Finite Group Databasebeta

← #6535601

Subgroups — #6535601

Order 512.

2343 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3276C212yesmaximal
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#34316C211yesmaximal, characteristic
4#29798C211yesmaximal, characteristic, complemented
5#29717C212yesmaximal, complemented
6#28581C212yesmaximal, complemented
7#28440C211yesmaximal, characteristic, complemented
8#28161C211yesmaximal, characteristic, complemented
9#25370C212yesmaximal, complemented
10#25095C214yesmaximal, complemented
11#24630C212yesmaximal, complemented
12#24374C214yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23432C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22430C212yesmaximal, complemented
18C22×#137C2214yes—
19C2×D4×Q8C2212yes—
20C2×#547C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22#3276—216no—
23C2×#494C2212yes—
24C2×#470C2214yes—
25C2×#462C2212yes—
26C2×#462C2214yes—
27C2×#462C2214yes—
28C2×#461C2212yes—
29C2×#452C2214yes—
30C2×#451C2214yes—
31C2×C4×SD16C2212yes—
32C2×#436C2212yes—
33C2×#419C2214yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2551—21no—
37#2551—21no—
38#2546C2212yes—
39#2440—22no—
40#2440—22no—
41#2434C2214yes—
42#2374—22no—
43#2365C2211yescharacteristic
44#2363—21no—
45#2362C2212yes—
46#2354C2211yescharacteristic
47#2354C2211yescharacteristic
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50#2345—21no—
51#2341C2212yes—
52#2335—22no—
53#2324—21no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C22⋊Q8C2212yes—
57#2297C2212yes—
58#2293C2211yescharacteristic
59#2274—21no—
60C42×D4C2211yescharacteristic
61#2084C2211yescharacteristic
62#2072C2212yes—
63#2071C2211yescharacteristic
64#2066C2212yes—
65#2056C2214yes—
66#2055C2214yes—
67#2055C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2211yescharacteristic
71#2016C2212yes—
72#2016C2214yes—
73#2016C2212yes—
74#2016C2212yes—
75#2013C2212yes—
76#2013C2212yes—
77#2013C2214yes—
78#2013C2211yescharacteristic
79#1981C2214yes—
80#1981C2212yes—
81#1956C2212yes—
82#1939C2212yes—
83#1927C2214yes—
84#1894C2214yes—
85#1878C2214yes—
86#1878C2214yes—
87#1873C2214yes—
88#1851C2214yes—
89#1831C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1728C2212yes—
96C22×#41C2312yes—
97C23×Q8C2314yes—
98C2×#137—216no—
99C2×#137—48no—
100C2×#137—48no—
101C22×Q16C2314yes—
102C22×Q16C2314yes—
103C22×Q16C2314yes—
104C22×SD16C2312yes—
105C22×SD16C2314yes—
106C22×M16C2314yes—
107D4×Q8—28no—
108D4×Q8—28no—
109#547—24no—
110#547D418yes—
111C2×C4⋊Q8—22no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—48no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2314yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2312yes—
121C2×#123—22no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2312yes—
125C2×#121—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—22no—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D411yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×#117—21no—
135C2×#117—21no—
136C2×#117C2312yes—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—22no—
140C22×C42C2312yes—
141#494—28no—
142#470—216no—
143#462—28no—
144#462—48no—
145#462—48no—
146#461—28no—
147#452—216no—
148#451—216no—
149C4×SD16—28no—
150#436—28no—
151C2×#106C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2314yes—
155#419—48no—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169#398—24no—
170#395—24no—
171#395—24no—
172#391C2312yes—
173#391C2312yes—
174#391C2314yes—
175#391C2312yes—
176#390—21no—
177#390—22no—
178#390—22no—
179#390C2311yescharacteristic
180#390C2312yes—
181#390C2311yescharacteristic
182#389—21no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389C2311yescharacteristic
192#389C2311yescharacteristic
193#388—22no—
194#388—22no—
195#388—22no—
196#388—21no—
197#388—21no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—24no—
206#385—24no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#380—22no—
214#380—22no—
215#380—21no—
216#380—22no—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—21no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4D412yes—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4C2312yes—
239C43C2311yescharacteristic
240#340C2314yes—
241#340C2312yes—
242#340C2312yes—
243#336C2311yescharacteristic
244#336C2312yes—
245#336C2311yescharacteristic
246#336C2312yes—
247#336C2314yes—
248C2×#41—24no—
249C2×#41—44no—
250C2×#41—44no—
251C2×#41C2×D418yes—
252C22×Q8—24no—
253C22×Q8—28no—
254C22×Q8—28no—
255C22×Q8—28no—
256C22×Q8—44no—
257C22×Q8—44no—
258C22×Q8—44no—
259C22×Q8C2414yes—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C22×D4—22no—
263C22×D4C2×D411yescharacteristic
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—24no—
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271#137—464no—
272C2×Q16—28no—
273C2×Q16—28no—
274C2×Q16—28no—
275C2×Q16—28no—
276C2×Q16—28no—
277C2×Q16—44no—
278C2×Q16—44no—
279C2×Q16—44no—
280C2×Q16—44no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—28no—
284C2×SD16—24no—
285C2×SD16—44no—
286C2×SD16—44no—
287C2×M16—28no—
288C2×M16—28no—
289C2×M16—44no—
290C22×C8C2414yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—24no—
294C4⋊Q8—28no—
295C4⋊Q8—24no—
296C4⋊Q8—22no—
297C4⋊Q8—24no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—48no—
302C4⋊Q8—48no—
303C4⋊Q8—48no—
304C4⋊Q8C2×D414yes—
305#123—216no—
306C22⋊Q8—24no—
307C22⋊Q8—28no—
308C22⋊Q8—28no—
309C22⋊Q8—416no—
310C22⋊Q8C2×D418yes—
311#121—28no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—22no—
319C4×D4—48no—
320C4×D4C2×D414yes—
321C4×D4C2×D414yes—
322C4×D4#4114yes—
323#117—44no—
324#117—44no—
325#117—44no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—48no—
351C2×C4⋊C4—48no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—44no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#106—24no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—44no—
395#103—28no—
396#103—24no—
397#103—24no—
398#103—28no—
399#103—24no—
400#103—28no—
401#103—44no—
402#103—44no—
403#103—44no—
404#103—44no—
405#102—24no—
406C22⋊C8—28no—
407C4×C8—24no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—21no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—21no—
419#95—24no—
420#95—24no—
421#95C2×D412yes—
422#95C2×D412yes—
423#95C2×D412yes—
424C24—21no—
425C24C22×D412yes—
426#41—416no—
427#41—88no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—48no—
443C2×Q8—416no—
444C2×Q8—416no—
445C2×Q8—48no—
446C2×Q8—84no—
447C2×Q8—88no—
448C2×Q8—88no—
449C2×Q8C22×D414yes—
450C2×Q8H(5,𝔽2)14yes—
451C2×D4—28no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—82no—
458C2×D4—82no—
459C2×D4#12114yes—
460C2×D4C22×D412yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—48no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—48no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—84no—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4114yes—
497C22×C4C2511yescharacteristic
498Q16—416no—
499Q16—416no—
500Q16—88no—
501SD16—216no—
502SD16—416no—
503M16—416no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8H(5,𝔽2)14yes—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—28no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4#12114yes—
544C4⋊C4#13714yescomplemented
545C4⋊C4C22×D412yes—
546C22⋊C4—28no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—81no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4#12118yes—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—24no—
564C42—22no—
565C42—42no—
566C42—42no—
567C42—41no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42—82no—
572C42—82no—
573C42—82no—
574C42#13714yescomplemented
575C42C22×D412yes—
576C42C2×#4112yes—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×#12114yes—
591C23C23×D411yescharacteristic
592Q8—24no—
593Q8—28no—
594Q8—24no—
595Q8—42no—
596Q8—42no—
597Q8—44no—
598Q8—44no—
599Q8—44no—
600Q8—44no—
601Q8—44no—
602Q8—816no—
603D4—24no—
604D4—28no—
605D4—42no—
606D4—88no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—48no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—88no—
655C2×C4—816no—
656C2×C4C2×#12112yes—
657C2×C4C2×#12114yes—
658C2×C4C2×#12112yes—
659C2×C4C2×#13714yes—
660C2×C4C2×#13712yes—
661C2×C4C2×#13712yes—
662C2×C4C2×#13712yes—
663C2×C4C2×#13712yes—
664C2×C4#57812yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)12yes—
668C2×C4C2×H(5,𝔽2)12yes—
669C8—24no—
670C8—44no—
671C8—42no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22C22×#12111yescharacteristic
687C22#348411yescharacteristic
688C22#349811yescharacteristic
689C22C22×#13711yescharacteristic
690C22C22×#13711yescharacteristic
691C22C2×#57811yescharacteristic
692C22#358611yescharacteristic
693C4—22no—
694C4—24no—
695C4—22no—
696C4—22no—
697C4—24no—
698C4—22no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—41no—
703C4—42no—
704C4—82no—
705C4—84no—
706C4—81no—
707C4—81no—
708C4—82no—
709C4#303914yes—
710C4#348412yes—
711C4#351912yes—
712C2—44no—
713C2—82no—
714C2#3569111yescharacteristic
715C2C2×#303911yescharacteristic
716C2#6043211yescharacteristic
717C2#6158411yescharacteristic
718C2#6160211yescharacteristic
719C2#6242411yescharacteristic
720C2#6245411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
2. D4×C4⋊Q8C2semidirect
4. #29798C2semidirect
5. #29717C2semidirect
6. #28581C2semidirect
7. #28440C2semidirect
8. #28161C2semidirect
9. #25370C2semidirect
10. #25095C2semidirect
11. #24630C2semidirect
12. #24374C2semidirect
13. #24212C2semidirect
14. #23677C2semidirect
15. #23432C2semidirect
16. #22700C2semidirect
17. #22430C2semidirect
544. C4⋊C4#137semidirect
574. C42#137semidirect

Maximal subgroups

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

subgroupindex
C2×#32762
D4×C4⋊Q82
#343162
#297982
#297172
#285812
#284402
#281612
#253702
#250952
#246302
#243742
#242122
#236772
#234322
#227002
#224302