Finite Group Databasebeta

← #6529753

Subgroups — #6529753

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic
3#34953C211yesmaximal, characteristic
4#34225C211yesmaximal, characteristic
5#28284C212yesmaximal, complemented
6#28283C212yesmaximal, complemented
7#28206C212yesmaximal, complemented
8#28205C212yesmaximal, complemented
9#25467C214yesmaximal, complemented
10#25290C214yesmaximal, complemented
11#24598C212yesmaximal
12#24596C212yesmaximal
13#24498C212yesmaximal
14#24496C212yesmaximal
15#21978C211yesmaximal, characteristic
16#21894C211yesmaximal, characteristic
17#20740C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3237—28no—
22C2×#501C2212yes—
23C2×#491C2212yes—
24C2×#485C2212yes—
25C2×#485C2212yes—
26#3078—28no—
27C2×#479C2214yes—
28C2×#477C2211yescharacteristic
29C2×#477C2212yes—
30C2×#477C2211yescharacteristic
31C2×#475C2211yescharacteristic
32C2×#475C2212yes—
33C2×#475C2211yescharacteristic
34C2×#432C2211yescharacteristic
35C2×#428C2212yes—
36C2×#423C2211yescharacteristic
37C2×#419C2212yes—
38#2584—21no—
39#2584—21no—
40#2583—22no—
41#2567C2212yes—
42#2546C2212yes—
43#2502C2211yescharacteristic
44#2502C2211yescharacteristic
45#2434C2211yescharacteristic
46#2434C2211yescharacteristic
47#2399—21no—
48#2399—21no—
49#2370C2214yes—
50#2362—22no—
51#2356—21no—
52#2351—21no—
53#2347C2212yes—
54#2346C2212yes—
55#2323—22no—
56#2320—21no—
57C4×C4⋊Q8C2212yes—
58C4×C4⋊Q8C2212yes—
59C4×C4⋊Q8C2212yes—
60C4×C4⋊Q8C2212yes—
61#2296—22no—
62#2285C2211yescharacteristic
63#2276—21no—
64C4×#117C2211yescharacteristic
65#2084C2214yes—
66#2083C2214yes—
67#2068C2214yes—
68#2068C2214yes—
69#2066C2214yes—
70#2066C2214yes—
71#2043C2214yes—
72#2043C2214yes—
73#1988C2212yes—
74#1981C2214yes—
75#1981C2214yes—
76#1981C2212yes—
77#1948C2214yes—
78#1947C2214yes—
79#1940C2214yes—
80#1928C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1894C2212yes—
85#1894C2212yes—
86#1851C2212yes—
87#1851C2212yes—
88#1842C2212yes—
89#1831C2212yes—
90#1823C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1803C2212yes—
94#1803C2212yes—
95#1787C2212yes—
96#1763C2214yes—
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99#533—24no—
100#533—24no—
101#533—24no—
102#533D418yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125—21no—
110C2×#125C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2312yes—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117D412yes—
132C2×#117C2312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#501—28no—
138#491—28no—
139#485—28no—
140#485—28no—
141#479—216no—
142#477—28no—
143#477—24no—
144#477—24no—
145#475—28no—
146#475—24no—
147#475—24no—
148#432—24no—
149#428—28no—
150C2×#107C2314yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2314yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156#423—24no—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2314yes—
163#419—28no—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×C22⋊C8C2312yes—
175C2×#97C2312yes—
176#398—22no—
177#398—22no—
178#398—21no—
179#398—21no—
180#398—21no—
181#398—21no—
182#395—22no—
183#395—22no—
184#395—21no—
185#395—21no—
186#395—21no—
187#395—21no—
188#391—22no—
189#391—22no—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#389—22no—
195#389—22no—
196#389—24no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—22no—
204#389—22no—
205#389C2314yes—
206#389C2314yes—
207#388—24no—
208#388—24no—
209#387—24no—
210#387—24no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#384C2312yes—
216#384C2312yes—
217#384C2312yes—
218#383—22no—
219#382—21no—
220#382—22no—
221#380—21no—
222#380C2312yes—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—24no—
232C4×C4⋊C4D411yescharacteristic
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4C2314yes—
235C4×C4⋊C4C2314yes—
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2312yes—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4C2311yescharacteristic
240C4×C22⋊C4C2311yescharacteristic
241#376—22no—
242#376D411yescharacteristic
243#376D411yescharacteristic
244C43C2312yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×Q8—24no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—42no—
259C22×Q8C2411yescharacteristic
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—42no—
281C4⋊Q8—42no—
282C4⋊Q8C2×D414yes—
283C4⋊Q8C2×D414yes—
284#125—24no—
285#125—44no—
286#125C2×D418yes—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—44no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—48no—
308C4×Q8—84no—
309#117—24no—
310#117—28no—
311#117—44no—
312#117—82no—
313#117C2×D418yes—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—44no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D414yes—
372C2×C42C2×D412yes—
373C2×C42#4114yes—
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#107—24no—
380#107—24no—
381#106—28no—
382#106—24no—
383#106—24no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—22no—
387C4⋊C8—22no—
388C4⋊C8—44no—
389C4⋊C8—44no—
390#103—28no—
391#103—28no—
392#103—28no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—44no—
398#103—44no—
399#103—44no—
400C22⋊C8—24no—
401#97—24no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95C2×D411yescharacteristic
417#95C2×D412yes—
418#95C2×D412yes—
419#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422C24C22×D411yescharacteristic
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—28no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—44no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q82−1+414yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4#12114yes—
470C22×C4#12114yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2511yescharacteristic
483M16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4C22×D414yes—
530C4⋊C4C22×D414yes—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—82no—
537C42—28no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—42no—
548C42—42no—
549C42—48no—
550C42—82no—
551C42—84no—
552C42#12114yes—
553C42#12114yes—
554C42#13718yes—
555C42C22×D412yes—
556C42C22×D412yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—22no—
563C23C2×#12112yes—
564C23C23×D411yescharacteristic
565Q8—24no—
566Q8—24no—
567Q8—48no—
568Q8—48no—
569Q8—42no—
570Q8—42no—
571Q8—48no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—42no—
577Q8—42no—
578Q8—84no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—84no—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#13714yes—
632C2×C4C2×#13714yes—
633C2×C4C2×#13714yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×2−1+412yes—
637C2×C4#58512yes—
638C8—42no—
639C8—42no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—42no—
645C22—42no—
646C22C22×#12111yescharacteristic
647C22#348912yes—
648C22C22×#13712yes—
649C22#358511yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—24no—
655C4—24no—
656C4—22no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—84no—
668C4#303914yes—
669C4#348611yescharacteristic
670C4#348711yescharacteristic
671C4#349512yes—
672C2—42no—
673C2#3566111yescharacteristic
674C2C2×#303911yescharacteristic
675C2#6042511yescharacteristic
676C2#6211312yes—
677C2#6211412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28284C2semidirect
6. #28283C2semidirect
7. #28206C2semidirect
8. #28205C2semidirect
9. #25467C2semidirect
10. #25290C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
C2×#30782
#349532
#342252
#282842
#282832
#282062
#282052
#254672
#252902
#245982
#245962
#244982
#244962
#219782
#218942
#207402