Finite Group Databasebeta

← #6561627

Subgroups — #6561627

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic, complemented
3#30365C212yesmaximal, complemented
4#30276C211yesmaximal, characteristic, complemented
5#29346C211yesmaximal, characteristic, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29163C212yesmaximal, complemented
8#28869C214yesmaximal, complemented
9#28712C212yesmaximal, complemented
10#28607C212yesmaximal
11#28092C214yesmaximal, complemented
12#27862C212yesmaximal, complemented
13#23837C211yesmaximal, characteristic, complemented
14#23761C212yesmaximal, complemented
15#23220C212yesmaximal, complemented
16#23161C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#556—21no—
20C2×#500C2212yes—
21C2×#486C2212yes—
22C2×#436C2212yes—
23#2868C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2212yes—
26#2854C2212yes—
27#2602—22no—
28#2592—21no—
29#2568—21no—
30#2557—22no—
31#2547—21no—
32#2547C2211yescharacteristic
33#2547C2212yes—
34#2544—21no—
35#2541C2211yescharacteristic
36#2499—22no—
37#2496—22no—
38#2496C2214yes—
39#2491C2212yes—
40#2422C2212yes—
41#2367—21no—
42#2366C2211yescharacteristic
43#2365—21no—
44#2365C2212yes—
45#2365C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2093C2212yes—
54#2093C2214yes—
55#2093C2212yes—
56#2084C2212yes—
57#2084C2211yescharacteristic
58#2084C2211yescharacteristic
59#2076C2214yes—
60#2076C2214yes—
61#2071C2212yes—
62#2058C2211yescharacteristic
63#2058C2212yes—
64#2058C2211yescharacteristic
65#2034C2214yes—
66#2034C2214yes—
67#2023C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1988C2212yes—
71#1988C2212yes—
72#1982C2212yes—
73#1943C2214yes—
74#1936C2214yes—
75#1934C2214yes—
76#1926C2214yes—
77#1921C2214yes—
78#1921C2214yes—
79#1895C2214yes—
80#1895C2214yes—
81#1864C2214yes—
82#1834C2212yes—
83#1809C2214yes—
84#1809C2214yes—
85#1776C2214yes—
86C4×#106C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×M16C2314yes—
96#559—48no—
97#556—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—21no—
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#124—22no—
108C2×#124—21no—
109C2×#124—22no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#123—22no—
113C2×#123—22no—
114C2×C22⋊Q8—22no—
115C2×#121C2312yes—
116C2×#121C2312yes—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C22×C42C2312yes—
129#500—28no—
130#486—28no—
131#436—28no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#400—24no—
148#397—22no—
149#397—24no—
150#397—24no—
151#397—22no—
152#394—22no—
153#394—22no—
154#393—22no—
155#392—22no—
156#391—22no—
157#391—22no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—22no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200#380C2314yes—
201#380C2314yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×M16—28no—
248C2×M16—44no—
249C2×M16—44no—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—41no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#127—24no—
262#127—44no—
263#126—416no—
264#124—48no—
265#124—48no—
266#124—44no—
267#124—44no—
268#124—44no—
269#123—48no—
270#123—48no—
271C22⋊Q8—48no—
272#121—24no—
273#121—42no—
274#121—44no—
275C4×Q8—48no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285#117—48no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
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—28no—
347#107—28no—
348#106—24no—
349#106—42no—
350#106—42no—
351#106—42no—
352C4⋊C8—24no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357C22⋊C8—28no—
358#97—44no—
359C4×C8—42no—
360C4×C8#4118yes—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—21no—
365#95—22no—
366#95—24no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24C22×D412yes—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—81no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442M16—88no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8C2×#4114yes—
451C2×C8C2×#4114yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—81no—
484C4⋊C42−1+412yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—21no—
502C42—21no—
503C42—24no—
504C42—22no—
505C42—42no—
506C42—42no—
507C42—41no—
508C42—41no—
509C42—48no—
510C42—44no—
511C42—42no—
512C42—44no—
513C42—42no—
514C42AGL(1,ℤ/8)14yes—
515C42AGL(1,ℤ/8)14yes—
516C42C2×#4114yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C23×D411yescharacteristic
532Q8—84no—
533Q8—84no—
534Q8—84no—
535Q8—88no—
536D4—44no—
537D4—44no—
538D4—44no—
539D4—88no—
540D4—84no—
541D4—162no—
542D4—164no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—84no—
590C2×C4—164no—
591C2×C4#53212yes—
592C2×C4#53212yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C22×#4112yes—
600C2×C4C2×2−1+411yescharacteristic
601C2×C4#58512yes—
602C2×C4#58511yescharacteristic
603C8—24no—
604C8—24no—
605C8—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×#53211yescharacteristic
621C22#348511yescharacteristic
622C22#348911yescharacteristic
623C22C22×AGL(1,ℤ/8)11yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22#358611yescharacteristic
626C22#358611yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—82no—
643C4#318414yes—
644C4#350712yes—
645C4#351012yes—
646C2—82no—
647C2—162no—
648C2#3595911yescharacteristic
649C2C2×#318411yescharacteristic
650C2#6072011yescharacteristic
651C2#6171711yescharacteristic
652C2#6172211yescharacteristic
653C2#6210911yescharacteristic
654C2#6211111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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
2. #36188C2semidirect
3. #30365C2semidirect
4. #30276C2semidirect
5. #29346C2semidirect
6. #29219C2semidirect
7. #29163C2semidirect
8. #28869C2semidirect
9. #28712C2semidirect
11. #28092C2semidirect
12. #27862C2semidirect
13. #23837C2semidirect
14. #23761C2semidirect
15. #23220C2semidirect
16. #23161C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#361882
#303652
#302762
#293462
#292192
#291632
#288692
#287122
#286072
#280922
#278622
#238372
#237612
#232202
#231612
#205682