Finite Group Databasebeta

← #6800895

Subgroups — #6800895

Order 512.

1465 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32248C214yesmaximal, complemented
5#30829C214yesmaximal
6#29983C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28799C212yesmaximal, complemented
9#28726C212yesmaximal
10#28284C211yesmaximal, characteristic, complemented
11#28204C212yesmaximal
12#27776C212yesmaximal
13#26320C212yesmaximal, complemented
14#24990C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22119C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic
18C2×Q82—21no—
19C2×#489C2214yes—
20C2×#477C2212yes—
21C2×#476C2211yescharacteristic
22C2×#476C2212yes—
23#2868C2212yes—
24#2868C2211yescharacteristic
25#2573—22no—
26#2571C2211yescharacteristic
27#2568C2211yescharacteristic
28#2567—22no—
29#2567—21no—
30#2551—22no—
31#2551—21no—
32#2551C2214yes—
33#2551C2212yes—
34#2547C2211yescharacteristic
35#2538C2211yescharacteristic
36#2514C2212yes—
37#2494—21no—
38#2447C2212yes—
39#2421—21no—
40#2401—22no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124C2212yes—
47C4×C22⋊Q8—21no—
48#2295C2212yes—
49#2283—21no—
50C42×Q8C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2099C2214yes—
53#2096C2214yes—
54#2085C2214yes—
55#2072C2214yes—
56#2071C2214yes—
57#2068C2214yes—
58#2058C2214yes—
59#2042C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#1993C2214yes—
63#1987C2211yescharacteristic
64#1987C2212yes—
65#1984C2212yes—
66#1981C2212yes—
67#1956C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1943C2214yes—
73#1935C2214yes—
74#1927C2214yes—
75#1927C2214yes—
76#1925C2214yes—
77#1920C2212yes—
78#1918C2214yes—
79#1917C2214yes—
80#1864C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95Q82—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×C4⋊Q8C2312yes—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C22×C42—21no—
119#489—48no—
120#477—28no—
121#476—24no—
122#476—44no—
123C2×#107C2314yes—
124C2×#106C2312yes—
125C2×#106C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#401—22no—
142#400—22no—
143#400—22no—
144#400—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#396—22no—
149#395—24no—
150#395—24no—
151#395—24no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#390C2312yes—
160#389—22no—
161#389—24no—
162#389—22no—
163#389—24no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—21no—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#384—21no—
180#384—22no—
181#384—22no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2314yes—
188#383—22no—
189#382—21no—
190#382—21no—
191#382C2312yes—
192#381—22no—
193#380—21no—
194#380—21no—
195#380—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217#376—22no—
218C43C2311yescharacteristic
219C43C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252C4⋊Q8—48no—
253C4⋊Q8—44no—
254C4⋊Q8—416no—
255C4⋊Q8—48no—
256C4⋊Q8—82no—
257C4⋊Q8—84no—
258C4⋊Q8#4114yes—
259#124—24no—
260#124—42no—
261C22⋊Q8—44no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8—48no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—41no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—28no—
338#106—24no—
339#106—42no—
340#106—44no—
341C4⋊C8—22no—
342C4⋊C8—24no—
343C4⋊C8—24no—
344C4⋊C8—22no—
345C4⋊C8—42no—
346#103—28no—
347#103—44no—
348#103—44no—
349#103—44no—
350#102—28no—
351#102—24no—
352#102—24no—
353#97—44no—
354C4×C8—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—82no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×Q8—82no—
395C2×Q8AGL(1,ℤ/8)14yes—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4H(5,𝔽2)12yes—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—42no—
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—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4C2×#4114yes—
479C22⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C42—24no—
487C42—22no—
488C42—24no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—21no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—22no—
497C42—41no—
498C42—42no—
499C42—44no—
500C42—48no—
501C42—44no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42#13714yes—
506C42C2×#4112yes—
507C422−1+412yes—
508C422−1+412yes—
509C422−1+412yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C23×D411yescharacteristic
518Q8—28no—
519Q8—48no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—164no—
525Q8—168no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4#53612yes—
578C2×C4#53612yes—
579C2×C4#54812yes—
580C2×C4#54812yes—
581C2×C4C2×AGL(1,ℤ/8)14yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4#57712yes—
587C2×C4C22×#4111yescharacteristic
588C2×C4C2×H(5,𝔽2)11yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4#58511yescharacteristic
593C8—42no—
594C8—44no—
595C8—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×#53611yescharacteristic
604C22C2×#54811yescharacteristic
605C22#348811yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C22×#13711yescharacteristic
608C22C2×#57711yescharacteristic
609C22#358611yescharacteristic
610C4—22no—
611C4—22no—
612C4—24no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—44no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—84no—
622C4—81no—
623C4—82no—
624C4—81no—
625C4—82no—
626C4#323814yes—
627C4#336914yes—
628C4#351811yescharacteristic
629C4#352211yescharacteristic
630C4#354411yescharacteristic
631C4#355511yescharacteristic
632C2—82no—
633C2#3912511yescharacteristic
634C2C2×#323811yescharacteristic
635C2#6088911yescharacteristic
636C2C2×#336911yescharacteristic
637C2#6131711yescharacteristic
638C2#6196011yescharacteristic
639C2#6199911yescharacteristic
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. Q8×C4⋊Q8C2semidirect
4. #32248C2semidirect
6. #29983C2semidirect
7. #29141C2semidirect
8. #28799C2semidirect
10. #28284C2semidirect
13. #26320C2semidirect
15. #22825C2semidirect
16. #22119C2semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#349452
#331972
#322482
#308292
#299832
#291412
#287992
#287262
#282842
#282042
#277762
#263202
#249902
#228252
#221192
C4×#4762