Finite Group Databasebeta

← #6485008

Subgroups — #6485008

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33099C212yesmaximal
4#33098C212yesmaximal
5#29913C212yesmaximal
6#29913C212yesmaximal
7#26319C212yesmaximal, complemented
8#26317C212yesmaximal, complemented
9#26283C212yesmaximal, complemented
10#26281C212yesmaximal, complemented
11#25071C214yesmaximal, complemented
12#25005C214yesmaximal, complemented
13#21987C211yesmaximal, characteristic
14#21860C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic
16#21394C211yesmaximal, characteristic
17#21361C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2855C2212yes—
23#2855C2212yes—
24#2634—22no—
25#2581C2212yes—
26#2581C2212yes—
27#2568C2211yescharacteristic
28#2568C2211yescharacteristic
29#2546C2211yescharacteristic
30#2546C2211yescharacteristic
31#2503—21no—
32#2484—22no—
33#2434—21no—
34#2355—21no—
35#2355—21no—
36#2355—22no—
37#2355C2212yes—
38#2355C2212yes—
39#2355C2214yes—
40#2327—21no—
41#2327—21no—
42C2×#391—21no—
43C2×#387—22no—
44C2×#384C2211yescharacteristic
45#2283C2211yescharacteristic
46C2×#382C2212yes—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2097C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2085C2214yes—
56#2074C2214yes—
57#2074C2214yes—
58#1990C2212yes—
59#1988C2212yes—
60#1985C2214yes—
61#1984C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2212yes—
65#1949C2212yes—
66#1949C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1927C2214yes—
73#1921C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1853C2211yescharacteristic
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1852C2211yescharacteristic
85#1852C2212yes—
86#1852C2211yescharacteristic
87#1847C2212yes—
88#1836C2212yes—
89#1817C2212yes—
90#1817C2212yes—
91#1803C2214yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1763C2214yes—
95C23×Q8D411yescharacteristic
96#533—24no—
97#533—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D412yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D412yes—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C42—21no—
124C22×C42—22no—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#432—24no—
128#431—24no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#398—22no—
151#398—22no—
152#395—22no—
153#395—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391—24no—
159#391—21no—
160#391—21no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#383—21no—
188#382—21no—
189#382—21no—
190#382—22no—
191#382—22no—
192#382C2311yescharacteristic
193#382C2311yescharacteristic
194#382C2314yes—
195#382C2314yes—
196#381—22no—
197#381—22no—
198#381—21no—
199#380C2311yescharacteristic
200#380C2311yescharacteristic
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215#376—21no—
216C2×#95—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2×D412yes—
241C22×Q8C2412yes—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259#125—44no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—44no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—28no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275#117—44no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—84no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4114yes—
333C2×C42#4114yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—24no—
339#107—44no—
340#107—42no—
341#106—24no—
342#106—44no—
343#106—42no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346#103—28no—
347#103—44no—
348#103—44no—
349#103—44no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—24no—
353#97—24no—
354C4×C8—24no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24C22×D411yescharacteristic
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—416no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4#12314yes—
432C22×C4#12314yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C42−1+414yes—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—44no—
499C42—44no—
500C42—44no—
501C42—42no—
502C42—42no—
503C42—84no—
504C42—84no—
505C42#12314yes—
506C42#12314yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23C2×#12312yes—
516C23#57712yes—
517C23#57812yes—
518C23C23×D411yescharacteristic
519Q8—48no—
520Q8—44no—
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—84no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—88no—
580C2×C4—84no—
581C2×C4C2×#12312yes—
582C2×C4C2×#12312yes—
583C2×C4C2×#12312yes—
584C2×C4C2×#12312yes—
585C2×C4C2×#12312yes—
586C2×C4#57512yes—
587C2×C4#57512yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×2−1+412yes—
591C2×C4#58512yes—
592C8—42no—
593C8—42no—
594C8—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22#309714yes—
602C22C22×#12311yescharacteristic
603C22#348611yescharacteristic
604C22#348711yescharacteristic
605C22C2×#57511yescharacteristic
606C22C2×#57511yescharacteristic
607C22C2×#57711yescharacteristic
608C22C2×#57811yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—82no—
619C4—82no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—84no—
624C4—81no—
625C4—82no—
626C4#309314yes—
627C4#346112yes—
628C4#350712yes—
629C2—24no—
630C2#3518211yescharacteristic
631C2C2×#309311yescharacteristic
632C2C2×#309711yescharacteristic
633C2#6184111yescharacteristic
634C2#6184211yescharacteristic
635C2#6190411yescharacteristic
636C2#6190511yescharacteristic
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. #26319C2semidirect
8. #26317C2semidirect
9. #26283C2semidirect
10. #26281C2semidirect
11. #25071C2semidirect
12. #25005C2semidirect

Maximal subgroups

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

subgroupindex
#352482
C2×#23552
#330992
#330982
#299132
#263192
#263172
#262832
#262812
#250712
#250052
#219872
#218602
#218202
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