Finite Group Databasebeta

← #6594215

Subgroups — #6594215

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#36164C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33057C212yesmaximal, complemented
5#29224C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#27979C212yesmaximal, complemented
8#27969C212yesmaximal, complemented
9#27730C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#27705C214yesmaximal, complemented
12#27703C214yesmaximal, complemented
13#22502C212yesmaximal, complemented
14#22500C212yesmaximal
15C4×#489C211yesmaximal, characteristic, complemented
16#20765C212yesmaximal, complemented
17#20494C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#489C2211yescharacteristic
21C2×#436C2212yes—
22C2×C4×M16C2212yes—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2571C2212yes—
28#2568C2211yescharacteristic
29#2563—21no—
30#2563C2211yescharacteristic
31#2538—21no—
32#2504C2212yes—
33#2503C2212yes—
34#2497—22no—
35#2492—22no—
36#2489—21no—
37#2489—21no—
38#2488—22no—
39#2488—22no—
40#2439C2214yes—
41#2369—21no—
42#2355—21no—
43#2354C2212yes—
44#2339—21no—
45#2339—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48#2285C2212yes—
49C2×#382C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2101C2214yes—
54#2074C2214yes—
55#2073C2214yes—
56#2024C2212yes—
57#2023C2212yes—
58#1993C2214yes—
59#1993C2212yes—
60#1993C2211yescharacteristic
61#1990C2212yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1959C2212yes—
66#1959C2211yescharacteristic
67#1936C2212yes—
68#1935C2212yes—
69#1934C2214yes—
70#1933C2214yes—
71#1895C2214yes—
72#1894C2214yes—
73#1875C2214yes—
74#1873C2214yes—
75#1836C2212yes—
76#1836C2212yes—
77#1836C2211yescharacteristic
78#1836C2212yes—
79#1836C2212yes—
80#1835C2214yes—
81#1835C2214yes—
82#1824C2214yes—
83#1823C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1753C2214yes—
91#1749C2212yes—
92#1736C2214yes—
93#1736C2214yes—
94C4×#97C2211yescharacteristic
95C22×#41—21no—
96C22×M16C2314yes—
97#554—28no—
98#547—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—21no—
106C2×#124C2311yescharacteristic
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C42C2312yes—
124#489—24no—
125#436—28no—
126C2×C4⋊C8C2312yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#102C2312yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×C22⋊C8C2314yes—
134C4×M16—28no—
135C2×#97C2312yes—
136C2×#97C2311yescharacteristic
137C2×#97C2312yes—
138C2×#97C2311yescharacteristic
139C2×C4×C8C2314yes—
140C2×C4×C8C2312yes—
141#400—22no—
142#400—24no—
143#400—22no—
144#400—22no—
145#397—22no—
146#397—22no—
147#396—24no—
148#393—24no—
149#393—22no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—22no—
156#391—21no—
157#391—21no—
158#391—24no—
159#391C2312yes—
160#391C2311yescharacteristic
161#390—22no—
162#390C2312yes—
163#390C2311yescharacteristic
164#389—22no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#388—24no—
170#388—22no—
171#388—22no—
172#388—21no—
173#388—21no—
174#387—24no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382—22no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#382C2314yes—
190#382C2312yes—
191#382C2311yescharacteristic
192#382C2312yes—
193#381—22no—
194#381—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380—21no—
200#380C2314yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—21no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214C2×#95—22no—
215C2×#95—22no—
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233C2×#41—44no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×M16—28no—
251C2×M16—28no—
252C2×M16—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#125—44no—
264#124—22no—
265#124—22no—
266#123—48no—
267C22⋊Q8—28no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270C22⋊Q8—84no—
271#121—28no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×D4—24no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—44no—
278C4×D4#4118yes—
279#117—44no—
280#117—44no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C4⋊C8—24no—
339#103—24no—
340#103—42no—
341#103—44no—
342#102—24no—
343#102—42no—
344#102—44no—
345C22⋊C8—28no—
346#97—24no—
347#97—22no—
348#97—42no—
349#97#4114yescomplemented
350C4×C8—28no—
351C4×C8—24no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—21no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D412yes—
377#41—816no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4#12414yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4118yes—
440C22×C4C2511yescharacteristic
441M16—416no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8C2×#4114yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4#12414yes—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—81no—
490C22⋊C4—82no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4#12418yes—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—42no—
504C42—44no—
505C42—84no—
506C42C2×#4114yes—
507C42C2×#4112yes—
508C422−1+412yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×#12414yes—
523C23C23×D411yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—84no—
527Q8—84no—
528Q8—162no—
529Q8—164no—
530D4—28no—
531D4—42no—
532D4—42no—
533D4—88no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4—84no—
584C2×C4—164no—
585C2×C4C2×#12412yes—
586C2×C4C2×#12414yes—
587C2×C4C2×#12412yes—
588C2×C4#54712yes—
589C2×C4#54712yes—
590C2×C4#57512yes—
591C2×C4#57512yes—
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4112yes—
594C2×C4C2×2−1+411yescharacteristic
595C2×C4#58512yes—
596C8—24no—
597C8—44no—
598C8—42no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22C22×#12411yescharacteristic
614C22C2×#54711yescharacteristic
615C22#350011yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57511yescharacteristic
618C22#358511yescharacteristic
619C22#358611yescharacteristic
620C4—24no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—82no—
630C4—84no—
631C4—82no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4#314014yes—
637C4#347512yes—
638C4#351512yes—
639C2—44no—
640C2—161no—
641C2#3616411yescharacteristic
642C2C2×#314011yescharacteristic
643C2#6062511yescharacteristic
644C2#6125611yescharacteristic
645C2#6125711yescharacteristic
646C2#6259011yescharacteristic
647C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33293C2semidirect
4. #33057C2semidirect
5. #29224C2semidirect
6. #29148C2semidirect
7. #27979C2semidirect
8. #27969C2semidirect
9. #27730C2semidirect
10. #27728C2semidirect
11. #27705C2semidirect
12. #27703C2semidirect
13. #22502C2semidirect
15. C4×#489C2semidirect
16. #20765C2semidirect
17. #20494C2semidirect
349. #97#41semidirect

Maximal subgroups

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

subgroupindex
#361942
#361642
#332932
#330572
#292242
#291482
#279792
#279692
#277302
#277282
#277052
#277032
#225022
#225002
C4×#4892
#207652
#204942