Finite Group Databasebeta

← #7510511

Subgroups — #7510511

Order 512.

1853 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35556C211yesmaximal, characteristic, complemented
2#35334C211yesmaximal, characteristic, complemented
3#32622C214yesmaximal, complemented
4#32617C214yesmaximal, complemented
5#32606C214yesmaximal, complemented
6#32601C214yesmaximal, complemented
7C2×#2092C211yesmaximal, characteristic, complemented
8#23631C211yesmaximal, characteristic, complemented
9#23610C211yesmaximal, characteristic, complemented
10#23544C211yesmaximal, characteristic, complemented
11#23511C211yesmaximal, characteristic, complemented
12#23426C211yesmaximal, characteristic
13#21304C211yesmaximal, characteristic, complemented
14#21230C211yesmaximal, characteristic, complemented
15#21089C211yesmaximal, characteristic, complemented
16#21068C211yesmaximal, characteristic, complemented
17#21037C211yesmaximal, characteristic, complemented
18#20928C211yesmaximal, characteristic, complemented
19#20924C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×#135C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#533C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2851—22no—
38#2628—22no—
39#2592C2214yes—
40#2581C2214yescomplemented
41#2580C2214yescomplemented
42#2564—24no—
43#2548C2214yes—
44#2491—22no—
45#2490—22no—
46#2472—24no—
47#2316C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2311C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2092C2214yes—
58#2092C2214yes—
59#2074C2214yescomplemented
60#2073C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yescomplemented
63#2055C2214yes—
64#2054C2214yes—
65#2042C2214yes—
66#2041C2214yes—
67#2015C2212yes—
68#2014C2212yes—
69#2012C2212yes—
70#2011C2212yes—
71#1985C2214yescomplemented
72#1984C2214yescomplemented
73#1973C2212yescomplemented
74#1973C2212yescomplemented
75#1970C2212yescomplemented
76#1969C2212yescomplemented
77#1936C2214yes—
78#1934C2214yes—
79#1927C2214yescomplemented
80#1926C2214yescomplemented
81#1921C2214yescomplemented
82#1918C2214yescomplemented
83#1880C2214yes—
84#1880C2212yescomplemented
85#1879C2214yes—
86#1879C2212yescomplemented
87#1878C2212yescomplemented
88#1877C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×#41C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—42no—
102C2×AGL(1,ℤ/8)—42no—
103C2×#135—24no—
104C2×#135—24no—
105C2×#135—42no—
106C22×Q16C2311yescharacteristic
107C22×SD16C2311yescharacteristic
108C22×SD16C2311yescharacteristic
109C22×SD16C2312yes—
110C22×D8C2311yescharacteristic
111C22×D8C2312yes—
112C22×M16C2311yescharacteristic
113C22×M16C2311yescharacteristic
114C23×C8C2311yescharacteristic
115C23×C8C2311yescharacteristic
116#535—42no—
117#535—42no—
118#533—42no—
119#533—42no—
120C2×C4⋊Q8D412yes—
121C2×#127C2312yes—
122C2×#125D412yes—
123C2×#124—24no—
124C2×#124C2312yes—
125C2×C22⋊Q8D414yes—
126C2×#121—22no—
127C2×#120—22no—
128C4×#41—42no—
129C4×#41—42no—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2314yes—
132C2×C4×D4—22no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—22no—
143C22×C42—21no—
144C22×C42C2311yescharacteristic
145C2×#107—24no—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#106—42no—
149C2×#106C2314yes—
150#424—42no—
151C2×C4⋊C8C2314yes—
152#418—42no—
153#417—42no—
154#417—42no—
155#417—42no—
156#416—24no—
157#416—42no—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#102—24no—
164C2×#102—42no—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2314yes—
173C2×#102C2312yes—
174#405—42no—
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#400—24no—
178#399—24no—
179#397—24no—
180#395—24no—
181#394—24no—
182#394—24no—
183#393—24no—
184#393—24no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#390—22no—
190#390—24no—
191#390—21no—
192#390—21no—
193#390C2314yes—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#390C2312yes—
198#390C2314yes—
199#389—22no—
200#389—22no—
201#389C2314yes—
202#389C2314yes—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#386—22no—
210#386—22no—
211#385—24no—
212#385—24no—
213#382—21no—
214#382—21no—
215#382C2314yes—
216#380—22no—
217#380—22no—
218#380—22no—
219#380—22no—
220#380—22no—
221#379—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2314yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C2×#95—22no—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41#4114yes—
252C22×Q8C2×D414yes—
253C22×Q8C2411yescharacteristic
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—24no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4C2412yes—
266C22×D4C2411yescharacteristic
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—22no—
274C23×C4C2×D412yes—
275C23×C4C2411yescharacteristic
276AGL(1,ℤ/8)—416no—
277#135—416no—
278C2×Q16—22no—
279C2×Q16—22no—
280C2×Q16—41no—
281C2×SD16—22no—
282C2×SD16—24no—
283C2×SD16—22no—
284C2×SD16—22no—
285C2×SD16—22no—
286C2×SD16—41no—
287C2×SD16—41no—
288C2×SD16—42no—
289C2×SD16—42no—
290C2×D8—22no—
291C2×D8—24no—
292C2×D8—22no—
293C2×D8—41no—
294C2×D8—42no—
295C2×D8—42no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8#4114yes—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C22×C8C2412yes—
311C22×C8C2411yescharacteristic
312C22×C8C2411yescharacteristic
313C4⋊Q8—44no—
314#127—42no—
315#127—42no—
316#125—44no—
317#124—42no—
318#124—42no—
319#124—88no—
320C22⋊Q8—48no—
321#121—48no—
322#120—48no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8—42no—
326C4×Q8—42no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4—48no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—44no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—84no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—24no—
383C2×C42—44no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389#107—24no—
390#107—24no—
391#106—44no—
392C4⋊C8—44no—
393#103—24no—
394#103—42no—
395#103—42no—
396#103—42no—
397#103—42no—
398#102—24no—
399#102—24no—
400#102—24no—
401#102—44no—
402#102—42no—
403#102—42no—
404#102—42no—
405#102—42no—
406#102—44no—
407#97—42no—
408C4×C8—42no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419C24—21no—
420C24—22no—
421C24—21no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—82no—
426C24—82no—
427C24C22×D411yescharacteristic
428#41—28no—
429#41—28no—
430#41—44no—
431#41—44no—
432#41—44no—
433#41—44no—
434#41—44no—
435#41—44no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8C2×#4112yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—48no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—84no—
483C2×D4—84no—
484C2×D4C2×#4112yes—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4C22×D412yes—
511C22×C4C22×D414yes—
512C22×C4C22×D412yes—
513C22×C4C22×D414yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C42−1+412yes—
521C22×C4C2511yescharacteristic
522Q16—44no—
523SD16—44no—
524SD16—44no—
525SD16—48no—
526D8—44no—
527D8—48no—
528M16—44no—
529M16—44no—
530C2×C8—28no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—44no—
545C2×C8C2×#4112yes—
546C2×C8C2×#4112yes—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—84no—
567C4⋊C4—84no—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—48no—
571C22⋊C4—44no—
572C22⋊C4—48no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—84no—
577C22⋊C4—82no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C22⋊C4—84no—
582C22⋊C4—84no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—44no—
592C42—42no—
593C42—42no—
594C42—84no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—42no—
599C23—42no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23—44no—
604C23—42no—
605C23—44no—
606C23—42no—
607C23—41no—
608C23—41no—
609C23—42no—
610C23—41no—
611C23—42no—
612C23—88no—
613C23—84no—
614C23—164no—
615C23—162no—
616C23—162no—
617C23#53412yes—
618C23#54812yes—
619C23#55012yes—
620C23C23×D411yescharacteristic
621Q8—22no—
622Q8—22no—
623Q8—41no—
624Q8—41no—
625Q8—41no—
626Q8—41no—
627Q8—41no—
628Q8—41no—
629Q8—88no—
630D4—22no—
631D4—22no—
632D4—48no—
633D4—42no—
634D4—41no—
635D4—41no—
636D4—42no—
637D4—41no—
638D4—41no—
639D4—42no—
640D4—41no—
641D4—42no—
642D4—41no—
643D4—84no—
644D4—84no—
645D4—168no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—48no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—84no—
678C2×C4—88no—
679C2×C4—84no—
680C2×C4—162no—
681C2×C4—162no—
682C2×C4—162no—
683C2×C4#53412yes—
684C2×C4#53814yes—
685C2×C4#54214yes—
686C2×C4#54812yes—
687C2×C4#55012yes—
688C2×C4#55114yes—
689C2×C4#56214yes—
690C2×C4C22×#4111yescharacteristic
691C2×C4C22×#4111yescharacteristic
692C2×C4C2×H(5,𝔽2)11yescharacteristic
693C2×C4C2×H(5,𝔽2)11yescharacteristic
694C2×C4C2×H(5,𝔽2)11yescharacteristic
695C2×C4C2×2−1+411yescharacteristic
696C8—22no—
697C8—22no—
698C8—42no—
699C8—42no—
700C8—42no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—44no—
706C22—42no—
707C22—82no—
708C22—81no—
709C22—81no—
710C22—82no—
711C22—84no—
712C22—82no—
713C22—84no—
714C22—81no—
715C22—82no—
716C22—81no—
717C22—82no—
718C22—81no—
719C22—82no—
720C22—168no—
721C22#273014yes—
722C22C2×#53411yescharacteristic
723C22C2×#54811yescharacteristic
724C22C2×#55011yescharacteristic
725C22#358412yes—
726C22#358512yes—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—44no—
731C4—42no—
732C4—82no—
733C4—82no—
734C4—81no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4#273014yes—
739C4C2×#53811yescharacteristic
740C4C2×#54211yescharacteristic
741C4C2×#55111yescharacteristic
742C4C2×#56211yescharacteristic
743C2—24no—
744C2—82no—
745C2—161no—
746C2—162no—
747C2C2×#273011yescharacteristic
748C2#6076712yes—
749C2#6134912yes—
750C2#6141311yescharacteristic
751C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23426C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35556C2semidirect
2. #35334C2semidirect
3. #32622C2semidirect
4. #32617C2semidirect
5. #32606C2semidirect
6. #32601C2semidirect
7. C2×#2092C2semidirect
8. #23631C2semidirect
9. #23610C2semidirect
10. #23544C2semidirect
11. #23511C2semidirect
13. #21304C2semidirect
14. #21230C2semidirect
15. #21089C2semidirect
16. #21068C2semidirect
17. #21037C2semidirect
18. #20928C2semidirect
19. #20924C2semidirect
40. #2581C22semidirect
41. #2580C22semidirect
59. #2074C22semidirect
60. #2073C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
71. #1985C22semidirect
72. #1984C22semidirect
73. #1973C22semidirect
74. #1973C22semidirect
75. #1970C22semidirect
76. #1969C22semidirect
79. #1927C22semidirect
80. #1926C22semidirect
81. #1921C22semidirect
82. #1918C22semidirect
84. #1880C22semidirect
86. #1879C22semidirect
87. #1878C22semidirect
88. #1877C22semidirect

Maximal subgroups

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

subgroupindex
#355562
#353342
#326222
#326172
#326062
#326012
C2×#20922
#236312
#236102
#235442
#235112
#234262
#213042
#212302
#210892
#210682
#210372
#209282
#209242