Finite Group Databasebeta

← #6447024

Subgroups — #6447024

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2D4×#117C211yesmaximal, characteristic, complemented
3#24978C214yesmaximal, complemented
4#24977C214yesmaximal, complemented
5#24974C214yesmaximal, complemented
6#24973C214yesmaximal, complemented
7#23642C212yesmaximal, complemented
8#23480C212yesmaximal, complemented
9#21190C211yesmaximal, characteristic, complemented
10#21172C211yesmaximal, characteristic, complemented
11#20912C212yesmaximal, complemented
12#20765C211yesmaximal, characteristic, complemented
13#20702C212yesmaximal, complemented
14#20646C211yesmaximal, characteristic, complemented
15C4×#418C211yesmaximal, characteristic, complemented
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#418C2211yescharacteristic
23C2×#418C2211yescharacteristic
24C2×#418C2212yes—
25C2×#407C2212yes—
26C2×#407C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2535—21no—
29#2535—22no—
30#2535—21no—
31#2535C2214yes—
32#2535C2212yes—
33#2535C2212yes—
34C2×#397—21no—
35C2×#392—21no—
36#2354C2212yescomplemented
37#2352C2212yescomplemented
38#2352C2212yescomplemented
39D4×C4⋊C4C2212yescomplemented
40#2344—21no—
41#2342—22no—
42D4×C22⋊C4—21no—
43#2293—21no—
44#2290—22no—
45C2×#386—22no—
46#2285C2211yescharacteristic
47C2×#382C2211yescharacteristic
48#2280C2212yes—
49C2×#380C2212yes—
50C42×D4—21no—
51C4×#117C2211yescharacteristic
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2076C2214yescomplemented
55#2075C2214yescomplemented
56#2034C2214yescomplemented
57#2034C2214yescomplemented
58#2033C2214yescomplemented
59#2033C2214yescomplemented
60#1955C2212yescomplemented
61#1955C2212yescomplemented
62#1954C2214yescomplemented
63#1949C2214yescomplemented
64#1949C2214yescomplemented
65#1875C2214yescomplemented
66#1875C2214yescomplemented
67#1874C2214yescomplemented
68#1874C2214yescomplemented
69#1836C2214yescomplemented
70#1835C2212yescomplemented
71#1835C2212yescomplemented
72#1825C2214yescomplemented
73#1825C2214yescomplemented
74#1824C2214yescomplemented
75#1823C2214yescomplemented
76#1804C2212yescomplemented
77#1804C2212yescomplemented
78#1802C2214yescomplemented
79#1762C2214yescomplemented
80C4×#102C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1736C2211yescharacteristic
84#1736C2212yes—
85#1736C2212yes—
86#1736C2211yescharacteristic
87C23×D4—22no—
88C23×D4D411yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C24×C4C2312yes—
92C22×M16C2311yescharacteristic
93C22×M16C2311yescharacteristic
94C22×M16C2312yes—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121—28no—
98C2×#121—44no—
99C2×#121C2314yes—
100C2×#121C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D412yes—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—28no—
119C2×#117—44no—
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D412yes—
135C22×C42—22no—
136C22×C42C2311yescharacteristic
137#428—44no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140#423—24no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143#418—24no—
144#418—24no—
145#418—44no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#407—44no—
151#407—44no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C4×M16—24no—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159#397—22no—
160#397—24no—
161#397—22no—
162#392—22no—
163#392—24no—
164#392—22no—
165#390—22no—
166#389—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—22no—
173#386—22no—
174#386—22no—
175#386—24no—
176#386—24no—
177#386—24no—
178#386—24no—
179#385—22no—
180#385—22no—
181#384C2312yes—
182#382—21no—
183#382—21no—
184#382C2311yescharacteristic
185#382C2311yescharacteristic
186#382C2312yes—
187#382C2312yes—
188#381—22no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199#379—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211#376—21no—
212C2×#95—21no—
213C2×#95—22no—
214C2×#95—21no—
215C43—21no—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25—21no—
227C25C2×D412yes—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—44no—
238C22×D4—48no—
239C22×D4—48no—
240C22×D4—84no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C22×D4#4114yes—
245C22×D4C2412yes—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—84no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—24no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—24no—
276C2×M16—24no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2414yes—
280#121—28no—
281#121—28no—
282#121—44no—
283#121—44no—
284#121—48no—
285#121—48no—
286C4×D4—24no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—48no—
300C4×D4—44no—
301C4×D4—44no—
302#117—22no—
303#117—24no—
304#117—22no—
305#117—24no—
306#117—22no—
307#117—24no—
308#117—24no—
309#117—48no—
310#117—48no—
311#117—42no—
312#117—42no—
313#117#4114yes—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—48no—
355C2×C22⋊C4—48no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—84no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—48no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42#4114yes—
373C2×C42#4114yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—44no—
377#106—44no—
378C4⋊C8—24no—
379C4⋊C8—24no—
380#102—28no—
381#102—28no—
382#102—44no—
383#102—44no—
384C22⋊C8—44no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387C22⋊C8—44no—
388#97—24no—
389C4×C8—24no—
390#95—24no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95—21no—
406C24—24no—
407C24—24no—
408C24—22no—
409C24—24no—
410C24—24no—
411C24—24no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—44no—
416C24—44no—
417C24—42no—
418C24—42no—
419C24C22×D412yes—
420C24C22×D412yes—
421C24C22×D411yescharacteristic
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—416no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—44no—
446C2×D4—84no—
447C2×D4—84no—
448C2×D4—832no—
449C2×D4C2×#4114yes—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—816no—
499C22×C4#12314yes—
500C22×C4#12314yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4C2511yescharacteristic
515M16—48no—
516M16—44no—
517M16—44no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—24no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—24no—
526C2×C8—24no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—82no—
547C4⋊C4C2×#4114yes—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—84no—
566C22⋊C4—84no—
567C42—24no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42—44no—
572C42—42no—
573C42—82no—
574C42#12314yes—
575C42#12314yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C42C2×#4112yes—
579C42C2×#4112yes—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—24no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—48no—
597C23—48no—
598C23—48no—
599C23—416no—
600C23—44no—
601C23—42no—
602C23—48no—
603C23—48no—
604C23—42no—
605C23—44no—
606C23—84no—
607C23—84no—
608C23—84no—
609C23—84no—
610C23—84no—
611C23—88no—
612C23C2×#12312yes—
613C23#54612yes—
614C23#54612yes—
615C23#57514yes—
616C23C23×D411yescharacteristic
617D4—28no—
618D4—24no—
619D4—24no—
620D4—48no—
621D4—44no—
622D4—48no—
623D4—42no—
624D4—42no—
625D4—816no—
626D4—816no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—48no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—88no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4—88no—
671C2×C4—84no—
672C2×C4—88no—
673C2×C4C2×#12312yes—
674C2×C4C2×#12312yes—
675C2×C4C2×#12312yes—
676C2×C4C2×#12312yes—
677C2×C4C2×#12312yes—
678C2×C4#52914yes—
679C2×C4#54014yes—
680C2×C4#54612yes—
681C2×C4#54612yes—
682C2×C4#55314yes—
683C2×C4#55514yes—
684C2×C4C22×#4111yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C8—44no—
691C8—42no—
692C8—42no—
693C22—22no—
694C22—22no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—42no—
709C22—48no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—816no—
714C22—816no—
715C22—82no—
716C22—82no—
717C22—84no—
718C22#242014yes—
719C22C22×#12311yescharacteristic
720C22C2×#54611yescharacteristic
721C22C2×#54611yescharacteristic
722C22C2×#57512yes—
723C22#358512yes—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—82no—
732C4—82no—
733C4—84no—
734C4—82no—
735C4—82no—
736C4#242014yes—
737C4C2×#52911yescharacteristic
738C4C2×#54011yescharacteristic
739C4C2×#55311yescharacteristic
740C4C2×#55511yescharacteristic
741C2—24no—
742C2—44no—
743C2—82no—
744C2—84no—
745C2C2×#242011yescharacteristic
746C2#6051111yescharacteristic
747C2#6051811yescharacteristic
748C2#6115612yes—
749C2#6116412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×#117C2semidirect
3. #24978C2semidirect
4. #24977C2semidirect
5. #24974C2semidirect
6. #24973C2semidirect
7. #23642C2semidirect
8. #23480C2semidirect
9. #21190C2semidirect
10. #21172C2semidirect
11. #20912C2semidirect
12. #20765C2semidirect
13. #20702C2semidirect
14. #20646C2semidirect
15. C4×#418C2semidirect
36. #2354C22semidirect
37. #2352C22semidirect
38. #2352C22semidirect
39. D4×C4⋊C4C22semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2075C22semidirect
56. #2034C22semidirect
57. #2034C22semidirect
58. #2033C22semidirect
59. #2033C22semidirect
60. #1955C22semidirect
61. #1955C22semidirect
62. #1954C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1875C22semidirect
66. #1875C22semidirect
67. #1874C22semidirect
68. #1874C22semidirect
69. #1836C22semidirect
70. #1835C22semidirect
71. #1835C22semidirect
72. #1825C22semidirect
73. #1825C22semidirect
74. #1824C22semidirect
75. #1823C22semidirect
76. #1804C22semidirect
77. #1804C22semidirect
78. #1802C22semidirect
79. #1762C22semidirect
80. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
D4×#1172
#249782
#249772
#249742
#249732
#236422
#234802
#211902
#211722
#209122
#207652
#207022
#206462
C4×#4182