Finite Group Databasebeta

← #6458630

Subgroups — #6458630

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#27635C214yesmaximal, complemented
4#27634C214yesmaximal, complemented
5#27633C214yesmaximal, complemented
6#27632C214yesmaximal, complemented
7#23642C212yesmaximal, complemented
8#23482C212yesmaximal
9#21172C211yesmaximal, characteristic, complemented
10#21172C211yesmaximal, characteristic, complemented
11#21059C212yesmaximal, complemented
12#21033C212yesmaximal, complemented
13#20649C211yesmaximal, characteristic, complemented
14#20649C211yesmaximal, characteristic, complemented
15#20626C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417C2212yes—
24C2×#417C2212yes—
25C2×#407C2212yes—
26C4×C22⋊Q8C2212yes—
27C4×C22⋊Q8C2212yes—
28C4×#123—21no—
29C4×#123—21no—
30C4×#121C2212yes—
31C4×#121C2212yes—
32C4×#120—21no—
33C4×#120—21no—
34#2297C2211yescharacteristic
35#2296C2211yescharacteristic
36#2290—22no—
37#2283—22no—
38#2283—22no—
39#2283C2214yes—
40#2283C2214yes—
41C2×#383—21no—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44#2280C2211yescharacteristic
45C2×#381—21no—
46C2×#380C2212yes—
47C42×D4—22no—
48C4×#117C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4—21no—
51C2×#376—21no—
52#2077C2214yes—
53#2077C2214yes—
54#2076C2214yes—
55#2075C2214yes—
56#2036C2214yes—
57#2035C2214yes—
58#2034C2214yes—
59#2033C2214yes—
60#1949C2214yescomplemented
61#1949C2214yescomplemented
62#1949C2214yescomplemented
63#1949C2214yescomplemented
64#1928C2214yescomplemented
65#1927C2214yescomplemented
66#1926C2214yescomplemented
67#1925C2214yescomplemented
68#1921C2214yescomplemented
69#1920C2214yescomplemented
70#1919C2214yescomplemented
71#1918C2214yescomplemented
72#1805C2212yescomplemented
73#1805C2212yescomplemented
74#1805C2212yescomplemented
75#1805C2212yescomplemented
76#1804C2212yescomplemented
77#1804C2212yescomplemented
78#1804C2212yescomplemented
79#1804C2212yescomplemented
80#1797C2214yes—
81#1797C2214yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1736C2212yes—
86#1728C2212yes—
87C22×#41C2311yescharacteristic
88C24×C4D411yescharacteristic
89C24×C4D411yescharacteristic
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C22×M16C2312yes—
93#525—24no—
94#525—24no—
95C2×#123—21no—
96C2×C22⋊Q8C2312yes—
97C2×#121C2312yes—
98C2×#120—21no—
99C4×#41—24no—
100C4×#41—24no—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×#117—21no—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—22no—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42—22no—
120C22×C42—21no—
121C22×C42D411yescharacteristic
122C22×C42D412yes—
123C22×C42D411yescharacteristic
124C22×C42D412yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#428—28no—
130#428—44no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135#423—24no—
136#423—24no—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141#417—28no—
142#417—44no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147#407—44no—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150#391C2312yes—
151#390C2312yes—
152#389C2312yes—
153#389C2312yes—
154#388—22no—
155#388—22no—
156#387—22no—
157#387—22no—
158#386—22no—
159#386—22no—
160#386—21no—
161#386—21no—
162#385—22no—
163#385—22no—
164#385—21no—
165#385—21no—
166#383—24no—
167#383—24no—
168#382—21no—
169#382—21no—
170#382—21no—
171#382—21no—
172#382C2312yes—
173#382C2312yes—
174#382C2312yes—
175#382C2312yes—
176#381—24no—
177#381—24no—
178#381—21no—
179#380—22no—
180#380—22no—
181#380C2312yes—
182#380C2314yes—
183#380C2312yes—
184#380C2314yes—
185#379—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—24no—
196C4×C22⋊C4—24no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205#376—24no—
206#376—24no—
207#376—21no—
208C2×#95—22no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—22no—
214C43—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2314yes—
224#336C2314yes—
225C25C2×D411yescharacteristic
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41#4114yes—
232C22×Q8C2411yescharacteristic
233C22×D4—22no—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C22×C8C2414yes—
270C22×C8C2414yes—
271#123—28no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274#121—24no—
275#121—24no—
276#120—28no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×D4—216no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4—44no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—44no—
296#117#4114yes—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—48no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—28no—
334C2×C42—28no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—48no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42#4114yes—
357C2×C42#4114yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#107—44no—
363#106—28no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369#103—28no—
370#103—44no—
371#102—28no—
372#102—44no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24—24no—
397C24—22no—
398C24—22no—
399C24—21no—
400C24—42no—
401C24C22×D412yes—
402C24C22×D411yescharacteristic
403#41—28no—
404#41—28no—
405#41—44no—
406#41—44no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×#4112yes—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4C2×#4112yes—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—416no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4#12314yes—
482C22×C4#12314yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C42−1+412yes—
497C22×C4C2511yescharacteristic
498M16—48no—
499M16—48no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4C2×#4114yes—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—48no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—48no—
554C42#12314yes—
555C42#12314yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C23—22no—
561C23—22no—
562C23—28no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—48no—
571C23—48no—
572C23—44no—
573C23—44no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23C2×#12312yes—
579C23#52512yes—
580C23#53712yes—
581C23#57514yes—
582C23C23×D411yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587D4—22no—
588D4—22no—
589D4—416no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—88no—
642C2×C4—88no—
643C2×C4C2×#12312yes—
644C2×C4C2×#12312yes—
645C2×C4C2×#12312yes—
646C2×C4C2×#12312yes—
647C2×C4C2×#12312yes—
648C2×C4#52512yes—
649C2×C4#53712yes—
650C2×C4#55214yes—
651C2×C4#55314yes—
652C2×C4#55514yes—
653C2×C4#55614yes—
654C2×C4#57514yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—44no—
662C8—44no—
663C22—22no—
664C22—22no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—416no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—82no—
680C22#248514yes—
681C22C22×#12311yescharacteristic
682C22C2×#52511yescharacteristic
683C22C2×#53711yescharacteristic
684C22C2×#57512yes—
685C22C2×#57512yes—
686C4—24no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—81no—
696C4—81no—
697C4—84no—
698C4—82no—
699C4—84no—
700C4#248514yes—
701C4C2×#55211yescharacteristic
702C4C2×#55311yescharacteristic
703C4C2×#55511yescharacteristic
704C4C2×#55611yescharacteristic
705C2—24no—
706C2—44no—
707C2—82no—
708C2C2×#248511yescharacteristic
709C2#6051111yescharacteristic
710C2#6051111yescharacteristic
711C2#6056312yes—
712C2#6091712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #27635C2semidirect
4. #27634C2semidirect
5. #27633C2semidirect
6. #27632C2semidirect
7. #23642C2semidirect
9. #21172C2semidirect
10. #21172C2semidirect
11. #21059C2semidirect
12. #21033C2semidirect
13. #20649C2semidirect
14. #20649C2semidirect
15. #20626C2semidirect
60. #1949C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1949C22semidirect
64. #1928C22semidirect
65. #1927C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
68. #1921C22semidirect
69. #1920C22semidirect
70. #1919C22semidirect
71. #1918C22semidirect
72. #1805C22semidirect
73. #1805C22semidirect
74. #1805C22semidirect
75. #1805C22semidirect
76. #1804C22semidirect
77. #1804C22semidirect
78. #1804C22semidirect
79. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#22832
#276352
#276342
#276332
#276322
#236422
#234822
#211722
#210592
#210332
#206492
#206262