Finite Group Databasebeta

← #6447648

Subgroups — #6447648

Order 512.

3075 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3504C211yesmaximal, characteristic, complemented
2C2×#3137C212yesmaximal, complemented
3#35308C211yesmaximal, characteristic
4#25916C214yesmaximal, complemented
5#25914C214yesmaximal, complemented
6#25912C212yesmaximal, complemented
7#25909C212yesmaximal, complemented
8#25907C212yesmaximal, complemented
9#25904C212yesmaximal, complemented
10#23455C211yesmaximal, characteristic, complemented
11#23453C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic, complemented
13C2×#2026C211yesmaximal, characteristic, complemented
14#21326C212yesmaximal, complemented
15#20915C212yesmaximal
16#20913C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic, complemented
18#3504—216no—
19C2×#557—22no—
20C2×#554—24no—
21C2×#554—22no—
22C2×#544C2214yes—
23C2×#544C2212yes—
24C2×#537C2211yescharacteristic
25C2×#536C2211yescharacteristic
26C2×#536C2211yescharacteristic
27C2×#533C2212yes—
28C2×#533C2211yescharacteristic
29C22×#125C2212yes—
30C22×#125C2211yescharacteristic
31C2×#520C2212yes—
32C2×C4×#41C2211yescharacteristic
33#3137—216no—
34C2×#491C2214yes—
35C2×#489C2214yes—
36C2×#488C2214yes—
37C2×#487C2214yes—
38C2×#417C2212yes—
39C2×#419C2212yes—
40C2×#416C2212yes—
41C22×#103C2212yes—
42C2×#408C2212yes—
43C2×#405C2212yes—
44#2857C2212yes—
45#2850C2212yes—
46#2621C2212yes—
47#2620C2212yes—
48#2429—22no—
49#2427—22no—
50#2425—24no—
51#2317C2211yescharacteristic
52#2297C2211yescharacteristic
53C2×#391C2211yescharacteristic
54#2026C2211yescharacteristic
55#2026C2212yes—
56#2026C2212yes—
57#2026C2212yes—
58#2026C2212yes—
59#2026C2212yes—
60#2026C2211yescharacteristic
61#2026C2212yes—
62#2026C2214yes—
63#2025C2211yescharacteristic
64#2025C2212yes—
65#2025C2211yescharacteristic
66#2025C2212yes—
67#2024C2212yes—
68#2024C2214yes—
69#2023C2212yes—
70#1993C2214yes—
71#1992C2214yes—
72#1991C2214yes—
73#1990C2214yes—
74#1875C2214yes—
75#1874C2214yes—
76#1873C2214yes—
77#1873C2214yes—
78#1873C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1750C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1735C2212yes—
88C2×#340C2212yes—
89#1728C2211yescharacteristic
90C2×#336C2211yescharacteristic
91C22×#41C2311yescharacteristic
92C22×#41C2311yescharacteristic
93C23×Q8C2311yescharacteristic
94C22×M16C2312yes—
95C23×C8C2312yes—
96#557—232no—
97#554—232no—
98#554—264no—
99#544—216no—
100#544—216no—
101#544—28no—
102#544D4116yescomplemented
103#537—28no—
104#536—28no—
105#536—28no—
106#533—216no—
107#533—28no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2314yes—
114C2×#125—216no—
115C2×#125—28no—
116C2×#125—24no—
117C2×#125—22no—
118C2×#125—22no—
119C2×#125—24no—
120C2×#125C2311yescharacteristic
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×#125C2311yescharacteristic
124C2×#125C2314yes—
125C2×#124—21no—
126C2×#124C2312yes—
127C2×#123—22no—
128C2×#123—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—24no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8—24no—
134C2×C22⋊Q8—24no—
135C2×C22⋊Q8—24no—
136C2×C22⋊Q8C2312yes—
137C2×C22⋊Q8C2312yes—
138C2×C22⋊Q8C2312yes—
139C2×#121—22no—
140C2×#121—21no—
141C2×#121C2312yes—
142#520—28no—
143#520—28no—
144C4×#41—28no—
145C2×C4×Q8—22no—
146C2×C4×Q8—22no—
147C2×C4×Q8—24no—
148C2×C4×Q8D412yes—
149C2×C4×D4—24no—
150C2×C4×D4—22no—
151C2×C4×D4D414yes—
152C2×C4×D4D412yes—
153C2×#117C2311yescharacteristic
154C2×#117C2314yes—
155C2×#117C2312yes—
156C2×#117C2312yes—
157C2×#117C2311yescharacteristic
158C22×C4⋊C4—22no—
159C22×C4⋊C4—24no—
160C22×C4⋊C4C2311yescharacteristic
161C22×C4⋊C4C2314yes—
162C22×C4⋊C4C2312yes—
163C22×C42C2312yes—
164C22×C42C2311yescharacteristic
165#491—216no—
166#489—216no—
167#488—216no—
168#487—216no—
169#419—28no—
170#417—28no—
171#416—28no—
172C2×#103—28no—
173C2×#103C2312yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2312yes—
180C2×#102C2312yes—
181C2×#102C2314yes—
182C2×#102C2312yes—
183#408—28no—
184C2×C22⋊C8C2314yes—
185C2×C22⋊C8C2314yes—
186#405—28no—
187C2×#97C2314yes—
188C2×C4×C8C2314yes—
189#398—24no—
190#397—24no—
191#396—24no—
192#394—24no—
193#391—22no—
194#391C2312yes—
195#391C2311yescharacteristic
196#391C2312yes—
197#391C2311yescharacteristic
198#390C2311yescharacteristic
199#390C2311yescharacteristic
200#389—22no—
201#389—24no—
202#388—24no—
203#386—24no—
204#386—22no—
205#385—22no—
206#384C2312yes—
207#382C2312yes—
208#380—24no—
209C4×C22⋊C4—22no—
210C2×#95—22no—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#336C2311yescharacteristic
222#336C2311yescharacteristic
223#336C2312yes—
224#336C2312yes—
225#336C2312yes—
226C2×#41—24no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41#4118yes—
234C22×Q8—22no—
235C22×Q8—24no—
236C22×Q8—21no—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4—21no—
245C22×D4—22no—
246C22×D4C2×D412yes—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D414yes—
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C4⋊Q8—232no—
268C4⋊Q8—216no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273C4⋊Q8—44no—
274C4⋊Q8C2×D4116yes—
275#125—232no—
276#125—216no—
277#125—232no—
278#125—216no—
279#125—24no—
280#125—28no—
281#125—28no—
282#125—24no—
283#125—28no—
284#125C2×D4116yes—
285#124—28no—
286#124—44no—
287#123—48no—
288#123—48no—
289C22⋊Q8—232no—
290C22⋊Q8—216no—
291C22⋊Q8—232no—
292C22⋊Q8—232no—
293C22⋊Q8—232no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—44no—
298C22⋊Q8—48no—
299#121—28no—
300#121—44no—
301#121—48no—
302C4×Q8—216no—
303C4×Q8—232no—
304C4×Q8—216no—
305C4×Q8—28no—
306C4×D4—232no—
307C4×D4—216no—
308C4×D4—28no—
309C4×D4—216no—
310#117—24no—
311#117—28no—
312#117—28no—
313#117—28no—
314#117—24no—
315#117C2×D418yes—
316#117C2×D418yes—
317C2×C4⋊C4—216no—
318C2×C4⋊C4—232no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2×D418yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×D414yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—28no—
364C2×C42—21no—
365C2×C42C2×D412yes—
366C2×C42C2×D414yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370#103—28no—
371#103—28no—
372#103—24no—
373#103—24no—
374#103—28no—
375#103—28no—
376#103—28no—
377#102—24no—
378#102—24no—
379#102—28no—
380C22⋊C8—28no—
381C22⋊C8—28no—
382#97—28no—
383C4×C8—28no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—24no—
389C24—21no—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392#41—216no—
393#41—216no—
394#41—48no—
395#41—48no—
396C2×Q8—28no—
397C2×Q8—22no—
398C2×Q8—216no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8#12414yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—84no—
431C2×D4#12418yes—
432C2×D4#12414yes—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4—82no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D414yes—
470C22×C4C2×#4118yes—
471C22×C4C2×#4114yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475M16—48no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—48no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—84no—
509C4⋊C4#12018yes—
510C4⋊C4C22×D414yes—
511C4⋊C4C22×D418yes—
512C4⋊C4C22×D414yes—
513C22⋊C4—216no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4#12018yes—
525C42—216no—
526C42—28no—
527C42—24no—
528C42—28no—
529C42—22no—
530C42—22no—
531C42—42no—
532C42C22×D418yes—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×#12012yes—
545C23C2×#12414yes—
546C23C2×#12412yes—
547C23#53612yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—44no—
551Q8—44no—
552Q8—42no—
553Q8—42no—
554Q8—816no—
555Q8—88no—
556Q8—84no—
557D4—216no—
558D4—28no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4—164no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—88no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4—816no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—162no—
603C2×C4C2×#12018yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12414yes—
606C2×C4C2×#12412yes—
607C2×C4#53612yes—
608C2×C4#54818yes—
609C2×C4#54818yes—
610C2×C4C23×D412yes—
611C2×C4C22×#4112yes—
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C8—42no—
615C8—42no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—44no—
625C22—44no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22—82no—
631C22#243614yes—
632C22C22×#12011yescharacteristic
633C22C22×#12411yescharacteristic
634C22C2×#53611yescharacteristic
635C22#358412yes—
636C22#358612yes—
637C4—24no—
638C4—28no—
639C4—42no—
640C4—42no—
641C4—48no—
642C4—44no—
643C4—44no—
644C4—81no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4#243614yes—
650C4C2×#54812yes—
651C4C2×#54812yes—
652C2—24no—
653C2—44no—
654C2—161no—
655C2C2×#243611yescharacteristic
656C2#6037712yes—
657C2#6062411yescharacteristic
658C2#6062711yescharacteristic
659C2#6089112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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×#3504C2semidirect
2. C2×#3137C2semidirect
4. #25916C2semidirect
5. #25914C2semidirect
6. #25912C2semidirect
7. #25909C2semidirect
8. #25907C2semidirect
9. #25904C2semidirect
10. #23455C2semidirect
11. #23453C2semidirect
12. #23451C2semidirect
13. C2×#2026C2semidirect
14. #21326C2semidirect
16. #20913C2semidirect
17. #20378C2semidirect
102. #544D4semidirect

Maximal subgroups

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

subgroupindex
C2×#35042
C2×#31372
#353082
#259162
#259142
#259122
#259092
#259072
#259042
#234552
#234532
#234512
C2×#20262
#213262
#209152
#209132
#203782