Finite Group Databasebeta

← #6529759

Subgroups — #6529759

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic, complemented
2C2×#3079C211yesmaximal, characteristic
3#34979C211yesmaximal, characteristic
4#34209C211yesmaximal, characteristic, complemented
5#28284C212yesmaximal, complemented
6#28282C212yesmaximal, complemented
7#28206C212yesmaximal, complemented
8#28204C212yesmaximal, complemented
9#25456C212yesmaximal, complemented
10#25449C212yesmaximal, complemented
11#25276C212yesmaximal, complemented
12#25275C212yesmaximal, complemented
13#24588C214yesmaximal, complemented
14#24489C214yesmaximal
15#21974C211yesmaximal, characteristic, complemented
16#21894C211yesmaximal, characteristic, complemented
17#20737C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#519—21no—
21C2×C4×#41C2211yescharacteristic
22#3230—28no—
23C2×#500C2212yes—
24C2×#487C2212yes—
25C2×#484C2214yes—
26#3079—28no—
27C2×#477C2212yes—
28C2×#477C2212yes—
29C2×#477C2212yes—
30C2×#475C2212yes—
31C2×#476C2212yes—
32C2×#474C2212yes—
33C2×#431C2211yescharacteristic
34C2×#428C2212yes—
35C2×#423C2211yescharacteristic
36C2×#417C2212yes—
37#2611—22no—
38#2568C2211yescharacteristic
39#2547C2211yescharacteristic
40#2546C2211yescharacteristic
41#2542C2211yescharacteristic
42#2499C2212yes—
43#2432C2212yes—
44#2415—21no—
45#2415—21no—
46#2395—22no—
47#2365—21no—
48#2365—21no—
49#2365C2212yes—
50#2365C2212yes—
51#2344—22no—
52#2344C2214yes—
53#2327—22no—
54#2310—21no—
55C4×C4⋊Q8C2212yes—
56C4×C4⋊Q8C2212yes—
57C4×C4⋊Q8C2212yes—
58C4×#124C2212yes—
59C4×C22⋊Q8—21no—
60C4×C22⋊Q8—21no—
61#2285C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2085C2214yes—
64#2083C2214yes—
65#2068C2214yes—
66#2068C2214yes—
67#2067C2214yes—
68#2066C2214yes—
69#2043C2214yes—
70#2042C2214yes—
71#1988C2212yes—
72#1983C2214yes—
73#1981C2214yes—
74#1981C2212yes—
75#1949C2214yes—
76#1947C2214yes—
77#1939C2212yes—
78#1939C2212yes—
79#1928C2214yes—
80#1928C2214yes—
81#1927C2214yes—
82#1925C2214yes—
83#1894C2214yes—
84#1851C2214yes—
85#1842C2212yes—
86#1831C2212yes—
87#1825C2212yes—
88#1803C2211yescharacteristic
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1803C2212yes—
92#1802C2211yescharacteristic
93#1802C2211yescharacteristic
94#1788C2212yes—
95C4×#103C2212yes—
96C4×#102C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8—21no—
99C22×M16C2312yes—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8D418yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#124C2312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117#519—48no—
118C4×#41—24no—
119C4×#41—24no—
120C2×C4×Q8—22no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—21no—
133C2×#117—21no—
134C2×#117—24no—
135C2×#117—21no—
136C2×#117D411yescharacteristic
137C2×#117D411yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#500—28no—
144#487—28no—
145#484—216no—
146#477—28no—
147#477—28no—
148#477—28no—
149#476—28no—
150#475—28no—
151#474—28no—
152#431—24no—
153#428—28no—
154C2×#107C2314yes—
155C2×#107C2314yes—
156C2×#107C2314yes—
157C2×#106C2314yes—
158#423—24no—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2314yes—
165#417—28no—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#102C2312yes—
174C2×#102C2314yes—
175C2×#102C2312yes—
176C2×C22⋊C8C2312yes—
177C2×C4×C8C2312yes—
178#400—22no—
179#399—22no—
180#398—22no—
181#397—22no—
182#397—22no—
183#397—22no—
184#394—22no—
185#394—22no—
186#391—22no—
187#391—22no—
188#391—21no—
189#391—21no—
190#391—22no—
191#391—22no—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#390C2312yes—
196#389—21no—
197#389—21no—
198#389—22no—
199#389—22no—
200#389—22no—
201#389—22no—
202#389C2314yes—
203#389C2314yes—
204#388—22no—
205#388—22no—
206#387—24no—
207#387—24no—
208#387—24no—
209#387—24no—
210#387—22no—
211#387—22no—
212#385—22no—
213#385—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2312yes—
217#382—21no—
218#382C2312yes—
219#381—24no—
220#380—21no—
221#380—21no—
222#380—21no—
223#380C2312yes—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4D411yescharacteristic
231C4×C4⋊C4C2314yes—
232C4×C4⋊C4C2314yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4C2312yes—
240#376D412yes—
241C43C2312yes—
242#340C2312yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2314yes—
247#340C2312yes—
248#336C2314yes—
249#336C2314yes—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—41no—
255C2×#41—41no—
256C22×Q8—28no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—21no—
260C22×Q8—44no—
261C22×Q8—44no—
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×D4—21no—
265C22×D4C2411yescharacteristic
266C23×C4—22no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×M16—24no—
274C2×M16—42no—
275C2×M16—42no—
276C22×C8C2414yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—24no—
280C4⋊Q8—28no—
281C4⋊Q8—28no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—42no—
289C4⋊Q8C2×D418yes—
290C4⋊Q8C2×D418yes—
291#125—44no—
292#124—24no—
293#124—42no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—24no—
305C4×Q8—48no—
306C4×Q8—48no—
307C4×Q8—48no—
308C4×Q8—48no—
309C4×Q8—44no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—44no—
315C4×D4—44no—
316#117—24no—
317#117—22no—
318#117—22no—
319#117—416no—
320#117—42no—
321#117—44no—
322#117—44no—
323#117—44no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D418yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—44no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D414yes—
383C2×C42#4114yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—28no—
389#107—28no—
390#107—28no—
391#106—28no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396C4⋊C8—44no—
397C4⋊C8—44no—
398#103—24no—
399#103—28no—
400#103—28no—
401#103—28no—
402#103—44no—
403#103—42no—
404#103—44no—
405#102—24no—
406#102—28no—
407#102—42no—
408C22⋊C8—24no—
409C4×C8—24no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—24no—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428C24—21no—
429C24C22×D411yescharacteristic
430#41—28no—
431#41—44no—
432#41—44no—
433#41—44no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—432no—
450C2×Q8—42no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—44no—
454C2×Q8—42no—
455C2×Q82−1+412yes—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D42−1+412yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—28no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507M16—48no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—24no—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—48no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4AGL(1,ℤ/8)14yes—
552C4⋊C4C22×D418yes—
553C22⋊C4—24no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—44no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—44no—
564C42—24no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—42no—
572C42—48no—
573C42—42no—
574C42—42no—
575C42—42no—
576C42—44no—
577C42—48no—
578C42—44no—
579C42#12114yes—
580C42#12114yes—
581C42C22×D412yes—
582C42C22×D412yes—
583C42C2×#4112yes—
584C42C2×#4112yes—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—22no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23C2×#12112yes—
599C23C23×D411yescharacteristic
600Q8—22no—
601Q8—22no—
602Q8—416no—
603Q8—41no—
604Q8—41no—
605Q8—416no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—84no—
611D4—22no—
612D4—22no—
613D4—48no—
614D4—41no—
615D4—41no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—48no—
642C2×C4—416no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4C2×#12112yes—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4C2×#12112yes—
672C2×C4C2×AGL(1,ℤ/8)14yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4#57512yes—
675C2×C4#57812yes—
676C2×C4#57812yes—
677C2×C4C23×D411yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C2×C4C2×2−1+411yescharacteristic
681C2×C4#58511yescharacteristic
682C2×C4#58511yescharacteristic
683C8—42no—
684C8—42no—
685C8—44no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22C22×#12111yescharacteristic
701C22#348911yescharacteristic
702C22#348911yescharacteristic
703C22C22×AGL(1,ℤ/8)11yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57811yescharacteristic
706C22C2×#57811yescharacteristic
707C4—22no—
708C4—22no—
709C4—24no—
710C4—22no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—48no—
716C4—81no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4—82no—
722C4#303614yes—
723C4#304414yes—
724C4#348611yescharacteristic
725C4#348711yescharacteristic
726C4#349511yescharacteristic
727C4#349511yescharacteristic
728C2—42no—
729C2—82no—
730C2#3566111yescharacteristic
731C2C2×#303611yescharacteristic
732C2C2×#304411yescharacteristic
733C2#6213211yescharacteristic
734C2#6213311yescharacteristic
735C2#6214611yescharacteristic
736C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
1. C2×#3230C2semidirect
4. #34209C2semidirect
5. #28284C2semidirect
6. #28282C2semidirect
7. #28206C2semidirect
8. #28204C2semidirect
9. #25456C2semidirect
10. #25449C2semidirect
11. #25276C2semidirect
12. #25275C2semidirect
13. #24588C2semidirect
15. #21974C2semidirect
16. #21894C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#30792
#349792
#342092
#282842
#282822
#282062
#282042
#254562
#254492
#252762
#252752
#245882
#244892
#219742
#218942
#207372