Finite Group Databasebeta

← #6556868

Subgroups — #6556868

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic, complemented
2#36496C211yesmaximal, characteristic
3#30216C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29350C212yesmaximal, complemented
7#29148C211yesmaximal, characteristic, complemented
8#29139C212yesmaximal, complemented
9#27763C214yesmaximal, complemented
10#27747C212yesmaximal
11#27739C214yesmaximal, complemented
12#27729C212yesmaximal, complemented
13#23822C212yesmaximal, complemented
14#23817C211yesmaximal, characteristic, complemented
15#22546C212yesmaximal, complemented
16#22522C212yesmaximal, complemented
17#20573C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#557—21no—
20C2×#554C2211yescharacteristic
21C2×#446C2212yes—
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2861C2212yes—
25#2858C2211yescharacteristic
26#2567C2211yescharacteristic
27#2566C2212yes—
28#2552—21no—
29#2552—22no—
30#2552C2212yes—
31#2552C2214yes—
32#2546—21no—
33#2546—22no—
34#2544C2212yes—
35#2541C2211yescharacteristic
36#2461—22no—
37#2461—22no—
38#2426—21no—
39#2426—21no—
40#2361—21no—
41#2361—21no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×#127C2212yes—
47C4×#124C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2101C2212yes—
50#2100C2214yes—
51#2072C2214yes—
52#2072C2211yescharacteristic
53#2071C2214yes—
54#2071C2211yescharacteristic
55#2071C2212yes—
56#2071C2214yes—
57#2042C2212yes—
58#2041C2214yes—
59#2024C2211yescharacteristic
60#2024C2212yes—
61#2023C2212yes—
62#2023C2212yes—
63#2023C2211yescharacteristic
64#2023C2212yes—
65#2023C2212yes—
66#1992C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1987C2212yes—
70#1983C2214yes—
71#1982C2212yes—
72#1959C2214yes—
73#1959C2212yes—
74#1957C2212yes—
75#1957C2214yes—
76#1936C2214yes—
77#1934C2214yes—
78#1933C2214yes—
79#1836C2212yes—
80#1834C2214yes—
81#1803C2212yes—
82#1802C2214yes—
83#1802C2212yes—
84#1802C2214yes—
85#1771C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2212yes—
89#1762C2214yes—
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95#559—28no—
96#557—48no—
97#554—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#127C2312yes—
107C2×#125—24no—
108C2×#125—22no—
109C2×#125C2312yes—
110C2×#124—21no—
111C2×#124—21no—
112C2×#124—21no—
113C2×#124C2312yes—
114C2×#124C2311yescharacteristic
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×#121—21no—
119C2×#121—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2311yescharacteristic
133C22×C4⋊C4—21no—
134#446—28no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#97C2314yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#398—24no—
152#398—24no—
153#398—24no—
154#396—22no—
155#396—22no—
156#395—22no—
157#394—22no—
158#394—22no—
159#392—22no—
160#391—22no—
161#391—21no—
162#391C2312yes—
163#391C2311yescharacteristic
164#390—22no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390C2311yescharacteristic
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—24no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389—22no—
179#389—21no—
180#389C2314yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—21no—
187#388—21no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2314yes—
199#384C2312yes—
200#384C2312yes—
201#384C2314yes—
202#382C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C43C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4C2412yes—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252C4⋊Q8—48no—
253C4⋊Q8—44no—
254C4⋊Q8—84no—
255C4⋊Q8—82no—
256#127—24no—
257#127—24no—
258#127—42no—
259#125—28no—
260#125—416no—
261#125—48no—
262#124—22no—
263#124—28no—
264#124—41no—
265#124—44no—
266#124—82no—
267#124—82no—
268#123—48no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271#121—44no—
272#121—44no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—48no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×Q8#4118yes—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—44no—
284C4×D4—84no—
285C4×D4—84no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C4⋊C8—24no—
351C4⋊C8—28no—
352#103—42no—
353#103—44no—
354#103—42no—
355#102—24no—
356#102—44no—
357#102—44no—
358#102—44no—
359#102—44no—
360#102—42no—
361#102—44no—
362#97—28no—
363#97—42no—
364C4×C8—24no—
365C4×C8—44no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8#12414yes—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D42−1+412yes—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4#12414yes—
485C4⋊C4#12418yes—
486C4⋊C4AGL(1,ℤ/8)14yes—
487C4⋊C4#13714yes—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—24no—
504C42—22no—
505C42—24no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—24no—
510C42—22no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—22no—
516C42—48no—
517C42—42no—
518C42—44no—
519C42C2×#4114yes—
520C42C2×#4118yes—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C23×D411yescharacteristic
536Q8—28no—
537Q8—42no—
538Q8—42no—
539Q8—88no—
540Q8—84no—
541Q8—162no—
542D4—24no—
543D4—44no—
544D4—42no—
545D4—44no—
546D4—88no—
547D4—88no—
548D4—84no—
549D4—162no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—164no—
593C2×C4—164no—
594C2×C4C2×#12414yes—
595C2×C4C2×#12414yes—
596C2×C4C2×#12412yes—
597C2×C4C2×#12412yes—
598C2×C4C2×AGL(1,ℤ/8)14yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×#13714yes—
601C2×C4C2×#13712yes—
602C2×C4#57612yes—
603C2×C4#57712yes—
604C2×C4C22×#4112yes—
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58512yes—
607C2×C4#58511yescharacteristic
608C8—42no—
609C8—42no—
610C8—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C22×#12411yescharacteristic
626C22#348711yescharacteristic
627C22#348911yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57611yescharacteristic
631C22C2×#57711yescharacteristic
632C4—24no—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4—81no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4#314114yes—
649C4#351612yes—
650C4#352412yes—
651C2—82no—
652C2—162no—
653C2#3585411yescharacteristic
654C2C2×#314111yescharacteristic
655C2#6062211yescharacteristic
656C2#6190611yescharacteristic
657C2#6190711yescharacteristic
658C2#6212711yescharacteristic
659C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36593C2semidirect
3. #30216C2semidirect
4. #30203C2semidirect
5. #29354C2semidirect
6. #29350C2semidirect
7. #29148C2semidirect
8. #29139C2semidirect
9. #27763C2semidirect
11. #27739C2semidirect
12. #27729C2semidirect
13. #23822C2semidirect
14. #23817C2semidirect
15. #22546C2semidirect
16. #22522C2semidirect
17. #20573C2semidirect

Maximal subgroups

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

subgroupindex
#365932
#364962
#302162
#302032
#293542
#293502
#291482
#291392
#277632
#277472
#277392
#277292
#238222
#238172
#225462
#225222
#205732