Finite Group Databasebeta

← #6848166

Subgroups — #6848166

Order 512.

1643 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#32228C214yesmaximal, complemented
5#32132C214yesmaximal, complemented
6#29641C212yesmaximal
7#28791C212yesmaximal, complemented
8#27960C211yesmaximal, characteristic
9#27815C212yesmaximal, complemented
10#27647C211yesmaximal, characteristic
11#27613C212yesmaximal
12#26442C212yesmaximal, complemented
13#26022C212yesmaximal, complemented
14#24601C212yesmaximal
15#22599C211yesmaximal, characteristic
16#21833C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#483C2211yescharacteristic
21C2×#483C2212yes—
22C2×#431C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2567C2211yescharacteristic
26#2552C2211yescharacteristic
27#2551—21no—
28#2551—21no—
29#2551—22no—
30#2551C2212yes—
31#2551C2212yes—
32#2551C2214yes—
33#2520—22no—
34#2511C2212yes—
35#2508—21no—
36#2507C2212yes—
37#2442—21no—
38#2438—22no—
39C2×#395—22no—
40#2370—21no—
41#2369C2211yescharacteristic
42#2355C2211yescharacteristic
43#2344—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47#2285C2212yes—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2088C2214yes—
54#2084C2214yes—
55#2084C2212yes—
56#2072C2212yes—
57#2066C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#1990C2212yes—
63#1990C2214yes—
64#1990C2214yes—
65#1988C2214yes—
66#1988C2214yes—
67#1987C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1941C2214yes—
71#1939C2212yes—
72#1933C2214yes—
73#1927C2212yes—
74#1920C2212yes—
75#1920C2214yes—
76#1919C2214yes—
77#1919C2214yes—
78#1916C2214yes—
79#1872C2211yescharacteristic
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1834C2212yes—
83#1823C2212yes—
84#1823C2212yes—
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1823C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2314yes—
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×#117—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#483—24no—
125#483—44no—
126#431—44no—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#400—22no—
143#398—22no—
144#398—22no—
145#398—22no—
146#395—22no—
147#395—24no—
148#395—24no—
149#395—24no—
150#395—24no—
151#391—21no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2311yescharacteristic
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—24no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2311yescharacteristic
171#389C2314yes—
172#389C2312yes—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—21no—
178#387—22no—
179#387—22no—
180#385—22no—
181#385—21no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#382—21no—
190#382—22no—
191#381—22no—
192#380—21no—
193#380—22no—
194#380C2312yes—
195#380C2311yescharacteristic
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95D412yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—21no—
237C22×Q8—48no—
238C22×Q8—48no—
239C22×Q8—84no—
240C22×Q8C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2411yescharacteristic
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—48no—
259C4⋊Q8—416no—
260C4⋊Q8—48no—
261C4⋊Q8—48no—
262#125—44no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—82no—
266C4×Q8—24no—
267#117—84no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4SD1618yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—44no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#106—24no—
339#106—28no—
340#106—24no—
341#106—44no—
342C4⋊C8—24no—
343C4⋊C8—44no—
344#103—24no—
345#103—24no—
346#103—44no—
347C22⋊C8—22no—
348C22⋊C8—22no—
349C22⋊C8—24no—
350C4×C8—24no—
351C4×C8—44no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—84no—
386C2×Q8—84no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×Q8—832no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4C2×SD1614yes—
421C22×C4C2×SD1618yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4C2511yescharacteristic
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4C2×SD1618yes—
480C4⋊C4C2×SD1614yes—
481C4⋊C4C2×SD1614yes—
482C4⋊C4#13714yes—
483C4⋊C4C2×#4114yes—
484C4⋊C42−1+414yes—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—44no—
499C42—44no—
500C42—42no—
501C42—82no—
502C42—82no—
503C422−1+414yes—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23#54612yes—
509C23C22×SD1612yes—
510C23#57612yes—
511C23C23×D411yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—816no—
515Q8—816no—
516Q8—162no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—216no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—88no—
571C2×C4#53612yes—
572C2×C4#53612yes—
573C2×C4#54612yes—
574C2×C4C22×SD1612yes—
575C2×C4C22×SD1614yes—
576C2×C4C22×SD1614yes—
577C2×C4C22×SD1612yes—
578C2×C4C2×#13714yes—
579C2×C4C2×#13712yes—
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×2−1+412yes—
583C2×C4C2×2−1+412yes—
584C8—42no—
585C8—42no—
586C8—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22#329514yes—
594C22C2×#53611yescharacteristic
595C22C2×#54611yescharacteristic
596C22#348611yescharacteristic
597C22C23×SD1611yescharacteristic
598C22C22×#13711yescharacteristic
599C22C2×#57611yescharacteristic
600C22#358611yescharacteristic
601C4—22no—
602C4—22no—
603C4—24no—
604C4—24no—
605C4—22no—
606C4—24no—
607C4—44no—
608C4—44no—
609C4—41no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—82no—
614C4—84no—
615C4—81no—
616C4—82no—
617C4—81no—
618C4#322914yes—
619C4#350912yes—
620C4#354412yes—
621C2—24no—
622C2#3946511yescharacteristic
623C2C2×#322911yescharacteristic
624C2#6087111yescharacteristic
625C2C2×#329511yescharacteristic
626C2#6115711yescharacteristic
627C2#6181811yescharacteristic
628C2#6184411yescharacteristic
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. #32228C2semidirect
5. #32132C2semidirect
7. #28791C2semidirect
9. #27815C2semidirect
12. #26442C2semidirect
13. #26022C2semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×#25512
#331482
#322282
#321322
#296412
#287912
#279602
#278152
#276472
#276132
#264422
#260222
#246012
#225992
#218332
C4×#4832