Finite Group Databasebeta

← #6802846

Subgroups — #6802846

Order 512.

1439 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33107C211yesmaximal, characteristic
4#32389C214yesmaximal, complemented
5#31585C214yesmaximal, complemented
6#29760C212yesmaximal, complemented
7#28828C212yesmaximal, complemented
8#27912C212yesmaximal, complemented
9#27909C211yesmaximal, characteristic
10#27750C212yesmaximal, complemented
11#27651C212yesmaximal, complemented
12#27647C211yesmaximal, characteristic, complemented
13#26533C212yesmaximal, complemented
14#26093C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22075C212yesmaximal, complemented
17C4×#484C211yesmaximal, characteristic, complemented
18C2×#484C2211yescharacteristic
19C2×#481C2212yes—
20C2×#483C2212yes—
21C2×#480C2212yes—
22C2×#480C2212yes—
23#2856C2211yescharacteristic
24#2850C2212yes—
25#2597—22no—
26#2571—22no—
27#2571—21no—
28#2571—21no—
29#2571C2214yes—
30#2571C2212yes—
31#2571C2212yes—
32#2571C2211yescharacteristic
33#2548C2211yescharacteristic
34#2513—21no—
35#2513C2212yes—
36#2504—21no—
37#2447C2212yes—
38#2444—22no—
39C2×#400—22no—
40#2366C2211yescharacteristic
41#2361—21no—
42#2357—21no—
43#2355C2211yescharacteristic
44C4×C22⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×#382—21no—
50C2×#382C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2093C2214yes—
55#2092C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2084C2212yes—
59#2073C2214yes—
60#2072C2212yes—
61#2068C2214yes—
62#2065C2214yes—
63#2042C2214yes—
64#1982C2212yes—
65#1958C2214yes—
66#1957C2212yes—
67#1949C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1942C2214yes—
71#1939C2212yes—
72#1934C2214yes—
73#1928C2212yes—
74#1926C2214yes—
75#1921C2212yes—
76#1918C2214yes—
77#1916C2214yes—
78#1874C2212yes—
79#1872C2211yescharacteristic
80#1862C2214yes—
81#1852C2211yescharacteristic
82#1846C2214yes—
83#1845C2212yes—
84#1824C2211yescharacteristic
85#1823C2212yes—
86#1804C2214yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C2×#127C2311yescharacteristic
96C2×#124C2312yes—
97C2×C22⋊Q8C2311yescharacteristic
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116#484—24no—
117#483—44no—
118#481—44no—
119#480—28no—
120#480—44no—
121C2×#107C2314yes—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2311yescharacteristic
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—24no—
142#400—24no—
143#400—24no—
144#400—24no—
145#400—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#397—22no—
150#394—22no—
151#391C2312yes—
152#391C2311yescharacteristic
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389—24no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—21no—
177#387—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#385—21no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#382—22no—
190#382—22no—
191#382—24no—
192#382—21no—
193#382—21no—
194#382—22no—
195#382—22no—
196#382C2314yes—
197#382C2314yes—
198#382C2312yes—
199#381—22no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8C2411yescharacteristic
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250#127—24no—
251#124—44no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254#121—24no—
255#121—24no—
256#121—44no—
257#121—82no—
258C4×Q8—24no—
259C4×D4—44no—
260C4×D4—44no—
261C4×D4—84no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—84no—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C42—24no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—28no—
329#107—24no—
330#107—42no—
331#107—44no—
332#106—44no—
333#106—44no—
334C4⋊C8—24no—
335C4⋊C8—44no—
336#103—24no—
337#103—42no—
338#103—44no—
339#102—28no—
340#102—44no—
341#102—44no—
342#102—44no—
343C22⋊C8—22no—
344C22⋊C8—24no—
345C22⋊C8#4114yes—
346C4×C8—24no—
347C4×C8—44no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—22no—
372C24C22×D411yescharacteristic
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—81no—
389C2×D4—81no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
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—24no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4#13514yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C4C2511yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8C2×#4114yes—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—42no—
499C42—44no—
500C42—42no—
501C42—82no—
502C422−1+414yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23#55012yes—
515C23C2×#13512yes—
516C23#57612yes—
517C23C23×D411yescharacteristic
518Q8—42no—
519Q8—42no—
520D4—44no—
521D4—44no—
522D4—88no—
523D4—84no—
524D4—162no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—88no—
579C2×C4—164no—
580C2×C4#53612yes—
581C2×C4#53612yes—
582C2×C4#55012yes—
583C2×C4C2×#13512yes—
584C2×C4C2×#13512yes—
585C2×C4#57512yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×2−1+412yes—
589C2×C4#58512yes—
590C8—24no—
591C8—42no—
592C8—44no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#339614yes—
607C22C2×#53611yescharacteristic
608C22C2×#55011yescharacteristic
609C22#348511yescharacteristic
610C22C22×#13511yescharacteristic
611C22C2×#57511yescharacteristic
612C22C2×#57611yescharacteristic
613C22#358611yescharacteristic
614C4—22no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—84no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4#323214yes—
631C4#351512yes—
632C4#355912yes—
633C2—24no—
634C2—162no—
635C2#3913411yescharacteristic
636C2C2×#323211yescharacteristic
637C2#6087111yescharacteristic
638C2C2×#339611yescharacteristic
639C2#6140311yescharacteristic
640C2#6170111yescharacteristic
641C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#2571C2semidirect
4. #32389C2semidirect
5. #31585C2semidirect
6. #29760C2semidirect
7. #28828C2semidirect
8. #27912C2semidirect
10. #27750C2semidirect
11. #27651C2semidirect
12. #27647C2semidirect
13. #26533C2semidirect
14. #26093C2semidirect
15. #22625C2semidirect
16. #22075C2semidirect
17. C4×#484C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25712
#331072
#323892
#315852
#297602
#288282
#279122
#279092
#277502
#276512
#276472
#265332
#260932
#226252
#220752
C4×#4842