Finite Group Databasebeta

← #6506769

Subgroups — #6506769

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#29303C212yesmaximal, complemented
4#29297C211yesmaximal, characteristic, complemented
5#29296C211yesmaximal, characteristic, complemented
6#28783C214yesmaximal, complemented
7#28781C214yesmaximal, complemented
8#27953C212yesmaximal, complemented
9#26074C214yesmaximal, complemented
10#26072C214yesmaximal, complemented
11#25813C211yesmaximal, characteristic, complemented
12#25811C211yesmaximal, characteristic, complemented
13#21944C212yesmaximal
14#20844C212yesmaximal
15#20524C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C4⋊C4C2211yescharacteristic
19C2×#500C2212yescomplemented
20C2×#498C2212yescomplemented
21C2×#428C2214yes—
22C2×#407C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2626—21no—
27#2626—21no—
28#2608—22no—
29#2607—22no—
30#2441C2212yes—
31#2441C2212yes—
32#2440—22no—
33#2440—22no—
34#2440C2214yes—
35#2440C2214yes—
36#2439C2212yes—
37#2439C2212yes—
38C2×#398—21no—
39C2×#395—21no—
40#2329—21no—
41#2329—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#384C2211yescharacteristic
46#2280C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2085C2214yescomplemented
50#2084C2214yescomplemented
51#2084C2212yescomplemented
52#2083C2214yescomplemented
53#2083C2212yescomplemented
54#2059C2214yescomplemented
55#2058C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2214yescomplemented
58#2057C2212yescomplemented
59#1992C2214yescomplemented
60#1990C2212yescomplemented
61#1990C2212yescomplemented
62#1944C2214yescomplemented
63#1943C2214yescomplemented
64#1942C2214yescomplemented
65#1941C2214yescomplemented
66#1854C2214yes—
67#1854C2214yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2214yes—
76#1852C2212yes—
77#1852C2212yes—
78#1847C2214yescomplemented
79#1809C2214yes—
80#1809C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86#1736C2214yes—
87C23×Q8—21no—
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×M16C2314yes—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—48no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—44no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#500—44no—
125#498—44no—
126#428—48no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135#407Q8116yescomplemented
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#399—22no—
143#399—22no—
144#398—24no—
145#398—24no—
146#396—22no—
147#396—22no—
148#396—22no—
149#396—22no—
150#395—24no—
151#395—24no—
152#393—22no—
153#393—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—22no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#385—24no—
181#385—24no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2314yes—
185#384C2311yescharacteristic
186#382C2312yes—
187#381—22no—
188#380—21no—
189#380—21no—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#379—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C4×C22⋊C4C2312yes—
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#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—82no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×Q818yes—
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C2×M16—44no—
247C2×M16—44no—
248C2×M16C2×Q8116yescomplemented
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259#125—44no—
260C22⋊Q8—416no—
261C22⋊Q8—416no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264#117—44no—
265#117—44no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—28no—
336#107—44no—
337#107—44no—
338#107—44no—
339#106—28no—
340#106—44no—
341#106—44no—
342#106—44no—
343C22⋊C8C2×Q818yes—
344C22⋊C8C2×Q818yes—
345C22⋊C8C2×Q8116yescomplemented
346#97—44no—
347C4×C8—42no—
348C4×C8—42no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
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#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24—22no—
368C24—22no—
369C24—41no—
370C24—41no—
371C24—41no—
372C24—41no—
373C24C4⋊Q814yes—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—82no—
383C2×Q8—82no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—816no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
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—416no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—84no—
431C22×C4—816no—
432C22×C4C4⋊Q818yes—
433C22×C4C4⋊Q814yes—
434C22×C4AGL(1,ℤ/8)14yes—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q814yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C2511yescharacteristic
448M16—416no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8C22×Q818yes—
454C2×C8C22×Q818yes—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4H(5,𝔽2)14yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—42no—
505C42—41no—
506C42—41no—
507C42—82no—
508C42—82no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—28no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C2×C4⋊Q814yes—
527C23C2×C4⋊Q812yes—
528C23C2×AGL(1,ℤ/8)12yes—
529C23C2×#13712yes—
530C23C23×D411yescharacteristic
531Q8—88no—
532Q8—88no—
533Q8—162no—
534Q8—162no—
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—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
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—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—816no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4C2×C4⋊Q814yes—
592C2×C4C2×C4⋊Q812yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C23×Q812yes—
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4#58512yes—
600C8—28no—
601C8—28no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—416no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22#316514yes—
617C22C22×C4⋊Q811yescharacteristic
618C22#348411yescharacteristic
619C22#348511yescharacteristic
620C22C22×AGL(1,ℤ/8)11yescharacteristic
621C22C22×#13711yescharacteristic
622C22#358511yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4#350114yes—
641C2—24no—
642C2—44no—
643C2#3539111yescharacteristic
644C2C2×#316511yescharacteristic
645C2#6066411yescharacteristic
646C2#6157111yescharacteristic
647C2#6157411yescharacteristic
648C2#6171711yescharacteristic
649C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29303C2semidirect
4. #29297C2semidirect
5. #29296C2semidirect
6. #28783C2semidirect
7. #28781C2semidirect
8. #27953C2semidirect
9. #26074C2semidirect
10. #26072C2semidirect
11. #25813C2semidirect
12. #25811C2semidirect
19. C2×#500C22semidirect
20. C2×#498C22semidirect
49. #2085C22semidirect
50. #2084C22semidirect
51. #2084C22semidirect
52. #2083C22semidirect
53. #2083C22semidirect
54. #2059C22semidirect
55. #2058C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #2057C22semidirect
59. #1992C22semidirect
60. #1990C22semidirect
61. #1990C22semidirect
62. #1944C22semidirect
63. #1943C22semidirect
64. #1942C22semidirect
65. #1941C22semidirect
78. #1847C22semidirect
135. #407Q8semidirect
248. C2×M16C2×Q8semidirect
345. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24402
#293032
#292972
#292962
#287832
#287812
#279532
#260742
#260722
#258132
#258112
#219442
#208442
#205242