Finite Group Databasebeta

← #6594221

Subgroups — #6594221

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#33057C212yesmaximal, complemented
5#29224C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#27978C212yesmaximal, complemented
8#27968C212yesmaximal, complemented
9#27729C212yesmaximal, complemented
10#27727C212yesmaximal, complemented
11#27706C214yesmaximal, complemented
12#27704C214yesmaximal, complemented
13#22504C212yesmaximal, complemented
14#22503C212yesmaximal
15C4×#489C211yesmaximal, characteristic, complemented
16#20764C212yesmaximal, complemented
17#20494C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#555C2211yescharacteristic
20C2×#554—21no—
21C2×#489C2211yescharacteristic
22C2×#436C2212yes—
23C2×C4×M16C2212yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2850C2212yes—
28#2567—21no—
29#2567C2211yescharacteristic
30#2552—21no—
31#2552C2212yes—
32#2542C2211yescharacteristic
33#2521—22no—
34#2521—22no—
35#2502—22no—
36#2500C2212yes—
37#2499C2212yes—
38#2440—22no—
39#2440C2214yes—
40#2427—21no—
41#2427—21no—
42#2362—21no—
43#2362—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2072C2214yes—
55#2071C2214yes—
56#2024C2212yes—
57#2023C2212yes—
58#1993C2214yes—
59#1993C2212yes—
60#1993C2211yescharacteristic
61#1990C2212yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1959C2212yes—
66#1959C2211yescharacteristic
67#1936C2212yes—
68#1935C2212yes—
69#1934C2214yes—
70#1933C2214yes—
71#1895C2214yes—
72#1894C2214yes—
73#1874C2214yes—
74#1872C2214yes—
75#1836C2212yes—
76#1836C2212yes—
77#1836C2211yescharacteristic
78#1836C2212yes—
79#1836C2212yes—
80#1834C2214yes—
81#1834C2214yes—
82#1825C2214yes—
83#1825C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1753C2214yes—
91#1749C2212yes—
92#1736C2214yes—
93#1736C2214yes—
94C4×#97C2211yescharacteristic
95C22×M16C2314yes—
96#556—28no—
97#555—28no—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#126—22no—
106C2×#125—22no—
107C2×#124—21no—
108C2×#124—22no—
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2314yes—
131C22×C42C2312yes—
132#489—24no—
133#436—28no—
134C2×C4⋊C8C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C4×M16—28no—
143C2×#97C2312yes—
144C2×#97C2311yescharacteristic
145C2×#97C2312yes—
146C2×#97C2311yescharacteristic
147C2×C4×C8C2314yes—
148C2×C4×C8C2312yes—
149#399—22no—
150#398—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#395—24no—
155#395—22no—
156#394—22no—
157#394—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2311yescharacteristic
166#390C2312yes—
167#390C2311yescharacteristic
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—21no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#385—24no—
187#385—24no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384—22no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#384C2314yes—
196#384C2312yes—
197#384C2312yes—
198#382C2312yes—
199#382C2311yescharacteristic
200#382C2312yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2314yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—24no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×M16—28no—
248C2×M16—28no—
249C2×M16—44no—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#126—48no—
261#125—48no—
262#124—22no—
263#124—22no—
264#124—44no—
265#124—48no—
266C22⋊Q8—28no—
267C22⋊Q8—416no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271#121—28no—
272#121—44no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—48no—
280C4×D4#4118yes—
281#117—28no—
282#117—28no—
283#117—44no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C4⋊C8—24no—
351#103—24no—
352#103—42no—
353#103—44no—
354#102—24no—
355#102—42no—
356#102—44no—
357C22⋊C8—28no—
358#97—24no—
359#97—22no—
360#97—42no—
361#97#4114yes—
362C4×C8—28no—
363C4×C8—24no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24C22×D412yes—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4#12414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—48no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4118yes—
442C22×C4C2511yescharacteristic
443M16—416no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8C2×#4114yes—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4#12414yes—
485C4⋊C4AGL(1,ℤ/8)14yes—
486C4⋊C4#13714yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4#12418yes—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—44no—
512C42—44no—
513C42—42no—
514C42—44no—
515C42C2×#4114yes—
516C42C2×#4112yes—
517C422−1+412yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×#12414yes—
532C23C23×D411yescharacteristic
533Q8—42no—
534Q8—42no—
535Q8—84no—
536Q8—84no—
537Q8—162no—
538Q8—164no—
539D4—28no—
540D4—42no—
541D4—42no—
542D4—164no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
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—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—168no—
591C2×C4C2×#12414yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#12412yes—
594C2×C4#54712yes—
595C2×C4#54712yes—
596C2×C4C2×AGL(1,ℤ/8)14yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4112yes—
602C2×C4C2×2−1+411yescharacteristic
603C2×C4#58512yes—
604C8—24no—
605C8—44no—
606C8—42no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C22×#12411yescharacteristic
622C22C2×#54711yescharacteristic
623C22#350011yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22C22×#13711yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C4—24no—
629C4—22no—
630C4—24no—
631C4—22no—
632C4—24no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4#314114yes—
645C4#347512yes—
646C4#351512yes—
647C2—44no—
648C2—161no—
649C2#3616411yescharacteristic
650C2C2×#314111yescharacteristic
651C2#6062511yescharacteristic
652C2#6125211yescharacteristic
653C2#6125411yescharacteristic
654C2#6258811yescharacteristic
655C2#6258911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36180C2semidirect
3. #33293C2semidirect
4. #33057C2semidirect
5. #29224C2semidirect
6. #29148C2semidirect
7. #27978C2semidirect
8. #27968C2semidirect
9. #27729C2semidirect
10. #27727C2semidirect
11. #27706C2semidirect
12. #27704C2semidirect
13. #22504C2semidirect
15. C4×#489C2semidirect
16. #20764C2semidirect
17. #20494C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#361802
#332932
#330572
#292242
#291482
#279782
#279682
#277292
#277272
#277062
#277042
#225042
#225032
C4×#4892
#207642
#204942