Finite Group Databasebeta

← #6857077

Subgroups — #6857077

Order 512.

1469 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic
4#32494C214yesmaximal, complemented
5#31822C214yesmaximal
6#29829C212yesmaximal
7#29355C212yesmaximal, complemented
8#29287C212yesmaximal, complemented
9#29219C211yesmaximal, characteristic, complemented
10#28283C211yesmaximal, characteristic, complemented
11#28195C212yesmaximal
12#26026C212yesmaximal, complemented
13#25479C212yesmaximal, complemented
14#24678C212yesmaximal
15#22722C211yesmaximal, characteristic, complemented
16#22116C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#475C2212yes—
20C2×#474C2212yes—
21C2×#461C2211yescharacteristic
22C2×#446C2212yes—
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2862C2212yes—
26#2582C2212yes—
27#2571—21no—
28#2570—22no—
29#2568—21no—
30#2556—22no—
31#2551C2211yescharacteristic
32#2546C2211yescharacteristic
33#2542C2211yescharacteristic
34#2538C2211yescharacteristic
35#2511C2214yes—
36#2506C2212yes—
37#2492—21no—
38#2452C2212yes—
39#2446—22no—
40#2429—21no—
41#2396—22no—
42#2365—21no—
43#2355—21no—
44#2355C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2099C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2055C2214yes—
61#2055C2214yes—
62#2024C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1987C2214yes—
68#1981C2212yes—
69#1981C2211yescharacteristic
70#1948C2212yes—
71#1948C2214yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1940C2214yes—
77#1939C2212yes—
78#1936C2214yes—
79#1935C2214yes—
80#1935C2214yes—
81#1927C2212yes—
82#1926C2214yes—
83#1921C2214yes—
84#1877C2212yes—
85#1842C2212yes—
86#1842C2211yescharacteristic
87#1834C2212yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#533—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42—21no—
122#475—44no—
123#474—28no—
124#461—24no—
125#446—44no—
126C4×SD16—44no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#400—24no—
148#400—22no—
149#399—22no—
150#398—24no—
151#398—22no—
152#398—22no—
153#396—22no—
154#396—22no—
155#395—22no—
156#393—22no—
157#391—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391C2312yes—
165#391C2314yes—
166#391C2314yes—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2314yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382—22no—
198#382—22no—
199#380—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C43C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C2×SD16—24no—
244C2×SD16—42no—
245C2×SD16—42no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261#125—48no—
262C22⋊Q8—48no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—48no—
273C4×Q8—44no—
274C4×Q8—84no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—41no—
280C4×D4#4114yes—
281#117—48no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—28no—
345#106—42no—
346#106—44no—
347C4⋊C8—22no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—42no—
352C4⋊C8—44no—
353C4⋊C8—44no—
354#103—24no—
355#103—24no—
356#103—44no—
357#103—42no—
358#103—44no—
359#102—28no—
360#102—42no—
361#97—44no—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—82no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×#4112yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436SD16—48no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4#13714yes—
486C4⋊C4C2×#4112yes—
487C4⋊C42−1+412yes—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—22no—
506C42—41no—
507C42—42no—
508C42—44no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—84no—
514C42AGL(1,ℤ/8)14yes—
515C42C2×#4112yes—
516C42H(5,𝔽2)12yes—
517C422−1+412yes—
518C422−1+412yes—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C23×D411yescharacteristic
527Q8—44no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—88no—
533Q8—88no—
534Q8—84no—
535D4—22no—
536D4—22no—
537D4—41no—
538D4—41no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#53712yes—
587C2×C4#53712yes—
588C2×C4#54812yes—
589C2×C4#54812yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×#13714yes—
594C2×C4C2×#13712yes—
595C2×C4#57712yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C2×C4#58511yescharacteristic
602C8—44no—
603C8—42no—
604C8—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#53711yescharacteristic
613C22C2×#54811yescharacteristic
614C22#348911yescharacteristic
615C22C22×AGL(1,ℤ/8)11yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57711yescharacteristic
618C22#358611yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—24no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4#325414yes—
636C4#337314yes—
637C4#350611yescharacteristic
638C4#350911yescharacteristic
639C4#355011yescharacteristic
640C4#355511yescharacteristic
641C2—82no—
642C2#3972711yescharacteristic
643C2C2×#325411yescharacteristic
644C2#6093111yescharacteristic
645C2C2×#337311yescharacteristic
646C2#6132211yescharacteristic
647C2#6210911yescharacteristic
648C2#6213111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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. #36598C2semidirect
4. #32494C2semidirect
7. #29355C2semidirect
8. #29287C2semidirect
9. #29219C2semidirect
10. #28283C2semidirect
12. #26026C2semidirect
13. #25479C2semidirect
15. #22722C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349292
#331952
#324942
#318222
#298292
#293552
#292872
#292192
#282832
#281952
#260262
#254792
#246782
#227222
#221162
C4×#4612