Finite Group Databasebeta

← #6660841

Subgroups — #6660841

Order 512.

1791 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32732C214yesmaximal, complemented
4#30325C212yesmaximal, complemented
5#30317C212yesmaximal, complemented
6#29910C212yesmaximal, complemented
7#29759C212yesmaximal, complemented
8#29714C212yesmaximal, complemented
9#29624C212yesmaximal, complemented
10#29341C212yesmaximal, complemented
11#28320C212yesmaximal, complemented
12#24518C214yesmaximal, complemented
13#24016C212yesmaximal, complemented
14#23926C212yesmaximal
15#20720C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#535C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic, complemented
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#448C2214yes—
22C2×#450C2214yes—
23C2×#433C2212yescomplemented
24#2857C2212yes—
25#2857C2212yescomplemented
26#2854C2212yes—
27#2854C2212yes—
28#2624C2212yes—
29#2624C2212yes—
30#2621C2212yes—
31#2621C2212yescomplemented
32#2547—22no—
33#2546—22no—
34#2486—22no—
35#2486—22no—
36#2448—22no—
37#2445—22no—
38#2418—22no—
39#2418—22no—
40#2316C2211yescharacteristic
41#2311C2211yescharacteristic
42#2310C2212yescomplemented
43#2309C2211yescharacteristic
44#2296C2211yescharacteristic
45C2×#390C2211yescharacteristic
46#2285C2211yescharacteristic, complemented
47#2101C2214yescomplemented
48#2092C2214yes—
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2071C2212yes—
52#2071C2212yes—
53#2068C2214yescomplemented
54#2066C2214yescomplemented
55#2055C2214yes—
56#2055C2214yes—
57#2043C2212yescomplemented
58#2043C2212yescomplemented
59#2043C2214yescomplemented
60#2036C2214yescomplemented
61#2035C2214yescomplemented
62#2013C2212yes—
63#2013C2212yes—
64#2013C2214yes—
65#2000C2214yes—
66#2000C2214yes—
67#1973C2214yes—
68#1968C2214yes—
69#1935C2214yescomplemented
70#1933C2214yescomplemented
71#1875C2214yes—
72#1873C2214yes—
73#1872C2214yes—
74#1854C2214yescomplemented
75#1817C2212yescomplemented
76#1798C2212yescomplemented
77#1797C2212yes—
78#1787C2214yescomplemented
79#1780C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C22×SD16C2314yes—
87C22×SD16C2314yes—
88#536—48no—
89#535—44no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×#127—21no—
96C2×#124—21no—
97C2×#124—22no—
98C2×#123—22no—
99C2×#123—22no—
100C2×#123—22no—
101C2×#123—22no—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2314yes—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—21no—
107C2×#120—21no—
108C4×#41—44no—
109C2×C4×Q8D412yes—
110C2×C4×D4D412yes—
111C2×C4×D4D412yes—
112C2×C4×D4D412yes—
113C2×#117C2311yescharacteristic
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—22no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#485—48no—
126#450—48no—
127#448—48no—
128#433—44no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146#399—22no—
147#399—22no—
148#398—24no—
149#398—24no—
150#397—22no—
151#397—22no—
152#397—24no—
153#397—24no—
154#394—22no—
155#394—22no—
156#394—22no—
157#394—22no—
158#391—21no—
159#391—24no—
160#391C2314yes—
161#391C2314yes—
162#390—21no—
163#390—22no—
164#390—22no—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#386—21no—
183#386—21no—
184#386—22no—
185#386—22no—
186#385—22no—
187#384C2312yes—
188#382C2312yes—
189#381—24no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2312yes—
198#379—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2312yes—
206#336C2312yes—
207#336C2314yes—
208C25—21no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—42no—
212C22×Q8—22no—
213C22×Q8—28no—
214C22×Q8—44no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8C2×D411yescharacteristic
218C22×Q8C2414yes—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—24no—
223C22×D4—42no—
224C22×D4—42no—
225C22×D4C2×D411yescharacteristic
226C22×D4C2×D411yescharacteristic
227C22×D4C2×D411yescharacteristic
228C22×D4C2412yes—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4C2×D412yes—
232C23×C4C2×D412yes—
233C23×C4C2×D412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×SD16—44no—
238C2×SD16—44no—
239C2×SD16—44no—
240C2×SD16—44no—
241C2×SD16—44no—
242C2×SD16—44no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—44no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—48no—
249C4⋊Q8—48no—
250#127—44no—
251#124—48no—
252#124—44no—
253#123—48no—
254#123—48no—
255#123—84no—
256#123—84no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261#121—28no—
262#121—44no—
263#120—44no—
264#120—44no—
265C4×Q8—44no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—44no—
269#117—24no—
270#117—24no—
271#117—42no—
272#117—42no—
273#117—42no—
274#117—44no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
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×C22⋊C4—22no—
313C2×C22⋊C4—84no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—44no—
331#106—44no—
332C4⋊C8—44no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339#102—44no—
340#102—44no—
341C22⋊C8—28no—
342C22⋊C8—28no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345#97—42no—
346C4×C8—42no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—82no—
363C24—82no—
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366C24C22×D411yescharacteristic
367#41—416no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—41no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—88no—
381C2×Q8—88no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—88no—
401C2×D4—82no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4—82no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4C2511yescharacteristic
441SD16—88no—
442SD16—88no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—42no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—42no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—84no—
521C23—88no—
522C23—162no—
523C23—164no—
524C23—162no—
525C23#53412yes—
526C23#53512yes—
527C23C23×D411yescharacteristic
528Q8—44no—
529Q8—44no—
530Q8—44no—
531Q8—44no—
532Q8—44no—
533Q8—88no—
534D4—44no—
535D4—44no—
536D4—44no—
537D4—44no—
538D4—44no—
539D4—84no—
540D4—164no—
541D4—164no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—88no—
582C2×C4—162no—
583C2×C4#53412yes—
584C2×C4#53512yes—
585C2×C4#57714yes—
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4#58512yes—
589C8—44no—
590C8—44no—
591C22—22no—
592C22—22no—
593C22—24no—
594C22—24no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—168no—
614C22C2×#53411yescharacteristic
615C22C2×#53511yescharacteristic
616C22#344911yescharacteristic
617C22C2×#57712yes—
618C22#358612yes—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—84no—
631C4#353414yes—
632C2—42no—
633C2—42no—
634C2—81no—
635C2—162no—
636C2#3744011yescharacteristic
637C2#6075812yes—
638C2#6083612yes—
639C2#6101211yescharacteristic
640C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35316C2semidirect
3. #32732C2semidirect
4. #30325C2semidirect
5. #30317C2semidirect
6. #29910C2semidirect
7. #29759C2semidirect
8. #29714C2semidirect
9. #29624C2semidirect
10. #29341C2semidirect
11. #28320C2semidirect
12. #24518C2semidirect
13. #24016C2semidirect
15. #20720C2semidirect
18. C22×C4⋊Q8C22semidirect
23. C2×#433C22semidirect
25. #2857C22semidirect
31. #2621C22semidirect
42. #2310C22semidirect
46. #2285C22semidirect
47. #2101C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
53. #2068C22semidirect
54. #2066C22semidirect
57. #2043C22semidirect
58. #2043C22semidirect
59. #2043C22semidirect
60. #2036C22semidirect
61. #2035C22semidirect
69. #1935C22semidirect
70. #1933C22semidirect
74. #1854C22semidirect
75. #1817C22semidirect
76. #1798C22semidirect
78. #1787C22semidirect
81. #1749C22semidirect
82. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353162
#352542
#327322
#303252
#303172
#299102
#297592
#297142
#296242
#293412
#283202
#245182
#240162
#239262
#207202