22no—
31#2502—21no—
32#2453C2212yes—
33#2365C2211yescharacteristic
34#2360—21no—
35#2355—22no—
36#2355—21no—
37#2355—21no—
38#2355—21no—
39#2355C2212yes—
40#2355C2214yes—
41#2355C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43C4×C22⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45C2×#391C2212yes—
46C2×#387—22no—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2083C2212yes—
54#2074C2214yes—
55#2072C2212yes—
56#2068C2214yes—
57#2056C2214yes—
58#2026C2214yes—
59#1990C2212yes—
60#1989C2214yes—
61#1949C2214yes—
62#1947C2212yes—
63#1939C2212yes—
64#1935C2214yes—
65#1935C2214yes—
66#1928C2214yes—
67#1927C2214yes—
68#1920C2214yes—
69#1918C2214yes—
70#1918C2214yes—
71#1917C2214yes—
72#1878C2212yes—
73#1878C2214yes—
74#1872C2211yescharacteristic
75#1865C2214yes—
76#1863C2212yes—
77#1853C2212yes—
78#1852C2211yescharacteristic
79#1834C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1797C2211yescharacteristic
86#1797C2212yes—
87#1787C2211yescharacteristic
88#1787C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2311yescharacteristic
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C42—21no—
117C22×C42—22no—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#484—28no—
121#475—28no—
122#451—24no—
123C4×Q16—28no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2311yescharacteristic
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#400—22no—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2314yes—
155#391C2311yescharacteristic
156#391C2314yes—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389C2312yes—
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—21no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#387—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—21no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#382—22no—
189#382—22no—
190#382—24no—
191#382—22no—
192#382C2312yes—
193#380—21no—
194#380C2311yescharacteristic
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—24no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2411yescharacteristic
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—42no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—24no—
256#125—44no—
257C22⋊Q8—22no—
258C22⋊Q8—44no—
259C22⋊Q8#4114yes—
260C4×Q8—28no—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×Q8#4118yes—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42—84no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334#107—24no—
335#107—28no—
336#106—28no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#103—24no—
340#103—28no—
341#103—28no—
342#103—24no—
343#103—24no—
344#103—44no—
345#103—42no—
346C22⋊C8—22no—
347C22⋊C8—24no—
348C22⋊C8—22no—
349#97—44no—
350C4×C8—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—21no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24C22×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×#4112yes—
394C2×Q8C2×#4114yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4#13714yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4112yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C2×#4112yes—
485C4⋊C4C2×#4114yes—
486C22⋊C4—24no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C42—28no—
490C42—28no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—24no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—42no—
500C42—42no—
501C42—84no—
502C42—84no—
503C42—82no—
504C42C2×#4114yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23#54812yes—
510C23C2×#13712yes—
511C23#57812yes—
512C23C23×D411yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—24no—
516Q8—48no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—84no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—88no—
578C2×C4#52512yes—
579C2×C4#52512yes—
580C2×C4#54812yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4#57512yes—
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4112yes—
586C2×C4C2×2−1+411yescharacteristic
587C2×C4#58512yes—
588C8—44no—
589C8—42no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22#337414yes—
598C22C2×#52511yescharacteristic
599C22C2×#54811yescharacteristic
600C22#348511yescharacteristic
601C22C22×#13711yescharacteristic
602C22C2×#57511yescharacteristic
603C22C2×#57811yescharacteristic
604C22#358511yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—44no—
614C4—42no—
615C4—41no—
616C4—44no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4—84no—
621C4—81no—
622C4#311714yes—
623C4#347512yes—
624C4#349512yes—
625C2—24no—
626C2#3651611yescharacteristic
627C2C2×#311711yescharacteristic
628C2#6058911yescharacteristic
629C2C2×#337411yescharacteristic
630C2#6132411yescharacteristic
631C2#6171911yescharacteristic
632C2#6173011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
9. #27444C2semidirect
10. #27378C2semidirect
11. #26023C2semidirect
12. #24683C2semidirect
13. #23068C2semidirect
14. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#23552
#331502
#329542
#291122
#281212
#279612
#276482
#274442
#273782
#260232
#246832
#230682
#227632
#226292
#215812
C4×#4512