Finite Group Databasebeta

← #6802527

Subgroups — #6802527

Order 512.

1834 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
1#36586C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#32906C211yesmaximal, characteristic, complemented
4#32184C214yesmaximal, complemented
5#30760C214yesmaximal, complemented
6#29719C212yesmaximal
7#28471C212yesmaximal, complemented
8#27757C212yesmaximal, complemented
9#25726C212yesmaximal
10#25722C211yesmaximal, characteristic, complemented
11#25370C212yesmaximal, complemented
12#24630C212yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#24209C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic, complemented
16#22050C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#497C2214yes—
22C2×#494C2212yes—
23C2×#462C2214yes—
24C2×#462C2212yes—
25C2×#461C2212yes—
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×#451C2212yes—
29C2×#448C2211yescharacteristic
30C2×#448C2212yes—
31C2×C4×SD16C2212yes—
32#2857C2211yescharacteristic
33#2850C2212yes—
34#2596—22no—
35#2571C2211yescharacteristic
36#2547C2211yescharacteristic
37#2546—22no—
38#2546—21no—
39#2546—21no—
40#2546C2212yes—
41#2546C2214yes—
42#2546C2212yes—
43#2514C2212yes—
44#2510—22no—
45#2494—21no—
46#2487—21no—
47#2448C2212yes—
48C2×#398—22no—
49#2365C2211yescharacteristic
50#2354C2211yescharacteristic
51#2352—21no—
52#2347—21no—
53C4×C22⋊Q8C2212yes—
54C4×#121C2211yescharacteristic
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#389—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2074C2214yes—
60#2056C2214yes—
61#2055C2214yes—
62#2043C2214yes—
63#2043C2212yes—
64#2016C2214yes—
65#2016C2214yes—
66#2016C2214yes—
67#2015C2214yes—
68#2013C2212yes—
69#2013C2214yes—
70#1981C2212yes—
71#1981C2214yes—
72#1971C2214yes—
73#1960C2214yes—
74#1956C2212yes—
75#1939C2212yes—
76#1935C2214yes—
77#1927C2214yes—
78#1917C2214yes—
79#1880C2212yes—
80#1878C2214yes—
81#1878C2214yes—
82#1874C2211yescharacteristic
83#1872C2212yes—
84#1865C2212yes—
85#1853C2211yescharacteristic
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1787C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×SD16C2312yes—
101C22×SD16C2314yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2312yes—
113C2×#124C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121C2311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—44no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#497—48no—
137#494—44no—
138#462—48no—
139#462—44no—
140#461—44no—
141#451—28no—
142#451—44no—
143#451—44no—
144#448—24no—
145#448—44no—
146C4×SD16—44no—
147C2×#106C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2311yescharacteristic
163C2×C22⋊C8C2311yescharacteristic
164C2×#97C2314yes—
165C2×C4×C8C2312yes—
166#401—22no—
167#400—22no—
168#400—22no—
169#399—22no—
170#398—24no—
171#398—24no—
172#398—24no—
173#398—24no—
174#397—22no—
175#397—22no—
176#391—22no—
177#391—22no—
178#391C2314yes—
179#391C2312yes—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#390C2312yes—
184#390C2311yescharacteristic
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—24no—
191#389—24no—
192#389—22no—
193#389—21no—
194#389—21no—
195#389C2312yes—
196#389C2311yescharacteristic
197#388—22no—
198#388—22no—
199#388—22no—
200#388—21no—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—22no—
207#386—22no—
208#386—21no—
209#385—22no—
210#384C2311yescharacteristic
211#383—22no—
212#382—21no—
213#382C2312yes—
214#380—21no—
215#380—21no—
216#380—21no—
217#380—21no—
218#380—21no—
219#380C2311yescharacteristic
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#336C2314yes—
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8C2412yes—
253C22×Q8C2414yes—
254C22×Q8C2412yes—
255C22×D4—21no—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—42no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—44no—
281C22×C8C2414yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4⋊Q8—28no—
285C4⋊Q8—28no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—48no—
295C4⋊Q8—48no—
296#124—44no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299#121—22no—
300#121—22no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—24no—
309C4×D4—44no—
310#117—44no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375#106—42no—
376C4⋊C8—24no—
377C4⋊C8—44no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—42no—
382#103—44no—
383#103—44no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#102—42no—
389C22⋊C8—24no—
390C22⋊C8—22no—
391C22⋊C8#4114yes—
392#97—44no—
393C4×C8—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415C24—21no—
416C24C22×D411yescharacteristic
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—88no—
441C2×Q8—88no—
442C2×Q8H(5,𝔽2)14yes—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4#13714yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4C2511yescharacteristic
496Q16—88no—
497Q16—88no—
498Q16—88no—
499SD16—48no—
500SD16—88no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8C2×#4114yes—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4#13714yes—
543C22⋊C4—44no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C42—22no—
554C42—22no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—42no—
559C42—42no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—42no—
564C42—82no—
565C42H(5,𝔽2)14yes—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23#55012yes—
578C23C2×#13712yes—
579C23#57812yes—
580C23C23×D411yescharacteristic
581Q8—24no—
582Q8—24no—
583Q8—48no—
584Q8—42no—
585Q8—42no—
586Q8—44no—
587Q8—44no—
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—88no—
593D4—42no—
594D4—42no—
595D4—84no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4#53512yes—
651C2×C4#53512yes—
652C2×C4#55012yes—
653C2×C4C2×#13714yes—
654C2×C4C2×#13712yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C2×H(5,𝔽2)12yes—
659C2×C4C2×H(5,𝔽2)12yes—
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C8—24no—
662C8—44no—
663C8—42no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22#339814yes—
678C22C2×#53511yescharacteristic
679C22C2×#55011yescharacteristic
680C22#348411yescharacteristic
681C22C22×#13711yescharacteristic
682C22C22×#13711yescharacteristic
683C22C2×#57811yescharacteristic
684C22#358611yescharacteristic
685C4—22no—
686C4—22no—
687C4—24no—
688C4—22no—
689C4—41no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—81no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4#321814yes—
702C4#351912yes—
703C4#353712yes—
704C2—24no—
705C2—161no—
706C2#3913211yescharacteristic
707C2C2×#321811yescharacteristic
708C2#6081411yescharacteristic
709C2C2×#339811yescharacteristic
710C2#6140411yescharacteristic
711C2#6158411yescharacteristic
712C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
3. #32906C2semidirect
4. #32184C2semidirect
5. #30760C2semidirect
7. #28471C2semidirect
8. #27757C2semidirect
10. #25722C2semidirect
11. #25370C2semidirect
12. #24630C2semidirect
13. #24212C2semidirect
15. #22629C2semidirect
16. #22050C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25462
#329062
#321842
#307602
#297192
#284712
#277572
#257262
#257222
#253702
#246302
#242122
#242092
#226292
#220502
C4×#4482