Finite Group Databasebeta

← #6617010

Subgroups — #6617010

Order 512.

1553 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33100C212yesmaximal
4#32954C211yesmaximal, characteristic
5#27950C211yesmaximal, characteristic
6#27680C211yesmaximal, characteristic
7#27589C212yesmaximal
8#27487C214yesmaximal, complemented
9#27361C214yesmaximal, complemented
10#26093C212yesmaximal, complemented
11#24683C212yesmaximal, complemented
12#24502C212yesmaximal
13#23062C212yesmaximal, complemented
14#22808C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic
16C4×#451C211yesmaximal, characteristic
17#21361C212yesmaximal
18C22×C4×Q8C2211yescharacteristic
19C2×#475C2212yes—
20C2×#451C2211yescharacteristic
21C2×C4×Q16C2212yes—
22C2×#431C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2571C2211yescharacteristic
26#2568C2211yescharacteristic
27#2533—22no—
28#2514—21no—
29#2514C2212yes—
30#2509C2212yes—
31#2503—21no—
32#2493—22no—
33#2365C2211yescharacteristic
34#2360—21no—
35#2356—21no—
36#2355—21no—
37#2355—21no—
38#2355—22no—
39#2355C2212yes—
40#2355C2211yescharacteristic
41#2355C2214yes—
42#2355C2212yes—
43C4×C22⋊Q8C2211yescharacteristic
44C2×#391C2212yes—
45C2×#387—22no—
46C2×#384C2211yescharacteristic
47#2283C2212yes—
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2212yes—
53#2097C2214yes—
54#2096C2212yes—
55#2083C2214yes—
56#2074C2214yes—
57#2068C2214yes—
58#2056C2214yes—
59#1990C2212yes—
60#1969C2214yes—
61#1949C2214yes—
62#1949C2214yes—
63#1948C2212yes—
64#1939C2212yes—
65#1933C2214yes—
66#1928C2214yes—
67#1928C2214yes—
68#1927C2214yes—
69#1927C2214yes—
70#1919C2214yes—
71#1918C2214yes—
72#1916C2214yes—
73#1878C2212yes—
74#1872C2212yes—
75#1872C2211yescharacteristic
76#1865C2214yes—
77#1863C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1852C2212yes—
82#1852C2211yescharacteristic
83#1845C2212yes—
84#1817C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2211yescharacteristic
90C4×#107C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2311yescharacteristic
100C2×C4×Q8—28no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C42—21no—
115C22×C42—22no—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118#475—28no—
119#451—24no—
120C4×Q16—28no—
121#431—28no—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#401—22no—
142#400—22no—
143#400—22no—
144#400—22no—
145#400—22no—
146#400—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2314yes—
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2314yes—
158#389—21no—
159#389—22no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2311yescharacteristic
165#387—22no—
166#387—22no—
167#387—22no—
168#387—21no—
169#387—24no—
170#387—24no—
171#387—24no—
172#387—24no—
173#387—22no—
174#387—22no—
175#387—22no—
176#385—22no—
177#385—21no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#383—22no—
184#383—22no—
185#382—21no—
186#382—22no—
187#382—22no—
188#382—24no—
189#382—21no—
190#382C2312yes—
191#382C2312yes—
192#381—22no—
193#381—22no—
194#380—21no—
195#380C2312yes—
196#380C2311yescharacteristic
197C4×C4⋊C4—24no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217#376—22no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—42no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8—44no—
259C4×Q8—28no—
260C4×Q8—28no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×Q8#4118yes—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C42—28no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—84no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328#107—24no—
329#107—44no—
330#107—44no—
331#106—28no—
332#106—44no—
333C4⋊C8—28no—
334C4⋊C8—24no—
335#103—24no—
336#103—28no—
337#103—24no—
338#103—24no—
339#103—42no—
340#103—44no—
341#103—44no—
342C22⋊C8—24no—
343C22⋊C8—22no—
344C22⋊C8#4114yes—
345C4×C8—28no—
346C4×C8—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
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#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24C22×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8C2×#4114yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4#13514yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4112yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439Q16—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8C2×#4114yes—
449C4⋊C4—24no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4#13714yes—
481C4⋊C4C2×#4114yes—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C42—28no—
486C42—28no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—82no—
500C42—84no—
501C42—84no—
502C42C2×#4114yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23#54712yes—
508C23C2×#13512yes—
509C23#57812yes—
510C23C23×D411yescharacteristic
511Q8—24no—
512Q8—24no—
513Q8—48no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—84no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—88no—
575C2×C4#52512yes—
576C2×C4#52512yes—
577C2×C4#54712yes—
578C2×C4C2×#13512yes—
579C2×C4C2×#13512yes—
580C2×C4C2×#13714yes—
581C2×C4C2×#13712yes—
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4112yes—
584C2×C4C2×2−1+411yescharacteristic
585C2×C4#58512yes—
586C8—24no—
587C8—44no—
588C8—42no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22#334214yes—
596C22C2×#52511yescharacteristic
597C22C2×#54711yescharacteristic
598C22#348711yescharacteristic
599C22C22×#13511yescharacteristic
600C22C22×#13711yescharacteristic
601C22C2×#57811yescharacteristic
602C22#358511yescharacteristic
603C4—24no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—44no—
611C4—41no—
612C4—44no—
613C4—81no—
614C4—81no—
615C4—82no—
616C4—81no—
617C4—82no—
618C4—84no—
619C4—81no—
620C4#310814yes—
621C4#347512yes—
622C4#349512yes—
623C2—24no—
624C2#3642711yescharacteristic
625C2C2×#310811yescharacteristic
626C2#6058911yescharacteristic
627C2C2×#334211yescharacteristic
628C2#6125411yescharacteristic
629C2#6188511yescharacteristic
630C2#6190011yescharacteristic
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
8. #27487C2semidirect
9. #27361C2semidirect
10. #26093C2semidirect
11. #24683C2semidirect
13. #23062C2semidirect
14. #22808C2semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#23552
#331002
#329542
#279502
#276802
#275892
#274872
#273612
#260932
#246832
#245022
#230622
#228082
#225992
C4×#4512
#213612