Finite Group Databasebeta

← #6570693

Subgroups — #6570693

Order 512.

1571 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36275C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29939C211yesmaximal, characteristic, complemented
4#29915C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27861C212yesmaximal, complemented
7#27680C211yesmaximal, characteristic, complemented
8#27610C212yesmaximal, complemented
9#26277C212yesmaximal, complemented
10#26221C214yesmaximal, complemented
11#24995C212yesmaximal, complemented
12#24956C214yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14#23747C212yesmaximal, complemented
15#22713C212yesmaximal, complemented
16#22502C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#436C2212yes—
21#2857C2211yescharacteristic
22#2855C2212yes—
23#2582C2211yescharacteristic
24#2567C2211yescharacteristic
25#2487C2212yes—
26#2453—21no—
27#2449—22no—
28#2441—21no—
29#2438—22no—
30#2417C2212yes—
31#2355—21no—
32#2355C2212yes—
33#2354—21no—
34#2354C2212yes—
35#2352—22no—
36#2352C2214yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2211yescharacteristic
39#2339—21no—
40#2314—22no—
41#2311—21no—
42C4×C22⋊Q8C2211yescharacteristic
43C2×#389C2212yes—
44#2285C2211yescharacteristic
45#2285C2212yes—
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48#2274—21no—
49#2101C2214yes—
50#2101C2214yes—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2099C2212yes—
54#2099C2212yes—
55#2099C2212yes—
56#2096C2212yes—
57#2095C2211yescharacteristic
58#2085C2214yes—
59#2083C2211yescharacteristic
60#2083C2212yes—
61#2076C2214yes—
62#2076C2212yes—
63#2074C2214yes—
64#2072C2211yescharacteristic
65#2036C2212yes—
66#1990C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1928C2214yes—
72#1919C2214yes—
73#1918C2214yes—
74#1918C2214yes—
75#1894C2214yes—
76#1873C2214yes—
77#1872C2212yes—
78#1870C2214yes—
79#1854C2214yes—
80#1853C2212yes—
81#1852C2214yes—
82#1834C2212yes—
83#1823C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1809C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#103C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2314yes—
97#554—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104C4×#41—28no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D412yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—22no—
116C2×#117D412yes—
117C2×#117D411yescharacteristic
118C2×#117C2314yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#436—28no—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#399—22no—
142#398—22no—
143#398—22no—
144#396—22no—
145#395—22no—
146#391—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391C2312yes—
151#391C2311yescharacteristic
152#390—22no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2311yescharacteristic
163#388—24no—
164#388—24no—
165#387—24no—
166#387—24no—
167#387—22no—
168#387—22no—
169#387—21no—
170#386—24no—
171#386—22no—
172#386—22no—
173#385—22no—
174#385—21no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#382—21no—
186#382—22no—
187#382C2314yes—
188#381—22no—
189#380—22no—
190#380—22no—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—24no—
212C2×#95—22no—
213#340C2314yes—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2314yes—
231C2×#41—24no—
232C2×#41—24no—
233C2×#41—42no—
234C22×Q8—21no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×D4C2×D412yes—
238C22×D4C2×D411yescharacteristic
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C2×M16—28no—
249C2×M16—44no—
250C2×M16—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255#125—44no—
256C22⋊Q8—28no—
257C22⋊Q8—22no—
258C22⋊Q8—41no—
259C22⋊Q8—48no—
260C22⋊Q8—82no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×D4—24no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4#4118yes—
270#117—24no—
271#117—28no—
272#117—44no—
273#117—44no—
274#117—44no—
275#117—84no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—84no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—24no—
343#107—44no—
344#107—42no—
345#106—44no—
346C4⋊C8—24no—
347#103—42no—
348#103—44no—
349#103—42no—
350C22⋊C8—28no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353C4×C8—24no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—21no—
361#95—24no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—24no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373C24C22×D412yes—
374C24C22×D411yescharacteristic
375#41—416no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—42no—
391C2×D4#12314yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4#13714yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4118yes—
438C22×C4C2511yescharacteristic
439M16—88no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—81no—
486C4⋊C4#12314yes—
487C4⋊C42−1+412yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—81no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C22⋊C4#12318yes—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—24no—
505C42—42no—
506C42—44no—
507C42—42no—
508C42—84no—
509C42—84no—
510C42C2×#4114yes—
511C23—21no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23C2×#12314yes—
524C23C2×#13712yes—
525C23C23×D411yescharacteristic
526Q8—44no—
527Q8—42no—
528Q8—42no—
529Q8—162no—
530D4—28no—
531D4—48no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—88no—
585C2×C4—164no—
586C2×C4—164no—
587C2×C4C2×#12314yes—
588C2×C4C2×#12312yes—
589C2×C4C2×#12312yes—
590C2×C4C2×#13712yes—
591C2×C4C2×#13712yes—
592C2×C4#57512yes—
593C2×C4#57612yes—
594C2×C4C22×#4112yes—
595C2×C4C2×2−1+411yescharacteristic
596C2×C4#58512yes—
597C2×C4#58511yescharacteristic
598C8—42no—
599C8—44no—
600C8—42no—
601C22—22no—
602C22—22no—
603C22—24no—
604C22—24no—
605C22—22no—
606C22—24no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—41no—
614C22—41no—
615C22C22×#12311yescharacteristic
616C22#348911yescharacteristic
617C22#350011yescharacteristic
618C22C22×#13711yescharacteristic
619C22C2×#57511yescharacteristic
620C22C2×#57611yescharacteristic
621C22#358611yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—84no—
631C4—81no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4#309514yes—
639C4#351512yes—
640C4#352112yes—
641C2—42no—
642C2—44no—
643C2#3606711yescharacteristic
644C2C2×#309511yescharacteristic
645C2#6052111yescharacteristic
646C2#6211311yescharacteristic
647C2#6214311yescharacteristic
648C2#6253711yescharacteristic
649C2#6259111yescharacteristic
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
3. #29939C2semidirect
4. #29915C2semidirect
5. #27960C2semidirect
6. #27861C2semidirect
7. #27680C2semidirect
8. #27610C2semidirect
9. #26277C2semidirect
10. #26221C2semidirect
11. #24995C2semidirect
12. #24956C2semidirect
13. #23757C2semidirect
14. #23747C2semidirect
15. #22713C2semidirect
16. #22502C2semidirect

Maximal subgroups

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

subgroupindex
#362752
#341882
#299392
#299152
#279602
#278612
#276802
#276102
#262772
#262212
#249952
#249562
#237572
#237472
#227132
#225022
#224732