Finite Group Databasebeta

← #6463618

Subgroups — #6463618

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2C4×#544C211yesmaximal, characteristic, complemented
3#29269C214yesmaximal, complemented
4#29267C214yesmaximal, complemented
5#29258C212yesmaximal, complemented
6#29257C212yesmaximal, complemented
7#29256C212yesmaximal, complemented
8#29255C212yesmaximal, complemented
9#23738C212yesmaximal, complemented
10#23458C212yesmaximal, complemented
11#21258C211yesmaximal, characteristic
12#21257C211yesmaximal, characteristic
13#21171C211yesmaximal, characteristic, complemented
14C2×#1948C211yesmaximal, characteristic, complemented
15#20882C212yesmaximal, complemented
16#20878C212yesmaximal, complemented
17#20755C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21C2×#428C2212yes—
22C22×#106C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#417C2212yes—
26C2×#417C2212yes—
27C2×#416C2212yes—
28C2×#416C2212yes—
29#2361—22no—
30#2361—22no—
31#2354—21no—
32#2354—21no—
33D4×C4⋊C4—21no—
34D4×C4⋊C4—21no—
35#2331—21no—
36#2331C2212yes—
37#2330—21no—
38#2330C2212yes—
39#2329—22no—
40#2326C2214yes—
41#2317C2211yescharacteristic
42#2317C2211yescharacteristic
43#2316C2211yescharacteristic
44#2311C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#127C2212yes—
49C4×#125—21no—
50C4×#125C2212yes—
51C4×C22⋊Q8—22no—
52C2×#384C2211yescharacteristic
53C4×#117C2211yescharacteristic
54C4×#117C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2214yes—
57#2096C2214yes—
58#2094C2214yes—
59#2092C2214yes—
60#2026C2214yes—
61#2025C2214yes—
62#2024C2214yes—
63#2023C2214yes—
64#1983C2212yes—
65#1983C2212yes—
66#1983C2214yes—
67#1982C2211yescharacteristic
68#1982C2211yescharacteristic
69#1982C2212yes—
70#1981C2211yescharacteristic
71#1981C2211yescharacteristic
72#1981C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1948C2214yes—
77#1948C2212yes—
78#1947C2212yes—
79#1947C2212yes—
80#1865C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1864C2214yes—
84#1864C2214yes—
85#1864C2214yes—
86#1863C2214yes—
87#1862C2214yes—
88#1832C2214yes—
89#1831C2212yes—
90#1831C2212yes—
91#1736C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C22×#41C2312yes—
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—24no—
99#544—24no—
100#532—24no—
101#532Q818yescomplemented
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#124C2312yes—
109C2×C22⋊Q8—22no—
110C2×#121—21no—
111C2×#121—21no—
112#520—44no—
113#520—44no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2314yes—
116C2×C4×D4—24no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4C2314yes—
122C2×#117—22no—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#428—44no—
136C2×#107C2314yes—
137C2×#106—44no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141#423—24no—
142C2×C4⋊C8—24no—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148#417—28no—
149#417—44no—
150#416—28no—
151#416—44no—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—24no—
185#388—24no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—24no—
193#385—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#382—21no—
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4D412yes—
229#376—21no—
230#376C2312yes—
231C2×#95—21no—
232C43—21no—
233C43C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2314yes—
239#336C2314yes—
240#336C2314yes—
241#336C2314yes—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41C2×Q818yes—
248C22×Q8—22no—
249C22×Q8C2412yes—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—24no—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8C2×Q814yes—
276#127—22no—
277#127C2×Q814yes—
278#125—28no—
279#125—24no—
280#125—24no—
281#124—24no—
282#124C2×Q818yes—
283C22⋊Q8—216no—
284#121—28no—
285#121—28no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—84no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—416no—
294C4×D4—84no—
295C4×D4—84no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—48no—
304#117#4114yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4#4114yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—82no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×Q814yes—
370C2×C42#4114yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—44no—
376#106—44no—
377#106—44no—
378#106—44no—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—22no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—24no—
385#103—24no—
386#103—28no—
387#103—42no—
388#103—42no—
389#103—44no—
390#102—24no—
391#102—24no—
392#102—28no—
393#102—42no—
394#102—42no—
395#102—44no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24—22no—
415C24C22×D411yescharacteristic
416#41—216no—
417#41—216no—
418#41—48no—
419#41—48no—
420C2×Q8—28no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8C22×Q814yes—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C22×Q814yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22⋊Q814yes—
472C22×C4C22⋊Q814yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489M16—48no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—48no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4C2×#4114yes—
535C4⋊C4C2×#4112yes—
536C4⋊C4C2×#4112yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—48no—
543C22⋊C4—42no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—42no—
554C42—42no—
555C42—48no—
556C42—42no—
557C42—42no—
558C42—44no—
559C42—42no—
560C42—81no—
561C42—81no—
562C42—82no—
563C42C22⋊Q814yes—
564C42C22⋊Q814yes—
565C42C22×Q812yes—
566C42C22×Q812yes—
567C42C2×#4112yes—
568C42C2×#4112yes—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—24no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23C2×C22⋊Q812yes—
581C23#52512yes—
582C23#53512yes—
583C23C23×D411yescharacteristic
584Q8—24no—
585Q8—24no—
586Q8—42no—
587Q8—42no—
588Q8—88no—
589D4—24no—
590D4—24no—
591D4—48no—
592D4—416no—
593D4—48no—
594D4—42no—
595D4—42no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—416no—
629C2×C4—41no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—168no—
644C2×C4—164no—
645C2×C4C2×C22⋊Q812yes—
646C2×C4C2×C22⋊Q812yes—
647C2×C4C2×C22⋊Q812yes—
648C2×C4C2×C22⋊Q812yes—
649C2×C4C2×C22⋊Q812yes—
650C2×C4#52512yes—
651C2×C4#53512yes—
652C2×C4#55414yes—
653C2×C4#55414yes—
654C2×C4#55914yes—
655C2×C4#55914yes—
656C2×C4#57712yes—
657C2×C4#57712yes—
658C2×C4C23×Q811yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C8—42no—
665C8—42no—
666C8—44no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22#258014yes—
681C22C22×C22⋊Q811yescharacteristic
682C22C2×#52511yescharacteristic
683C22C2×#53511yescharacteristic
684C22C2×#57711yescharacteristic
685C22C2×#57711yescharacteristic
686C22#358511yescharacteristic
687C22#358611yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—81no—
700C4—84no—
701C4—81no—
702C4—82no—
703C4#258014yes—
704C4#308414yes—
705C4C2×#55411yescharacteristic
706C4C2×#55411yescharacteristic
707C4C2×#55911yescharacteristic
708C4C2×#55911yescharacteristic
709C2—24no—
710C2—84no—
711C2C2×#258011yescharacteristic
712C2C2×#308411yescharacteristic
713C2#6049211yescharacteristic
714C2#6059711yescharacteristic
715C2#6059911yescharacteristic
716C2#6083311yescharacteristic
717C2#6083611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#544C2semidirect
3. #29269C2semidirect
4. #29267C2semidirect
5. #29258C2semidirect
6. #29257C2semidirect
7. #29256C2semidirect
8. #29255C2semidirect
9. #23738C2semidirect
10. #23458C2semidirect
13. #21171C2semidirect
14. C2×#1948C2semidirect
15. #20882C2semidirect
16. #20878C2semidirect
17. #20755C2semidirect
101. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#342072
C4×#5442
#292692
#292672
#292582
#292572
#292562
#292552
#237382
#234582
#212582
#212572
#211712
C2×#19482
#208822
#208782
#207552