Finite Group Databasebeta

← #6461982

Subgroups — #6461982

Order 512.

1582 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#35286C211yesmaximal, characteristic, complemented
2#34253C211yesmaximal, characteristic
3#29589C214yesmaximal, complemented
4#29584C214yesmaximal, complemented
5#29580C212yesmaximal, complemented
6#29579C212yesmaximal, complemented
7#29574C212yesmaximal, complemented
8#29573C212yesmaximal, complemented
9C2×#2100C211yesmaximal, characteristic, complemented
10#23734C211yesmaximal, characteristic
11#23657C211yesmaximal, characteristic, complemented
12#23627C211yesmaximal, characteristic
13#21160C212yesmaximal, complemented
14#21087C212yesmaximal
15#21036C212yesmaximal, complemented
16#20878C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#520C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
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#2100C2212yes—
54#2100C2212yes—
55#2100C2214yes—
56#2096C2212yes—
57#2095C2212yes—
58#2094C2212yes—
59#2093C2212yes—
60#2085C2214yes—
61#2084C2212yes—
62#2083C2212yes—
63#2074C2212yes—
64#2073C2212yes—
65#2072C2211yescharacteristic
66#2072C2211yescharacteristic
67#2071C2211yescharacteristic
68#2071C2211yescharacteristic
69#1944C2214yes—
70#1943C2214yes—
71#1942C2214yes—
72#1941C2214yes—
73#1936C2214yes—
74#1935C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1865C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1862C2214yes—
85#1746C2214yes—
86#1745C2212yes—
87#1744C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90C2×#340C2212yes—
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#428—28no—
120#428—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125#417—28no—
126#416—28no—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133#405—28no—
134C2×#97C2314yes—
135C2×C4×C8C2314yes—
136#400—24no—
137#400—24no—
138#398—24no—
139#398—24no—
140#391C2312yes—
141#391C2311yescharacteristic
142#391C2311yescharacteristic
143#390C2312yes—
144#390C2311yescharacteristic
145#390C2311yescharacteristic
146#389—22no—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#386—21no—
173#386—21no—
174#386—22no—
175#385—24no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#383—22no—
184#382—22no—
185#382—22no—
186#382—22no—
187#382—22no—
188#382C2312yes—
189#381—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—22no—
195#380—24no—
196#380—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214#376—22no—
215#376C2312yes—
216C2×#95—21no—
217C43C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
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#4118yes—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
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—28no—
342#107—28no—
343#106—28no—
344#106—28no—
345#103—24no—
346#103—24no—
347#103—28no—
348#102—24no—
349#102—24no—
350#102—28no—
351#97—28no—
352C4×C8—28no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D411yescharacteristic
377#41—216no—
378#41—48no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8C2×#4112yes—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C2×#4112yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C42−1+414yes—
432C22×C4C2511yescharacteristic
433M16—216no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—48no—
443C2×C8C2×#4118yes—
444C4⋊C4—28no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4C2×#4114yes—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—82no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C42—28no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—42no—
504C42—82no—
505C42—82no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23#52512yes—
518C23#53212yes—
519C23#53612yes—
520C23C23×D411yescharacteristic
521Q8—24no—
522Q8—42no—
523D4—24no—
524D4—48no—
525D4—42no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—84no—
576C2×C4—88no—
577C2×C4—164no—
578C2×C4—162no—
579C2×C4#52512yes—
580C2×C4#53212yes—
581C2×C4#53612yes—
582C2×C4#55218yes—
583C2×C4#55218yes—
584C2×C4#57712yes—
585C2×C4#57812yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C22×#4112yes—
589C2×C4C2×2−1+412yes—
590C8—28no—
591C8—42no—
592C8—42no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22#261114yes—
607C22C2×#52511yescharacteristic
608C22C2×#53211yescharacteristic
609C22C2×#53611yescharacteristic
610C22C2×#57711yescharacteristic
611C22C2×#57811yescharacteristic
612C22#358511yescharacteristic
613C22#358511yescharacteristic
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#261114yes—
631C4C2×#55212yes—
632C4C2×#55212yes—
633C2—24no—
634C2—82no—
635C2C2×#261111yescharacteristic
636C2#6059711yescharacteristic
637C2#6060011yescharacteristic
638C2#6072311yescharacteristic
639C2#6072511yescharacteristic
640C2#6087311yescharacteristic
641C2#6087411yescharacteristic
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. #29589C2semidirect
4. #29584C2semidirect
5. #29580C2semidirect
6. #29579C2semidirect
7. #29574C2semidirect
8. #29573C2semidirect
9. C2×#2100C2semidirect
11. #23657C2semidirect
13. #21160C2semidirect
15. #21036C2semidirect
17. #20368C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#342532
#295892
#295842
#295802
#295792
#295742
#295732
C2×#21002
#237342
#236572
#236272
#211602
#210872
#210362
#208782
#203682