Finite Group Databasebeta

← #6623904

Subgroups — #6623904

Order 512.

1753 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2C4×#559C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#33024C212yesmaximal, complemented
5#28508C211yesmaximal, characteristic, complemented
6#28383C212yesmaximal, complemented
7#28251C211yesmaximal, characteristic
8#28203C212yesmaximal, complemented
9#27514C214yesmaximal, complemented
10#27086C214yesmaximal, complemented
11#25057C212yesmaximal, complemented
12#24186C212yesmaximal, complemented
13#23099C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22811C212yesmaximal, complemented
16#21613C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#556—21no—
20C2×#475C2212yes—
21C2×#474C2212yes—
22C2×#474C2212yes—
23C2×#460C2211yescharacteristic
24C2×#459C2212yes—
25C2×#459C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2547—21no—
30#2545—22no—
31#2538—21no—
32#2538C2212yes—
33#2512—22no—
34#2512C2214yes—
35#2374—22no—
36#2367—21no—
37#2367C2212yes—
38#2366C2211yescharacteristic
39#2357—21no—
40#2357C2212yes—
41#2355C2211yescharacteristic
42#2355C2211yescharacteristic
43#2354—21no—
44#2354C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#124—21no—
49C4×#124C2212yes—
50C4×#123—22no—
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53C42×D4—21no—
54C42×D4C2212yes—
55#2093C2214yes—
56#2092C2214yes—
57#2085C2214yes—
58#2054C2214yes—
59#2042C2214yes—
60#2012C2214yes—
61#1988C2212yes—
62#1984C2212yes—
63#1982C2211yescharacteristic
64#1973C2214yes—
65#1972C2214yes—
66#1957C2212yes—
67#1948C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1928C2214yes—
71#1926C2214yes—
72#1926C2212yes—
73#1925C2214yes—
74#1925C2214yes—
75#1921C2214yes—
76#1917C2214yes—
77#1880C2212yes—
78#1879C2214yes—
79#1879C2214yes—
80#1864C2214yes—
81#1845C2212yes—
82#1842C2211yescharacteristic
83#1831C2212yes—
84#1803C2212yes—
85#1803C2211yescharacteristic
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C22×SD16C2312yes—
96C22×D8C2314yes—
97#559—24no—
98#559—24no—
99#556—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#126—24no—
106C2×#124—21no—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117—22no—
130C22×C42—22no—
131#475—44no—
132#474—28no—
133#474—44no—
134#460—24no—
135#459—28no—
136#459—44no—
137C4×SD16—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#397—24no—
159#396—24no—
160#396—24no—
161#391—21no—
162#391—22no—
163#391C2312yes—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390C2312yes—
172#390C2312yes—
173#390C2314yes—
174#390C2314yes—
175#390C2312yes—
176#389—22no—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—24no—
184#388—24no—
185#388—22no—
186#387—22no—
187#387—22no—
188#386—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#382C2312yes—
198#381—22no—
199#380—22no—
200#380—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C43—21no—
227C43C2311yescharacteristic
228C43C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—42no—
254C2×D8—28no—
255C2×D8—44no—
256C2×D8—44no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264#127—22no—
265#127—22no—
266#127—22no—
267#127#4114yes—
268#126—416no—
269#124—28no—
270#124—24no—
271#124—24no—
272#124—48no—
273#124—84no—
274#123—216no—
275C22⋊Q8—48no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—42no—
284C4×Q8—41no—
285C4×Q8—48no—
286C4×Q8#4114yes—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—82no—
301#117—48no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—84no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—28no—
367#107—44no—
368#106—24no—
369#106—44no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—42no—
374C4⋊C8#4114yes—
375#103—24no—
376#103—44no—
377#102—28no—
378#102—24no—
379#102—28no—
380#102—24no—
381#102—24no—
382#102—44no—
383#102—44no—
384#97—44no—
385C4×C8—24no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—24no—
396#95—24no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8C2×#4112yes—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—81no—
436C2×D4—81no—
437C2×D4C2×#4114yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—28no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468SD16—48no—
469D8—88no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8C2×#4114yes—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4C2×#4112yes—
509C4⋊C42−1+412yes—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—22no—
536C42—21no—
537C42—41no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—48no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—81no—
546C42—82no—
547C42—82no—
548C42AGL(1,ℤ/8)14yes—
549C42C2×#4112yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C42H(5,𝔽2)12yes—
553C23—24no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C23×D411yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—44no—
571Q8—41no—
572Q8—41no—
573Q8—88no—
574D4—24no—
575D4—24no—
576D4—48no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—84no—
584D4—84no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—88no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4#52512yes—
633C2×C4#52512yes—
634C2×C4#55012yes—
635C2×C4#55012yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4#57512yes—
640C2×C4#57612yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C2×C4#58511yescharacteristic
647C8—24no—
648C8—42no—
649C8—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22C2×#52511yescharacteristic
665C22C2×#55011yescharacteristic
666C22#348911yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C2×#57511yescharacteristic
669C22C2×#57611yescharacteristic
670C22#358611yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4—81no—
686C4#311514yes—
687C4#339914yes—
688C4#347511yescharacteristic
689C4#348211yescharacteristic
690C4#351111yescharacteristic
691C4#351411yescharacteristic
692C2—84no—
693C2—162no—
694C2#3650411yescharacteristic
695C2C2×#311511yescharacteristic
696C2#6059611yescharacteristic
697C2C2×#339911yescharacteristic
698C2#6140311yescharacteristic
699C2#6211111yescharacteristic
700C2#6214311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. C4×#559C2semidirect
3. #33196C2semidirect
4. #33024C2semidirect
5. #28508C2semidirect
6. #28383C2semidirect
8. #28203C2semidirect
9. #27514C2semidirect
10. #27086C2semidirect
11. #25057C2semidirect
12. #24186C2semidirect
13. #23099C2semidirect
14. #22825C2semidirect
15. #22811C2semidirect
16. #21613C2semidirect
17. C4×#460C2semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×#5592
#331962
#330242
#285082
#283832
#282512
#282032
#275142
#270862
#250572
#241862
#230992
#228252
#228112
#216132
C4×#4602