Finite Group Databasebeta

← #6565931

Subgroups — #6565931

Order 512.

1597 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34322C211yesmaximal, characteristic, complemented
3#30045C212yesmaximal
4#29980C211yesmaximal, characteristic, complemented
5#29561C212yesmaximal, complemented
6#29394C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28157C211yesmaximal, characteristic, complemented
9#27578C214yesmaximal, complemented
10#27086C214yesmaximal, complemented
11#24993C212yesmaximal, complemented
12#24185C212yesmaximal, complemented
13#23821C212yesmaximal, complemented
14#23676C211yesmaximal, characteristic
15#23100C212yesmaximal, complemented
16#22832C211yesmaximal, characteristic, complemented
17#22758C212yesmaximal, complemented
18C2×#556—21no—
19C2×#556C2211yescharacteristic
20C2×#461C2212yes—
21C2×C4×D8C2212yes—
22#2868C2211yescharacteristic
23#2858C2212yes—
24#2547—21no—
25#2545—22no—
26#2538—21no—
27#2538C2212yes—
28#2512—22no—
29#2512C2214yes—
30#2373—22no—
31#2367C2212yes—
32#2365C2211yescharacteristic
33#2360—21no—
34#2355C2211yescharacteristic
35#2354—21no—
36#2354C2212yes—
37#2351C2212yes—
38D4×C4⋊C4C2211yescharacteristic
39#2346—21no—
40#2334—22no—
41#2323—21no—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#124C2212yes—
46#2295C2212yes—
47#2293C2211yescharacteristic
48#2274—21no—
49C42×Q8C2211yescharacteristic
50#2101C2214yes—
51#2099C2212yes—
52#2093C2212yes—
53#2083C2211yescharacteristic
54#2073C2214yes—
55#2072C2212yes—
56#2071C2211yescharacteristic
57#2067C2212yes—
58#2057C2212yes—
59#2054C2212yes—
60#2042C2214yes—
61#2041C2211yescharacteristic
62#2024C2212yes—
63#2024C2212yes—
64#2023C2212yes—
65#2014C2212yes—
66#2012C2214yes—
67#2011C2211yescharacteristic
68#1984C2212yes—
69#1973C2214yes—
70#1956C2212yes—
71#1947C2212yes—
72#1936C2214yes—
73#1928C2214yes—
74#1925C2214yes—
75#1921C2214yes—
76#1921C2214yes—
77#1917C2214yes—
78#1880C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1879C2214yes—
82#1863C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1831C2211yescharacteristic
86#1803C2214yes—
87#1802C2212yes—
88#1802C2212yes—
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91#1763C2214yes—
92#1762C2214yes—
93C4×#102C2212yes—
94#1745C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—24no—
98#556—24no—
99#556—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—22no—
106C2×#126—24no—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117—22no—
126C2×#117—21no—
127C22×C4⋊C4—21no—
128#461—28no—
129C4×D8—28no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#397—24no—
146#396—24no—
147#396—24no—
148#391—21no—
149#391—22no—
150#391C2312yes—
151#391C2312yes—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#390C2311yescharacteristic
158#390C2311yescharacteristic
159#390C2314yes—
160#390C2314yes—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—21no—
174#388—22no—
175#388—21no—
176#388—22no—
177#388—24no—
178#388—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382C2312yes—
193#380—21no—
194#380—22no—
195#380—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216#376—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2412yes—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—22no—
243C2×SD16—28no—
244C2×SD16—44no—
245C2×SD16—44no—
246C2×D8—24no—
247C2×D8—42no—
248C2×D8—42no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8#4114yes—
257#127—42no—
258#126—216no—
259#126—416no—
260#124—24no—
261#124—24no—
262#124—48no—
263#124—84no—
264C22⋊Q8—28no—
265C22⋊Q8—48no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—42no—
272C4×Q8—48no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4#4114yes—
282#117—28no—
283#117—48no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—21no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—44no—
352C4⋊C8—24no—
353C4⋊C8—42no—
354#103—28no—
355#103—44no—
356#103—44no—
357#102—24no—
358#102—24no—
359#102—44no—
360#102—42no—
361#102—44no—
362#102—42no—
363#97—44no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—21no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8C2×#4114yes—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C2×#4112yes—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436SD16—88no—
437D8—48no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4112yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—81no—
492C22⋊C4—82no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—28no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—42no—
506C42—42no—
507C42—41no—
508C42—48no—
509C42—42no—
510C42—82no—
511C42—82no—
512C42—82no—
513C42AGL(1,ℤ/8)14yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×D411yescharacteristic
531Q8—28no—
532Q8—44no—
533Q8—44no—
534Q8—88no—
535D4—24no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—84no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—88no—
589C2×C4—164no—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4#57512yes—
596C2×C4#57812yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4#58512yes—
600C2×C4#58512yes—
601C8—42no—
602C8—42no—
603C8—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#52511yescharacteristic
619C22#348711yescharacteristic
620C22#350011yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57811yescharacteristic
624C22#358511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—81no—
640C4—84no—
641C4#311514yes—
642C4#349312yes—
643C4#352112yes—
644C2—82no—
645C2—162no—
646C2#3601911yescharacteristic
647C2C2×#311511yescharacteristic
648C2#6060011yescharacteristic
649C2#6189011yescharacteristic
650C2#6190511yescharacteristic
651C2#6253811yescharacteristic
652C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2011C22
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
1. #36594C2semidirect
2. #34322C2semidirect
4. #29980C2semidirect
5. #29561C2semidirect
6. #29394C2semidirect
7. #29141C2semidirect
8. #28157C2semidirect
9. #27578C2semidirect
10. #27086C2semidirect
11. #24993C2semidirect
12. #24185C2semidirect
13. #23821C2semidirect
15. #23100C2semidirect
16. #22832C2semidirect
17. #22758C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#343222
#300452
#299802
#295612
#293942
#291412
#281572
#275782
#270862
#249932
#241852
#238212
#236762
#231002
#228322
#227582