Finite Group Databasebeta

← #6484733

Subgroups — #6484733

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C212yesmaximal
2#36645C211yesmaximal, characteristic
3C2×C4×#125C211yesmaximal, characteristic
4#33109C212yesmaximal
5#33108C212yesmaximal
6#29757C212yesmaximal
7#29756C212yesmaximal
8#28755C212yesmaximal, complemented
9#28694C212yesmaximal, complemented
10#28660C214yesmaximal, complemented
11#28287C214yesmaximal, complemented
12#28209C214yesmaximal, complemented
13#21866C212yesmaximal
14#20810C211yesmaximal, characteristic
15C2×#533C2212yes—
16C22×#125C2211yescharacteristic
17#3236—216no—
18C2×#501C2214yes—
19C2×#491C2214yes—
20C2×#484C2214yes—
21C2×#483C2214yes—
22C2×#479C2218yes—
23C2×#432C2212yes—
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2862C2212yes—
27#2859—21no—
28#2859—21no—
29#2857—21no—
30#2857—21no—
31#2857C2212yes—
32#2857C2212yes—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×#125—21no—
38C4×#125—21no—
39C4×#125—21no—
40C4×#125—21no—
41C4×#125C2212yes—
42C4×#125C2212yes—
43C4×#125C2212yes—
44C4×#125C2212yes—
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49#2285—21no—
50#2285—21no—
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#2084C2214yes—
58#2083C2214yes—
59#2068C2218yes—
60#2066C2218yes—
61#2043C2214yes—
62#2043C2214yes—
63#1993C2214yes—
64#1981C2214yes—
65#1981C2214yes—
66#1960C2214yes—
67#1948C2214yes—
68#1947C2214yes—
69#1928C2218yes—
70#1927C2218yes—
71#1872C2214yes—
72#1872C2214yes—
73#1842C2212yes—
74#1831C2212yes—
75#1805C2218yes—
76#1797C2212yes—
77#1797C2212yes—
78#1767C2212yes—
79#1767C2212yes—
80#533—28no—
81#533—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—21no—
89C2×#125—21no—
90C2×#125—22no—
91C2×#125—24no—
92C2×#125D418yes—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8C2314yes—
99C2×#117—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C42—21no—
109C22×C42—21no—
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#501—216no—
118#491—216no—
119#484—216no—
120#483—216no—
121#479—232no—
122#432—28no—
123C2×#107C2318yes—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2314yes—
134#391—22no—
135#391—22no—
136#391C2314yes—
137#391C2314yes—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#384—21no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384C2312yes—
169#384C2312yes—
170#382—21no—
171#382—21no—
172#382—22no—
173#382—22no—
174#382C2312yes—
175#382C2312yes—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2318yes—
215#340C2318yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—22no—
219C22×Q8C2414yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8C2418yes—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—24no—
236C4⋊Q8—28no—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240#125—28no—
241#125—28no—
242#125—24no—
243#125—24no—
244#125—48no—
245#125C2×D418yes—
246#125C2×D418yes—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—48no—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×Q8—48no—
253#117—48no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
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×D414yes—
331C2×C42#4114yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—216no—
339#106—216no—
340C4⋊C8—28no—
341C4⋊C8—28no—
342C4⋊C8—44no—
343#103—216no—
344#103—216no—
345#103—48no—
346#103—48no—
347C22⋊C8—28no—
348#97—28no—
349#95—22no—
350#95—22no—
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—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365C24C22×D411yescharacteristic
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4#12114yes—
408C22×C4#12114yes—
409C22×C4#13714yes—
410C22×C4#13714yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427C2×C8—28no—
428C2×C8—28no—
429C2×C8—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4C22×D414yes—
462C4⋊C4C22×D414yes—
463C22⋊C4—48no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—44no—
488C42—48no—
489C42—44no—
490C42—44no—
491C42—44no—
492C42#12114yes—
493C42#12114yes—
494C42C22×D412yes—
495C42C22×D412yes—
496C42C2×#4112yes—
497C42C2×#4112yes—
498C422−1+414yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23C2×#12112yes—
504C23C2×#13712yes—
505C23C2×#13712yes—
506C23C23×D411yescharacteristic
507Q8—48no—
508Q8—48no—
509Q8—88no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4C2×#12112yes—
567C2×C4C2×#12112yes—
568C2×C4C2×#12112yes—
569C2×C4C2×#12112yes—
570C2×C4C2×#12112yes—
571C2×C4#53612yes—
572C2×C4#53612yes—
573C2×C4C2×#13712yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4#57512yes—
578C2×C4#57512yes—
579C2×C4C23×D411yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×2−1+412yes—
582C2×C4#58512yes—
583C8—48no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22#303914yes—
591C22C22×#12111yescharacteristic
592C22C2×#53611yescharacteristic
593C22#344911yescharacteristic
594C22C22×#13711yescharacteristic
595C22C22×#13711yescharacteristic
596C22C2×#57511yescharacteristic
597C22C2×#57511yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—44no—
611C4—44no—
612C4—82no—
613C4—82no—
614C4—84no—
615C4—84no—
616C4#323614yescomplemented
617C4#349712yes—
618C4#350712yes—
619C2—24no—
620C2#3517811yescharacteristic
621C2C2×#303911yescharacteristic
622C2#6041511yescharacteristic
623C2C2×#323611yescharacteristic
624C2#6086511yescharacteristic
625C2#6095011yescharacteristic
626C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
8. #28755C2semidirect
9. #28694C2semidirect
10. #28660C2semidirect
11. #28287C2semidirect
12. #28209C2semidirect
616. C4#3236semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
#366452
C2×C4×#1252
#331092
#331082
#297572
#297562
#287552
#286942
#286602
#282872
#282092
#218662
#208102