Finite Group Databasebeta

← #6629696

Subgroups — #6629696

Order 512.

2282 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3097C212yesmaximal
2#35316C211yesmaximal, characteristic
3C4×#520C211yesmaximal, characteristic
4#33119C214yesmaximal, complemented
5#33117C214yesmaximal, complemented
6#33114C212yesmaximal, complemented
7#33114C212yesmaximal, complemented
8#33109C212yesmaximal, complemented
9#33108C212yesmaximal, complemented
10#21259C211yesmaximal, characteristic, complemented
11C2×#1981C211yesmaximal, characteristic, complemented
12#21170C211yesmaximal, characteristic, complemented
13#21169C211yesmaximal, characteristic, complemented
14#21066C212yesmaximal, complemented
15#21064C212yesmaximal, complemented
16#20913C212yesmaximal
17#20629C211yesmaximal, characteristic
18C2×#536C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23#3097—216no—
24C2×#484C2214yes—
25C2×#485C2214yes—
26C2×#485C2214yes—
27C2×#483C2214yes—
28C2×#428C2212yes—
29C2×#427C2212yes—
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C2×#419C2212yes—
33C2×#419C2212yes—
34C22×#103C2212yes—
35C22×#103C2212yes—
36C2×#408C2212yes—
37#2857C2212yes—
38#2857C2212yes—
39#2621C2214yes—
40#2547—21no—
41#2547—21no—
42#2546—21no—
43#2546—21no—
44#2486—22no—
45#2486—22no—
46#2310C2212yes—
47#2285C2211yescharacteristic
48#2276—21no—
49#2275—22no—
50#2275—21no—
51#2274C2211yescharacteristic
52#2274C2211yescharacteristic
53#2274C2212yes—
54#2274C2212yes—
55C42×Q8—21no—
56C42×D4—22no—
57C42×D4—21no—
58C4×#117C2211yescharacteristic
59C4×#117C2212yes—
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2212yes—
62#1981C2212yes—
63#1981C2212yes—
64#1981C2214yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1949C2212yes—
70#1949C2214yes—
71#1949C2212yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1947C2211yescharacteristic
76#1947C2212yes—
77#1947C2211yescharacteristic
78#1928C2214yes—
79#1928C2214yes—
80#1928C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1927C2214yes—
85#1927C2214yes—
86#1873C2214yes—
87#1873C2214yes—
88#1872C2214yes—
89#1872C2214yes—
90#1798C2214yes—
91#1797C2212yes—
92#1797C2212yes—
93#1736C2212yes—
94#1735C2212yes—
95#1729C2212yes—
96C22×#41C2311yescharacteristic
97C23×Q8C2312yes—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#536—28no—
101#536—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#124—22no—
110C2×#123—24no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2314yes—
113#520—24no—
114#520—24no—
115#520—24no—
116#520D418yes—
117C4×#41—24no—
118C4×#41—24no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4D412yes—
128C2×C4×D4D412yes—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×#117C2312yes—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C42—22no—
145C22×C42—21no—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148#485—216no—
149#485—216no—
150#484—216no—
151#483—216no—
152#428—28no—
153#427—28no—
154C2×#107C2312yes—
155C2×#107C2314yes—
156C2×#107C2312yes—
157C2×#106C2312yes—
158C2×#106C2314yes—
159C2×#106C2312yes—
160#423—24no—
161C2×C4⋊C8—24no—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167#419—28no—
168#419—28no—
169C2×#103—28no—
170C2×#103—28no—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179#408—28no—
180C2×C22⋊C8C2314yes—
181C2×C22⋊C8C2314yes—
182#398—24no—
183#398—24no—
184#397—24no—
185#397—24no—
186#391—22no—
187#391—22no—
188#391C2314yes—
189#391C2314yes—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2312yes—
197#381—24no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—21no—
203#380—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238#376—22no—
239#376—21no—
240C2×#95—22no—
241C2×#95—21no—
242C43—22no—
243C43—21no—
244#340C2312yes—
245#340C2312yes—
246#340C2314yes—
247#340C2314yes—
248#340C2312yes—
249#340C2312yes—
250#336C2314yes—
251#336C2314yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41C2×D414yes—
257C2×#41#4114yes—
258C22×Q8—22no—
259C22×Q8—24no—
260C22×Q8—24no—
261C22×Q8—24no—
262C22×Q8—24no—
263C22×Q8—24no—
264C22×Q8—24no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2414yes—
267C22×D4C2×D412yes—
268C22×D4C2×D411yescharacteristic
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—22no—
274C23×C4C2×D412yes—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C2×M16—24no—
281C2×M16—24no—
282C2×M16—24no—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—48no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296#124—48no—
297#123—416no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8—28no—
301C22⋊Q8—28no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×D4—216no—
307C4×D4—28no—
308C4×D4—216no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—24no—
313C4×D4—24no—
314#117—22no—
315#117—22no—
316#117—24no—
317#117—22no—
318#117—22no—
319#117—24no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117—24no—
324#117—24no—
325#117C2×D414yes—
326#117C2×D418yes—
327#117#4114yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D418yes—
355C2×C4⋊C4#4114yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—82no—
392C2×C42—84no—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42#4114yes—
398C2×C42#4114yes—
399C2×C42C2412yes—
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—24no—
403#107—24no—
404#107—28no—
405#106—24no—
406#106—24no—
407#106—28no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—28no—
414#103—28no—
415#103—28no—
416#103—28no—
417#103—28no—
418#103—28no—
419#103—28no—
420#103—28no—
421C22⋊C8—28no—
422C22⋊C8—28no—
423#95—21no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—21no—
429#95—22no—
430#95—22no—
431#95—21no—
432#95—21no—
433#95—22no—
434#95—22no—
435#95—24no—
436#95—24no—
437#95—24no—
438#95—24no—
439C24C22×D412yes—
440C24C22×D411yescharacteristic
441#41—28no—
442#41—28no—
443#41—28no—
444#41—28no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—48no—
457C2×Q8—48no—
458C2×Q8—48no—
459C2×Q8—48no—
460C2×Q8#12114yes—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4#12118yes—
466C2×D4#12114yes—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4#12314yes—
506C22×C4#12314yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D414yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C2×#4114yes—
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4C2×#4112yes—
523C22×C4C2511yescharacteristic
524M16—48no—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—24no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—48no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—48no—
556C4⋊C4—48no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4C22×D414yes—
564C4⋊C4C22×D412yes—
565C4⋊C4C22×D412yes—
566C4⋊C4C2×#4112yes—
567C4⋊C4C2×#4114yes—
568C4⋊C4C2×#4112yes—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—44no—
577C22⋊C4—48no—
578C22⋊C4—48no—
579C42—28no—
580C42—28no—
581C42—24no—
582C42—24no—
583C42—24no—
584C42—24no—
585C42—22no—
586C42—22no—
587C42—44no—
588C42—44no—
589C42—42no—
590C42—42no—
591C42—81no—
592C42—81no—
593C42—82no—
594C42—82no—
595C42—82no—
596C42—82no—
597C42—82no—
598C42#12314yes—
599C42#12314yes—
600C42C2×#4112yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C42C2×#4112yes—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—22no—
612C23—22no—
613C23—22no—
614C23C2×#12114yes—
615C23C2×#12112yes—
616C23C2×#12312yes—
617C23#52512yes—
618C23C23×D411yescharacteristic
619Q8—28no—
620Q8—44no—
621Q8—44no—
622Q8—44no—
623Q8—44no—
624Q8—88no—
625D4—216no—
626D4—28no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—48no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—88no—
678C2×C4—84no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4C2×#12114yes—
682C2×C4C2×#12112yes—
683C2×C4C2×#12312yes—
684C2×C4C2×#12312yes—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4C2×#12312yes—
688C2×C4#52512yes—
689C2×C4#52914yes—
690C2×C4#52914yes—
691C2×C4#53214yes—
692C2×C4#53214yes—
693C2×C4#57712yes—
694C2×C4#57812yes—
695C2×C4C23×D411yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C22×#4111yescharacteristic
701C8—42no—
702C8—42no—
703C8—44no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—24no—
711C22—24no—
712C22—24no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22#285414yes—
718C22C22×#12111yescharacteristic
719C22C22×#12311yescharacteristic
720C22C2×#52511yescharacteristic
721C22C2×#57711yescharacteristic
722C22C2×#57811yescharacteristic
723C22#358611yescharacteristic
724C22#358611yescharacteristic
725C4—24no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—44no—
735C4—42no—
736C4—42no—
737C4—84no—
738C4—82no—
739C4—82no—
740C4#285414yes—
741C4#309714yescomplemented
742C4C2×#52911yescharacteristic
743C4C2×#52911yescharacteristic
744C4C2×#53211yescharacteristic
745C4C2×#53211yescharacteristic
746C2—24no—
747C2—44no—
748C2C2×#285411yescharacteristic
749C2#6042911yescharacteristic
750C2#6043211yescharacteristic
751C2C2×#309711yescharacteristic
752C2#6051911yescharacteristic
753C2#6059111yescharacteristic
754C2#6059411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33119C2semidirect
5. #33117C2semidirect
6. #33114C2semidirect
7. #33114C2semidirect
8. #33109C2semidirect
9. #33108C2semidirect
10. #21259C2semidirect
11. C2×#1981C2semidirect
12. #21170C2semidirect
13. #21169C2semidirect
14. #21066C2semidirect
15. #21064C2semidirect
741. C4#3097semidirect

Maximal subgroups

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

subgroupindex
C2×#30972
#353162
C4×#5202
#331192
#331172
#331142
#331092
#331082
#212592
C2×#19812
#211702
#211692
#210662
#210642
#209132
#206292