Finite Group Databasebeta

← #6571402

Subgroups — #6571402

Order 512.

1592 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#36315C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#30064C212yesmaximal, complemented
4#29922C211yesmaximal, characteristic, complemented
5#29394C212yesmaximal, complemented
6#29352C212yesmaximal, complemented
7#27408C214yesmaximal
8#27342C214yesmaximal, complemented
9#26447C212yesmaximal, complemented
10#25902C211yesmaximal, characteristic, complemented
11#25275C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic
13#23783C212yesmaximal, complemented
14#23754C211yesmaximal, characteristic
15#23222C212yesmaximal, complemented
16#23066C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519—21no—
20C2×#487C2212yes—
21C2×#474C2212yes—
22#2861C2212yes—
23#2857C2211yescharacteristic
24#2552C2211yescharacteristic
25#2547C2211yescharacteristic
26#2531—22no—
27#2529—22no—
28#2497—21no—
29#2489C2212yes—
30#2485—21no—
31#2426C2212yes—
32#2365—21no—
33#2365—21no—
34#2365C2212yes—
35#2365C2212yes—
36#2365C2211yescharacteristic
37#2354—21no—
38#2354C2211yescharacteristic
39#2346—21no—
40#2344—22no—
41#2344C2214yes—
42#2327—22no—
43#2310—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#121C2211yescharacteristic
47#2295C2212yes—
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2211yescharacteristic
51#2096C2212yes—
52#2093C2211yescharacteristic
53#2088C2214yes—
54#2083C2211yescharacteristic
55#2075C2212yes—
56#2072C2212yes—
57#2071C2211yescharacteristic
58#2068C2214yes—
59#2059C2214yes—
60#2057C2212yes—
61#2042C2214yes—
62#2035C2214yes—
63#2033C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1988C2212yes—
70#1983C2214yes—
71#1981C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1928C2214yes—
77#1928C2214yes—
78#1926C2214yes—
79#1924C2214yes—
80#1921C2214yes—
81#1921C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1863C2214yes—
85#1852C2212yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×#121C2311yescharacteristic
106#519—48no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—21no—
116C2×#117—24no—
117C2×#117D411yescharacteristic
118C2×#117D411yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122#487—28no—
123#474—28no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#400—22no—
142#398—22no—
143#397—22no—
144#397—22no—
145#396—22no—
146#394—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#390—22no—
155#390C2311yescharacteristic
156#390C2312yes—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389C2314yes—
164#389C2314yes—
165#389C2311yescharacteristic
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—21no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—22no—
182#386—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—21no—
188#382—22no—
189#381—24no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2311yescharacteristic
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376—22no—
216#376D412yes—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—44no—
236C22×Q8—44no—
237C22×Q8C2412yes—
238C22×D4—21no—
239C22×D4C2411yescharacteristic
240C23×C4—22no—
241C23×C4—21no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8#4118yes—
254#125—44no—
255#124—44no—
256C22⋊Q8—44no—
257#121—22no—
258#121—41no—
259C4×Q8—24no—
260C4×Q8—42no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×Q8—48no—
265C4×D4—24no—
266C4×D4—44no—
267#117—24no—
268#117—416no—
269#117—42no—
270#117—44no—
271#117—44no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—24no—
336#107—42no—
337#106—44no—
338C4⋊C8—24no—
339C4⋊C8—44no—
340#103—28no—
341#103—28no—
342#103—44no—
343#103—44no—
344#102—28no—
345#102—42no—
346#102—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349#97—44no—
350C4×C8—24no—
351#95—21no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—21no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—21no—
373C24C22×D411yescharacteristic
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—44no—
381C2×Q8—432no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—81no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4AGL(1,ℤ/8)14yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4C2×#4118yes—
468C4⋊C42−1+412yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—42no—
491C42—42no—
492C42—44no—
493C42—44no—
494C42—48no—
495C42—48no—
496C42—44no—
497C42—42no—
498C42C2×#4114yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C2×AGL(1,ℤ/8)12yes—
513C23C23×D411yescharacteristic
514Q8—416no—
515Q8—416no—
516Q8—44no—
517Q8—44no—
518D4—42no—
519D4—42no—
520D4—84no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—416no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—84no—
572C2×C4#52512yes—
573C2×C4#52512yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4#57512yes—
577C2×C4#57812yes—
578C2×C4C22×#4112yes—
579C2×C4C2×2−1+411yescharacteristic
580C2×C4#58512yes—
581C2×C4#58511yescharacteristic
582C8—42no—
583C8—42no—
584C8—44no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22C2×#52511yescharacteristic
600C22#348911yescharacteristic
601C22#350011yescharacteristic
602C22C22×AGL(1,ℤ/8)11yescharacteristic
603C22C2×#57511yescharacteristic
604C22C2×#57811yescharacteristic
605C22#358611yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—48no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—81no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#311414yes—
623C4#351812yes—
624C4#352412yes—
625C2—42no—
626C2—161no—
627C2#3607711yescharacteristic
628C2C2×#311411yescharacteristic
629C2#6058311yescharacteristic
630C2#6210711yescharacteristic
631C2#6214611yescharacteristic
632C2#6253811yescharacteristic
633C2#6258711yescharacteristic
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. #34209C2semidirect
3. #30064C2semidirect
4. #29922C2semidirect
5. #29394C2semidirect
6. #29352C2semidirect
8. #27342C2semidirect
9. #26447C2semidirect
10. #25902C2semidirect
11. #25275C2semidirect
13. #23783C2semidirect
15. #23222C2semidirect
16. #23066C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363152
#342092
#300642
#299222
#293942
#293522
#274082
#273422
#264472
#259022
#252752
#242982
#237832
#237542
#232222
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