Finite Group Databasebeta

← #6795351

Subgroups — #6795351

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32389C214yesmaximal, complemented
5#32325C214yesmaximal, complemented
6#29912C212yesmaximal
7#28830C212yesmaximal, complemented
8#27948C211yesmaximal, characteristic, complemented
9#27728C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic
11#27617C212yesmaximal, complemented
12#26511C212yesmaximal, complemented
13#26091C212yesmaximal, complemented
14#24603C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21983C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C2×#481C2211yescharacteristic
19C2×#431C2212yes—
20#2857C2211yescharacteristic
21#2854C2212yes—
22#2571—21no—
23#2571—21no—
24#2571—22no—
25#2571C2214yes—
26#2571C2212yes—
27#2571C2211yescharacteristic
28#2571C2212yes—
29#2563C2211yescharacteristic
30#2532—22no—
31#2513C2212yes—
32#2508C2212yes—
33#2495—21no—
34#2488—22no—
35#2486—21no—
36C2×#400—22no—
37#2369C2211yescharacteristic
38#2354C2211yescharacteristic
39#2352—21no—
40#2346—21no—
41C4×#121C2211yescharacteristic
42C2×#389—21no—
43C2×#389C2212yes—
44C2×#384C2211yescharacteristic
45#2283C2212yes—
46C2×#382—21no—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2099C2212yes—
53#2093C2212yes—
54#2092C2214yes—
55#2088C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2084C2214yes—
59#2073C2214yes—
60#2065C2214yes—
61#1990C2212yes—
62#1987C2212yes—
63#1949C2214yes—
64#1949C2214yes—
65#1947C2212yes—
66#1947C2212yes—
67#1941C2214yes—
68#1939C2212yes—
69#1934C2214yes—
70#1926C2212yes—
71#1921C2214yes—
72#1919C2212yes—
73#1919C2214yes—
74#1918C2214yes—
75#1916C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1853C2212yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1846C2214yes—
84#1844C2212yes—
85#1835C2214yes—
86#1824C2212yes—
87#1824C2211yescharacteristic
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×#121C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4C2311yescharacteristic
101C2×#117—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113#481—24no—
114#431—28no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2312yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#102C2312yes—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2311yescharacteristic
128C2×C22⋊C8C2311yescharacteristic
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131#400—24no—
132#400—24no—
133#400—24no—
134#400—24no—
135#400—22no—
136#400—22no—
137#398—22no—
138#397—22no—
139#397—22no—
140#393—22no—
141#391—21no—
142#391—22no—
143#390C2312yes—
144#390C2312yes—
145#390C2311yescharacteristic
146#389—22no—
147#389—22no—
148#389—24no—
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389C2314yes—
155#389C2314yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2311yescharacteristic
160#388—22no—
161#388—22no—
162#388—22no—
163#388—21no—
164#388—22no—
165#388—22no—
166#387—22no—
167#386—22no—
168#386—21no—
169#384—21no—
170#384—21no—
171#384C2312yes—
172#384C2312yes—
173#384C2311yescharacteristic
174#383—22no—
175#382—22no—
176#382—22no—
177#382—24no—
178#382—21no—
179#382—22no—
180#382—22no—
181#382—22no—
182#382C2312yes—
183#382C2314yes—
184#382C2314yes—
185#382C2312yes—
186#381—22no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192#380C2311yescharacteristic
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4D411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—21no—
209#376—22no—
210C2×#95—21no—
211C2×#95—22no—
212C2×#95—22no—
213C2×#95—22no—
214C2×#95D412yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230C22×Q8—21no—
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2411yescharacteristic
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C4⋊Q8—24no—
246#123—84no—
247C22⋊Q8—82no—
248#121—22no—
249#121—22no—
250C4×D4—24no—
251C4×D4—44no—
252#117—44no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—44no—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319#107—44no—
320#107—44no—
321#106—24no—
322#106—44no—
323#106—42no—
324#106—44no—
325C4⋊C8—28no—
326C4⋊C8—24no—
327#102—24no—
328#102—42no—
329#102—44no—
330C22⋊C8—24no—
331C22⋊C8—22no—
332C22⋊C8#4114yes—
333C4×C8—28no—
334C4×C8—24no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—21no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—21no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—21no—
361C24C22×D411yescharacteristic
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4#13514yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C42−1+412yes—
417C22×C4C2511yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8C2×#4114yes—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4—81no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—44no—
481C42—44no—
482C42—42no—
483C42—44no—
484C42—82no—
485C422−1+414yes—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#54712yes—
498C23C2×#13512yes—
499C23#57712yes—
500C23C23×D411yescharacteristic
501Q8—162no—
502D4—42no—
503D4—42no—
504D4—84no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—88no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4#53612yes—
561C2×C4#53612yes—
562C2×C4#54712yes—
563C2×C4C2×#13512yes—
564C2×C4C2×#13512yes—
565C2×C4#57512yes—
566C2×C4C22×#4111yescharacteristic
567C2×C4C2×2−1+412yes—
568C2×C4C2×2−1+411yescharacteristic
569C2×C4#58512yes—
570C8—24no—
571C8—44no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#334014yes—
587C22C2×#53611yescharacteristic
588C22C2×#54711yescharacteristic
589C22#348711yescharacteristic
590C22C22×#13511yescharacteristic
591C22C2×#57511yescharacteristic
592C22C2×#57711yescharacteristic
593C22#358511yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—82no—
604C4—81no—
605C4—81no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4#323214yes—
611C4#351512yes—
612C4#356312yes—
613C2—24no—
614C2—161no—
615C2#3891911yescharacteristic
616C2C2×#323211yescharacteristic
617C2#6087311yescharacteristic
618C2C2×#334011yescharacteristic
619C2#6125611yescharacteristic
620C2#6188511yescharacteristic
621C2#6189911yescharacteristic
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×#2571C2semidirect
3. #33149C2semidirect
4. #32389C2semidirect
5. #32325C2semidirect
7. #28830C2semidirect
8. #27948C2semidirect
9. #27728C2semidirect
11. #27617C2semidirect
12. #26511C2semidirect
13. #26091C2semidirect
15. #22599C2semidirect
16. #21983C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25712
#331492
#323892
#323252
#299122
#288302
#279482
#277282
#276772
#276172
#265112
#260912
#246032
#225992
#219832
C4×#4812