Finite Group Databasebeta

← #6447023

Subgroups — #6447023

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2D4×#117C211yesmaximal, characteristic
3#24980C214yesmaximal, complemented
4#24979C214yesmaximal, complemented
5#24976C214yesmaximal, complemented
6#24975C214yesmaximal, complemented
7#23642C212yesmaximal, complemented
8#23481C212yesmaximal, complemented
9#21190C211yesmaximal, characteristic
10#21172C211yesmaximal, characteristic, complemented
11#20911C212yesmaximal, complemented
12#20765C211yesmaximal, characteristic, complemented
13#20702C212yesmaximal, complemented
14#20647C211yesmaximal, characteristic, complemented
15C4×#419C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#419C2211yescharacteristic
23C2×#419C2211yescharacteristic
24C2×#419C2212yes—
25C2×#407C2212yes—
26C2×#407C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2536—21no—
29#2536—22no—
30#2536—21no—
31#2536C2214yes—
32#2536C2212yes—
33#2536C2212yes—
34C2×#398—21no—
35C2×#394—21no—
36#2354C2212yes—
37#2352C2212yes—
38#2352C2212yes—
39D4×C4⋊C4C2212yes—
40#2344—21no—
41#2342—22no—
42D4×C22⋊C4—21no—
43#2293—21no—
44#2290—22no—
45C2×#388—22no—
46#2285C2211yescharacteristic
47C2×#382C2211yescharacteristic
48#2280C2212yes—
49C2×#380C2212yes—
50C42×D4—21no—
51C4×#117C2211yescharacteristic
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2076C2214yescomplemented
55#2075C2214yescomplemented
56#2036C2214yescomplemented
57#2036C2214yescomplemented
58#2035C2214yescomplemented
59#2035C2214yescomplemented
60#1955C2212yes—
61#1955C2212yes—
62#1954C2214yes—
63#1949C2214yescomplemented
64#1949C2214yescomplemented
65#1873C2214yescomplemented
66#1873C2214yescomplemented
67#1872C2214yescomplemented
68#1872C2214yescomplemented
69#1836C2214yescomplemented
70#1835C2212yescomplemented
71#1835C2212yescomplemented
72#1825C2214yescomplemented
73#1825C2214yescomplemented
74#1824C2214yescomplemented
75#1823C2214yescomplemented
76#1805C2212yescomplemented
77#1805C2212yescomplemented
78#1803C2214yescomplemented
79#1763C2214yescomplemented
80C4×#103C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1736C2211yescharacteristic
84#1736C2212yes—
85#1736C2212yes—
86#1736C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4C2312yes—
90C22×M16C2311yescharacteristic
91C22×M16C2311yescharacteristic
92C22×M16C2312yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D412yes—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—28no—
117C2×#117—44no—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4D412yes—
134C22×C42—22no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#428—44no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140#423—24no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143#419—24no—
144#419—24no—
145#419—44no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
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#398—22no—
160#398—24no—
161#398—22no—
162#394—22no—
163#394—24no—
164#394—22no—
165#390—22no—
166#389—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—24no—
172#388—24no—
173#388—24no—
174#388—24no—
175#387—22no—
176#387—22no—
177#386—22no—
178#386—22no—
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—
226C25C2×D412yes—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8#4114yes—
234C22×Q8C2412yes—
235C22×D4—24no—
236C22×D4—44no—
237C22×D4C2×D411yescharacteristic
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—24no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—22no—
292C4×D4—48no—
293#117—22no—
294#117—24no—
295#117—22no—
296#117—24no—
297#117—22no—
298#117—24no—
299#117—24no—
300#117—48no—
301#117—48no—
302#117—42no—
303#117—42no—
304#117#4114yes—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4—48no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—48no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42#4114yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—44no—
374#106—44no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381C22⋊C8—44no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385#97—24no—
386C4×C8—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—21no—
403C24—24no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24C22×D412yes—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—28no—
416C2×Q8—28no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8C2×#4114yes—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—416no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4#12314yes—
483C22×C4#12314yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4C2511yescharacteristic
498M16—48no—
499M16—44no—
500M16—44no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—82no—
534C4⋊C4C2×#4114yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—42no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—24no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—82no—
561C42#12314yes—
562C42#12314yes—
563C42C2×#4112yes—
564C42C2×#4112yes—
565C42C2×#4112yes—
566C42C2×#4112yes—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23—48no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23C2×#12312yes—
590C23#54612yes—
591C23#54612yes—
592C23#57514yes—
593C23C23×D411yescharacteristic
594Q8—24no—
595Q8—24no—
596Q8—48no—
597Q8—42no—
598Q8—42no—
599D4—28no—
600D4—48no—
601D4—44no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4C2×#12312yes—
653C2×C4C2×#12312yes—
654C2×C4C2×#12312yes—
655C2×C4#52914yes—
656C2×C4#54014yes—
657C2×C4#54612yes—
658C2×C4#54612yes—
659C2×C4#55314yes—
660C2×C4#55514yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C8—44no—
668C8—42no—
669C8—42no—
670C22—22no—
671C22—22no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—816no—
689C22#242014yes—
690C22C22×#12311yescharacteristic
691C22C2×#54611yescharacteristic
692C22C2×#54611yescharacteristic
693C22C2×#57512yes—
694C22#358612yes—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—44no—
703C4—82no—
704C4—82no—
705C4—84no—
706C4—82no—
707C4—82no—
708C4#242014yes—
709C4C2×#52911yescharacteristic
710C4C2×#54011yescharacteristic
711C4C2×#55311yescharacteristic
712C4C2×#55511yescharacteristic
713C2—24no—
714C2—44no—
715C2—82no—
716C2C2×#242011yescharacteristic
717C2#6051111yescharacteristic
718C2#6051911yescharacteristic
719C2#6115812yes—
720C2#6116312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
3. #24980C2semidirect
4. #24979C2semidirect
5. #24976C2semidirect
6. #24975C2semidirect
7. #23642C2semidirect
8. #23481C2semidirect
10. #21172C2semidirect
11. #20911C2semidirect
12. #20765C2semidirect
13. #20702C2semidirect
14. #20647C2semidirect
15. C4×#419C2semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2075C22semidirect
56. #2036C22semidirect
57. #2036C22semidirect
58. #2035C22semidirect
59. #2035C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1873C22semidirect
66. #1873C22semidirect
67. #1872C22semidirect
68. #1872C22semidirect
69. #1836C22semidirect
70. #1835C22semidirect
71. #1835C22semidirect
72. #1825C22semidirect
73. #1825C22semidirect
74. #1824C22semidirect
75. #1823C22semidirect
76. #1805C22semidirect
77. #1805C22semidirect
78. #1803C22semidirect
79. #1763C22semidirect
80. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
D4×#1172
#249802
#249792
#249762
#249752
#236422
#234812
#211902
#211722
#209112
#207652
#207022
#206472
C4×#4192