Finite Group Databasebeta

← #6506763

Subgroups — #6506763

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#29303C212yesmaximal, complemented
4#29303C212yesmaximal, complemented
5#28776C214yesmaximal, complemented
6#28775C214yesmaximal, complemented
7#27953C212yesmaximal, complemented
8#26071C214yesmaximal, complemented
9#26068C214yesmaximal, complemented
10#25816C212yesmaximal, complemented
11#21940C211yesmaximal, characteristic
12#21938C211yesmaximal, characteristic
13C2×#1853C211yesmaximal, characteristic
14C2×#1852C211yesmaximal, characteristic
15#20526C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C4⋊C4C2211yescharacteristic
19C2×#501C2214yescomplemented
20C22×#107C2212yes—
21C22×#106C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2851C2212yes—
25#2851C2212yes—
26#2646—21no—
27#2646—21no—
28#2569—22no—
29#2569—22no—
30#2442C2212yes—
31#2442C2212yes—
32#2442C2212yes—
33#2442C2212yes—
34#2440—22no—
35#2440—22no—
36#2440C2214yes—
37#2440C2214yes—
38C2×#398—21no—
39C2×#395—21no—
40#2315—21no—
41#2315—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#384C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2084C2214yescomplemented
53#2083C2214yescomplemented
54#2059C2214yescomplemented
55#2059C2214yescomplemented
56#2058C2214yescomplemented
57#2057C2214yescomplemented
58#1993C2214yescomplemented
59#1992C2214yescomplemented
60#1944C2214yescomplemented
61#1944C2214yescomplemented
62#1943C2214yescomplemented
63#1943C2214yescomplemented
64#1853C2212yes—
65#1853C2212yes—
66#1853C2212yes—
67#1853C2214yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2214yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1847C2214yescomplemented
79#1808C2214yes—
80#1807C2214yes—
81#1752C2214yes—
82#1752C2214yes—
83#1749C2212yes—
84#1749C2212yes—
85C2×#340C2212yes—
86C2×#340C2212yes—
87C23×Q8—21no—
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C23×C8C2312yes—
91C23×C8C2312yes—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2314yes—
95C2×#125C2312yes—
96C2×#125C2312yes—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—48no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#501—48no—
127C2×#107—44no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106—44no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C22⋊C8Q818yescomplemented
138C2×C22⋊C8Q818yescomplemented
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2314yes—
144C2×#97C2314yes—
145#400—22no—
146#400—22no—
147#400—22no—
148#400—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#398—24no—
154#398—24no—
155#395—24no—
156#395—24no—
157#389—22no—
158#389—22no—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#389C2312yes—
164#389C2312yes—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—24no—
178#385—24no—
179#384—22no—
180#384C2314yes—
181#382C2312yes—
182#382C2312yes—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C4×C22⋊C4C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C22×Q8—24no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8—82no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×Q814yes—
248C23×C4C2×Q814yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8—42no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8C2×Q818yescomplemented
257C22×C8C2×Q818yescomplemented
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263#125—44no—
264#125—44no—
265C22⋊Q8—416no—
266C22⋊Q8—416no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—28no—
340#107—44no—
341#107—44no—
342#107—44no—
343#106—28no—
344#106—44no—
345#106—44no—
346#106—44no—
347C22⋊C8C2×Q818yes—
348C22⋊C8C2×Q818yes—
349C22⋊C8C2×Q818yescomplemented
350C22⋊C8C2×Q818yescomplemented
351#97—44no—
352#97—44no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24C4⋊Q814yes—
378C24AGL(1,ℤ/8)14yes—
379C24#13714yes—
380C24C22×D411yescharacteristic
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—84no—
386C2×Q8—816no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
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—22no—
405C22×C4—22no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—816no—
452C22×C4C4⋊Q814yes—
453C22×C4C4⋊Q814yes—
454C22×C4C4⋊Q814yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q812yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q812yes—
465C22×C4C22×Q812yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C2511yescharacteristic
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—48no—
474C2×C8—48no—
475C2×C8C22×Q818yes—
476C2×C8C22×Q818yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4H(5,𝔽2)14yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—42no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23#501116yescomplemented
543C23C2×C4⋊Q812yes—
544C23C2×C4⋊Q812yes—
545C23C2×C4⋊Q812yes—
546C23C2×AGL(1,ℤ/8)12yes—
547C23C2×AGL(1,ℤ/8)12yes—
548C23C2×AGL(1,ℤ/8)12yes—
549C23C2×#13712yes—
550C23C2×#13712yes—
551C23C2×#13712yes—
552C23C23×D411yescharacteristic
553Q8—816no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—162no—
616C2×C4—162no—
617C2×C4—162no—
618C2×C4—162no—
619C2×C4C2×C4⋊Q812yes—
620C2×C4C2×C4⋊Q812yes—
621C2×C4C2×C4⋊Q812yes—
622C2×C4C23×Q811yescharacteristic
623C2×C4C23×Q811yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4#58512yes—
626C8—28no—
627C8—28no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—416no—
647C22C2×#50114yes—
648C22C2×#50114yes—
649C22C2×#50114yes—
650C22C22×C4⋊Q811yescharacteristic
651C22#348411yescharacteristic
652C22#348511yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C22×#13711yescharacteristic
655C22#358511yescharacteristic
656C22#358611yescharacteristic
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#350112yes—
673C4#350112yes—
674C2—24no—
675C2—24no—
676C2—24no—
677C2#3539111yescharacteristic
678C2C22×#50111yescharacteristic
679C2#6066411yescharacteristic
680C2#6157111yescharacteristic
681C2#6157411yescharacteristic
682C2#6170611yescharacteristic
683C2#6170911yescharacteristic
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. #29303C2semidirect
5. #28776C2semidirect
6. #28775C2semidirect
7. #27953C2semidirect
8. #26071C2semidirect
9. #26068C2semidirect
10. #25816C2semidirect
19. C2×#501C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
54. #2059C22semidirect
55. #2059C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #1993C22semidirect
59. #1992C22semidirect
60. #1944C22semidirect
61. #1944C22semidirect
62. #1943C22semidirect
63. #1943C22semidirect
78. #1847C22semidirect
137. C2×C22⋊C8Q8semidirect
138. C2×C22⋊C8Q8semidirect
256. C22×C8C2×Q8semidirect
257. C22×C8C2×Q8semidirect
349. C22⋊C8C2×Q8semidirect
350. C22⋊C8C2×Q8semidirect
542. C23#501semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#24402
#293032
#287762
#287752
#279532
#260712
#260682
#258162
#219402
#219382
C2×#18532
C2×#18522
#205262