Finite Group Databasebeta

← #6498463

Subgroups — #6498463

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3240C211yesmaximal, characteristic
2C2×#3218C211yesmaximal, characteristic
3#35320C211yesmaximal, characteristic, complemented
4#34227C211yesmaximal, characteristic
5#33041C212yesmaximal, complemented
6#33040C212yesmaximal, complemented
7#29837C212yesmaximal, complemented
8#29836C212yesmaximal, complemented
9#28660C212yesmaximal, complemented
10#28660C212yesmaximal, complemented
11#28620C212yesmaximal, complemented
12#28620C212yesmaximal, complemented
13#28132C214yesmaximal, complemented
14#28128C214yesmaximal
15#21842C211yesmaximal, characteristic, complemented
16#21811C211yesmaximal, characteristic, complemented
17#20811C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#529C2211yescharacteristic
21#3240—28no—
22#3218—28no—
23C2×#501C2211yescharacteristic
24C2×#501C2211yescharacteristic
25C2×#497C2212yes—
26C2×#491C2212yes—
27C2×#489C2211yescharacteristic
28C2×#489C2211yescharacteristic
29C2×#485C2212yes—
30C2×#482C2212yes—
31C2×#479C2212yes—
32C2×#478C2212yes—
33C2×#462C2212yes—
34C2×#461C2212yes—
35C2×#451C2212yes—
36C2×#448C2212yes—
37C2×#432C2211yescharacteristic
38C2×#432C2211yescharacteristic
39#2862C2212yes—
40#2862C2212yes—
41#2642C2212yes—
42#2642C2212yes—
43#2568—21no—
44#2568—21no—
45#2568—22no—
46#2503—21no—
47#2503—21no—
48#2503—22no—
49#2374—22no—
50#2373—22no—
51#2357—21no—
52#2353—21no—
53#2347—21no—
54#2343—21no—
55#2324C2212yes—
56#2323C2212yes—
57#2320C2211yescharacteristic
58#2319C2211yescharacteristic
59#2310C2214yes—
60C4×#125C2212yes—
61C4×#124C2212yes—
62#2283C2211yescharacteristic
63#2276C2211yescharacteristic
64#2275C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2068C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2066C2214yes—
70#2056C2214yes—
71#2056C2214yes—
72#2055C2214yes—
73#2055C2214yes—
74#1993C2212yes—
75#1993C2212yes—
76#1989C2212yes—
77#1989C2212yes—
78#1981C2212yes—
79#1981C2212yes—
80#1960C2212yes—
81#1960C2212yes—
82#1928C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1927C2214yes—
86#1878C2214yes—
87#1878C2214yes—
88#1877C2214yes—
89#1877C2214yes—
90#1843C2212yes—
91#1832C2212yes—
92#1825C2214yes—
93#1798C2214yes—
94#1787C2214yes—
95#1767C2212yes—
96#1767C2212yes—
97C23×Q8C2312yes—
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110#529—24no—
111#529D418yes—
112C2×#126—21no—
113C2×#125C2312yes—
114C2×#125C2311yescharacteristic
115C2×#124C2311yescharacteristic
116C2×#123—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2311yescharacteristic
120C2×#121C2311yescharacteristic
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117C2312yes—
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C42C2311yescharacteristic
134#501—24no—
135#501—24no—
136#497—28no—
137#491—28no—
138#489—24no—
139#489—24no—
140#485—28no—
141#482—28no—
142#479—28no—
143#478—28no—
144#462—28no—
145#461—28no—
146#451—28no—
147#448—28no—
148#432—24no—
149#432—24no—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#106C2312yes—
153C2×#106C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C2×#97C2311yescharacteristic
171C2×#97C2311yescharacteristic
172C2×#97C2312yes—
173#400—22no—
174#400—24no—
175#400—22no—
176#400—22no—
177#400—22no—
178#400—24no—
179#391—24no—
180#391—24no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391—22no—
185#391C2314yes—
186#391C2314yes—
187#391C2314yes—
188#391C2314yes—
189#391C2312yes—
190#390—22no—
191#390C2312yes—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#386—22no—
206#385—22no—
207#383—21no—
208#383—21no—
209#382—22no—
210#382—22no—
211#382—22no—
212#382—22no—
213#382—21no—
214#382C2312yes—
215#381—21no—
216#381—24no—
217#381—24no—
218#381—21no—
219#380—21no—
220#380C2312yes—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D412yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C22⋊C4—21no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4D411yescharacteristic
238C4×C22⋊C4D411yescharacteristic
239C4×C22⋊C4D412yes—
240#376—21no—
241#376—21no—
242#376C2311yescharacteristic
243#376C2311yescharacteristic
244C2×#95—21no—
245C43C2312yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—24no—
256C22×Q8—24no—
257C22×Q8—44no—
258C22×Q8—44no—
259C22×Q8C2414yes—
260C22×Q8C2411yescharacteristic
261C22×D4—21no—
262C22×D4C2411yescharacteristic
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×Q16—28no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×SD16—28no—
273C2×SD16—44no—
274C2×SD16—44no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288#126—28no—
289#125—24no—
290#125—22no—
291#125—24no—
292#125C2×D414yes—
293#124—22no—
294#124C2×D414yes—
295#123—28no—
296C22⋊Q8—22no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8C2×D414yes—
300#121—22no—
301#121C2×D414yes—
302C4×Q8—22no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8C2×D414yes—
307C4×D4—28no—
308C4×D4—22no—
309C4×D4—44no—
310C4×D4C2×D414yes—
311#117—28no—
312#117—24no—
313#117—24no—
314#117—44no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D414yes—
366C2×C42#4114yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—24no—
372#107—24no—
373#106—24no—
374#106—24no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377C4⋊C8—44no—
378C4⋊C8—44no—
379#103—24no—
380#103—24no—
381#103—28no—
382#103—28no—
383#103—44no—
384#103—44no—
385#103—44no—
386#103—44no—
387#102—24no—
388#102—24no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391#97—22no—
392#97—22no—
393#97—24no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95C2×D411yescharacteristic
414#95C2×D412yes—
415#95C2×D411yescharacteristic
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D412yes—
419C24—21no—
420C24C22×D411yescharacteristic
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—48no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—88no—
437C2×Q8—84no—
438C2×Q8—84no—
439C2×Q8C22×D412yes—
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—24no—
443C2×D4—81no—
444C2×D4—81no—
445C2×D4C22×D412yes—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4#12114yes—
466C22×C4#12114yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479Q16—88no—
480SD16—88no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4C22×D412yes—
514C4⋊C4C22×D412yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42—81no—
537C42—81no—
538C42—82no—
539C42—82no—
540C42#12114yes—
541C42#12114yes—
542C42C22×D412yes—
543C42C22×D412yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23C2×#12112yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—42no—
559Q8—44no—
560Q8—44no—
561Q8—44no—
562Q8—44no—
563Q8—84no—
564Q8—84no—
565Q8—88no—
566D4—24no—
567D4—42no—
568D4—84no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—164no—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4#58514yes—
624C8—42no—
625C8—42no—
626C8—44no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—82no—
636C22—82no—
637C22C22×#12111yescharacteristic
638C22#344912yes—
639C22#358611yescharacteristic
640C22#358612yes—
641C22#358611yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4#350012yes—
660C4#351012yes—
661C2—42no—
662C2—82no—
663C2#3532711yescharacteristic
664C2#6042711yescharacteristic
665C2#6042711yescharacteristic
666C2#6101812yes—
667C2#6102012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
3. #35320C2semidirect
5. #33041C2semidirect
6. #33040C2semidirect
7. #29837C2semidirect
8. #29836C2semidirect
9. #28660C2semidirect
10. #28660C2semidirect
11. #28620C2semidirect
12. #28620C2semidirect
13. #28132C2semidirect
15. #21842C2semidirect
16. #21811C2semidirect
17. #20811C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32402
C2×#32182
#353202
#342272
#330412
#330402
#298372
#298362
#286602
#286202
#281322
#281282
#218422
#218112
#208112