Finite Group Databasebeta

← #6852642

Subgroups — #6852642

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic
4#32372C214yesmaximal, complemented
5#31546C214yesmaximal
6#29986C212yesmaximal
7#29259C212yesmaximal, complemented
8#29212C212yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28540C212yesmaximal
11#28283C211yesmaximal, characteristic, complemented
12#27823C212yesmaximal, complemented
13#25840C212yesmaximal, complemented
14#24328C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22152C212yesmaximal
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#533—21no—
20C2×#500C2212yes—
21C2×#475C2212yes—
22C2×#474C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2568C2212yes—
28#2565C2214yes—
29#2556—22no—
30#2551C2211yescharacteristic
31#2546—21no—
32#2546—21no—
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2538C2211yescharacteristic
36#2506—22no—
37#2506—22no—
38#2506C2212yes—
39#2492—21no—
40#2452C2212yes—
41#2429—21no—
42#2396—22no—
43#2365—21no—
44#2365—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2095C2214yes—
55#2092C2214yes—
56#2083C2214yes—
57#2066C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2059C2214yes—
61#2025C2214yes—
62#2024C2214yes—
63#1993C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1981C2212yes—
69#1973C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1935C2214yes—
77#1927C2212yes—
78#1917C2214yes—
79#1880C2214yes—
80#1877C2212yes—
81#1863C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#557—44no—
97#557—44no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×#125—24no—
110C2×#125C2314yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C22×C42—21no—
128#500—44no—
129#475—44no—
130#474—44no—
131#461Q818yescomplemented
132C4×SD16—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—24no—
153#400—22no—
154#399—22no—
155#398—24no—
156#398—24no—
157#398—22no—
158#398—22no—
159#396—22no—
160#396—22no—
161#395—22no—
162#393—22no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391C2312yes—
169#391C2314yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#387—24no—
191#387—24no—
192#387—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382C2314yes—
202#380—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C2×SD16—42no—
246C2×SD16—42no—
247C2×SD16C2×Q818yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—48no—
262C4⋊Q8—44no—
263C4⋊Q8C2×Q814yes—
264#125—44no—
265#125—44no—
266#125—48no—
267#125—416no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—48no—
282C4×Q8—44no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—41no—
288C4×D4C2×Q814yes—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—21no—
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—22no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—44no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—44no—
355#107—44no—
356#107—44no—
357#106—42no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—42no—
363C4⋊C8C2×Q814yes—
364#103—24no—
365#103—44no—
366#103—42no—
367#103—44no—
368#103C2×Q818yes—
369#102—42no—
370#102—44no—
371#97—44no—
372C4×C8—24no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8C22×Q814yes—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4C22×Q812yes—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444SD16—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8C22×Q814yes—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4C22×Q812yes—
497C4⋊C42−1+412yes—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—82no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—48no—
519C42—44no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—44no—
524C42AGL(1,ℤ/8)14yes—
525C42C22×Q812yes—
526C42H(5,𝔽2)12yes—
527C42H(5,𝔽2)12yes—
528C422−1+412yes—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—24no—
539Q8—44no—
540Q8—42no—
541Q8—42no—
542Q8—88no—
543Q8—88no—
544Q8—84no—
545D4—22no—
546D4—22no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4#53512yes—
598C2×C4#53512yes—
599C2×C4D4×Q812yes—
600C2×C4D4×Q812yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4#57512yes—
605C2×C4#57812yes—
606C2×C4C23×Q811yescharacteristic
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4#58511yescharacteristic
612C8—24no—
613C8—42no—
614C8—44no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#53511yescharacteristic
623C22C2×D4×Q811yescharacteristic
624C22#348911yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#321414yes—
646C4#338614yes—
647C4#350111yescharacteristic
648C4#350311yescharacteristic
649C4#355011yescharacteristic
650C4#355411yescharacteristic
651C2—82no—
652C2#3959611yescharacteristic
653C2C2×#321411yescharacteristic
654C2#6083711yescharacteristic
655C2C2×#338611yescharacteristic
656C2#6136911yescharacteristic
657C2#6210911yescharacteristic
658C2#6214611yescharacteristic
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. #36560C2semidirect
4. #32372C2semidirect
7. #29259C2semidirect
8. #29212C2semidirect
9. #28689C2semidirect
11. #28283C2semidirect
12. #27823C2semidirect
13. #25840C2semidirect
15. #22825C2semidirect
131. #461Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
#349292
#331952
#323722
#315462
#299862
#292592
#292122
#286892
#285402
#282832
#278232
#258402
#243282
#228252
#221522
C4×#4612