Finite Group Databasebeta

← #6856573

Subgroups — #6856573

Order 512.

1401 conjugacy classes of subgroups.

The subgroup lattice has 632 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#32340C214yesmaximal, complemented
5#31482C214yesmaximal
6#29865C212yesmaximal
7#29287C211yesmaximal, characteristic, complemented
8#29276C212yesmaximal, complemented
9#29224C212yesmaximal, complemented
10#29219C211yesmaximal, characteristic, complemented
11#28799C212yesmaximal, complemented
12#27736C212yesmaximal
13#26281C212yesmaximal, complemented
14#24991C212yesmaximal
15#22825C211yesmaximal, characteristic, complemented
16#22116C212yesmaximal
17C4×#476C211yesmaximal, characteristic
18C2×#533—21no—
19C2×#476C2211yescharacteristic
20C2×#474C2212yes—
21#2868C2211yescharacteristic
22#2862C2212yes—
23#2582—21no—
24#2571C2212yes—
25#2570C2214yes—
26#2568—21no—
27#2556—22no—
28#2551C2211yescharacteristic
29#2546C2211yescharacteristic
30#2542C2211yescharacteristic
31#2538C2211yescharacteristic
32#2511—22no—
33#2506C2212yes—
34#2492—21no—
35#2452C2212yes—
36#2446—22no—
37#2429—21no—
38#2396—22no—
39#2365—21no—
40#2355—21no—
41#2355C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8—21no—
47#2293C2212yes—
48C42×Q8C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2073C2214yes—
58#2073C2214yes—
59#2058C2214yes—
60#1993C2214yes—
61#1990C2212yes—
62#1988C2212yes—
63#1988C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1987C2211yescharacteristic
67#1987C2212yes—
68#1984C2212yes—
69#1948C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1943C2214yes—
75#1936C2214yes—
76#1926C2214yes—
77#1921C2214yes—
78#1920C2212yes—
79#1920C2214yes—
80#1920C2214yes—
81#1917C2214yes—
82#1864C2212yes—
83#1842C2211yescharacteristic
84#1836C2214yes—
85#1834C2212yes—
86#1831C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#533—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C22×C42—21no—
121#476—24no—
122#474—28no—
123C2×#107C2314yes—
124C2×#106C2312yes—
125C2×#106C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137#400—24no—
138#400—24no—
139#400—22no—
140#399—22no—
141#398—24no—
142#398—22no—
143#398—22no—
144#396—22no—
145#396—22no—
146#395—22no—
147#393—22no—
148#391—22no—
149#391—21no—
150#391—22no—
151#391—21no—
152#391—22no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2314yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#382—22no—
188#382C2314yes—
189#382C2314yes—
190#380—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C43C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2×D411yescharacteristic
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255#125—48no—
256C22⋊Q8—44no—
257C22⋊Q8—48no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—24no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—42no—
264C4×Q8—44no—
265C4×Q8—48no—
266C4×Q8—44no—
267C4×Q8—84no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—41no—
273C4×D4#4114yes—
274#117—48no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
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—24no—
340#106—44no—
341C4⋊C8—24no—
342C4⋊C8—22no—
343C4⋊C8—24no—
344C4⋊C8—22no—
345C4⋊C8—42no—
346#102—28no—
347#102—24no—
348#102—42no—
349#97—44no—
350C4×C8—24no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—82no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×#4112yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—81no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2511yescharacteristic
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C2×#4112yes—
472C4⋊C42−1+412yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—82no—
480C42—24no—
481C42—22no—
482C42—24no—
483C42—22no—
484C42—21no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—21no—
490C42—41no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—44no—
497C42—44no—
498C42—84no—
499C42AGL(1,ℤ/8)14yes—
500C42C2×#4112yes—
501C422−1+412yes—
502C422−1+412yes—
503C422−1+412yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C23×D411yescharacteristic
512Q8—44no—
513Q8—84no—
514Q8—88no—
515Q8—88no—
516Q8—84no—
517D4—22no—
518D4—22no—
519D4—41no—
520D4—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4#53612yes—
570C2×C4#53612yes—
571C2×C4#54812yes—
572C2×C4#54812yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4#57512yes—
577C2×C4#57712yes—
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×2−1+411yescharacteristic
580C2×C4C2×2−1+411yescharacteristic
581C2×C4C2×2−1+411yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C2×C4#58511yescharacteristic
584C8—44no—
585C8—42no—
586C8—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C2×#53611yescharacteristic
595C22C2×#54811yescharacteristic
596C22#348911yescharacteristic
597C22C22×AGL(1,ℤ/8)11yescharacteristic
598C22C2×#57511yescharacteristic
599C22C2×#57711yescharacteristic
600C22#358611yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—84no—
612C4—82no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4#323514yes—
618C4#337314yes—
619C4#350711yescharacteristic
620C4#350911yescharacteristic
621C4#355511yescharacteristic
622C4#355611yescharacteristic
623C2—82no—
624C2#3972411yescharacteristic
625C2C2×#323511yescharacteristic
626C2#6088911yescharacteristic
627C2C2×#337311yescharacteristic
628C2#6132311yescharacteristic
629C2#6210911yescharacteristic
630C2#6214511yescharacteristic
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. #32340C2semidirect
7. #29287C2semidirect
8. #29276C2semidirect
9. #29224C2semidirect
10. #29219C2semidirect
11. #28799C2semidirect
13. #26281C2semidirect
15. #22825C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349292
#331952
#323402
#314822
#298652
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