Finite Group Databasebeta

← #6528580

Subgroups — #6528580

Order 512.

1599 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#29335C212yesmaximal, complemented
4#29330C212yesmaximal
5#27978C212yesmaximal
6#27972C212yesmaximal, complemented
7#27765C214yesmaximal, complemented
8#27756C212yesmaximal, complemented
9#27749C212yesmaximal, complemented
10#27741C214yesmaximal, complemented
11#27737C212yesmaximal, complemented
12#27727C212yesmaximal, complemented
13#21867C211yesmaximal, characteristic
14#21860C211yesmaximal, characteristic, complemented
15#21394C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic
17#20535C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×#125—21no—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2212yes—
24#2850C2212yes—
25#2664—22no—
26#2568C2211yescharacteristic
27#2565C2212yes—
28#2552—21no—
29#2552—21no—
30#2552—22no—
31#2552C2212yes—
32#2552C2212yes—
33#2552C2214yes—
34#2547C2211yescharacteristic
35#2546C2211yescharacteristic
36#2545C2212yes—
37#2542C2211yescharacteristic
38#2498—22no—
39#2417—21no—
40#2417—21no—
41C2×#398—22no—
42#2323—21no—
43#2323—21no—
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2099C2214yes—
52#2072C2214yes—
53#2072C2214yes—
54#2072C2214yes—
55#2071C2214yes—
56#2043C2214yes—
57#2042C2214yes—
58#1992C2212yes—
59#1990C2212yes—
60#1989C2212yes—
61#1988C2212yes—
62#1983C2212yes—
63#1981C2212yes—
64#1959C2214yes—
65#1959C2212yes—
66#1956C2214yes—
67#1956C2212yes—
68#1935C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1933C2214yes—
72#1874C2211yescharacteristic
73#1874C2212yes—
74#1874C2211yescharacteristic
75#1872C2211yescharacteristic
76#1872C2212yes—
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1836C2214yes—
83#1834C2214yes—
84#1803C2212yes—
85#1803C2214yes—
86#1802C2212yes—
87#1771C2214yes—
88#1770C2212yes—
89C8×C4⋊C4C2212yes—
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95#533—24no—
96#533—24no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100#529—24no—
101#529—24no—
102C2×#126—21no—
103C2×#125—22no—
104C2×#125—24no—
105C2×#125—22no—
106C2×#125—21no—
107C2×#125—48no—
108C2×#125C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×#124C2312yes—
111C2×#123—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2311yescharacteristic
117C2×#121C2311yescharacteristic
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#432—24no—
139#431—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2312yes—
155C2×#97C2312yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—24no—
164#398—24no—
165#398—24no—
166#398—24no—
167#398—22no—
168#397—22no—
169#396—22no—
170#396—22no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—24no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#386—22no—
195#386—22no—
196#384—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#384C2314yes—
204#384C2314yes—
205#382C2312yes—
206#380—22no—
207#380—22no—
208#380—21no—
209#380—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C2×#95—21no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×Q8C2412yes—
238C22×D4—21no—
239C22×D4C2411yescharacteristic
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257#126—44no—
258#125—22no—
259#125—22no—
260#125—416no—
261#125—48no—
262#125—48no—
263#125—44no—
264#124—22no—
265#124—22no—
266#124—44no—
267#123—44no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275#121—22no—
276#121—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—44no—
285#117—44no—
286#117—44no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4—84no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42#4114yes—
356C2×C42#4114yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C4⋊C8—24no—
362C4⋊C8—24no—
363#103—42no—
364#103—44no—
365#103—42no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#102—42no—
371#102—44no—
372#102—42no—
373C22⋊C8—22no—
374C22⋊C8—22no—
375C22⋊C8—24no—
376#97—24no—
377#97—44no—
378C4×C8—24no—
379C4×C8—44no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D411yescharacteristic
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4#12414yes—
449C22×C4#12414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4AGL(1,ℤ/8)14yes—
514C4⋊C4#13714yes—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42#12414yes—
534C42#12414yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C2×#12412yes—
551C23#57712yes—
552C23#57812yes—
553C23C23×D411yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—44no—
557Q8—44no—
558Q8—84no—
559D4—42no—
560D4—42no—
561D4—84no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—84no—
614C2×C4C2×#12412yes—
615C2×C4C2×#12412yes—
616C2×C4C2×#12412yes—
617C2×C4C2×#12412yes—
618C2×C4C2×#12412yes—
619C2×C4C2×AGL(1,ℤ/8)14yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×#13714yes—
622C2×C4C2×#13712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4#58512yes—
626C2×C4#58512yes—
627C8—42no—
628C8—42no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#313714yes—
644C22C22×#12411yescharacteristic
645C22#348511yescharacteristic
646C22#348711yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57711yescharacteristic
650C22C2×#57811yescharacteristic
651C4—24no—
652C4—24no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4#314114yes—
668C4#351212yes—
669C4#351212yes—
670C2—24no—
671C2—161no—
672C2#3564911yescharacteristic
673C2C2×#313711yescharacteristic
674C2C2×#314111yescharacteristic
675C2#6173511yescharacteristic
676C2#6173711yescharacteristic
677C2#6190011yescharacteristic
678C2#6190111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #29335C2semidirect
6. #27972C2semidirect
7. #27765C2semidirect
8. #27756C2semidirect
9. #27749C2semidirect
10. #27741C2semidirect
11. #27737C2semidirect
12. #27727C2semidirect
14. #21860C2semidirect
15. #21394C2semidirect
17. #20535C2semidirect

Maximal subgroups

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

subgroupindex
#352412
C2×#25522
#293352
#293302
#279782
#279722
#277652
#277562
#277492
#277412
#277372
#277272
#218672
#218602
#213942
#213882
#205352