Finite Group Databasebeta

← #6795363

Subgroups — #6795363

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32394C214yesmaximal, complemented
5#32332C214yesmaximal, complemented
6#29906C212yesmaximal
7#28835C212yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27735C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic
11#27617C212yesmaximal, complemented
12#26508C212yesmaximal, complemented
13#26086C212yesmaximal, complemented
14#24607C212yesmaximal
15#22603C211yesmaximal, characteristic, complemented
16#21987C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C22×#125—21no—
19C2×#481C2211yescharacteristic
20C2×#432C2212yes—
21#2857C2211yescharacteristic
22#2854C2212yes—
23#2571C2211yescharacteristic
24#2563C2211yescharacteristic
25#2552—21no—
26#2552—21no—
27#2552—22no—
28#2552C2214yes—
29#2552C2212yes—
30#2552C2212yes—
31#2532—22no—
32#2513C2212yes—
33#2508C2212yes—
34#2495—21no—
35#2488—22no—
36#2486—21no—
37C2×#398—22no—
38#2369C2211yescharacteristic
39#2354C2211yescharacteristic
40#2352—21no—
41#2346—21no—
42C4×#121C2211yescharacteristic
43C2×#389—21no—
44C2×#389C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47#2283C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2100C2212yes—
52#2099C2214yes—
53#2094C2212yes—
54#2093C2214yes—
55#2088C2214yes—
56#2085C2214yes—
57#2083C2214yes—
58#2083C2214yes—
59#2071C2214yes—
60#2067C2214yes—
61#1990C2212yes—
62#1987C2212yes—
63#1948C2212yes—
64#1948C2214yes—
65#1947C2214yes—
66#1947C2212yes—
67#1943C2214yes—
68#1939C2212yes—
69#1936C2214yes—
70#1926C2212yes—
71#1921C2214yes—
72#1919C2212yes—
73#1919C2214yes—
74#1918C2214yes—
75#1917C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1853C2212yes—
79#1853C2211yescharacteristic
80#1852C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1847C2214yes—
84#1845C2212yes—
85#1836C2214yes—
86#1824C2212yes—
87#1824C2211yescharacteristic
88#1817C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92#1770C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—22no—
97C2×#125—24no—
98C2×#125—22no—
99C2×#125—48no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×#121C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4C2311yescharacteristic
105C2×#117—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#481—24no—
120#432—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2312yes—
125C2×#106C2314yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#102C2312yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2311yescharacteristic
134C2×C22⋊C8C2311yescharacteristic
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137#400—22no—
138#400—22no—
139#398—22no—
140#398—24no—
141#398—24no—
142#398—24no—
143#398—24no—
144#397—22no—
145#397—22no—
146#393—22no—
147#391—21no—
148#391—22no—
149#390C2312yes—
150#390C2312yes—
151#390C2311yescharacteristic
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389—24no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2314yes—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2311yescharacteristic
166#388—22no—
167#388—22no—
168#388—22no—
169#388—21no—
170#388—22no—
171#388—22no—
172#387—22no—
173#386—22no—
174#386—21no—
175#384—22no—
176#384—21no—
177#384—21no—
178#384—22no—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2314yes—
182#384C2312yes—
183#384C2314yes—
184#383—22no—
185#382—21no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195#380C2311yescharacteristic
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212#376—22no—
213C2×#95—21no—
214C2×#95—22no—
215C2×#95D412yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2312yes—
231C22×Q8—21no—
232C22×D4—21no—
233C22×D4C2411yescharacteristic
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247#125—416no—
248#125—48no—
249#125—48no—
250#123—84no—
251C22⋊Q8—82no—
252#121—22no—
253#121—22no—
254C4×D4—24no—
255C4×D4—44no—
256#117—44no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
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—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—84no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—44no—
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329#107—44no—
330#107—44no—
331#107—44no—
332#106—24no—
333#106—42no—
334#106—44no—
335C4⋊C8—28no—
336C4⋊C8—24no—
337#102—24no—
338#102—42no—
339#102—44no—
340C22⋊C8—24no—
341C22⋊C8—22no—
342C22⋊C8#4114yes—
343#97—28no—
344C4×C8—24no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—21no—
367C24C22×D411yescharacteristic
368C2×Q8—42no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4#13714yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4112yes—
422C22×C42−1+412yes—
423C22×C4C2511yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8C2×#4114yes—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—82no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4AGL(1,ℤ/8)14yes—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C42—22no—
482C42—22no—
483C42—24no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—44no—
491C42—44no—
492C42—44no—
493C42—42no—
494C42—82no—
495C422−1+414yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23#54712yes—
508C23C2×#13712yes—
509C23#57612yes—
510C23C23×D411yescharacteristic
511Q8—162no—
512D4—42no—
513D4—42no—
514D4—84no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—84no—
569C2×C4#53612yes—
570C2×C4#53612yes—
571C2×C4#54712yes—
572C2×C4C2×AGL(1,ℤ/8)14yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×2−1+412yes—
578C2×C4C2×2−1+411yescharacteristic
579C2×C4#58512yes—
580C8—24no—
581C8—44no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22#335714yes—
597C22C2×#53611yescharacteristic
598C22C2×#54711yescharacteristic
599C22#348711yescharacteristic
600C22C22×AGL(1,ℤ/8)11yescharacteristic
601C22C22×#13711yescharacteristic
602C22C2×#57611yescharacteristic
603C22#358611yescharacteristic
604C4—22no—
605C4—24no—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—81no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—84no—
620C4#323914yes—
621C4#351512yes—
622C4#356312yes—
623C2—24no—
624C2—161no—
625C2#3891911yescharacteristic
626C2C2×#323911yescharacteristic
627C2#6087111yescharacteristic
628C2C2×#335711yescharacteristic
629C2#6125211yescharacteristic
630C2#6189311yescharacteristic
631C2#6190711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #33149C2semidirect
4. #32394C2semidirect
5. #32332C2semidirect
7. #28835C2semidirect
8. #27950C2semidirect
9. #27735C2semidirect
11. #27617C2semidirect
12. #26508C2semidirect
13. #26086C2semidirect
15. #22603C2semidirect
16. #21987C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25522
#331492
#323942
#323322
#299062
#288352
#279502
#277352
#276782
#276172
#265082
#260862
#246072
#226032
#219872
C4×#4812