Finite Group Databasebeta

← #6434243

Subgroups — #6434243

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic, complemented
3#30139C214yesmaximal, complemented
4#30137C214yesmaximal, complemented
5#30134C214yesmaximal, complemented
6#30133C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23611C211yesmaximal, characteristic, complemented
10#23610C211yesmaximal, characteristic, complemented
11#23544C211yesmaximal, characteristic, complemented
12#23544C211yesmaximal, characteristic, complemented
13#23504C211yesmaximal, characteristic, complemented
14#23428C211yesmaximal, characteristic
15#20932C211yesmaximal, characteristic, complemented
16#20932C211yesmaximal, characteristic, complemented
17#20882C211yesmaximal, characteristic, complemented
18#20881C211yesmaximal, characteristic, complemented
19#20754C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×AGL(1,ℤ/8)C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37#2860C2214yes—
38#2644C2214yes—
39#2513—22no—
40#2512—22no—
41#2480—24no—
42#2364—24no—
43#2352—22no—
44#2352—22no—
45#2328C2212yescomplemented
46#2327C2212yescomplemented
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53C4×#124C2212yescomplemented
54C4×#124C2212yescomplemented
55#2296C2211yescharacteristic
56C2×#390C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2214yescomplemented
60#2101C2214yescomplemented
61#2092C2214yes—
62#2092C2214yes—
63#2068C2214yescomplemented
64#2067C2214yescomplemented
65#2066C2214yescomplemented
66#2065C2214yescomplemented
67#2055C2214yes—
68#2055C2214yes—
69#2054C2214yes—
70#2054C2214yes—
71#2043C2212yescomplemented
72#2042C2212yescomplemented
73#2042C2212yescomplemented
74#2041C2212yescomplemented
75#2013C2212yes—
76#2012C2212yes—
77#2012C2212yes—
78#2011C2212yes—
79#1880C2214yes—
80#1880C2214yes—
81#1879C2214yes—
82#1879C2214yes—
83#1865C2214yescomplemented
84#1864C2214yescomplemented
85#1863C2214yescomplemented
86#1862C2214yescomplemented
87#1832C2214yescomplemented
88#1831C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
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)—24no—
102C2×AGL(1,ℤ/8)—42no—
103C2×AGL(1,ℤ/8)—42no—
104C2×AGL(1,ℤ/8)—42no—
105C2×AGL(1,ℤ/8)—42no—
106C22×SD16C2312yes—
107C22×SD16C2312yes—
108C22×D8C2312yes—
109C22×D8C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#532—24no—
115#532—24no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×#127C2311yescharacteristic
118C2×#126—24no—
119C2×#124C2311yescharacteristic
120C2×#124C2311yescharacteristic
121C2×#121—22no—
122#520—42no—
123#520—42no—
124#520—42no—
125#520—42no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4—22no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C22⋊C4—22no—
145C22×C42C2311yescharacteristic
146C2×#107—42no—
147C2×#107C2314yes—
148C2×#106—42no—
149C2×#106C2314yes—
150#424—42no—
151#424—42no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154#417—42no—
155#417—42no—
156#416—24no—
157#416—24no—
158#416—42no—
159#416—42no—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#102—42no—
167C2×#102—42no—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176#397—24no—
177#397—24no—
178#397—24no—
179#396—24no—
180#391C2312yes—
181#391C2312yes—
182#390—21no—
183#390—22no—
184#390—21no—
185#390—22no—
186#390C2314yes—
187#390C2314yes—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—22no—
193#389C2314yes—
194#389C2314yes—
195#389C2312yes—
196#389C2312yes—
197#388—24no—
198#388—24no—
199#388—24no—
200#388—24no—
201#388—24no—
202#387—24no—
203#387—24no—
204#386—22no—
205#386—22no—
206#386—24no—
207#386—24no—
208#385—22no—
209#385—22no—
210#384—21no—
211#382—21no—
212#382—22no—
213#382C2312yes—
214#382C2312yes—
215#380—22no—
216#380—22no—
217#380—22no—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D412yes—
223C4×C22⋊C4—24no—
224C4×C22⋊C4—24no—
225C4×C22⋊C4D414yes—
226#376C2312yes—
227C2×#95—22no—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
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#4114yes—
251C2×#41#4114yes—
252C22×Q8C2411yescharacteristic
253C22×Q8C2411yescharacteristic
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2412yes—
265C22×D4C2411yescharacteristic
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4C2×D412yes—
275C23×C4C2411yescharacteristic
276AGL(1,ℤ/8)—416no—
277AGL(1,ℤ/8)—416no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—42no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—42no—
284C2×D8—24no—
285C2×D8—24no—
286C2×D8—42no—
287C2×D8—42no—
288C2×D8—42no—
289C2×D8—42no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308#127—22no—
309#127—22no—
310#126—88no—
311#124—22no—
312#124—22no—
313#124—22no—
314#124—22no—
315#121—216no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×D4—216no—
321C4×D4—42no—
322C4×D4—42no—
323C4×D4—42no—
324C4×D4—42no—
325C4×D4—48no—
326C4×D4—48no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—44no—
336#117—44no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—84no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C2×D412yes—
387C2×C42C2×D414yes—
388C2×C42C2×D414yes—
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—44no—
394#106—44no—
395C4⋊C8—44no—
396C4⋊C8—44no—
397#103—24no—
398#103—24no—
399#103—42no—
400#103—42no—
401#103—42no—
402#103—42no—
403#102—24no—
404#102—24no—
405#102—44no—
406#102—44no—
407#102—42no—
408#102—42no—
409#102—42no—
410#102—42no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95C2×D414yes—
420#95C2×D414yes—
421C24—21no—
422C24—21no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—82no—
428C24—82no—
429C24C22×D411yescharacteristic
430#41—28no—
431#41—28no—
432#41—28no—
433#41—28no—
434#41—44no—
435#41—44no—
436#41—44no—
437#41—44no—
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—21no—
448C2×Q8C2×#4112yes—
449C2×Q8C2×#4112yes—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—28no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—42no—
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—44no—
479C2×D4—44no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4C2×#4112yes—
483C2×D4C2×#4112yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
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—44no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4C22×D412yes—
511C22×C4C22×D414yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C4H(5,𝔽2)12yes—
521C22×C42−1+412yes—
522C22×C4C2511yescharacteristic
523SD16—48no—
524SD16—48no—
525D8—48no—
526D8—48no—
527M16—44no—
528M16—44no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—44no—
542C2×C8—44no—
543C4⋊C4—28no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
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—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—82no—
564C4⋊C4—84no—
565C4⋊C4—84no—
566C22⋊C4—28no—
567C22⋊C4—28no—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—48no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—84no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C22⋊C4—84no—
582C42—24no—
583C42—21no—
584C42—21no—
585C42—21no—
586C42—21no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—42no—
596C42—42no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—88no—
615C23—84no—
616C23—162no—
617C23—162no—
618C23—164no—
619C23#52512yes—
620C23#53512yes—
621C23#53712yes—
622C23C23×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—22no—
626Q8—22no—
627Q8—41no—
628Q8—41no—
629Q8—41no—
630Q8—41no—
631D4—22no—
632D4—22no—
633D4—22no—
634D4—22no—
635D4—48no—
636D4—48no—
637D4—41no—
638D4—41no—
639D4—42no—
640D4—42no—
641D4—41no—
642D4—41no—
643D4—42no—
644D4—42no—
645D4—88no—
646D4—84no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—84no—
681C2×C4—164no—
682C2×C4—164no—
683C2×C4—162no—
684C2×C4—162no—
685C2×C4—162no—
686C2×C4#52512yes—
687C2×C4#53512yes—
688C2×C4#53712yes—
689C2×C4#53814yes—
690C2×C4#53814yes—
691C2×C4#53914yes—
692C2×C4#53914yes—
693C2×C4C22×#4111yescharacteristic
694C2×C4C22×#4111yescharacteristic
695C2×C4C2×H(5,𝔽2)11yescharacteristic
696C2×C4C2×H(5,𝔽2)11yescharacteristic
697C2×C4C2×H(5,𝔽2)11yescharacteristic
698C2×C4C2×2−1+411yescharacteristic
699C8—42no—
700C8—42no—
701C8—42no—
702C8—42no—
703C22—24no—
704C22—22no—
705C22—22no—
706C22—24no—
707C22—42no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—82no—
712C22—82no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—84no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—84no—
722C22—168no—
723C22#263614yes—
724C22C2×#52511yescharacteristic
725C22C2×#53511yescharacteristic
726C22C2×#53711yescharacteristic
727C22#358412yes—
728C22#358512yes—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4#263614yes—
742C4C2×#53811yescharacteristic
743C4C2×#53811yescharacteristic
744C4C2×#53911yescharacteristic
745C4C2×#53911yescharacteristic
746C2—24no—
747C2—82no—
748C2—82no—
749C2—162no—
750C2C2×#263611yescharacteristic
751C2#6058712yes—
752C2#6081911yescharacteristic
753C2#6082511yescharacteristic
754C2#6093312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
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
1. #35266C2semidirect
2. #34208C2semidirect
3. #30139C2semidirect
4. #30137C2semidirect
5. #30134C2semidirect
6. #30133C2semidirect
7. #23746C2semidirect
8. C2×#2092C2semidirect
9. #23611C2semidirect
10. #23610C2semidirect
11. #23544C2semidirect
12. #23544C2semidirect
13. #23504C2semidirect
15. #20932C2semidirect
16. #20932C2semidirect
17. #20882C2semidirect
18. #20881C2semidirect
19. #20754C2semidirect
45. #2328C22semidirect
46. #2327C22semidirect
53. C4×#124C22semidirect
54. C4×#124C22semidirect
59. #2101C22semidirect
60. #2101C22semidirect
63. #2068C22semidirect
64. #2067C22semidirect
65. #2066C22semidirect
66. #2065C22semidirect
71. #2043C22semidirect
72. #2042C22semidirect
73. #2042C22semidirect
74. #2041C22semidirect
83. #1865C22semidirect
84. #1864C22semidirect
85. #1863C22semidirect
86. #1862C22semidirect
87. #1832C22semidirect
88. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342082
#301392
#301372
#301342
#301332
#237462
C2×#20922
#236112
#236102
#235442
#235042
#234282
#209322
#208822
#208812
#207542