Finite Group Databasebeta

← #6495507

Subgroups — #6495507

Order 512.

1742 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
1#36645C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33294C211yesmaximal, characteristic, complemented
5#33238C211yesmaximal, characteristic, complemented
6#33230C211yesmaximal, characteristic, complemented
7#29845C212yesmaximal, complemented
8#29842C212yesmaximal
9#29198C214yesmaximal, complemented
10#28658C214yesmaximal, complemented
11#27894C214yesmaximal, complemented
12#25679C214yesmaximal, complemented
13#22221C212yesmaximal, complemented
14#22219C212yesmaximal, complemented
15#20463C211yesmaximal, characteristic, complemented
16C2×#533—21no—
17C2×#529C2212yes—
18C2×#501C2214yes—
19C2×#489C2214yes—
20C2×#487C2212yes—
21C2×#486C2212yes—
22C2×#431C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2862—21no—
26#2861C2212yes—
27#2860—21no—
28#2860—21no—
29#2859—21no—
30#2859C2212yes—
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857—21no—
34#2857—21no—
35#2857C2212yes—
36#2857C2212yes—
37#2857C2211yescharacteristic
38#2856C2211yescharacteristic
39#2854—21no—
40#2854—21no—
41#2850C2212yes—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#124C2212yes—
49C4×#121—21no—
50C4×#121—21no—
51#2293—21no—
52#2293—21no—
53#2285—21no—
54#2285C2212yes—
55C2×C43C2211yescharacteristic
56#2067C2218yes—
57#2066C2218yes—
58#2035C2218yes—
59#2033C2218yes—
60#1992C2214yes—
61#1990C2212yes—
62#1990C2212yes—
63#1959C2214yes—
64#1959C2214yes—
65#1959C2214yes—
66#1957C2212yes—
67#1956C2212yes—
68#1928C2218yes—
69#1926C2218yes—
70#1864C2218yes—
71#1863C2218yes—
72#1847C2214yes—
73#1845C2212yes—
74#1844C2212yes—
75#1836C2214yes—
76#1767C2212yes—
77#1767C2212yes—
78#1749C2212yes—
79#1749C2212yes—
80#1744C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#533—48no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92#529—48no—
93C2×#126—22no—
94C2×#125—22no—
95C2×#125C2312yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117—22no—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#501—48no—
122#489—48no—
123#487—28no—
124#486—28no—
125#431—28no—
126C2×#107C2318yes—
127C2×#106C2318yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#102C2318yes—
132C2×#102C2318yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2312yes—
145#391C2312yes—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389C2314yes—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#386—22no—
181#386—22no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382C2312yes—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2314yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2314yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C43D411yescharacteristic
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217#340C2318yes—
218#340C2318yes—
219#336C2314yes—
220#336C2314yes—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8C2412yes—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2412yes—
228C23×C4—22no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—24no—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—28no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246#127—24no—
247#126—48no—
248#125—28no—
249#125—48no—
250#124—28no—
251#124—28no—
252#123—48no—
253C22⋊Q8—44no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256#121—44no—
257#121—44no—
258#121—44no—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—48no—
265C4×D4—44no—
266#117—48no—
267#117—48no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2414yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—48no—
344#106—48no—
345C4⋊C8—28no—
346#103—216no—
347#103—48no—
348#102—216no—
349#102—48no—
350C22⋊C8—28no—
351#97—24no—
352#97—24no—
353#97—24no—
354#97—24no—
355C4×C8—24no—
356C4×C8#4118yes—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367C24—22no—
368C24C22×D411yescharacteristic
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C22×C4—22no—
391C22×C4—22no—
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—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4AGL(1,ℤ/8)14yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8C2×#4118yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—44no—
503C42—48no—
504C42#13714yes—
505C42C2×#4114yes—
506C42H(5,𝔽2)14yes—
507C422−1+414yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C2×AGL(1,ℤ/8)12yes—
522C23C23×D411yescharacteristic
523Q8—48no—
524Q8—88no—
525Q8—88no—
526D4—48no—
527D4—88no—
528D4—84no—
529D4—84no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—88no—
579C2×C4#53212yes—
580C2×C4#53212yes—
581C2×C4#53512yes—
582C2×C4#53512yes—
583C2×C4#53612yes—
584C2×C4#53612yes—
585C2×C4C2×AGL(1,ℤ/8)12yes—
586C2×C4C2×AGL(1,ℤ/8)12yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4#57512yes—
591C2×C4#57512yes—
592C2×C4C22×#4112yes—
593C2×C4C2×H(5,𝔽2)12yes—
594C2×C4C2×2−1+412yes—
595C8—28no—
596C8—44no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22C2×#53211yescharacteristic
612C22C2×#53511yescharacteristic
613C22C2×#53611yescharacteristic
614C22C22×AGL(1,ℤ/8)11yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57511yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—84no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—82no—
634C4#317214yes—
635C4#318014yes—
636C4#351114yes—
637C2—42no—
638C2—162no—
639C2#3529811yescharacteristic
640C2C2×#317211yescharacteristic
641C2C2×#318011yescharacteristic
642C2#6080311yescharacteristic
643C2#6080411yescharacteristic
644C2#6086011yescharacteristic
645C2#6086211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33316C2semidirect
4. #33294C2semidirect
5. #33238C2semidirect
6. #33230C2semidirect
7. #29845C2semidirect
9. #29198C2semidirect
10. #28658C2semidirect
11. #27894C2semidirect
12. #25679C2semidirect
13. #22221C2semidirect
14. #22219C2semidirect
15. #20463C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366312
#333162
#332942
#332382
#332302
#298452
#298422
#291982
#286582
#278942
#256792
#222212
#222192
#204632