Finite Group Databasebeta

← #6514195

Subgroups — #6514195

Order 512.

1962 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3255C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic
3#34946C211yesmaximal, characteristic
4#34225C211yesmaximal, characteristic, complemented
5#28281C212yesmaximal, complemented
6#28254C212yesmaximal, complemented
7#28203C212yesmaximal, complemented
8#28159C212yesmaximal, complemented
9#25229C214yesmaximal, complemented
10#25105C214yesmaximal
11#24828C212yesmaximal, complemented
12#24743C212yesmaximal, complemented
13#24695C212yesmaximal, complemented
14#24645C212yesmaximal, complemented
15#21978C211yesmaximal, characteristic, complemented
16#21870C211yesmaximal, characteristic, complemented
17#20646C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3255—28no—
22#3110—28no—
23C2×#484C2212yes—
24C2×#475C2212yescomplemented
25C2×#475C2212yescomplemented
26C2×#474C2211yescharacteristic
27C2×#474C2211yescharacteristic
28C2×#470C2214yes—
29C2×#464C2214yes—
30C2×#462C2212yescomplemented
31C2×#462C2212yescomplemented
32C2×#461C2211yescharacteristic
33C2×#461C2211yescharacteristic
34C2×#450C2212yes—
35C2×#440C2212yes—
36C2×#440C2212yes—
37C2×#432C2211yescharacteristic
38C2×#423C2211yescharacteristic
39C2×#418C2212yes—
40#2563C2211yescharacteristic
41#2563C2211yescharacteristic
42#2562—22no—
43#2561—21no—
44#2561—21no—
45#2547C2211yescharacteristic
46#2547C2211yescharacteristic
47#2503C2212yes—
48#2435C2212yes—
49#2398—21no—
50#2398—21no—
51#2370—22no—
52#2362C2214yes—
53#2356C2212yescomplemented
54#2351C2212yescomplemented
55#2347—21no—
56#2346—21no—
57#2323—22no—
58#2320—21no—
59C4×C4⋊Q8C2212yescomplemented
60C4×C4⋊Q8C2212yescomplemented
61C4×#124C2212yes—
62C4×#124C2212yes—
63#2285C2211yescharacteristic
64#2280—22no—
65#2276—21no—
66C4×#117C2211yescharacteristic
67#2085C2214yescomplemented
68#2074C2214yescomplemented
69#2068C2214yescomplemented
70#2065C2214yes—
71#2056C2214yescomplemented
72#2055C2214yes—
73#2042C2214yes—
74#2015C2214yes—
75#1993C2212yescomplemented
76#1983C2214yes—
77#1960C2212yescomplemented
78#1949C2214yescomplemented
79#1940C2212yescomplemented
80#1940C2212yescomplemented
81#1928C2214yescomplemented
82#1926C2214yes—
83#1880C2214yes—
84#1878C2214yescomplemented
85#1873C2214yes—
86#1851C2214yes—
87#1842C2212yescomplemented
88#1831C2212yescomplemented
89#1805C2214yescomplemented
90#1804C2212yes—
91#1804C2212yes—
92#1803C2212yescomplemented
93#1803C2212yescomplemented
94#1802C2212yes—
95#1802C2212yes—
96#1736C2212yes—
97C23×D4C2311yescharacteristic
98C24×C4—21no—
99C22×Q16C2314yes—
100C22×SD16C2312yes—
101C22×SD16C2312yes—
102C22×M16C2312yes—
103#533—24no—
104#533—24no—
105#533—24no—
106#533D418yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×#125—21no—
113C2×#124C2312yes—
114C2×#124C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2314yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117—21no—
135C2×#117D412yes—
136C2×#117C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C42C2311yescharacteristic
140#484—28no—
141#475—28no—
142#475—28no—
143#474—24no—
144#474—24no—
145#470—216no—
146#464—48no—
147#462—28no—
148#462—28no—
149#461—24no—
150#461—24no—
151#450—28no—
152#440—28no—
153#440—28no—
154#432—24no—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#106C2314yes—
158#423—24no—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2314yes—
165#418—44no—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#102C2314yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2314yes—
176C2×C22⋊C8C2312yes—
177C2×#97C2312yes—
178#400—22no—
179#400—22no—
180#397—21no—
181#397—21no—
182#397—21no—
183#397—21no—
184#394—22no—
185#394—22no—
186#393—21no—
187#393—21no—
188#393—21no—
189#393—21no—
190#391—22no—
191#391C2314yes—
192#391C2312yes—
193#391C2312yes—
194#390C2312yes—
195#390C2312yes—
196#389—24no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389C2314yes—
201#388—24no—
202#388—24no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#387—24no—
208#387—24no—
209#384C2312yes—
210#383—22no—
211#382—22no—
212#382C2312yes—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#381—22no—
217#380—21no—
218#380—21no—
219#380—22no—
220#380—22no—
221#380—22no—
222#380—22no—
223#379—22no—
224#379—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—24no—
232C4×C4⋊C4D412yes—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2311yescharacteristic
235C4×C4⋊C4C2311yescharacteristic
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4D411yescharacteristic
241C4×C22⋊C4D411yescharacteristic
242C4×C22⋊C4C2312yes—
243#376—22no—
244#376D411yescharacteristic
245#376D411yescharacteristic
246C43C2312yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253C25—21no—
254C22×Q8—21no—
255C22×Q8—21no—
256C22×Q8C2412yes—
257C22×Q8C2412yes—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—22no—
263C22×D4—41no—
264C22×D4—41no—
265C22×D4C2411yescharacteristic
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—48no—
273C23×C4—44no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×Q16—28no—
278C2×Q16—44no—
279C2×Q16—44no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×M16—24no—
287C2×M16—42no—
288C2×M16—42no—
289C22×C8C2414yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8C2×D414yes—
299C4⋊Q8C2×D414yes—
300#125—28no—
301#125—44no—
302#124—24no—
303#124—24no—
304#124—42no—
305#124—42no—
306C22⋊Q8—28no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—44no—
311C22⋊Q8C2×D418yes—
312C4×Q8—28no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—48no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8C2×D418yes—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—44no—
330C4×D4—44no—
331#117—28no—
332#117—44no—
333#117—44no—
334#117—42no—
335#117—42no—
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—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—24no—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D414yes—
390C2×C42#4114yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—24no—
396#107—24no—
397#106—28no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—44no—
403C4⋊C8—44no—
404#103—24no—
405#103—24no—
406#103—28no—
407#103—28no—
408#103—44no—
409#103—44no—
410#102—28no—
411#102—24no—
412#102—24no—
413#102—44no—
414C22⋊C8—24no—
415#97—24no—
416#95—22no—
417#95—24no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D412yes—
435#95C2×D412yes—
436C24—21no—
437C24—21no—
438C24—42no—
439C24—42no—
440C24—44no—
441C24—42no—
442C24—82no—
443C24C22×D411yescharacteristic
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8—82no—
459C2×Q8C22×D414yes—
460C2×D4—28no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—44no—
476C2×D4—42no—
477C2×D4H(5,𝔽2)12yes—
478C2×D42−1+412yes—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—816no—
503C22×C4#12114yes—
504C22×C4#12114yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4112yes—
516C22×C4C2511yescharacteristic
517Q16—88no—
518SD16—48no—
519SD16—48no—
520M16—84no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—48no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—84no—
556C4⋊C4C22×D414yes—
557C22⋊C4—28no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C42—28no—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—42no—
582C42—42no—
583C42—48no—
584C42—44no—
585C42—82no—
586C42—84no—
587C42#12114yes—
588C42#12114yes—
589C42C22×D412yes—
590C42C22×D412yes—
591C42C2×#4112yes—
592C42C2×#4112yes—
593C23—22no—
594C23—21no—
595C23—21no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—48no—
600C23—42no—
601C23—42no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—42no—
607C23—42no—
608C23—84no—
609C23—84no—
610C23—88no—
611C23—88no—
612C23C2×#12112yes—
613C23C23×D411yescharacteristic
614Q8—24no—
615Q8—24no—
616Q8—42no—
617Q8—42no—
618Q8—48no—
619Q8—42no—
620Q8—42no—
621Q8—42no—
622Q8—42no—
623Q8—84no—
624D4—22no—
625D4—22no—
626D4—22no—
627D4—22no—
628D4—48no—
629D4—41no—
630D4—41no—
631D4—41no—
632D4—41no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—28no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4C2×#12112yes—
680C2×C4C2×#12112yes—
681C2×C4C2×#12112yes—
682C2×C4C2×#12112yes—
683C2×C4C2×#12112yes—
684C2×C4#57514yes—
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×2−1+411yescharacteristic
689C2×C4#58511yescharacteristic
690C2×C4#58511yescharacteristic
691C8—44no—
692C8—42no—
693C8—42no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—42no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22—816no—
704C22—81no—
705C22—81no—
706C22—82no—
707C22—82no—
708C22—81no—
709C22—81no—
710C22C22×#12111yescharacteristic
711C22#348811yescharacteristic
712C22#348911yescharacteristic
713C22C2×#57512yes—
714C22#358612yes—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—44no—
724C4—44no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4—81no—
729C4—81no—
730C4#304014yes—
731C4#348811yescharacteristic
732C4#348911yescharacteristic
733C4#349211yescharacteristic
734C4#349311yescharacteristic
735C2—42no—
736C2—82no—
737C2—82no—
738C2#3551811yescharacteristic
739C2C2×#304011yescharacteristic
740C2#6042411yescharacteristic
741C2#6196712yes—
742C2#6211612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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×#3255C2semidirect
4. #34225C2semidirect
5. #28281C2semidirect
6. #28254C2semidirect
7. #28203C2semidirect
8. #28159C2semidirect
9. #25229C2semidirect
11. #24828C2semidirect
12. #24743C2semidirect
13. #24695C2semidirect
14. #24645C2semidirect
15. #21978C2semidirect
16. #21870C2semidirect
24. C2×#475C22semidirect
25. C2×#475C22semidirect
30. C2×#462C22semidirect
31. C2×#462C22semidirect
53. #2356C22semidirect
54. #2351C22semidirect
59. C4×C4⋊Q8C22semidirect
60. C4×C4⋊Q8C22semidirect
67. #2085C22semidirect
68. #2074C22semidirect
69. #2068C22semidirect
71. #2056C22semidirect
75. #1993C22semidirect
77. #1960C22semidirect
78. #1949C22semidirect
79. #1940C22semidirect
80. #1940C22semidirect
81. #1928C22semidirect
84. #1878C22semidirect
87. #1842C22semidirect
88. #1831C22semidirect
89. #1805C22semidirect
92. #1803C22semidirect
93. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32552
C2×#31102
#349462
#342252
#282812
#282542
#282032
#281592
#252292
#251052
#248282
#247432
#246952
#246452
#219782
#218702
#206462