Finite Group Databasebeta

← #6565770

Subgroups — #6565770

Order 512.

1775 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3372C212yesmaximal
2#36593C211yesmaximal, characteristic, complemented
3#34317C211yesmaximal, characteristic
4#29798C211yesmaximal, characteristic
5#29798C212yesmaximal
6#28755C212yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28209C212yesmaximal, complemented
9#28206C211yesmaximal, characteristic, complemented
10#25290C214yesmaximal, complemented
11#25278C212yesmaximal, complemented
12#24646C214yesmaximal
13#24630C212yesmaximal
14#23683C212yesmaximal
15#23677C211yesmaximal, characteristic
16#22736C211yesmaximal, characteristic, complemented
17#22547C212yesmaximal, complemented
18C2×#557C2212yes—
19C2×#554C2211yescharacteristic
20#3372—216no—
21C2×#491C2214yes—
22C2×#487C2212yes—
23C2×#479C2214yes—
24C2×#477C2212yes—
25C2×#461C2212yes—
26C2×#461C2214yes—
27C2×#446C2212yes—
28C2×#440C2214yes—
29C2×C4×SD16C2212yes—
30#2868C2211yescharacteristic
31#2868C2212yes—
32#2552—22no—
33#2552—21no—
34#2552—22no—
35#2552—21no—
36#2546C2212yes—
37#2546C2214yes—
38#2370C2212yes—
39#2369C2211yescharacteristic
40#2362—22no—
41#2362—21no—
42#2357C2212yes—
43#2355C2211yescharacteristic
44#2354C2211yescharacteristic
45Q8×C4⋊C4—21no—
46Q8×C4⋊C4—21no—
47#2344—21no—
48#2332—22no—
49#2323—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×#125C2212yes—
54#2293C2211yescharacteristic
55#2293C2212yes—
56#2274—21no—
57C42×Q8C2211yescharacteristic
58#2084C2211yescharacteristic
59#2083C2212yes—
60#2071C2211yescharacteristic
61#2068C2214yes—
62#2066C2212yes—
63#2055C2214yes—
64#2055C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2043C2211yescharacteristic
70#2043C2214yes—
71#2043C2212yes—
72#2013C2214yes—
73#2013C2212yes—
74#2013C2211yescharacteristic
75#2013C2212yes—
76#1981C2212yes—
77#1981C2214yes—
78#1981C2214yes—
79#1981C2214yes—
80#1981C2214yes—
81#1960C2214yes—
82#1956C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1928C2214yes—
86#1928C2214yes—
87#1927C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1805C2214yes—
91#1803C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#103C2212yes—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#557—28no—
99#557—28no—
100#554—24no—
101#554D418yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125—22no—
110C2×#125—24no—
111C2×#125C2312yes—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117—22no—
126C22×C4⋊C4—21no—
127#491—216no—
128#487—28no—
129#479—216no—
130#477—28no—
131#461—216no—
132#461—28no—
133#446—28no—
134#440—216no—
135C4×SD16—28no—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#102C2312yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#398—24no—
158#398—24no—
159#398—24no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391C2312yes—
164#391C2314yes—
165#390C2312yes—
166#390C2311yescharacteristic
167#390C2311yescharacteristic
168#389—21no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389—21no—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2311yescharacteristic
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—21no—
187#388—21no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#384—21no—
196#384—22no—
197#384—22no—
198#384—21no—
199#384—22no—
200#384—21no—
201#384—22no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382C2312yes—
206#381—22no—
207#380—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—22no—
232C43C2312yes—
233C43C2311yescharacteristic
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2314yes—
241#336C2312yes—
242C22×Q8—21no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×Q8C2414yes—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—22no—
249C2×SD16—24no—
250C2×SD16—28no—
251C2×SD16—42no—
252C2×SD16—44no—
253C2×SD16—42no—
254C2×SD16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—28no—
260C4⋊Q8—22no—
261C4⋊Q8—28no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8C2×D414yes—
269#125—24no—
270#125—48no—
271#125—416no—
272#125C2×D418yes—
273C22⋊Q8—28no—
274C22⋊Q8—216no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—24no—
281C4×Q8—48no—
282C4×Q8—82no—
283C4×Q8C2×D414yes—
284C4×Q8#4114yes—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4C2×D414yes—
288C4×D4C2×D418yes—
289#117—44no—
290#117—44no—
291#117—84no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358#107—28no—
359#106—24no—
360C4⋊C8—24no—
361C4⋊C8—28no—
362C4⋊C8—42no—
363C4⋊C8—44no—
364#103—24no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—44no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—42no—
375#103—44no—
376#102—24no—
377#97—28no—
378C4×C8—24no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—81no—
414C2×Q8—81no—
415C2×Q8#12114yes—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C22×D412yes—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445SD16—48no—
446SD16—416no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4#12114yes—
492C4⋊C4#12118yes—
493C4⋊C4#13714yes—
494C4⋊C4C22×D412yes—
495C4⋊C4C22×D414yes—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—81no—
503C22⋊C4—82no—
504C42—24no—
505C42—22no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—22no—
512C42—42no—
513C42—41no—
514C42—48no—
515C42—44no—
516C42—44no—
517C42—42no—
518C42—82no—
519C42—82no—
520C42—82no—
521C42#13714yes—
522C42C22×D412yes—
523C42C22×D414yes—
524C42C2×#4114yes—
525C42C2×#4112yes—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—42no—
536Q8—44no—
537Q8—42no—
538Q8—44no—
539Q8—44no—
540Q8—42no—
541Q8—44no—
542Q8—42no—
543Q8—84no—
544D4—24no—
545D4—42no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—88no—
592C2×C4C2×#12114yes—
593C2×C4C2×#12112yes—
594C2×C4C2×#12114yes—
595C2×C4C2×#12112yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13714yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4#57712yes—
602C2×C4C23×D411yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4#58512yes—
605C2×C4#58512yes—
606C8—42no—
607C8—42no—
608C8—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C22×#12111yescharacteristic
617C22#348711yescharacteristic
618C22#350011yescharacteristic
619C22C22×#13711yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358611yescharacteristic
623C4—24no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—22no—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—82no—
635C4—81no—
636C4—84no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4#303914yes—
641C4#348712yes—
642C4#352412yes—
643C2—82no—
644C2#3601811yescharacteristic
645C2C2×#303911yescharacteristic
646C2#6042911yescharacteristic
647C2#6189311yescharacteristic
648C2#6190611yescharacteristic
649C2#6254011yescharacteristic
650C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36593C2semidirect
6. #28755C2semidirect
7. #28748C2semidirect
8. #28209C2semidirect
9. #28206C2semidirect
10. #25290C2semidirect
11. #25278C2semidirect
16. #22736C2semidirect
17. #22547C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33722
#365932
#343172
#297982
#287552
#287482
#282092
#282062
#252902
#252782
#246462
#246302
#236832
#236772
#227362
#225472