Finite Group Databasebeta

← #6606558

Subgroups — #6606558

Order 512.

1941 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3338C212yesmaximal
2#36180C211yesmaximal, characteristic
3C4×#554C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#33109C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28284C211yesmaximal, characteristic, complemented
8#27648C212yesmaximal, complemented
9#25455C212yesmaximal, complemented
10#25206C214yesmaximal, complemented
11#24752C212yesmaximal, complemented
12#24682C212yesmaximal, complemented
13#24582C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic
15#22487C212yesmaximal
16C4×#462C211yesmaximal, characteristic, complemented
17C2×#1947C212yesmaximal, complemented
18C2×#554C2212yes—
19C2×#554C2211yescharacteristic
20#3338—216no—
21C2×#500C2212yes—
22C2×#484C2214yes—
23C2×#484C2214yes—
24C2×#477C2212yes—
25C2×#477C2212yes—
26C2×#467C2214yes—
27C2×#462C2211yescharacteristic
28C2×C4×Q16C2212yes—
29C2×C8×D4C2212yes—
30C22×#107C2214yes—
31C22×C4⋊C8C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2567C2212yes—
35#2552—21no—
36#2552—21no—
37#2502C2214yes—
38#2440—22no—
39#2440—22no—
40#2370—21no—
41#2369C2211yescharacteristic
42#2368—22no—
43#2360C2212yes—
44#2355C2211yescharacteristic
45#2349—21no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2211yescharacteristic
49#2345C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#125—21no—
53C4×C22⋊Q8—22no—
54C4×C22⋊Q8C2212yes—
55C2×#384C2212yes—
56C42×Q8C2211yescharacteristic
57C42×D4—21no—
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4C2212yes—
60#2083C2214yes—
61#2083C2214yes—
62#2072C2214yes—
63#2068C2214yes—
64#2068C2214yes—
65#2056C2214yes—
66#1990C2212yes—
67#1987C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2214yes—
74#1947C2212yes—
75#1947C2212yes—
76#1947C2214yes—
77#1947C2211yescharacteristic
78#1939C2212yes—
79#1928C2212yes—
80#1928C2214yes—
81#1928C2214yes—
82#1892C2214yes—
83#1878C2212yes—
84#1872C2214yes—
85#1850C2214yes—
86#1842C2211yescharacteristic
87#1824C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×#340C2214yes—
96C22×Q16C2312yes—
97C23×C8C2314yes—
98#554—24no—
99#554—28no—
100#554—28no—
101#554D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C42—22no—
130C22×C42C2312yes—
131#500—28no—
132#484—216no—
133#484—216no—
134#477—28no—
135#477—28no—
136#467—216no—
137#462—24no—
138C4×Q16—28no—
139C8×D4—28no—
140C2×#107—216no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×C4⋊C8—28no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C4×C8C2312yes—
162#398—24no—
163#398—24no—
164#395—24no—
165#391—21no—
166#391—22no—
167#391C2312yes—
168#391C2314yes—
169#391C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384—22no—
195#384—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2314yes—
199#384C2312yes—
200#384C2312yes—
201#382—21no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4C2312yes—
232C2×#95—22no—
233C43—21no—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—42no—
258C22×C8—28no—
259C22×C8—28no—
260C22×C8—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—82no—
270C4⋊Q8—82no—
271C4⋊Q8C2×D414yes—
272#125—28no—
273#125—48no—
274C22⋊Q8—24no—
275C22⋊Q8—216no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—416no—
279C22⋊Q8C2×D418yes—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—41no—
287C4×Q8—82no—
288C4×Q8C2×D414yes—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4Q1618yescomplemented
294C4×D4C2×D414yes—
295C4×D4#4114yes—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—84no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D418yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—84no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—24no—
369#107—28no—
370#107—28no—
371#107—28no—
372#107—28no—
373C4⋊C8—28no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C4⋊C8—22no—
377C4⋊C8—42no—
378C4⋊C8#4114yes—
379#103—24no—
380#103—28no—
381#103—24no—
382#103—24no—
383#103—28no—
384#103—24no—
385#103—44no—
386C22⋊C8—28no—
387C4×C8—24no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24C22×D412yes—
407C2×Q8—24no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—44no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8C22×D412yes—
425C2×D4#12114yes—
426C2×D4C2×Q1614yes—
427C2×D4#13714yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C2×Q1618yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4114yes—
467C22×C4C2511yescharacteristic
468Q16—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—416no—
478C2×C8C2×#4114yes—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4#12114yes—
517C4⋊C4C2×Q1614yes—
518C4⋊C4C22×D412yes—
519C4⋊C4C22×D414yes—
520C4⋊C42−1+412yes—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4#12118yes—
526C22⋊C4C2×Q1618yes—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—41no—
537C42—42no—
538C42—44no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—81no—
544C42—82no—
545C42—82no—
546C42C2×Q1614yes—
547C42C22×D412yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C422−1+412yes—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23C2×#12114yes—
556C23C22×Q1614yes—
557C23C2×#13714yes—
558C23C23×D411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—84no—
568Q8—164no—
569D4#462116yescomplemented
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—216no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—168no—
622C2×C4C2×#12112yes—
623C2×C4C2×#12114yes—
624C2×C4C2×#12112yes—
625C2×C4#54712yes—
626C2×C4#54712yes—
627C2×C4C22×Q1612yes—
628C2×C4C22×Q1614yes—
629C2×C4C22×Q1612yes—
630C2×C4C22×Q1612yes—
631C2×C4C2×#13712yes—
632C2×C4#57612yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C8—24no—
640C8—44no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22C2×#46218yes—
649C22C22×#12111yescharacteristic
650C22C2×#54711yescharacteristic
651C22#348911yescharacteristic
652C22C23×Q1611yescharacteristic
653C22C22×#13711yescharacteristic
654C22C2×#57611yescharacteristic
655C22#358511yescharacteristic
656C4—22no—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—82no—
668C4—84no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4C2×#46214yes—
673C4#333814yescomplemented
674C4D4×#4111yescharacteristic
675C4#348211yescharacteristic
676C4#349911yescharacteristic
677C4#351511yescharacteristic
678C2—28no—
679C2#3625911yescharacteristic
680C2C22×#46211yescharacteristic
681C2#6042811yescharacteristic
682C2C2×#333811yescharacteristic
683C2#6125411yescharacteristic
684C2#6210011yescharacteristic
685C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33198C2semidirect
5. #33109C2semidirect
6. #28689C2semidirect
7. #28284C2semidirect
8. #27648C2semidirect
9. #25455C2semidirect
10. #25206C2semidirect
11. #24752C2semidirect
12. #24682C2semidirect
13. #24582C2semidirect
16. C4×#462C2semidirect
17. C2×#1947C2semidirect
293. C4×D4Q16semidirect
569. D4#462semidirect
673. C4#3338semidirect

Maximal subgroups

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

subgroupindex
C2×#33382
#361802
C4×#5542
#331982
#331092
#286892
#282842
#276482
#254552
#252062
#247522
#246822
#245822
D4×C4⋊C82
#224872
C4×#4622
C2×#19472