Finite Group Databasebeta

← #6795344

Subgroups — #6795344

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#32374C214yesmaximal, complemented
5#32310C214yesmaximal, complemented
6#29906C212yesmaximal
7#28827C212yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27727C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic
11#27610C212yesmaximal, complemented
12#26506C212yesmaximal, complemented
13#26086C212yesmaximal, complemented
14#24600C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21982C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic
18C2×#552—21no—
19C22×#125—21no—
20C2×#480C2211yescharacteristic
21C2×#431C2212yes—
22#2857C2211yescharacteristic
23#2854C2212yes—
24#2552—21no—
25#2552—21no—
26#2552—22no—
27#2552C2214yes—
28#2552C2212yes—
29#2552C2211yescharacteristic
30#2552C2212yes—
31#2542C2211yescharacteristic
32#2519—22no—
33#2512C2212yes—
34#2506C2212yes—
35#2498—22no—
36#2489—21no—
37#2417—21no—
38C2×#398—22no—
39#2365C2211yescharacteristic
40#2362—21no—
41#2354C2211yescharacteristic
42C4×#121C2211yescharacteristic
43C2×#389—21no—
44C2×#389C2212yes—
45#2285C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2099C2212yes—
51#2099C2214yes—
52#2099C2214yes—
53#2093C2212yes—
54#2093C2214yes—
55#2086C2214yes—
56#2083C2214yes—
57#2083C2214yes—
58#2083C2214yes—
59#2071C2214yes—
60#2067C2214yes—
61#1990C2212yes—
62#1988C2212yes—
63#1947C2212yes—
64#1947C2214yes—
65#1947C2214yes—
66#1947C2212yes—
67#1942C2214yes—
68#1939C2212yes—
69#1934C2214yes—
70#1925C2212yes—
71#1918C2212yes—
72#1918C2214yes—
73#1918C2214yes—
74#1918C2214yes—
75#1916C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1852C2212yes—
79#1852C2212yes—
80#1852C2212yes—
81#1852C2211yescharacteristic
82#1852C2212yes—
83#1845C2212yes—
84#1845C2214yes—
85#1834C2214yes—
86#1823C2212yes—
87#1823C2211yescharacteristic
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#552—48no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#126—22no—
98C2×#125—22no—
99C2×#125—22no—
100C2×#125—24no—
101C2×#125—21no—
102C2×#125—22no—
103C2×#125—48no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121C2311yescharacteristic
109C2×C4×Q8—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2311yescharacteristic
112C2×#117—22no—
113C2×#117—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#480—24no—
128#431—28no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×C4×C8C2314yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#399—22no—
147#398—22no—
148#398—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#398—22no—
153#397—22no—
154#396—22no—
155#391—21no—
156#391—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2311yescharacteristic
160#389—21no—
161#389—22no—
162#389—24no—
163#389—22no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#389C2311yescharacteristic
173#388—22no—
174#388—22no—
175#388—21no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—21no—
182#384—21no—
183#384—21no—
184#384—22no—
185#384—22no—
186#384C2312yes—
187#384C2314yes—
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2314yes—
193#382—22no—
194#380—21no—
195#380—22no—
196#380—21no—
197#380C2312yes—
198#380C2311yescharacteristic
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95D412yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×D4—21no—
234C22×D4C2411yescharacteristic
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—24no—
248#126—48no—
249#125—48no—
250#125—48no—
251#125—416no—
252#125—48no—
253#125—44no—
254#123—48no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258#121—22no—
259#121—22no—
260C4×Q8—44no—
261C4×D4—24no—
262C4×D4—44no—
263#117—48no—
264#117—44no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—84no—
299C2×C4⋊C4D818yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—44no—
343#107—42no—
344#107—44no—
345#107—44no—
346#107—44no—
347C4⋊C8—28no—
348C4⋊C8—24no—
349#102—24no—
350#102—44no—
351#102—42no—
352C22⋊C8—24no—
353C22⋊C8—22no—
354C22⋊C8#4114yes—
355C4×C8—28no—
356C4×C8—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
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—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4C2×D814yes—
419C22×C4C2×D818yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430C22×C42−1+412yes—
431C22×C4C2511yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8C2×#4114yes—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4—81no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4C2×D814yes—
480C4⋊C4C2×D814yes—
481C4⋊C4C2×D818yes—
482C4⋊C4#13714yes—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—81no—
492C22⋊C4—82no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—44no—
503C42—44no—
504C42—44no—
505C42—44no—
506C42—42no—
507C422−1+414yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#54712yes—
520C23C22×D812yes—
521C23#57812yes—
522C23C23×D411yescharacteristic
523Q8—84no—
524D4—42no—
525D4—42no—
526D4—84no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—216no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—88no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4#53612yes—
580C2×C4#53612yes—
581C2×C4#54712yes—
582C2×C4C22×D814yes—
583C2×C4C22×D812yes—
584C2×C4C22×D814yes—
585C2×C4C22×D812yes—
586C2×C4C2×#13714yes—
587C2×C4C2×#13712yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×2−1+412yes—
590C2×C4C2×2−1+411yescharacteristic
591C2×C4#58512yes—
592C8—24no—
593C8—44no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22#333414yes—
609C22C2×#53611yescharacteristic
610C22C2×#54711yescharacteristic
611C22#348711yescharacteristic
612C22C23×D811yescharacteristic
613C22C22×#13711yescharacteristic
614C22C2×#57811yescharacteristic
615C22#358511yescharacteristic
616C4—22no—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—41no—
624C4—82no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4#322614yes—
633C4#351512yes—
634C4#356312yes—
635C2—24no—
636C2—161no—
637C2#3891911yescharacteristic
638C2C2×#322611yescharacteristic
639C2#6087411yescharacteristic
640C2C2×#333411yescharacteristic
641C2#6125411yescharacteristic
642C2#6188011yescharacteristic
643C2#6190011yescharacteristic
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. #33146C2semidirect
4. #32374C2semidirect
5. #32310C2semidirect
7. #28827C2semidirect
8. #27950C2semidirect
9. #27727C2semidirect
11. #27610C2semidirect
12. #26506C2semidirect
13. #26086C2semidirect
15. #22599C2semidirect
16. #21982C2semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#25522
#331462
#323742
#323102
#299062
#288272
#279502
#277272
#276772
#276102
#265062
#260862
#246002
#225992
#219822
C4×#4802