Finite Group Databasebeta

← #6570675

Subgroups — #6570675

Order 512.

1555 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36405C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29937C211yesmaximal, characteristic, complemented
4#29915C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27855C212yesmaximal, complemented
7#27680C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#26283C212yesmaximal, complemented
10#26205C214yesmaximal, complemented
11#24997C212yesmaximal, complemented
12#24938C214yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14#23744C212yesmaximal, complemented
15#22708C212yesmaximal, complemented
16#22490C212yesmaximal, complemented
17#22469C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×C8×D4C2212yes—
21#2857C2211yescharacteristic
22#2855C2212yes—
23#2582C2211yescharacteristic
24#2546C2211yescharacteristic
25#2485C2212yes—
26#2478—22no—
27#2468—22no—
28#2447—21no—
29#2432—21no—
30#2415C2212yes—
31#2365—21no—
32#2362—21no—
33#2355—21no—
34#2355C2212yes—
35#2354—21no—
36#2354C2211yescharacteristic
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40#2314—22no—
41#2311—21no—
42C4×C22⋊Q8C2211yescharacteristic
43#2296C2212yes—
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#2084C2212yes—
60#2083C2211yescharacteristic
61#2077C2214yes—
62#2076C2212yes—
63#2074C2214yes—
64#2072C2211yescharacteristic
65#2036C2212yes—
66#1990C2212yes—
67#1988C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1927C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1919C2214yes—
75#1892C2214yes—
76#1873C2214yes—
77#1872C2212yes—
78#1869C2214yes—
79#1854C2214yes—
80#1854C2214yes—
81#1853C2212yes—
82#1834C2212yes—
83#1825C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1807C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#103C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92#1735C2214yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×C8C2314yes—
97#556—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—22no—
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105C4×#41—28no—
106C2×C4×Q8—21no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4D412yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—22no—
116C2×#117D412yes—
117C2×#117D411yescharacteristic
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2314yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C8×D4—28no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2312yes—
139#398—22no—
140#398—22no—
141#397—22no—
142#397—22no—
143#394—22no—
144#394—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391C2312yes—
150#391C2311yescharacteristic
151#390—22no—
152#390—21no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389C2312yes—
157#389C2311yescharacteristic
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—24no—
162#388—24no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—24no—
167#387—24no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—21no—
172#386—24no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—21no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#382—21no—
185#382—22no—
186#382C2314yes—
187#381—22no—
188#380—22no—
189#380—22no—
190#380—21no—
191#380—21no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195#380C2311yescharacteristic
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
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⋊C4—22no—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—24no—
213#340C2314yes—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
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×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C22×C8—28no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—44no—
254#126—48no—
255#124—44no—
256C22⋊Q8—28no—
257C22⋊Q8—22no—
258C22⋊Q8—41no—
259C22⋊Q8—44no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—44no—
264C4×D4—24no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4#4118yes—
268#117—24no—
269#117—28no—
270#117—28no—
271#117—24no—
272#117—42no—
273#117—44no—
274#117—44no—
275#117—44no—
276#117—84no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
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—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—28no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—42no—
343#106—44no—
344#106—44no—
345C4⋊C8—24no—
346#103—42no—
347#103—44no—
348#103—42no—
349C22⋊C8—28no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C4×C8—24no—
353#95—24no—
354#95—24no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372C24C22×D412yes—
373C24C22×D411yescharacteristic
374#41—416no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4#12314yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4#13514yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4118yes—
431C22×C4C2511yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—88no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4#12314yes—
478C4⋊C42−1+412yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4#12318yes—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—44no—
499C42—44no—
500C42—42no—
501C42—44no—
502C42—42no—
503C42—84no—
504C42—84no—
505C42C2×#4114yes—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23C2×#12314yes—
519C23C2×#13512yes—
520C23C23×D411yescharacteristic
521Q8—44no—
522Q8—42no—
523Q8—42no—
524Q8—84no—
525D4—28no—
526D4—48no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—164no—
580C2×C4C2×#12314yes—
581C2×C4C2×#12312yes—
582C2×C4C2×#12312yes—
583C2×C4C2×#13512yes—
584C2×C4C2×#13512yes—
585C2×C4#57512yes—
586C2×C4#57712yes—
587C2×C4C22×#4112yes—
588C2×C4C2×2−1+411yescharacteristic
589C2×C4#58512yes—
590C2×C4#58511yescharacteristic
591C8—42no—
592C8—44no—
593C8—42no—
594C22—22no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22C22×#12311yescharacteristic
609C22#348911yescharacteristic
610C22#350011yescharacteristic
611C22C22×#13511yescharacteristic
612C22C2×#57511yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358611yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—24no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—84no—
624C4—82no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4#309114yes—
632C4#351512yes—
633C4#352112yes—
634C2—42no—
635C2—44no—
636C2#3606711yescharacteristic
637C2C2×#309111yescharacteristic
638C2#6052211yescharacteristic
639C2#6210111yescharacteristic
640C2#6213411yescharacteristic
641C2#6253111yescharacteristic
642C2#6258111yescharacteristic
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. #29937C2semidirect
4. #29915C2semidirect
5. #27960C2semidirect
6. #27855C2semidirect
7. #27680C2semidirect
8. #27615C2semidirect
9. #26283C2semidirect
10. #26205C2semidirect
11. #24997C2semidirect
12. #24938C2semidirect
13. #23757C2semidirect
14. #23744C2semidirect
15. #22708C2semidirect
16. #22490C2semidirect

Maximal subgroups

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

subgroupindex
#364052
#341882
#299372
#299152
#279602
#278552
#276802
#276152
#262832
#262052
#249972
#249382
#237572
#237442
#227082
#224902
#224692