Finite Group Databasebeta

← #6593803

Subgroups — #6593803

Order 512.

1685 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33147C211yesmaximal, characteristic
4#33010C212yesmaximal
5#27589C212yesmaximal
6#27082C214yesmaximal, complemented
7#26774C214yesmaximal, complemented
8#26527C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic
10#25323C212yesmaximal, complemented
11#24508C212yesmaximal
12#24212C211yesmaximal, characteristic
13#23222C212yesmaximal, complemented
14#23102C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic
16C4×#484C211yesmaximal, characteristic
17#20741C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#484C2211yescharacteristic
22C2×#462C2212yes—
23C2×#451C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2571C2211yescharacteristic
28#2568C2211yescharacteristic
29#2533—22no—
30#2514—21no—
31#2514C2212yes—
32#2509C2212yes—
33#2503—21no—
34#2493—22no—
35#2365—21no—
36#2365—21no—
37#2365—22no—
38#2365C2211yescharacteristic
39#2365C2212yes—
40#2365C2214yes—
41#2365C2212yes—
42#2360—21no—
43#2356—21no—
44#2355C2211yescharacteristic
45C4×C22⋊Q8C2211yescharacteristic
46C2×#391C2212yes—
47C2×#389—21no—
48C2×#387—22no—
49#2283C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2074C2214yes—
55#2066C2214yes—
56#2056C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2016C2212yes—
60#1988C2212yes—
61#1981C2212yes—
62#1971C2214yes—
63#1956C2212yes—
64#1948C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1928C2212yes—
68#1928C2214yes—
69#1927C2214yes—
70#1921C2212yes—
71#1919C2214yes—
72#1917C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1872C2211yescharacteristic
76#1863C2214yes—
77#1863C2214yes—
78#1853C2212yes—
79#1852C2211yescharacteristic
80#1824C2211yescharacteristic
81#1824C2212yes—
82#1817C2212yes—
83#1805C2214yes—
84#1805C2214yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—28no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42—22no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#487—28no—
125#484—24no—
126#462—28no—
127#451—44no—
128#432—44no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2311yescharacteristic
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#401—22no—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#391C2314yes—
162#391C2312yes—
163#391C2314yes—
164#391C2311yescharacteristic
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—24no—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#387—22no—
176#387—22no—
177#387—22no—
178#387—21no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—22no—
186#385—22no—
187#385—21no—
188#384C2311yescharacteristic
189#383—22no—
190#383—22no—
191#382—21no—
192#382—21no—
193#382C2312yes—
194#382C2312yes—
195#381—22no—
196#381—22no—
197#380—21no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#376—22no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8C2411yescharacteristic
243C22×Q8C2414yes—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8#4118yes—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—44no—
267C4×Q8—28no—
268C4×Q8—28no—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×Q8—48no—
274C4×Q8—44no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
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—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—48no—
328C2×C42—44no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—24no—
338#107—42no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—24no—
346#103—44no—
347#103—44no—
348#103—44no—
349#103—42no—
350#103—44no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—24no—
354#97—44no—
355C4×C8—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—28no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—416no—
397C2×Q8—416no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C2×#4118yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4C2511yescharacteristic
448Q16—88no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4C2×#4114yes—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—48no—
504C42—48no—
505C42—44no—
506C42—44no—
507C42—42no—
508C42—44no—
509C42—42no—
510C42—82no—
511C42C2×#4114yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#54612yes—
517C23C2×#13712yes—
518C23#57812yes—
519C23C23×D411yescharacteristic
520Q8—28no—
521Q8—28no—
522Q8—416no—
523Q8—44no—
524Q8—44no—
525Q8—42no—
526Q8—42no—
527Q8—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—416no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#52512yes—
586C2×C4#52512yes—
587C2×C4#54612yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4#57512yes—
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4112yes—
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4#58512yes—
595C8—42no—
596C8—42no—
597C8—44no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22#332814yes—
605C22C2×#52511yescharacteristic
606C22C2×#54611yescharacteristic
607C22#348711yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57811yescharacteristic
611C22#358611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—48no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—81no—
623C4—82no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4#311714yes—
630C4#347512yes—
631C4#351312yes—
632C2—24no—
633C2#3616211yescharacteristic
634C2C2×#311711yescharacteristic
635C2#6058311yescharacteristic
636C2C2×#332811yescharacteristic
637C2#6115911yescharacteristic
638C2#6189311yescharacteristic
639C2#6190511yescharacteristic
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
6. #27082C2semidirect
7. #26774C2semidirect
8. #26527C2semidirect
10. #25323C2semidirect
13. #23222C2semidirect
14. #23102C2semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#23652
#331472
#330102
#275892
#270822
#267742
#265272
#258562
#253232
#245082
#242122
#232222
#231022
#226032
C4×#4842
#207412