Finite Group Databasebeta

← #6989357

Subgroups — #6989357

Order 512.

1431 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36449C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#32837C212yesmaximal, complemented
4#31380C214yesmaximal, complemented
5#30655C214yesmaximal
6#30382C212yesmaximal, complemented
7#30082C212yesmaximal, complemented
8#29923C211yesmaximal, characteristic, complemented
9#28799C212yesmaximal, complemented
10#28726C212yesmaximal, complemented
11#27972C211yesmaximal, characteristic, complemented
12#27776C212yesmaximal, complemented
13#27645C211yesmaximal, characteristic
14#26528C212yesmaximal, complemented
15#26106C212yesmaximal, complemented
16#23755C211yesmaximal, characteristic
17#22643C211yesmaximal, characteristic, complemented
18C2×#544—21no—
19C2×#489C2214yes—
20C2×#481C2212yes—
21#2868C2212yes—
22#2863C2212yes—
23#2859C2212yes—
24#2857C2211yescharacteristic
25#2660—22no—
26#2578—22no—
27#2571C2211yescharacteristic
28#2552—21no—
29#2552C2212yes—
30#2551—21no—
31#2551C2212yes—
32#2547C2211yescharacteristic
33#2530—22no—
34#2495—21no—
35#2485—21no—
36#2440—22no—
37#2440C2214yes—
38#2427C2212yes—
39#2421C2212yes—
40#2365C2211yescharacteristic
41#2352—21no—
42D4×C4⋊C4C2211yescharacteristic
43#2345—21no—
44#2317—21no—
45C4×#124C2212yes—
46C4×#121C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2100C2211yescharacteristic
51#2099C2214yes—
52#2096C2214yes—
53#2095C2212yes—
54#2094C2211yescharacteristic
55#2086C2214yes—
56#2085C2214yes—
57#2084C2211yescharacteristic
58#2084C2212yes—
59#2077C2212yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2071C2212yes—
65#2066C2214yes—
66#2058C2214yes—
67#2042C2214yes—
68#2036C2214yes—
69#2034C2212yes—
70#2024C2214yes—
71#1993C2214yes—
72#1981C2212yes—
73#1956C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1943C2214yes—
77#1939C2212yes—
78#1935C2214yes—
79#1924C2214yes—
80#1872C2214yes—
81#1865C2214yes—
82#1862C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1824C2212yes—
86#1803C2214yes—
87#1802C2214yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#544—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×#125—22no—
102C2×#125—22no—
103C2×#124C2312yes—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C2312yes—
106C2×#121C2311yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×D4—21no—
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2312yes—
111C2×#117—21no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#489—48no—
120#481—44no—
121C2×#107C2314yes—
122C2×#106C2312yes—
123C2×#106C2314yes—
124C2×#106C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#398—24no—
141#398—24no—
142#397—22no—
143#397—22no—
144#397—22no—
145#395—24no—
146#395—24no—
147#394—22no—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#390C2311yescharacteristic
152#389—21no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2314yes—
162#389C2311yescharacteristic
163#389C2314yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—21no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#386—21no—
175#386—22no—
176#385—24no—
177#385—22no—
178#384—22no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384C2314yes—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#383—22no—
189#382—21no—
190#382—22no—
191#382—21no—
192#382C2312yes—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380C2311yescharacteristic
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231C22×Q8—24no—
232C22×Q8C2412yes—
233C22×D4—21no—
234C22×D4C2411yescharacteristic
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—48no—
248C4⋊Q8—48no—
249C4⋊Q8—84no—
250#125—48no—
251#125—48no—
252#124—24no—
253#124—42no—
254C22⋊Q8—416no—
255C22⋊Q8—44no—
256#121—22no—
257#121—41no—
258C4×Q8—44no—
259C4×D4—24no—
260C4×D4—44no—
261C4×D4—44no—
262#117—24no—
263#117—24no—
264#117—44no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—44no—
331#106—42no—
332#106—42no—
333#106—44no—
334C4⋊C8—24no—
335C4⋊C8—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#102—28no—
341#102—24no—
342#102—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345#97—44no—
346C4×C8—24no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—21no—
369C24C22×D411yescharacteristic
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—48no—
375C2×Q8—84no—
376C2×Q8—84no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—81no—
383C2×D4H(5,𝔽2)12yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4#13714yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
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—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4AGL(1,ℤ/8)14yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C42—22no—
481C42—22no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—42no—
490C42—44no—
491C42—42no—
492C42—82no—
493C422−1+414yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C2×#13712yes—
508C23C23×D411yescharacteristic
509Q8—44no—
510Q8—44no—
511Q8—168no—
512D4—24no—
513D4—42no—
514D4—84no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—84no—
564C2×C4—164no—
565C2×C4—164no—
566C2×C4#53612yes—
567C2×C4#53612yes—
568C2×C4C2×AGL(1,ℤ/8)14yes—
569C2×C4C2×AGL(1,ℤ/8)12yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13712yes—
572C2×C4#57712yes—
573C2×C4C2×H(5,𝔽2)11yescharacteristic
574C2×C4C2×2−1+412yes—
575C2×C4#58512yes—
576C2×C4#58511yescharacteristic
577C8—42no—
578C8—44no—
579C8—42no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22C2×#53611yescharacteristic
595C22#348811yescharacteristic
596C22#350011yescharacteristic
597C22C22×AGL(1,ℤ/8)11yescharacteristic
598C22C22×#13711yescharacteristic
599C22C2×#57711yescharacteristic
600C22#358611yescharacteristic
601C4—24no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—81no—
610C4—84no—
611C4—82no—
612C4—81no—
613C4—82no—
614C4—82no—
615C4—81no—
616C4—82no—
617C4#323914yes—
618C4#355912yes—
619C4#356512yes—
620C2—42no—
621C2—82no—
622C2#4294411yescharacteristic
623C2C2×#323911yescharacteristic
624C2#6087211yescharacteristic
625C2#6196011yescharacteristic
626C2#6199911yescharacteristic
627C2#6254211yescharacteristic
628C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #35275C2semidirect
3. #32837C2semidirect
4. #31380C2semidirect
6. #30382C2semidirect
7. #30082C2semidirect
8. #29923C2semidirect
9. #28799C2semidirect
10. #28726C2semidirect
11. #27972C2semidirect
12. #27776C2semidirect
14. #26528C2semidirect
15. #26106C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#364492
#352752
#328372
#313802
#306552
#303822
#300822
#299232
#287992
#287262
#279722
#277762
#276452
#265282
#261062
#237552
#226432