Finite Group Databasebeta

← #6461975

Subgroups — #6461975

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34253C211yesmaximal, characteristic
3#29591C214yesmaximal, complemented
4#29590C214yesmaximal, complemented
5#29579C212yesmaximal, complemented
6#29578C212yesmaximal, complemented
7#29575C212yesmaximal, complemented
8#29574C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic
11C2×#2084C211yesmaximal, characteristic, complemented
12#23630C211yesmaximal, characteristic
13#21157C212yesmaximal, complemented
14#21092C212yesmaximal
15#21039C212yesmaximal, complemented
16#20882C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#520C2211yescharacteristic
19C2×#427C2212yes—
20C22×#106C2212yes—
21C2×#417C2212yes—
22C2×#416C2212yes—
23C2×#405C2212yes—
24#2859C2212yes—
25#2859C2212yes—
26#2647C2214yes—
27#2571—21no—
28#2571—21no—
29#2552—21no—
30#2552—21no—
31#2442—22no—
32#2442—22no—
33#2353—22no—
34#2352—22no—
35#2347—21no—
36#2346—21no—
37#2345—21no—
38#2344—21no—
39#2320C2212yes—
40#2319C2212yes—
41#2317C2212yes—
42#2316C2212yes—
43#2316C2211yescharacteristic
44#2311C2211yescharacteristic
45#2297C2211yescharacteristic
46#2295C2212yes—
47#2293C2212yes—
48C2×#389C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2214yes—
54#2100C2212yes—
55#2099C2212yes—
56#2097C2212yes—
57#2096C2211yescharacteristic
58#2095C2211yescharacteristic
59#2094C2211yescharacteristic
60#2093C2211yescharacteristic
61#2092C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2084C2214yes—
65#2072C2212yes—
66#2072C2212yes—
67#2071C2212yes—
68#2071C2212yes—
69#1944C2214yes—
70#1944C2214yes—
71#1941C2214yes—
72#1941C2214yes—
73#1936C2214yes—
74#1935C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1864C2214yes—
84#1864C2214yes—
85#1746C2214yes—
86#1745C2212yes—
87#1744C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×#125—22no—
99C2×#125—22no—
100#520—24no—
101#520—24no—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C2×#117—21no—
106C2×#117—22no—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#427—28no—
120C2×#107C2312yes—
121C2×#107C2312yes—
122C2×#106—28no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127#417—28no—
128#416—28no—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135#405—28no—
136C2×#97C2314yes—
137C2×C4×C8C2314yes—
138#400—24no—
139#400—24no—
140#398—24no—
141#398—24no—
142#391C2312yes—
143#391C2311yescharacteristic
144#391C2311yescharacteristic
145#390C2312yes—
146#390C2311yescharacteristic
147#390C2311yescharacteristic
148#389—22no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#386—21no—
175#386—21no—
176#386—22no—
177#385—24no—
178#384—22no—
179#384—21no—
180#384—21no—
181#384C2312yes—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#383—22no—
186#382—22no—
187#382—22no—
188#382—22no—
189#382—22no—
190#382C2312yes—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—24no—
198#380—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216#376—22no—
217#376C2312yes—
218C2×#95—21no—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—22no—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41#4114yes—
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—24no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8#4118yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259#125—84no—
260#125—84no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×D4—216no—
264C4×D4—24no—
265C4×D4—24no—
266#117—28no—
267#117—22no—
268#117—22no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—24no—
273#117—84no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—24no—
343#106—24no—
344#106—24no—
345#106—28no—
346#106—28no—
347#103—28no—
348#103—24no—
349#103—24no—
350#102—28no—
351#102—24no—
352#102—24no—
353#97—28no—
354C4×C8—28no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D411yescharacteristic
379#41—216no—
380#41—48no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8C2×#4112yes—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4C2×#4112yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—81no—
419C22×C4—82no—
420C22×C4—81no—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4114yes—
433C22×C42−1+414yes—
434C22×C4C2511yescharacteristic
435M16—48no—
436C2×C8—216no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8C2×#4114yes—
445C2×C8C2×#4114yes—
446C4⋊C4—28no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4C2×#4114yes—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—82no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—82no—
492C42—28no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—82no—
507C42—82no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#52512yes—
520C23#53212yes—
521C23#53612yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—42no—
525D4—24no—
526D4—48no—
527D4—42no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—88no—
579C2×C4—164no—
580C2×C4—162no—
581C2×C4#52512yes—
582C2×C4#53212yes—
583C2×C4#53612yes—
584C2×C4#55218yes—
585C2×C4#55218yes—
586C2×C4#57712yes—
587C2×C4#57712yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C22×#4112yes—
591C2×C4C2×2−1+412yes—
592C8—24no—
593C8—24no—
594C8—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22#261114yes—
609C22C2×#52511yescharacteristic
610C22C2×#53211yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#57711yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358511yescharacteristic
615C22#358611yescharacteristic
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—82no—
628C4—81no—
629C4—82no—
630C4—84no—
631C4—82no—
632C4#261114yes—
633C4C2×#55212yes—
634C4C2×#55212yes—
635C2—24no—
636C2—82no—
637C2C2×#261111yescharacteristic
638C2#6059111yescharacteristic
639C2#6059311yescharacteristic
640C2#6071611yescharacteristic
641C2#6072111yescharacteristic
642C2#6087211yescharacteristic
643C2#6087311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. #35286C2semidirect
3. #29591C2semidirect
4. #29590C2semidirect
5. #29579C2semidirect
6. #29578C2semidirect
7. #29575C2semidirect
8. #29574C2semidirect
9. #23747C2semidirect
11. C2×#2084C2semidirect
13. #21157C2semidirect
15. #21039C2semidirect
17. #20368C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#342532
#295912
#295902
#295792
#295782
#295752
#295742
#237472
#237302
C2×#20842
#236302
#211572
#210922
#210392
#208822
#203682