Finite Group Databasebeta

← #6447975

Subgroups — #6447975

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#25009C214yesmaximal, complemented
4#25008C214yesmaximal, complemented
5#24997C212yesmaximal, complemented
6#24996C212yesmaximal, complemented
7#24995C212yesmaximal, complemented
8#24994C212yesmaximal, complemented
9#23747C212yesmaximal
10#23629C212yesmaximal
11#21303C211yesmaximal, characteristic
12#21171C211yesmaximal, characteristic
13#21061C212yesmaximal
14#21034C212yesmaximal
15#20764C211yesmaximal, characteristic
16#20647C211yesmaximal, characteristic
17C4×#419C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C2×#419C2211yescharacteristic
23C2×#419C2212yes—
24C2×#419C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2582—21no—
27#2582—21no—
28#2582—22no—
29#2582C2212yes—
30#2582C2212yes—
31#2582C2214yes—
32C2×#398—22no—
33#2355C2211yescharacteristic
34#2355C2211yescharacteristic
35#2355C2212yes—
36#2355C2212yes—
37Q8×C4⋊C4C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2345—22no—
40#2345—21no—
41Q8×C22⋊C4—21no—
42#2295—22no—
43#2295—21no—
44C2×#391—21no—
45C2×#391C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48#2285C2211yescharacteristic
49#2285C2212yes—
50C2×#384C2211yescharacteristic
51C42×Q8—21no—
52C4×#117C2212yes—
53C4×#117C2211yescharacteristic
54#2101C2214yes—
55#2101C2214yes—
56#2100C2214yes—
57#2099C2214yes—
58#2074C2214yes—
59#2074C2214yes—
60#2072C2214yes—
61#2072C2214yes—
62#1985C2214yes—
63#1984C2212yes—
64#1984C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1927C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1919C2214yes—
76#1918C2214yes—
77#1836C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1805C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1763C2212yes—
84#1763C2212yes—
85C4×#103C2212yes—
86C4×#103C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2211yescharacteristic
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2312yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D412yes—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4D412yes—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#428—28no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133#423—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136#419—28no—
137#419—24no—
138#419—24no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C4×M16—24no—
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148#398—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#391—22no—
153#391—24no—
154#391—22no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391C2314yes—
162#391C2314yes—
163#389—22no—
164#389—24no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#384C2312yes—
188#382C2312yes—
189#382C2312yes—
190#382C2312yes—
191#380—22no—
192#380—21no—
193#380—22no—
194#380—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215#376—22no—
216#376—21no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95—22no—
220C43—22no—
221C43—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—42no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D412yes—
249C22×Q8C2×D411yescharacteristic
250C22×Q8#4114yes—
251C22×Q8C2412yes—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—24no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—22no—
268C2×M16—22no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C4×Q8—24no—
273C4×Q8—28no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289#117—22no—
290#117—24no—
291#117—22no—
292#117—24no—
293#117—22no—
294#117#4114yes—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4—84no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—44no—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42#4114yes—
362C2×C42#4114yes—
363C2×C42C2412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—28no—
368#106—28no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371#103—28no—
372#103—28no—
373#103—24no—
374#103—24no—
375#103—24no—
376#103—24no—
377#97—24no—
378C4×C8—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—42no—
416C2×Q8—416no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8#13714yes—
422C2×Q8C2×#4114yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469M16—48no—
470M16—44no—
471M16—44no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C2×#4112yes—
521C4⋊C4C2×#4112yes—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—44no—
533C42—44no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—44no—
538C42—42no—
539C42—42no—
540C42—81no—
541C42—81no—
542C42—82no—
543C42—82no—
544C42#12314yes—
545C42#12314yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23C2×#12312yes—
555C23#54812yes—
556C23#54812yes—
557C23C23×D411yescharacteristic
558Q8—28no—
559Q8—24no—
560Q8—24no—
561Q8—48no—
562Q8—44no—
563Q8—48no—
564Q8—42no—
565Q8—42no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4C2×#12312yes—
618C2×C4C2×#12312yes—
619C2×C4C2×#12312yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#12312yes—
622C2×C4#52914yes—
623C2×C4#54114yes—
624C2×C4#54812yes—
625C2×C4#54812yes—
626C2×C4#55214yes—
627C2×C4#55614yes—
628C2×C4C2×#13714yes—
629C2×C4C2×#13712yes—
630C2×C4#57512yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C8—44no—
638C8—42no—
639C8—42no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22#242114yes—
647C22C22×#12311yescharacteristic
648C22C2×#54811yescharacteristic
649C22C2×#54811yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57511yescharacteristic
652C22#358511yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—24no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4—81no—
669C4—84no—
670C4#242114yes—
671C4#309514yes—
672C4C2×#52911yescharacteristic
673C4C2×#54111yescharacteristic
674C4C2×#55211yescharacteristic
675C4C2×#55611yescharacteristic
676C2—24no—
677C2C2×#242111yescharacteristic
678C2C2×#309511yescharacteristic
679C2#6052011yescharacteristic
680C2#6131911yescharacteristic
681C2#6132011yescharacteristic
682C2#6132211yescharacteristic
683C2#6132411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #25009C2semidirect
4. #25008C2semidirect
5. #24997C2semidirect
6. #24996C2semidirect
7. #24995C2semidirect
8. #24994C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
Q8×#1172
#250092
#250082
#249972
#249962
#249952
#249942
#237472
#236292
#213032
#211712
#210612
#210342
#207642
#206472
C4×#4192