Finite Group Databasebeta

← #6435255

Subgroups — #6435255

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35273C211yesmaximal, characteristic
2#34060C211yesmaximal, characteristic, complemented
3#29935C214yesmaximal, complemented
4#29931C214yesmaximal, complemented
5#29926C212yesmaximal, complemented
6#29925C212yesmaximal, complemented
7#29923C212yesmaximal, complemented
8#29920C212yesmaximal, complemented
9#23737C212yesmaximal, complemented
10#23634C212yesmaximal, complemented
11#23488C212yesmaximal, complemented
12#23485C212yesmaximal, complemented
13#21258C211yesmaximal, characteristic, complemented
14#21257C211yesmaximal, characteristic, complemented
15#21171C211yesmaximal, characteristic, complemented
16C2×#1947C211yesmaximal, characteristic, complemented
17#20629C211yesmaximal, characteristic, complemented
18C2×#537C2212yes—
19C2×#532C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C22×#107C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#417C2212yes—
26C2×#417C2212yes—
27C2×#416C2212yes—
28C2×#416C2212yes—
29C2×#408C2212yes—
30#2857C2212yes—
31#2856C2212yes—
32#2620C2214yes—
33#2538—21no—
34#2538—21no—
35#2537—21no—
36#2537—21no—
37#2416—22no—
38#2416—22no—
39#2320—21no—
40#2319—21no—
41#2318—22no—
42#2317C2211yescharacteristic
43#2317C2211yescharacteristic
44#2314C2212yes—
45#2314C2212yes—
46#2311C2212yes—
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2295—21no—
50#2293—21no—
51#2290—22no—
52#2285C2212yes—
53C2×#384C2211yescharacteristic
54#2283C2212yes—
55C4×#117C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2097C2214yes—
58#2095C2214yes—
59#2093C2214yes—
60#2092C2214yes—
61#2074C2214yes—
62#2073C2214yes—
63#2072C2214yes—
64#2071C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2214yes—
68#2035C2214yes—
69#2034C2214yes—
70#2033C2214yes—
71#2033C2214yes—
72#2033C2214yes—
73#1983C2212yes—
74#1983C2212yes—
75#1983C2214yes—
76#1982C2211yescharacteristic
77#1982C2211yescharacteristic
78#1982C2212yes—
79#1981C2211yescharacteristic
80#1981C2211yescharacteristic
81#1981C2212yes—
82#1948C2212yes—
83#1948C2212yes—
84#1947C2212yes—
85#1947C2212yes—
86#1947C2214yes—
87#1947C2212yes—
88#1947C2212yes—
89#1798C2214yes—
90#1797C2212yes—
91#1797C2212yes—
92#1736C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2312yes—
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—44no—
100#537—44no—
101#532—24no—
102#532—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124C2312yes—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2314yes—
108C2×#121—24no—
109C2×#121—22no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C4×#41—24no—
116C4×#41—24no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D412yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D412yes—
124C2×C4×D4D411yescharacteristic
125C2×#117—22no—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#428—44no—
139C2×#107—44no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144#423—24no—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151#417—44no—
152#417—44no—
153#416—44no—
154#416—44no—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167#408—44no—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2314yes—
170#396—24no—
171#396—24no—
172#394—24no—
173#394—24no—
174#391—21no—
175#391—21no—
176#391C2314yes—
177#390—21no—
178#390—21no—
179#390—22no—
180#390—22no—
181#390C2314yes—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—22no—
198#386—24no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384C2312yes—
204#384C2312yes—
205#384C2312yes—
206#384C2312yes—
207#383—22no—
208#382—22no—
209#382C2312yes—
210#382C2312yes—
211#381—22no—
212#380—22no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380C2312yes—
218#380C2312yes—
219#380C2312yes—
220#380C2312yes—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233#376—21no—
234#376—22no—
235C2×#95—21no—
236C43—21no—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—24no—
253C2×#41—24no—
254C2×#41—24no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2412yes—
257C22×D4—22no—
258C22×D4—24no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C2×D412yes—
262C22×D4C2×D411yescharacteristic
263C22×D4C2412yes—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C2×M16—24no—
275C2×M16—24no—
276C2×M16—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285#127—22no—
286#127—22no—
287#124—24no—
288#124—24no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C22⋊Q8—48no—
292#121—44no—
293#121—44no—
294#121—416no—
295#121—48no—
296#121—44no—
297#121—44no—
298#121—84no—
299#121—84no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—44no—
307C4×D4—44no—
308#117—22no—
309#117—22no—
310#117—24no—
311#117—24no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—84no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—82no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42#4114yes—
381C2×C42#4114yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—44no—
386#107—44no—
387#107—44no—
388#106—44no—
389C4⋊C8—22no—
390C4⋊C8—22no—
391C4⋊C8—24no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394#103—42no—
395#103—42no—
396#103—42no—
397#103—42no—
398#103—44no—
399#103—44no—
400#102—42no—
401#102—42no—
402#102—42no—
403#102—42no—
404#102—44no—
405#102—44no—
406C22⋊C8—44no—
407C22⋊C8—44no—
408#95—22no—
409#95—22no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—24no—
424C24—22no—
425C24C22×D412yes—
426C24C22×D411yescharacteristic
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431#41—48no—
432#41—48no—
433#41—48no—
434#41—48no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—48no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—84no—
467C2×D4—84no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4#12314yes—
505C22×C4#12314yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+412yes—
520C22×C4C2511yescharacteristic
521M16—48no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—24no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—48no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—44no—
551C4⋊C4—48no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—48no—
568C22⋊C4—48no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—84no—
574C22⋊C4—84no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C42—24no—
580C42—24no—
581C42—22no—
582C42—22no—
583C42—42no—
584C42—42no—
585C42—44no—
586C42—42no—
587C42—42no—
588C42—81no—
589C42—81no—
590C42—82no—
591C42#12314yes—
592C42#12314yes—
593C42C2×#4112yes—
594C42C2×#4112yes—
595C42C2×#4112yes—
596C42C2×#4112yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23C2×#12312yes—
616C23#53512yes—
617C23#53712yes—
618C23C23×D411yescharacteristic
619Q8—44no—
620Q8—42no—
621Q8—42no—
622Q8—42no—
623Q8—42no—
624D4—48no—
625D4—44no—
626D4—42no—
627D4—42no—
628D4—42no—
629D4—42no—
630D4—84no—
631D4—88no—
632D4—84no—
633D4—168no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—28no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—44no—
651C2×C4—48no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—44no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—88no—
684C2×C4—164no—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4C2×#12312yes—
688C2×C4C2×#12312yes—
689C2×C4C2×#12312yes—
690C2×C4#53512yes—
691C2×C4#53712yes—
692C2×C4#54014yes—
693C2×C4#54014yes—
694C2×C4#54214yes—
695C2×C4#54214yes—
696C2×C4#57612yes—
697C2×C4#57812yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)11yescharacteristic
701C2×C4C2×H(5,𝔽2)11yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C8—42no—
705C8—42no—
706C8—44no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22#262414yes—
728C22C22×#12311yescharacteristic
729C22C2×#53511yescharacteristic
730C22C2×#53711yescharacteristic
731C22C2×#57611yescharacteristic
732C22C2×#57811yescharacteristic
733C22#358511yescharacteristic
734C22#358611yescharacteristic
735C4—22no—
736C4—22no—
737C4—22no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—81no—
742C4—82no—
743C4—81no—
744C4—82no—
745C4—81no—
746C4—82no—
747C4—84no—
748C4—81no—
749C4#262414yes—
750C4#309814yes—
751C4C2×#54011yescharacteristic
752C4C2×#54011yescharacteristic
753C4C2×#54211yescharacteristic
754C4C2×#54211yescharacteristic
755C2—24no—
756C2—82no—
757C2—162no—
758C2C2×#262411yescharacteristic
759C2C2×#309811yescharacteristic
760C2#6052011yescharacteristic
761C2#6083211yescharacteristic
762C2#6083711yescharacteristic
763C2#6092811yescharacteristic
764C2#6093011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34060C2semidirect
3. #29935C2semidirect
4. #29931C2semidirect
5. #29926C2semidirect
6. #29925C2semidirect
7. #29923C2semidirect
8. #29920C2semidirect
9. #23737C2semidirect
10. #23634C2semidirect
11. #23488C2semidirect
12. #23485C2semidirect
13. #21258C2semidirect
14. #21257C2semidirect
15. #21171C2semidirect
16. C2×#1947C2semidirect
17. #20629C2semidirect

Maximal subgroups

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

subgroupindex
#352732
#340602
#299352
#299312
#299262
#299252
#299232
#299202
#237372
#236342
#234882
#234852
#212582
#212572
#211712
C2×#19472
#206292