Finite Group Databasebeta

← #6556249

Subgroups — #6556249

Order 512.

1615 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30200C212yesmaximal, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29105C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#27976C212yesmaximal, complemented
9#27748C212yesmaximal, complemented
10#27730C212yesmaximal, complemented
11#27705C214yesmaximal, complemented
12#27285C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14#23629C212yesmaximal, complemented
15#22500C212yesmaximal
16#22430C212yesmaximal, complemented
17#20567C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#489C2212yes—
21C2×#436C2212yes—
22C2×#419C2214yes—
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2857C2212yes—
26#2850C2212yes—
27#2571C2212yes—
28#2568C2211yescharacteristic
29#2563—21no—
30#2547C2211yescharacteristic
31#2538—21no—
32#2503C2212yes—
33#2497—22no—
34#2492—21no—
35#2492—21no—
36#2492—22no—
37#2488—22no—
38#2488—22no—
39#2439C2214yes—
40#2436C2212yes—
41#2369—21no—
42#2355—21no—
43#2354C2212yes—
44#2339—21no—
45#2339—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C2×#391C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2101C2214yes—
53#2074C2214yes—
54#2074C2214yes—
55#2072C2211yescharacteristic
56#2072C2212yes—
57#2072C2212yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2071C2212yes—
62#2042C2212yes—
63#2042C2214yes—
64#2024C2211yescharacteristic
65#2024C2214yes—
66#2024C2212yes—
67#2023C2211yescharacteristic
68#1993C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1981C2212yes—
72#1958C2214yes—
73#1956C2212yes—
74#1935C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1895C2214yes—
78#1894C2214yes—
79#1875C2214yes—
80#1873C2214yes—
81#1835C2214yes—
82#1824C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1787C2214yes—
86C8×C4⋊C4C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1753C2214yes—
91C4×C22⋊C8C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41—21no—
96C23×Q8C2312yes—
97C22×M16C2314yes—
98D4×Q8—28no—
99#547—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#124C2311yescharacteristic
108C2×#124C2312yes—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C42C2312yes—
125#489—44no—
126#436—28no—
127C2×C4⋊C8C2312yes—
128#419—48no—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—24no—
145#400—22no—
146#400—22no—
147#397—22no—
148#396—24no—
149#393—24no—
150#393—22no—
151#393—22no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—24no—
161#391C2312yes—
162#391C2312yes—
163#391C2311yescharacteristic
164#391C2312yes—
165#391C2312yes—
166#390—22no—
167#390C2311yescharacteristic
168#389—22no—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2314yes—
173#388—24no—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—21no—
178#387—24no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#382—22no—
189#382—21no—
190#382—21no—
191#382C2312yes—
192#382C2314yes—
193#382C2311yescharacteristic
194#381—22no—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C2×#95—22no—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2312yes—
233C2×#41—44no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—24no—
241C22×Q8—28no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269#124—22no—
270#124—41no—
271#124—44no—
272#123—48no—
273C22⋊Q8—28no—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276C22⋊Q8—84no—
277#121—28no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4#4118yes—
285#117—44no—
286#117—44no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C4⋊C8—24no—
342#103—24no—
343#103—28no—
344#103—42no—
345#103—44no—
346#103—44no—
347#103—44no—
348#103—44no—
349#102—42no—
350#102—42no—
351#102—44no—
352C22⋊C8—28no—
353#97—42no—
354C4×C8—24no—
355C4×C8—44no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D412yes—
381#41—816no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—416no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—88no—
403C2×Q8H(5,𝔽2)12yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4#12414yes—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4118yes—
450C22×C4C2511yescharacteristic
451M16—88no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4#12414yes—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4#12418yes—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—24no—
509C42—22no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—44no—
515C42—84no—
516C42C2×#4114yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C2×#12414yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—42no—
534Q8—44no—
535Q8—44no—
536Q8—88no—
537Q8—84no—
538Q8—84no—
539Q8—164no—
540D4—28no—
541D4—42no—
542D4—42no—
543D4—88no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4C2×#12412yes—
596C2×C4C2×#12414yes—
597C2×C4C2×#12412yes—
598C2×C4#57512yes—
599C2×C4#57512yes—
600C2×C4#57712yes—
601C2×C4#57812yes—
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4#58512yes—
605C2×C4#58511yescharacteristic
606C8—44no—
607C8—42no—
608C8—42no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C22×#12411yescharacteristic
624C22#348511yescharacteristic
625C22#348811yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57711yescharacteristic
629C22C2×#57811yescharacteristic
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—82no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4#314014yes—
647C4#351312yes—
648C4#352112yes—
649C2—44no—
650C2—161no—
651C2#3584711yescharacteristic
652C2C2×#314011yescharacteristic
653C2#6062311yescharacteristic
654C2#6173011yescharacteristic
655C2#6173111yescharacteristic
656C2#6199711yescharacteristic
657C2#6199811yescharacteristic
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
2. #36194C2semidirect
3. #30205C2semidirect
4. #30200C2semidirect
5. #29355C2semidirect
6. #29105C2semidirect
7. #28726C2semidirect
8. #27976C2semidirect
9. #27748C2semidirect
10. #27730C2semidirect
11. #27705C2semidirect
12. #27285C2semidirect
13. #23817C2semidirect
14. #23629C2semidirect
16. #22430C2semidirect
17. #20567C2semidirect

Maximal subgroups

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

subgroupindex
#363032
#361942
#302052
#302002
#293552
#291052
#287262
#279762
#277482
#277302
#277052
#272852
#238172
#236292
#225002
#224302
#205672