Finite Group Databasebeta

← #6461504

Subgroups — #6461504

Order 512.

2068 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic, complemented
3#29378C214yesmaximal, complemented
4#29376C214yesmaximal, complemented
5#29373C214yesmaximal, complemented
6#29372C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23504C211yesmaximal, characteristic, complemented
10#23428C211yesmaximal, characteristic
11#21068C211yesmaximal, characteristic, complemented
12#21067C211yesmaximal, characteristic, complemented
13#20932C211yesmaximal, characteristic, complemented
14#20932C211yesmaximal, characteristic, complemented
15#20929C211yesmaximal, characteristic, complemented
16#20929C211yesmaximal, characteristic, complemented
17#20882C211yesmaximal, characteristic, complemented
18#20881C211yesmaximal, characteristic, complemented
19#20774C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×AGL(1,ℤ/8)C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36#2364—24no—
37#2357—22no—
38#2354—22no—
39#2352—22no—
40#2352—22no—
41#2342—24no—
42#2328C2212yescomplemented
43#2327C2212yescomplemented
44#2319C2214yes—
45#2317C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2309C2211yescharacteristic
49C4×#124C2212yescomplemented
50C4×#124C2212yescomplemented
51#2296C2211yescharacteristic
52#2293C2214yes—
53C2×#390C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2101C2214yescomplemented
58#2101C2214yescomplemented
59#2092C2214yes—
60#2092C2214yes—
61#2043C2212yescomplemented
62#2042C2212yescomplemented
63#2042C2212yescomplemented
64#2041C2212yescomplemented
65#2013C2212yes—
66#2012C2212yes—
67#2012C2212yes—
68#2011C2212yes—
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1926C2214yescomplemented
72#1925C2214yescomplemented
73#1880C2214yes—
74#1880C2214yes—
75#1879C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1879C2214yes—
79#1877C2214yes—
80#1877C2214yes—
81#1865C2214yescomplemented
82#1864C2214yescomplemented
83#1863C2214yescomplemented
84#1862C2214yescomplemented
85#1840C2214yescomplemented
86#1839C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1729C2211yescharacteristic
92#1729C2211yescharacteristic
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)—24no—
103C2×AGL(1,ℤ/8)—24no—
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×#124C2311yescharacteristic
119C2×#124C2311yescharacteristic
120C2×#121—22no—
121#520—42no—
122#520—42no—
123#520—42no—
124#520—42no—
125#518—24no—
126#518—24no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—22no—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4—22no—
133C2×C4×D4C2314yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117—21no—
138C2×#117—24no—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C22⋊C4—22no—
149C22×C22⋊C4—22no—
150C22×C42C2311yescharacteristic
151C2×#107—42no—
152C2×#107C2314yes—
153C2×#106—42no—
154C2×#106C2314yes—
155#417—42no—
156#417—42no—
157#416—24no—
158#416—24no—
159#416—24no—
160#416—24no—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#102—42no—
168C2×#102—42no—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2314yes—
174C2×#102C2314yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390—21no—
180#390—21no—
181#390—22no—
182#390C2314yes—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#388—24no—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#388—24no—
196#387—24no—
197#387—24no—
198#386—22no—
199#386—22no—
200#386—24no—
201#386—24no—
202#385—22no—
203#385—22no—
204#384—21no—
205#382—21no—
206#382—22no—
207#382C2312yes—
208#382C2312yes—
209#381—24no—
210#380—22no—
211#380—22no—
212#380—24no—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4D414yes—
224#376C2312yes—
225C2×#95—22no—
226C43C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
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—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4#4114yes—
264C22×D4C2411yescharacteristic
265C22×D4C2411yescharacteristic
266C22×D4C2412yes—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—24no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4C2×D412yes—
276C23×C4C2411yescharacteristic
277AGL(1,ℤ/8)—416no—
278AGL(1,ℤ/8)—416no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—42no—
284C2×SD16—42no—
285C2×D8—24no—
286C2×D8—24no—
287C2×D8—24no—
288C2×D8—24no—
289C2×D8—42no—
290C2×D8—42no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309#127—22no—
310#127—22no—
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—216no—
322C4×D4—28no—
323C4×D4—28no—
324C4×D4—42no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—48no—
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—44no—
338#117—44no—
339#117—88no—
340#117#4114yes—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—216no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—84no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—21no—
389C2×C42—24no—
390C2×C42C2×D412yes—
391C2×C42C2×D414yes—
392C2×C42C2×D414yes—
393C2×C42C2412yes—
394C2×C42C2412yes—
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—44no—
398#106—44no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—42no—
404#103—42no—
405#102—24no—
406#102—24no—
407#102—24no—
408#102—24no—
409#102—44no—
410#102—44no—
411#102—42no—
412#102—42no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—22no—
421#95C2×D414yes—
422#95C2×D414yes—
423C24—22no—
424C24—21no—
425C24—21no—
426C24—24no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24C22×D411yescharacteristic
432#41—28no—
433#41—28no—
434#41—28no—
435#41—28no—
436#41—44no—
437#41—44no—
438#41—44no—
439#41—44no—
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×Q8—21no—
449C2×Q8—21no—
450C2×Q8C2×#4112yes—
451C2×Q8C2×#4112yes—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—28no—
455C2×D4—24no—
456C2×D4—28no—
457C2×D4—28no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—44no—
480C2×D4—44no—
481C2×D4—82no—
482C2×D4—82no—
483C2×D4C2×#4112yes—
484C2×D4C2×#4112yes—
485C2×D4C2×#4114yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—216no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—82no—
513C22×C4—82no—
514C22×C4—82no—
515C22×C4C22×D412yes—
516C22×C4C22×D414yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C2×#4112yes—
522C22×C4C2×#4112yes—
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C4H(5,𝔽2)12yes—
526C22×C4H(5,𝔽2)12yes—
527C22×C4C2511yescharacteristic
528SD16—48no—
529SD16—48no—
530D8—48no—
531D8—48no—
532M16—44no—
533M16—44no—
534C2×C8—22no—
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—22no—
545C2×C8—22no—
546C2×C8—44no—
547C2×C8—44no—
548C4⋊C4—28no—
549C4⋊C4—24no—
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—22no—
560C4⋊C4—22no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—82no—
568C4⋊C4—84no—
569C4⋊C4C2×#4114yes—
570C22⋊C4—216no—
571C22⋊C4—28no—
572C22⋊C4—28no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—48no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—42no—
580C22⋊C4—84no—
581C22⋊C4—84no—
582C22⋊C4—84no—
583C22⋊C4—84no—
584C22⋊C4—84no—
585C42—24no—
586C42—24no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—21no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—44no—
596C42—44no—
597C42—44no—
598C42—44no—
599C42—42no—
600C42—84no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—22no—
605C23—22no—
606C23—42no—
607C23—44no—
608C23—42no—
609C23—44no—
610C23—42no—
611C23—42no—
612C23—41no—
613C23—41no—
614C23—42no—
615C23—41no—
616C23—41no—
617C23—42no—
618C23—48no—
619C23—88no—
620C23—162no—
621C23—162no—
622C23—164no—
623C23#52512yes—
624C23#52512yes—
625C23#53512yes—
626C23C23×D411yescharacteristic
627Q8—22no—
628Q8—22no—
629Q8—22no—
630Q8—22no—
631Q8—41no—
632Q8—41no—
633Q8—41no—
634Q8—41no—
635D4—22no—
636D4—22no—
637D4—24no—
638D4—24no—
639D4—22no—
640D4—22no—
641D4—48no—
642D4—48no—
643D4—48no—
644D4—41no—
645D4—41no—
646D4—42no—
647D4—42no—
648D4—41no—
649D4—41no—
650D4—88no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—48no—
674C2×C4—48no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—84no—
685C2×C4—816no—
686C2×C4—164no—
687C2×C4—164no—
688C2×C4—162no—
689C2×C4—162no—
690C2×C4—162no—
691C2×C4#52512yes—
692C2×C4#52512yes—
693C2×C4#53512yes—
694C2×C4#55114yes—
695C2×C4#55114yes—
696C2×C4#55314yes—
697C2×C4#55314yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C22×#4111yescharacteristic
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×H(5,𝔽2)11yescharacteristic
704C8—42no—
705C8—42no—
706C8—42no—
707C8—42no—
708C22—22no—
709C22—22no—
710C22—24no—
711C22—24no—
712C22—42no—
713C22—42no—
714C22—44no—
715C22—44no—
716C22—44no—
717C22—48no—
718C22—81no—
719C22—81no—
720C22—82no—
721C22—81no—
722C22—81no—
723C22—82no—
724C22—84no—
725C22—82no—
726C22—82no—
727C22—168no—
728C22#259814yes—
729C22C2×#52511yescharacteristic
730C22C2×#52511yescharacteristic
731C22C2×#53511yescharacteristic
732C22#358412yes—
733C22#358512yes—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—44no—
741C4—44no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4#259814yes—
747C4C2×#55111yescharacteristic
748C4C2×#55111yescharacteristic
749C4C2×#55311yescharacteristic
750C4C2×#55311yescharacteristic
751C2—24no—
752C2—82no—
753C2—82no—
754C2—84no—
755C2C2×#259811yescharacteristic
756C2#6058712yes—
757C2#6060112yes—
758C2#6081911yescharacteristic
759C2#6082511yescharacteristic
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. #34255C2semidirect
2. #34208C2semidirect
3. #29378C2semidirect
4. #29376C2semidirect
5. #29373C2semidirect
6. #29372C2semidirect
7. #23746C2semidirect
8. C2×#2092C2semidirect
9. #23504C2semidirect
11. #21068C2semidirect
12. #21067C2semidirect
13. #20932C2semidirect
14. #20932C2semidirect
15. #20929C2semidirect
16. #20929C2semidirect
17. #20882C2semidirect
18. #20881C2semidirect
19. #20774C2semidirect
42. #2328C22semidirect
43. #2327C22semidirect
49. C4×#124C22semidirect
50. C4×#124C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
61. #2043C22semidirect
62. #2042C22semidirect
63. #2042C22semidirect
64. #2041C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
71. #1926C22semidirect
72. #1925C22semidirect
81. #1865C22semidirect
82. #1864C22semidirect
83. #1863C22semidirect
84. #1862C22semidirect
85. #1840C22semidirect
86. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#342082
#293782
#293762
#293732
#293722
#237462
C2×#20922
#235042
#234282
#210682
#210672
#209322
#209292
#208822
#208812
#207742