Finite Group Databasebeta

← #6450686

Subgroups — #6450686

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34186C211yesmaximal, characteristic
3#26103C214yesmaximal, complemented
4#26102C214yesmaximal, complemented
5#26091C212yesmaximal, complemented
6#26090C212yesmaximal, complemented
7#26087C212yesmaximal, complemented
8#26086C212yesmaximal, complemented
9#23735C212yesmaximal
10#23614C212yesmaximal
11#21175C212yesmaximal, complemented
12#21169C211yesmaximal, characteristic, complemented
13#21145C211yesmaximal, characteristic, complemented
14#21036C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#424C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C2×#418C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C22×#102C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2859—21no—
29#2859—21no—
30#2647—22no—
31#2571C2212yescomplemented
32#2571C2212yescomplemented
33#2552C2212yescomplemented
34#2552C2212yescomplemented
35#2442—22no—
36#2442—22no—
37#2357—22no—
38#2357C2214yes—
39#2354—21no—
40#2354—21no—
41#2354C2212yes—
42#2354C2212yes—
43#2317C2212yes—
44#2316C2212yes—
45#2312—22no—
46#2309C2212yes—
47#2285C2211yescharacteristic
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50#2280—22no—
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2094C2214yes—
54#2093C2214yes—
55#2092C2214yes—
56#2092C2214yes—
57#2067C2214yes—
58#2067C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#1949C2212yescomplemented
62#1949C2212yescomplemented
63#1949C2214yescomplemented
64#1949C2214yescomplemented
65#1948C2211yescharacteristic, complemented
66#1948C2211yescharacteristic, complemented
67#1948C2214yescomplemented
68#1947C2211yescharacteristic, complemented
69#1947C2211yescharacteristic, complemented
70#1947C2214yescomplemented
71#1940C2214yescomplemented
72#1939C2212yescomplemented
73#1939C2212yescomplemented
74#1921C2214yescomplemented
75#1920C2214yescomplemented
76#1919C2214yescomplemented
77#1918C2214yescomplemented
78#1847C2214yescomplemented
79#1845C2212yescomplemented
80#1844C2212yescomplemented
81#1802C2212yes—
82#1802C2212yes—
83#1802C2214yes—
84#1762C2212yes—
85#1762C2212yes—
86C4×#102C2212yes—
87C4×#102C2212yes—
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98C2×#125—22no—
99C2×#125—22no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—24no—
103C2×C4×D4Q818yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C42C2311yescharacteristic
118#428—28no—
119#428—44no—
120#427—44no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×#106C2312yes—
129#424—28no—
130C2×C4⋊C8—24no—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136#418—24no—
137#418—24no—
138C2×#102—44no—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C4×M16—24no—
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#400—24no—
150#398—24no—
151#398—24no—
152#390—22no—
153#390—22no—
154#390C2314yes—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#388—24no—
167#388—24no—
168#388—24no—
169#388—24no—
170#385—24no—
171#384—21no—
172#384—21no—
173#384D411yescharacteristic
174#384D411yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#384C2314yes—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382—22no—
181#382—22no—
182#382D412yes—
183#382C2314yes—
184#382C2312yes—
185#381—24no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380—24no—
191#380D412yes—
192#380D412yes—
193#379—24no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—24no—
209C4×C22⋊C4—24no—
210#376—21no—
211C43—21no—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#336C2314yes—
223#336C2314yes—
224C25—21no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C2×Q814yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—28no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—82no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16#4114yes—
248C2×M16#4114yes—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256#125—84no—
257#125—84no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4C2×Q818yes—
263C4×D4C2×Q818yes—
264#117—24no—
265#117—24no—
266#117—22no—
267#117—22no—
268#117—24no—
269#117—22no—
270#117—24no—
271#117—22no—
272#117—84no—
273#117—84no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×Q814yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4C2×D414yes—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×Q814yes—
331C2×C42#4114yes—
332C2×C42C2414yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—28no—
336#107—42no—
337#107—42no—
338#107—44no—
339#106—28no—
340#106—42no—
341#106—42no—
342#106—44no—
343C4⋊C8—28no—
344C4⋊C8—28no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8—24no—
348#102—24no—
349#102—24no—
350#102—24no—
351#102—24no—
352#102—44no—
353#102—44no—
354#97—24no—
355C4×C8—24no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—22no—
371C24—24no—
372C24—42no—
373C24—42no—
374C24—81no—
375C24—81no—
376C24—81no—
377C24—81no—
378C24C22×D411yescharacteristic
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4C22×Q814yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—816no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4C22⋊Q814yes—
418C22×C4C22⋊Q814yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q812yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C22×C4C2511yescharacteristic
433M16—28no—
434M16—28no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—48no—
446C2×C8C2×#4114yes—
447C2×C8C2×#4114yes—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4C22×Q814yes—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—42no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C42—24no—
495C42—24no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—44no—
501C42—42no—
502C42—42no—
503C42—84no—
504C42—84no—
505C42C22⋊Q814yes—
506C42C22⋊Q814yes—
507C42C22×Q812yes—
508C42C22×Q812yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23C2×C22⋊Q812yes—
530C23#54712yes—
531C23#55012yes—
532C23C23×D411yescharacteristic
533D4—24no—
534D4—24no—
535D4—48no—
536D4—42no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—416no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4—164no—
580C2×C4C2×C22⋊Q812yes—
581C2×C4C2×C22⋊Q812yes—
582C2×C4C2×C22⋊Q812yes—
583C2×C4C2×C22⋊Q812yes—
584C2×C4C2×C22⋊Q812yes—
585C2×C4#53314yes—
586C2×C4#54414yes—
587C2×C4#54712yes—
588C2×C4#55012yes—
589C2×C4#55214yes—
590C2×C4#55314yes—
591C2×C4#57612yes—
592C2×C4#57712yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C8—24no—
600C8—24no—
601C8—42no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—816no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#244114yes—
618C22C22×C22⋊Q811yescharacteristic
619C22C2×#54711yescharacteristic
620C22C2×#55011yescharacteristic
621C22C2×#57611yescharacteristic
622C22C2×#57711yescharacteristic
623C22#358511yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—82no—
635C4—84no—
636C4—81no—
637C4—82no—
638C4#244114yes—
639C4#308114yes—
640C4C2×#53311yescharacteristic
641C4C2×#54411yescharacteristic
642C4C2×#55211yescharacteristic
643C4C2×#55311yescharacteristic
644C2—24no—
645C2—84no—
646C2C2×#244111yescharacteristic
647C2C2×#308111yescharacteristic
648C2#6049311yescharacteristic
649C2#6127811yescharacteristic
650C2#6127911yescharacteristic
651C2#6141911yescharacteristic
652C2#6142011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #26103C2semidirect
4. #26102C2semidirect
5. #26091C2semidirect
6. #26090C2semidirect
7. #26087C2semidirect
8. #26086C2semidirect
11. #21175C2semidirect
12. #21169C2semidirect
13. #21145C2semidirect
14. #21036C2semidirect
15. #20800C2semidirect
31. #2571C22semidirect
32. #2571C22semidirect
33. #2552C22semidirect
34. #2552C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1948C22semidirect
68. #1947C22semidirect
69. #1947C22semidirect
70. #1947C22semidirect
71. #1940C22semidirect
72. #1939C22semidirect
73. #1939C22semidirect
74. #1921C22semidirect
75. #1920C22semidirect
76. #1919C22semidirect
77. #1918C22semidirect
78. #1847C22semidirect
79. #1845C22semidirect
80. #1844C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#341862
#261032
#261022
#260912
#260902
#260872
#260862
#237352
#236142
#211752
#211692
#211452
#210362
#208002
C2×#18022
C4×#4182