Finite Group Databasebeta

← #6497487

Subgroups — #6497487

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic, complemented
2C2×#2970C211yesmaximal, characteristic
3#35297C211yesmaximal, characteristic
4#35138C211yesmaximal, characteristic
5#33217C212yesmaximal, complemented
6#33217C212yesmaximal, complemented
7#29884C212yesmaximal, complemented
8#29883C212yesmaximal, complemented
9#28708C212yesmaximal, complemented
10#28707C212yesmaximal, complemented
11#28681C212yesmaximal, complemented
12#28680C212yesmaximal, complemented
13#27669C214yesmaximal, complemented
14#27668C214yesmaximal, complemented
15#21842C211yesmaximal, characteristic, complemented
16#21838C211yesmaximal, characteristic, complemented
17#20819C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3166—28no—
22C2×#501C2211yescharacteristic
23C2×#501C2211yescharacteristic
24C2×#501C2211yescharacteristic
25C2×#501C2211yescharacteristic
26C2×#499C2212yes—
27C2×#499C2212yes—
28#2970—28no—
29C2×#446C2212yes—
30C2×#435C2214yes—
31C2×C8×D4C2212yes—
32C2×#432C2211yescharacteristic
33C2×#432C2212yes—
34C2×#428C2212yes—
35C2×#428C2212yes—
36C2×C4×M16C2211yescharacteristic
37C22×#97C2212yes—
38#2857C2212yes—
39#2857C2212yes—
40#2851C2212yes—
41#2851C2212yes—
42#2634—21no—
43#2633—21no—
44#2625—21no—
45#2616—21no—
46#2611—22no—
47#2584C2211yescharacteristic
48#2584C2212yes—
49#2584C2211yescharacteristic
50#2581—21no—
51#2562C2211yescharacteristic
52#2562C2212yes—
53#2562C2211yescharacteristic
54#2536—21no—
55#2487C2212yes—
56#2486C2212yes—
57#2485C2214yes—
58#2484—21no—
59#2415—22no—
60#2406—21no—
61#2323—22no—
62#2322—22no—
63#2285C2211yescharacteristic
64#2283C2211yescharacteristic
65#2088C2214yes—
66#2088C2214yes—
67#2087C2214yes—
68#2086C2214yes—
69#2077C2214yes—
70#2077C2214yes—
71#2076C2214yes—
72#2075C2214yes—
73#1993C2212yes—
74#1993C2212yes—
75#1992C2212yes—
76#1992C2212yes—
77#1989C2212yes—
78#1989C2212yes—
79#1989C2212yes—
80#1989C2212yes—
81#1944C2214yes—
82#1944C2214yes—
83#1943C2214yes—
84#1943C2214yes—
85#1847C2212yes—
86#1836C2212yes—
87#1825C2214yes—
88#1824C2212yes—
89#1823C2212yes—
90#1809C2214yes—
91#1808C2214yes—
92#1808C2214yes—
93#1807C2214yes—
94#1807C2214yes—
95C4×#107C2212yes—
96C4×#106C2212yes—
97C22×#41D411yescharacteristic
98C22×M16C2312yes—
99C23×C8C2314yes—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—24no—
107C2×#125—24no—
108C2×#125C2311yescharacteristic
109C2×#125C2311yescharacteristic
110C2×#125C2312yes—
111C2×#125C2311yescharacteristic
112C2×#125C2311yescharacteristic
113C2×#123—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C4×#41—24no—
119C4×#41—24no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117D412yes—
130C2×#117D411yescharacteristic
131C2×#117C2312yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#501—24no—
139#501—24no—
140#501—24no—
141#501—24no—
142#499—28no—
143#499—28no—
144#446—28no—
145#435—216no—
146C8×D4—28no—
147#432—24no—
148#432—28no—
149#428—28no—
150#428—28no—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#106C2314yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2312yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2312yes—
165C4×M16D418yes—
166C2×#97—28no—
167C2×#97C2312yes—
168C2×#97C2314yes—
169C2×#97C2311yescharacteristic
170C2×#97C2311yescharacteristic
171C2×C4×C8C2312yes—
172#400—22no—
173#400—21no—
174#400—21no—
175#400—22no—
176#399—22no—
177#398—22no—
178#398—21no—
179#398—21no—
180#398—22no—
181#397—22no—
182#397—24no—
183#395—22no—
184#395—21no—
185#395—21no—
186#394—22no—
187#394—21no—
188#394—21no—
189#394—24no—
190#391—22no—
191#391—22no—
192#389—22no—
193#389—22no—
194#389—24no—
195#389—22no—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2314yes—
202#389C2314yes—
203#388—22no—
204#388—22no—
205#388—24no—
206#388—24no—
207#387—24no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#386—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#384C2312yes—
219#383—21no—
220#382—21no—
221#382C2312yes—
222#381—21no—
223#381—22no—
224#380—22no—
225#380—22no—
226#380—21no—
227#380C2312yes—
228#380C2312yes—
229#380C2314yes—
230#380C2314yes—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234#380C2312yes—
235C4×C4⋊C4—21no—
236C4×C4⋊C4C2311yescharacteristic
237C4×C4⋊C4C2311yescharacteristic
238C4×C4⋊C4C2312yes—
239C4×C22⋊C4—21no—
240#376—21no—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251C2×#41—24no—
252C2×#41—44no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8C2×D411yescharacteristic
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D412yes—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—28no—
267C2×M16C2×D418yes—
268C22×C8—28no—
269C22×C8—28no—
270C22×C8—28no—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278#125—22no—
279#125—22no—
280#125—22no—
281#125—22no—
282#125—24no—
283#125—22no—
284#125—22no—
285#125—22no—
286#125—22no—
287#125—24no—
288#125—44no—
289#125—44no—
290#123—48no—
291C22⋊Q8—48no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294C22⋊Q8—84no—
295C4×Q8—24no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×D4—28no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—22no—
306#117—24no—
307#117—28no—
308#117—24no—
309#117—24no—
310#117—82no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2414yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D414yes—
375C2×C42#4114yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—24no—
381#107—24no—
382#107—24no—
383#107—28no—
384#106—24no—
385#106—24no—
386#106—24no—
387#106—24no—
388#106—28no—
389C4⋊C8—28no—
390C4⋊C8—24no—
391C22⋊C8—24no—
392C22⋊C8—28no—
393#97—28no—
394#97—24no—
395#97C2×D414yes—
396#97C2×D414yes—
397C4×C8C2×D418yes—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—21no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417C24C22×D411yescharacteristic
418C24C22×D412yes—
419#41—416no—
420C2×Q8—22no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—42no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473M16—216no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—416no—
479C2×C8C22×D414yes—
480C2×C8C22×D414yes—
481C2×C82−1+418yes—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—84no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—44no—
541C42—81no—
542C42—81no—
543C42—82no—
544C42#12114yes—
545C42#12114yes—
546C42C22×D412yes—
547C42C22×D412yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C23—22no—
551C23—21no—
552C23—22no—
553C23—21no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23C2×#12112yes—
561C23#57514yes—
562C23C23×D411yescharacteristic
563Q8—44no—
564Q8—84no—
565Q8—84no—
566D4—28no—
567D4—48no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
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—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×2−1+414yes—
626C8—24no—
627C8—44no—
628C8#53618yes—
629C22—22no—
630C22—24no—
631C22—22no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22C22×#12111yescharacteristic
642C22#349912yes—
643C22C2×#57512yes—
644C22#358511yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4C2×#53612yes—
664C4#350912yes—
665C2—42no—
666C2—44no—
667C2#3531611yescharacteristic
668C2#6044411yescharacteristic
669C2#6045611yescharacteristic
670C2#6250512yes—
671C2#6250612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3166C2semidirect
5. #33217C2semidirect
6. #33217C2semidirect
7. #29884C2semidirect
8. #29883C2semidirect
9. #28708C2semidirect
10. #28707C2semidirect
11. #28681C2semidirect
12. #28680C2semidirect
13. #27669C2semidirect
14. #27668C2semidirect
15. #21842C2semidirect
16. #21838C2semidirect
17. #20819C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
C2×#29702
#352972
#351382
#332172
#298842
#298832
#287082
#287072
#286812
#286802
#276692
#276682
#218422
#218382
#208192