Finite Group Databasebeta

← #6792922

Subgroups — #6792922

Order 512.

1445 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4#31595C214yesmaximal, complemented
5#31481C214yesmaximal, complemented
6#29913C212yesmaximal
7#28799C212yesmaximal, complemented
8#27972C211yesmaximal, characteristic
9#27757C212yesmaximal, complemented
10#27681C211yesmaximal, characteristic
11#26531C212yesmaximal, complemented
12#26113C212yesmaximal, complemented
13#24609C212yesmaximal
14#24063C212yesmaximal
15#22603C211yesmaximal, characteristic
16#21866C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#483C2211yescharacteristic
20C2×#483C2212yes—
21C2×#432C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2571—21no—
25#2571—22no—
26#2571—21no—
27#2571C2212yes—
28#2571C2214yes—
29#2571C2212yes—
30#2551C2211yescharacteristic
31#2546C2211yescharacteristic
32#2506—21no—
33#2462—22no—
34#2451C2212yes—
35#2446C2212yes—
36#2440—21no—
37#2438—22no—
38C2×#400—22no—
39#2365—21no—
40#2365C2211yescharacteristic
41Q8×C4⋊C4C2211yescharacteristic
42#2344—21no—
43C4×C22⋊Q8C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382—21no—
48C2×#382C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2097C2214yes—
54#2095C2212yes—
55#2088C2214yes—
56#2085C2214yes—
57#2074C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2058C2214yes—
61#2043C2214yes—
62#1993C2214yes—
63#1981C2212yes—
64#1960C2214yes—
65#1956C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1943C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1927C2212yes—
72#1927C2214yes—
73#1920C2212yes—
74#1917C2214yes—
75#1872C2212yes—
76#1872C2212yes—
77#1872C2212yes—
78#1872C2211yescharacteristic
79#1872C2212yes—
80#1865C2214yes—
81#1865C2214yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1834C2212yes—
85#1823C2211yescharacteristic
86#1817C2212yes—
87#1805C2214yes—
88#1805C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C4×Q8—22no—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×#117—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#483—24no—
122#483—44no—
123#432—44no—
124C2×#107C2314yes—
125C2×#106C2312yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#400—24no—
144#400—24no—
145#400—24no—
146#400—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#395—22no—
151#395—22no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2311yescharacteristic
157#389—22no—
158#389—24no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#389C2311yescharacteristic
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—21no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—24no—
192#382—22no—
193#382—22no—
194#382C2314yes—
195#382C2314yes—
196#381—22no—
197#380—21no—
198#380—21no—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2411yescharacteristic
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253#125—48no—
254C22⋊Q8—22no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—48no—
258C22⋊Q8#4114yes—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—42no—
263C4×Q8—42no—
264C4×Q8—48no—
265#117—48no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—84no—
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⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—44no—
334#106—24no—
335#106—42no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#103—24no—
340#103—44no—
341#103—44no—
342#103—44no—
343#103—42no—
344#103—44no—
345#103—44no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—24no—
349#97—44no—
350C4×C8—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—22no—
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—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—82no—
387C2×Q8C2×#4112yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4#13714yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4C2511yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
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—22no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4C2×#4112yes—
483C22⋊C4—24no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—44no—
496C42—44no—
497C42—42no—
498C42—44no—
499C42—44no—
500C42—42no—
501C422−1+414yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23#54612yes—
507C23C2×#13712yes—
508C23#57712yes—
509C23C23×D411yescharacteristic
510Q8—24no—
511Q8—42no—
512Q8—44no—
513Q8—44no—
514Q8—88no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—88no—
571C2×C4—84no—
572C2×C4#53612yes—
573C2×C4#53612yes—
574C2×C4#54612yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4#57512yes—
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C2×C4C2×2−1+412yes—
581C2×C4#58512yes—
582C8—42no—
583C8—42no—
584C8—44no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22#332914yes—
592C22C2×#53611yescharacteristic
593C22C2×#54611yescharacteristic
594C22#348711yescharacteristic
595C22C22×#13711yescharacteristic
596C22C2×#57511yescharacteristic
597C22C2×#57711yescharacteristic
598C22#358611yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4#323614yes—
617C4#351512yes—
618C4#355912yes—
619C2—24no—
620C2#3890711yescharacteristic
621C2C2×#323611yescharacteristic
622C2#6087211yescharacteristic
623C2C2×#332911yescharacteristic
624C2#6116511yescharacteristic
625C2#6188911yescharacteristic
626C2#6190411yescharacteristic
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
4. #31595C2semidirect
5. #31481C2semidirect
7. #28799C2semidirect
9. #27757C2semidirect
11. #26531C2semidirect
12. #26113C2semidirect

Maximal subgroups

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

subgroupindex
#361332
C2×#25712
#331082
#315952
#314812
#299132
#287992
#279722
#277572
#276812
#265312
#261132
#246092
#240632
#226032
#218662
C4×#4832