Finite Group Databasebeta

← #6528575

Subgroups — #6528575

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29330C212yesmaximal, complemented
5#27978C212yesmaximal, complemented
6#27977C212yesmaximal, complemented
7#27766C214yesmaximal, complemented
8#27764C214yesmaximal, complemented
9#27742C214yesmaximal, complemented
10#27740C214yesmaximal, complemented
11#21866C211yesmaximal, characteristic, complemented
12#21865C211yesmaximal, characteristic
13#21395C211yesmaximal, characteristic
14#21394C211yesmaximal, characteristic, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2850C2212yes—
21#2850C2212yes—
22#2639—21no—
23#2639—21no—
24#2592—22no—
25#2570—22no—
26#2570—22no—
27#2570C2214yes—
28#2570C2214yes—
29#2568C2212yes—
30#2563C2212yes—
31#2562—22no—
32#2548C2212yes—
33#2547C2212yes—
34C2×#400—21no—
35C2×#398—21no—
36#2311—21no—
37#2311—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#387—21no—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2101C2214yes—
48#2101C2214yes—
49#2074C2214yes—
50#2073C2214yes—
51#2072C2214yes—
52#2071C2214yes—
53#2042C2214yes—
54#2042C2214yes—
55#1993C2212yes—
56#1993C2212yes—
57#1988C2212yes—
58#1987C2212yes—
59#1982C2212yes—
60#1981C2212yes—
61#1961C2212yes—
62#1960C2212yes—
63#1959C2214yes—
64#1958C2214yes—
65#1936C2214yes—
66#1935C2214yes—
67#1934C2214yes—
68#1933C2214yes—
69#1874C2212yes—
70#1874C2212yes—
71#1874C2212yes—
72#1874C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2212yes—
77#1836C2214yes—
78#1835C2214yes—
79#1805C2214yes—
80#1804C2214yes—
81#1771C2214yes—
82#1771C2214yes—
83#1763C2214yes—
84#1762C2214yes—
85#1749C2212yes—
86C4×C22⋊C8C2212yes—
87C22×#41—21no—
88#533—24no—
89#533—24no—
90#532—24no—
91#532—24no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#125—21no—
97C2×#124—21no—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#432—24no—
124#432—24no—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2314yes—
141#400—24no—
142#400—24no—
143#400—22no—
144#400—22no—
145#398—24no—
146#398—24no—
147#397—22no—
148#397—22no—
149#394—22no—
150#394—22no—
151#393—22no—
152#393—22no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#387—24no—
170#387—24no—
171#387—24no—
172#387—24no—
173#386—22no—
174#386—22no—
175#386—22no—
176#386—22no—
177#384—22no—
178#384C2314yes—
179#382—21no—
180#382—21no—
181#382—22no—
182#382C2312yes—
183#382C2312yes—
184#382C2312yes—
185#382C2312yes—
186#382C2314yes—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218C2×#41—44no—
219C2×#41—44no—
220C2×#41—44no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8C2412yes—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248#127—22no—
249#127—22no—
250#125—44no—
251#124—44no—
252#124—44no—
253#124—44no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—44no—
257#121—24no—
258#121—24no—
259#121—44no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—44no—
266#117—44no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—82no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42#4114yes—
323C2×C42#4114yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C4⋊C8—24no—
329C4⋊C8—24no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334#102—44no—
335#102—44no—
336#102—44no—
337#102—44no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340#97—24no—
341#97—24no—
342#97—44no—
343C4×C8—44no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—22no—
367C24C22×D411yescharacteristic
368#41—816no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—81no—
391C2×D4—81no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4#12414yes—
433C22×C4#12414yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2511yescharacteristic
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—24no—
492C42—24no—
493C42—42no—
494C42—42no—
495C42—44no—
496C42—82no—
497C42—82no—
498C42#12414yes—
499C42#12414yes—
500C42C2×#4112yes—
501C42C2×#4112yes—
502C42C2×#4112yes—
503C42C2×#4112yes—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C2×#12412yes—
516C23#57512yes—
517C23#57512yes—
518C23C23×D411yescharacteristic
519Q8—44no—
520Q8—44no—
521Q8—84no—
522Q8—84no—
523D4—44no—
524D4—44no—
525D4—84no—
526D4—84no—
527D4—88no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—28no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
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—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—88no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4C2×#12412yes—
585C2×C4C2×#12412yes—
586C2×C4C2×#12412yes—
587C2×C4C2×#12412yes—
588C2×C4C2×#12412yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4#58512yes—
592C2×C4#58512yes—
593C8—44no—
594C8—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22#314014yes—
609C22C22×#12411yescharacteristic
610C22#348511yescharacteristic
611C22#348711yescharacteristic
612C22C2×#57511yescharacteristic
613C22C2×#57511yescharacteristic
614C22#358511yescharacteristic
615C22#358611yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4#351212yes—
633C4#351212yes—
634C2—24no—
635C2—162no—
636C2#3564911yescharacteristic
637C2C2×#314011yescharacteristic
638C2#6062111yescharacteristic
639C2#6171111yescharacteristic
640C2#6171311yescharacteristic
641C2#6189411yescharacteristic
642C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. C2×#2570C2semidirect
3. #29336C2semidirect
4. #29330C2semidirect
5. #27978C2semidirect
6. #27977C2semidirect
7. #27766C2semidirect
8. #27764C2semidirect
9. #27742C2semidirect
10. #27740C2semidirect
11. #21866C2semidirect
14. #21394C2semidirect
15. #20534C2semidirect

Maximal subgroups

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

subgroupindex
#351802
C2×#25702
#293362
#293302
#279782
#279772
#277662
#277642
#277422
#277402
#218662
#218652
#213952
#213942
#205342