Finite Group Databasebeta

← #6568683

Subgroups — #6568683

Order 512.

1772 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#36416C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#30228C211yesmaximal, characteristic, complemented
4#30201C212yesmaximal, complemented
5#29337C212yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29121C211yesmaximal, characteristic, complemented
8#27965C212yesmaximal, complemented
9#27822C212yesmaximal, complemented
10#27736C212yesmaximal, complemented
11#27719C214yesmaximal, complemented
12#27709C214yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23745C212yesmaximal, complemented
15#22500C212yesmaximal, complemented
16#22500C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×#436C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2856C2212yes—
25#2851C2212yes—
26#2589—22no—
27#2587—22no—
28#2580—21no—
29#2571C2211yescharacteristic
30#2566—21no—
31#2552C2211yescharacteristic
32#2542—21no—
33#2542—21no—
34#2542C2212yes—
35#2501—22no—
36#2501—22no—
37#2501C2214yes—
38#2441C2212yes—
39#2439C2212yes—
40#2370—21no—
41#2355—21no—
42#2354C2211yescharacteristic
43#2354C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2296C2212yes—
47C2×#384C2212yes—
48C42×D4C2211yescharacteristic
49#2100C2214yes—
50#2100C2214yes—
51#2099C2211yescharacteristic
52#2099C2211yescharacteristic
53#2099C2214yes—
54#2099C2212yes—
55#2099C2212yes—
56#2099C2212yes—
57#2099C2212yes—
58#2099C2212yes—
59#2073C2214yes—
60#2071C2212yes—
61#2041C2211yescharacteristic
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2025C2214yes—
65#2023C2212yes—
66#1992C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1988C2212yes—
72#1936C2214yes—
73#1936C2214yes—
74#1934C2214yes—
75#1895C2214yes—
76#1895C2214yes—
77#1875C2214yes—
78#1875C2214yes—
79#1836C2214yes—
80#1834C2212yes—
81#1824C2214yes—
82#1824C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1770C2212yes—
86C4×C4⋊C8C2211yescharacteristic
87C4×#102C2212yes—
88C4×#102C2212yes—
89#1753C2214yes—
90#1749C2212yes—
91#1744C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41—22no—
96C22×M16C2314yes—
97#547—28no—
98#547—28no—
99#547—48no—
100#547—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#123—24no—
109C2×#123—24no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2314yes—
129C22×C42C2312yes—
130#436—28no—
131C2×C4⋊C8C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#399—22no—
144#399—24no—
145#399—24no—
146#399—24no—
147#398—22no—
148#396—22no—
149#396—22no—
150#393—22no—
151#391—21no—
152#391—22no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2314yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384—22no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#384C2314yes—
195#382—21no—
196#382—21no—
197#382C2312yes—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2312yes—
229#336C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233C2×#41—48no—
234C2×#41—48no—
235C2×#41—48no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2412yes—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×M16—28no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—41no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260#127—42no—
261#125—44no—
262#125—44no—
263#123—416no—
264#123—416no—
265#123—416no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269#121—28no—
270#121—28no—
271#121—48no—
272#121—48no—
273C4×Q8—44no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4#4118yes—
281#117—24no—
282#117—42no—
283#117—44no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C4⋊C8—24no—
353#102—24no—
354#102—44no—
355#102—42no—
356#102—42no—
357#102—44no—
358#102—42no—
359C22⋊C8—28no—
360#97—44no—
361C4×C8—24no—
362C4×C8—42no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24C22×D412yes—
380#41—832no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—81no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4#12414yes—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4118yes—
438C22×C4C2511yescharacteristic
439M16—88no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4#12414yes—
489C4⋊C42−1+412yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4#12418yes—
503C42—21no—
504C42—21no—
505C42—24no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42—44no—
512C42—42no—
513C42—42no—
514C42—82no—
515C42AGL(1,ℤ/8)14yes—
516C42AGL(1,ℤ/8)14yes—
517C42C2×#4114yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×#12414yes—
532C23C23×D411yescharacteristic
533Q8—84no—
534Q8—88no—
535D4—28no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—84no—
541D4—84no—
542D4—816no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—816no—
590C2×C4—88no—
591C2×C4C2×#12412yes—
592C2×C4C2×#12414yes—
593C2×C4C2×#12412yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C22×#4112yes—
601C2×C4C2×2−1+411yescharacteristic
602C2×C4#58512yes—
603C2×C4#58511yescharacteristic
604C8—42no—
605C8—44no—
606C8—42no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C22×#12411yescharacteristic
622C22#348511yescharacteristic
623C22#348911yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22#358611yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4#313814yes—
645C4#351512yes—
646C4#352112yes—
647C2—44no—
648C2—162no—
649C2#3604211yescharacteristic
650C2C2×#313811yescharacteristic
651C2#6062711yescharacteristic
652C2#6171711yescharacteristic
653C2#6171711yescharacteristic
654C2#6210711yescharacteristic
655C2#6211111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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
1. #36416C2semidirect
3. #30228C2semidirect
4. #30201C2semidirect
5. #29337C2semidirect
6. #29219C2semidirect
7. #29121C2semidirect
8. #27965C2semidirect
9. #27822C2semidirect
10. #27736C2semidirect
11. #27719C2semidirect
12. #27709C2semidirect
13. #23815C2semidirect
14. #23745C2semidirect
15. #22500C2semidirect
16. #22500C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364162
#361792
#302282
#302012
#293372
#292192
#291212
#279652
#278222
#277362
#277192
#277092
#238152
#237452
#225002
#205682