Finite Group Databasebeta

← #6855921

Subgroups — #6855921

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4#32255C214yesmaximal, complemented
5#30834C214yesmaximal, complemented
6#29756C212yesmaximal
7#28800C212yesmaximal, complemented
8#27972C211yesmaximal, characteristic
9#27777C212yesmaximal, complemented
10#27681C211yesmaximal, characteristic
11#26465C212yesmaximal, complemented
12#26027C212yesmaximal, complemented
13#25912C212yesmaximal
14#24307C212yesmaximal
15#22629C211yesmaximal, characteristic
16#22084C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C22×#125C2212yes—
19C2×#491C2214yes—
20C2×#483C2211yescharacteristic
21C2×#483C2212yes—
22C2×#479C2214yes—
23#2857C2212yes—
24#2857C2211yescharacteristic
25#2585—22no—
26#2569—22no—
27#2552—22no—
28#2552—21no—
29#2552—21no—
30#2552C2211yescharacteristic
31#2552C2212yes—
32#2552C2214yes—
33#2552C2212yes—
34#2546C2211yescharacteristic
35#2511C2212yes—
36#2509—21no—
37#2446C2212yes—
38#2442—21no—
39C2×#398—22no—
40#2370—21no—
41#2365C2211yescharacteristic
42#2355C2211yescharacteristic
43#2345—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384—21no—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yes—
53#2095C2212yes—
54#2084C2214yes—
55#2072C2214yes—
56#2066C2214yes—
57#2066C2214yes—
58#2059C2214yes—
59#2043C2214yes—
60#2026C2214yes—
61#2026C2214yes—
62#2026C2214yes—
63#1992C2214yes—
64#1989C2214yes—
65#1981C2212yes—
66#1956C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1944C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1927C2212yes—
73#1927C2214yes—
74#1920C2214yes—
75#1920C2212yes—
76#1917C2214yes—
77#1872C2212yes—
78#1872C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1863C2214yes—
83#1853C2211yescharacteristic
84#1852C2212yes—
85#1834C2212yes—
86#1823C2211yescharacteristic
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×#125—21no—
98C2×#125—22no—
99C2×#125—22no—
100C2×#125—48no—
101C2×#125C2314yes—
102C2×#125C2314yes—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2312yes—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#491—48no—
121#483—24no—
122#483—28no—
123#479—48no—
124C2×#107C2314yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#400—22no—
144#400—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#398—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#391C2311yescharacteristic
156#391C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—24no—
163#389—22no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2314yes—
172#389C2314yes—
173#389C2311yescharacteristic
174#387—22no—
175#387—22no—
176#387—21no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—21no—
183#385—22no—
184#384—22no—
185#384—22no—
186#384—24no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#382—21no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380C2311yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2412yes—
234C22×Q8C2411yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2411yescharacteristic
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—44no—
248#125—28no—
249#125—28no—
250#125—44no—
251#125—44no—
252#125—44no—
253#125—48no—
254#125—48no—
255C22⋊Q8—22no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258C22⋊Q8—24no—
259C4×Q8—24no—
260C4×Q8—44no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—84no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4#4114yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—24no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—44no—
333#106—24no—
334#106—28no—
335#106—24no—
336C4⋊C8—24no—
337C4⋊C8—44no—
338#103—24no—
339#103—28no—
340#103—28no—
341#103—24no—
342#103—44no—
343#103—44no—
344#103—44no—
345C22⋊C8—24no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348#97—44no—
349C4×C8—24no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—21no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24C22×D411yescharacteristic
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4#13714yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424C22×C42−1+412yes—
425C22×C4C2511yescharacteristic
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4C2×#4114yes—
479C4⋊C4H(5,𝔽2)14yes—
480C22⋊C4—44no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—24no—
487C42—22no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—44no—
492C42—44no—
493C42—42no—
494C42—82no—
495C42—82no—
496C422−1+414yes—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23#54812yes—
502C23C2×#13712yes—
503C23#57712yes—
504C23C23×D411yescharacteristic
505Q8—44no—
506Q8—42no—
507Q8—44no—
508Q8—42no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—88no—
566C2×C4#53612yes—
567C2×C4#53612yes—
568C2×C4#54812yes—
569C2×C4C2×#13712yes—
570C2×C4C2×#13712yes—
571C2×C4#57512yes—
572C2×C4C22×#4111yescharacteristic
573C2×C4C2×H(5,𝔽2)12yes—
574C2×C4C2×2−1+412yes—
575C2×C4C2×2−1+411yescharacteristic
576C8—44no—
577C8—42no—
578C8—42no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22#337214yes—
586C22C2×#53611yescharacteristic
587C22C2×#54811yescharacteristic
588C22#348411yescharacteristic
589C22C22×#13711yescharacteristic
590C22C2×#57511yescharacteristic
591C22C2×#57711yescharacteristic
592C22#358611yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—82no—
605C4—82no—
606C4—81no—
607C4—81no—
608C4—84no—
609C4—81no—
610C4#323614yes—
611C4#351112yes—
612C4#355212yes—
613C2—24no—
614C2#3972011yescharacteristic
615C2C2×#323611yescharacteristic
616C2#6087211yescharacteristic
617C2C2×#337211yescharacteristic
618C2#6132011yescharacteristic
619C2#6158611yescharacteristic
620C2#6159611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32255C2semidirect
5. #30834C2semidirect
7. #28800C2semidirect
9. #27777C2semidirect
11. #26465C2semidirect
12. #26027C2semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#25522
#331082
#322552
#308342
#297562
#288002
#279722
#277772
#276812
#264652
#260272
#259122
#243072
#226292
#220842
C4×#4832