Finite Group Databasebeta

← #6536234

Subgroups — #6536234

Order 512.

1617 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34324C211yesmaximal, characteristic
3#30176C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29299C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#28205C211yesmaximal, characteristic, complemented
8#27173C214yesmaximal, complemented
9#26560C214yesmaximal, complemented
10#25017C212yesmaximal
11#24123C212yesmaximal
12#24065C212yesmaximal, complemented
13#23679C211yesmaximal, characteristic, complemented
14#23571C212yesmaximal, complemented
15#22964C212yesmaximal, complemented
16#22835C211yesmaximal, characteristic
17C4⋊C4×Q16C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#533C2212yes—
20C2×#485C2214yes—
21C2×#475C2212yes—
22C2×C4×Q16C2212yes—
23C2×C4×Q16C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542—21no—
28#2538C2212yes—
29#2441—22no—
30#2429—22no—
31#2423C2214yes—
32#2373—22no—
33#2367—21no—
34#2365—21no—
35#2365C2211yescharacteristic
36#2362C2212yes—
37#2355—21no—
38#2354—21no—
39#2354C2211yescharacteristic
40Q8×C4⋊C4C2211yescharacteristic
41Q8×C4⋊C4C2212yes—
42#2346C2212yes—
43#2334—22no—
44#2326—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2296C2212yes—
49#2295C2211yescharacteristic
50#2276—21no—
51C42×D4C2211yescharacteristic
52#2099C2212yes—
53#2096C2212yes—
54#2084C2212yes—
55#2083C2211yescharacteristic
56#2075C2214yes—
57#2072C2211yescharacteristic
58#2068C2212yes—
59#2058C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2056C2212yes—
63#2043C2211yescharacteristic
64#2043C2212yes—
65#2036C2214yes—
66#2035C2214yes—
67#2024C2212yes—
68#2016C2211yescharacteristic
69#2001C2214yes—
70#1990C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1947C2214yes—
74#1942C2214yes—
75#1941C2214yes—
76#1927C2214yes—
77#1923C2214yes—
78#1920C2214yes—
79#1873C2214yes—
80#1865C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1845C2212yes—
86#1831C2211yescharacteristic
87#1817C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2212yes—
95C22×Q16C2312yes—
96#556—24no—
97#556—24no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×#125—22no—
106C2×#124—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2314yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C22×C42C2312yes—
126#485—48no—
127#475—28no—
128C4×Q16—28no—
129C4×Q16—48no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#399—24no—
149#398—24no—
150#396—24no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#391C2311yescharacteristic
156#391C2311yescharacteristic
157#390—21no—
158#390—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389C2312yes—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—24no—
178#387—21no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—24no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376—21no—
221#376—22no—
222C43C2311yescharacteristic
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2411yescharacteristic
238C22×Q8C2412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—24no—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—82no—
254C4⋊Q8#4114yes—
255#126—216no—
256#125—48no—
257#124—28no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—28no—
261C22⋊Q8—44no—
262C22⋊Q8—48no—
263C22⋊Q8—84no—
264C4×Q8—28no—
265C4×Q8—22no—
266C4×Q8—28no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—48no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8#4114yes—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279#117—24no—
280#117—24no—
281#117—48no—
282#117—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—24no—
347#107—44no—
348#107—44no—
349#106—28no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353#103—24no—
354#103—28no—
355#103—24no—
356#103—24no—
357#103—24no—
358#103—28no—
359#103—44no—
360C22⋊C8—28no—
361C22⋊C8—44no—
362C4×C8—24no—
363C4×C8—44no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24C22×D412yes—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8C2×#4112yes—
400C2×Q8H(5,𝔽2)14yes—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433Q16—216no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C82−1+414yes—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4#13714yes—
479C4⋊C4C2×#4114yes—
480C4⋊C4C2×#4112yes—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—48no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—84no—
490C42—28no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—48no—
501C42—44no—
502C42—44no—
503C42—44no—
504C42—42no—
505C42—82no—
506C42—82no—
507C42#13514yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—24no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—88no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—84no—
577C2×C4#52512yes—
578C2×C4#52512yes—
579C2×C4C2×#13512yes—
580C2×C4C2×#13512yes—
581C2×C4C2×#13512yes—
582C2×C4C2×#13714yes—
583C2×C4C2×#13712yes—
584C2×C4#57812yes—
585C2×C4C22×#4111yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4C2×2−1+412yes—
589C8—24no—
590C8—42no—
591C8—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×#52511yescharacteristic
600C22#348411yescharacteristic
601C22#349911yescharacteristic
602C22C22×#13511yescharacteristic
603C22C22×#13711yescharacteristic
604C22C2×#57811yescharacteristic
605C22#358511yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—24no—
610C4—22no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—44no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4—82no—
620C4—84no—
621C4—81no—
622C4—82no—
623C4#310714yes—
624C4#349212yes—
625C4#352212yes—
626C2—82no—
627C2#3569311yescharacteristic
628C2C2×#310711yescharacteristic
629C2#6060011yescharacteristic
630C2#6156611yescharacteristic
631C2#6160211yescharacteristic
632C2#6247311yescharacteristic
633C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30176C2semidirect
4. #29983C2semidirect
5. #29299C2semidirect
6. #29208C2semidirect
7. #28205C2semidirect
8. #27173C2semidirect
9. #26560C2semidirect
12. #24065C2semidirect
13. #23679C2semidirect
14. #23571C2semidirect
15. #22964C2semidirect
17. C4⋊C4×Q16C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343242
#301762
#299832
#292992
#292082
#282052
#271732
#265602
#250172
#241232
#240652
#236792
#235712
#229642
#228352
C4⋊C4×Q162