Finite Group Databasebeta

← #6532014

Subgroups — #6532014

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2#34196C211yesmaximal, characteristic
3#27616C212yesmaximal
4#27610C212yesmaximal
5#27589C212yesmaximal
6#27586C212yesmaximal
7#25075C214yesmaximal, complemented
8#25063C212yesmaximal, complemented
9#25062C212yesmaximal, complemented
10#25008C214yesmaximal, complemented
11#24995C212yesmaximal, complemented
12#24994C212yesmaximal, complemented
13#21987C211yesmaximal, characteristic
14#21982C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic
16#20740C211yesmaximal, characteristic
17#20735C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2582—21no—
23#2582—22no—
24#2582—21no—
25#2582C2212yes—
26#2582C2212yes—
27#2582C2214yes—
28C2×#398—22no—
29#2355C2211yescharacteristic
30#2355C2212yes—
31#2355C2211yescharacteristic
32#2355C2212yes—
33Q8×C4⋊C4C2211yescharacteristic
34Q8×C4⋊C4C2211yescharacteristic
35#2344—22no—
36#2344—21no—
37Q8×C22⋊C4—21no—
38#2320—22no—
39#2320—21no—
40C2×#391—21no—
41C2×#391C2212yes—
42C2×#389—21no—
43C2×#389C2212yes—
44#2285C2211yescharacteristic
45#2285C2212yes—
46C2×#384C2211yescharacteristic
47#2283C2212yes—
48#2283C2211yescharacteristic
49#2276—21no—
50#2101C2214yes—
51#2099C2214yes—
52#2097C2214yes—
53#2095C2214yes—
54#2085C2214yes—
55#2083C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#1985C2214yes—
59#1984C2212yes—
60#1984C2212yes—
61#1948C2212yes—
62#1947C2212yes—
63#1947C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1928C2214yes—
67#1927C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1918C2214yes—
71#1918C2214yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1847C2212yes—
77#1845C2212yes—
78#1836C2212yes—
79#1834C2212yes—
80#1823C2212yes—
81#1823C2211yescharacteristic
82#1823C2211yescharacteristic
83#1817C2212yes—
84#1817C2212yes—
85#1805C2212yes—
86#1805C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1787C2212yes—
90#1787C2211yescharacteristic
91#1787C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94C4×#103C2212yes—
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8D412yes—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×#117—22no—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D412yes—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#432—24no—
127#431—24no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2312yes—
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—24no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#383—22no—
186#383—21no—
187#382—22no—
188#382—21no—
189#382C2311yescharacteristic
190#382C2311yescharacteristic
191#382C2312yes—
192#382C2312yes—
193#381—22no—
194#381—21no—
195#380—22no—
196#380—21no—
197#380C2312yes—
198#380C2311yescharacteristic
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215#376—22no—
216#376—21no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2×D412yes—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2412yes—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C4×Q8—24no—
258C4×Q8—28no—
259C4×Q8—24no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—24no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274#117—82no—
275#117—84no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
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—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—84no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42#4114yes—
338C2×C42#4114yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—44no—
344#107—44no—
345#107—44no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349#103—28no—
350#103—24no—
351#103—24no—
352#103—42no—
353#103—44no—
354#103—42no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C22⋊C8—24no—
358#97—24no—
359C4×C8—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—416no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8#13714yes—
405C2×Q82−1+414yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4#12314yes—
437C22×C4#12314yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—44no—
504C42—44no—
505C42—44no—
506C42—44no—
507C42—42no—
508C42—42no—
509C42—82no—
510C42—82no—
511C42—82no—
512C42#12314yes—
513C42#12314yes—
514C42C2×#4112yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23C2×#12312yes—
523C23#57612yes—
524C23#57812yes—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—24no—
528Q8—24no—
529Q8—48no—
530Q8—48no—
531Q8—44no—
532Q8—42no—
533Q8—42no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—88no—
588C2×C4—84no—
589C2×C4C2×#12312yes—
590C2×C4C2×#12312yes—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#13714yes—
595C2×C4C2×#13712yes—
596C2×C4#57512yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×2−1+412yes—
600C2×C4#58512yes—
601C8—42no—
602C8—42no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#309814yes—
611C22C22×#12311yescharacteristic
612C22#348611yescharacteristic
613C22#348711yescharacteristic
614C22C22×#13711yescharacteristic
615C22C2×#57511yescharacteristic
616C22C2×#57611yescharacteristic
617C22C2×#57811yescharacteristic
618C4—22no—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—22no—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—81no—
633C4—84no—
634C4—81no—
635C4#309514yes—
636C4#349512yes—
637C4#349512yes—
638C2—24no—
639C2#3567211yescharacteristic
640C2C2×#309511yescharacteristic
641C2C2×#309811yescharacteristic
642C2#6184211yescharacteristic
643C2#6184411yescharacteristic
644C2#6189811yescharacteristic
645C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #25075C2semidirect
8. #25063C2semidirect
9. #25062C2semidirect
10. #25008C2semidirect
11. #24995C2semidirect
12. #24994C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#341962
#276162
#276102
#275892
#275862
#250752
#250632
#250622
#250082
#249952
#249942
#219872
#219822
#218202
#207402
#207352