Finite Group Databasebeta

← #6449855

Subgroups — #6449855

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic, complemented
2C2×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
3#35333C211yesmaximal, characteristic, complemented
4#34186C211yesmaximal, characteristic, complemented
5#24690C214yesmaximal, complemented
6#24688C214yesmaximal, complemented
7#24686C214yesmaximal, complemented
8#24684C214yesmaximal, complemented
9#23632C211yesmaximal, characteristic, complemented
10#23632C211yesmaximal, characteristic, complemented
11#23544C211yesmaximal, characteristic
12#23544C211yesmaximal, characteristic
13#21325C211yesmaximal, characteristic, complemented
14#21174C211yesmaximal, characteristic, complemented
15#21173C211yesmaximal, characteristic, complemented
16#21145C211yesmaximal, characteristic, complemented
17#21068C211yesmaximal, characteristic, complemented
18#21067C211yesmaximal, characteristic, complemented
19#20645C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#478—28no—
25C2×#478C2214yes—
26C2×#478C2214yes—
27C4×AGL(1,ℤ/8)—28no—
28C2×#442C2212yes—
29C2×#440C2212yes—
30C2×C4×SD16C2212yes—
31C2×C4×D8C2212yes—
32C2×#428C2211yescharacteristic
33C22×#107C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C2×#424C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#416C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C2×C4×M16C2211yescharacteristic
45#2851—22no—
46#2629—22no—
47#2579—22no—
48#2579—22no—
49#2566C2212yescomplemented
50#2566C2212yescomplemented
51#2565C2212yescomplemented
52#2565C2212yescomplemented
53#2357—22no—
54#2357C2214yes—
55#2354—22no—
56#2354C2214yes—
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2312—22no—
60#2311C2211yescharacteristic
61#2311C2211yescharacteristic
62#2309C2211yescharacteristic
63#2309C2211yescharacteristic
64#2285C2211yescharacteristic
65#2285C2211yescharacteristic
66#2280—22no—
67#2274C2211yescharacteristic
68#2274C2211yescharacteristic
69#2073C2214yes—
70#2073C2214yes—
71#2071C2214yes—
72#2071C2214yes—
73#2055C2214yes—
74#2055C2214yes—
75#2054C2214yes—
76#2054C2214yes—
77#1992C2214yescomplemented
78#1991C2214yescomplemented
79#1949C2214yescomplemented
80#1949C2214yescomplemented
81#1948C2214yescomplemented
82#1947C2214yescomplemented
83#1940C2214yescomplemented
84#1939C2214yescomplemented
85#1928C2214yescomplemented
86#1927C2214yescomplemented
87#1926C2214yescomplemented
88#1925C2214yescomplemented
89#1805C2212yescomplemented
90#1804C2212yescomplemented
91#1803C2212yescomplemented
92#1802C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C2×AGL(1,ℤ/8)—24no—
105C2×AGL(1,ℤ/8)—24no—
106C2×AGL(1,ℤ/8)—24no—
107C22×SD16C2312yes—
108C22×D8C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—24no—
116C2×#125—24no—
117C2×#125C2312yes—
118C2×#125C2312yes—
119C4×#41—24no—
120C4×#41—24no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—24no—
128C2×C4×D4D418yes—
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—24no—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#478—216no—
149#478—216no—
150#442—28no—
151#440—28no—
152C4×SD16—28no—
153C4×D8—28no—
154#428—24no—
155C2×#107—24no—
156C2×#107C2314yes—
157C2×#107C2312yes—
158C2×#106—24no—
159C2×#106C2314yes—
160C2×#106C2312yes—
161#424—24no—
162#424—24no—
163C2×C4⋊C8—24no—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2314yes—
167C2×C4⋊C8C2314yes—
168#418—24no—
169#417—24no—
170#416—24no—
171#416—24no—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#102—24no—
176C2×#102—24no—
177C2×#102C2314yes—
178C2×#102C2314yes—
179C2×#102C2312yes—
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C4×M16—24no—
185C2×#97C2312yes—
186C2×C4×C8C2312yes—
187#398—24no—
188#398—24no—
189#396—24no—
190#396—24no—
191#391C2312yes—
192#391C2312yes—
193#390—22no—
194#390—22no—
195#390C2314yes—
196#390C2314yes—
197#390C2312yes—
198#390C2312yes—
199#389—22no—
200#389—22no—
201#388—24no—
202#388—24no—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#385—24no—
212#385—24no—
213#384D412yes—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#382D412yes—
218#382C2312yes—
219#382C2312yes—
220#382C2312yes—
221#381—24no—
222#380—21no—
223#380—21no—
224#380—22no—
225#380—22no—
226#380—24no—
227#380D412yes—
228#380D412yes—
229#379—22no—
230#379—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C2314yes—
236C4×C4⋊C4C2314yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241C4×C22⋊C4—24no—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C25—21no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41#4114yes—
259C22×Q8C2411yescharacteristic
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4C2×D414yes—
268C22×D4C2412yes—
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—21no—
273C23×C4—28no—
274C23×C4—22no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—84no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283AGL(1,ℤ/8)—232no—
284C2×SD16—24no—
285C2×SD16—24no—
286C2×SD16—24no—
287C2×D8—24no—
288C2×D8—24no—
289C2×D8—24no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16#4114yes—
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—
306#125—24no—
307#125—24no—
308#125—24no—
309#125—24no—
310#125—44no—
311#125—44no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×D4—28no—
317C4×D4—28no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4C2×D418yes—
327C4×D4C2×D418yes—
328#117—22no—
329#117—22no—
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—
339#117—88no—
340C2×C4⋊C4—21no—
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—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—24no—
394C2×C42—24no—
395C2×C42—44no—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D414yes—
399C2×C42#4114yes—
400C2×C42C2412yes—
401C2×C42C2412yes—
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404#107—28no—
405#107—24no—
406#106—28no—
407#106—24no—
408C4⋊C8—28no—
409C4⋊C8—28no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#102—28no—
416#102—28no—
417#102—24no—
418#102—24no—
419#102—24no—
420#102—24no—
421#102—24no—
422#97—24no—
423C4×C8—24no—
424#95—22no—
425#95—24no—
426#95—24no—
427#95—22no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—21no—
435C24—22no—
436C24—24no—
437C24—42no—
438C24—42no—
439C24—82no—
440C24—82no—
441C24C22×D411yescharacteristic
442#41—28no—
443#41—28no—
444#41—44no—
445#41—44no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×#4112yes—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—28no—
455C2×D4—28no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4C22×D414yes—
473C2×D4C2×#4112yes—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—28no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—48no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—84no—
498C22×C4—816no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4#12114yes—
503C22×C4#12114yes—
504C22×C4C22×D414yes—
505C22×C4C22×D414yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C42−1+412yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518SD16—216no—
519D8—216no—
520M16—28no—
521M16—44no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—44no—
534C2×C8—44no—
535C2×C8C2×#4114yes—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—82no—
564C4⋊C4—84no—
565C4⋊C4—84no—
566C4⋊C4C22×D414yes—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C22⋊C4—84no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—42no—
587C42—42no—
588C42—42no—
589C42—42no—
590C42—42no—
591C42—84no—
592C42—84no—
593C42#12114yes—
594C42#12114yes—
595C42C22×D412yes—
596C42C22×D412yes—
597C42C2×#4112yes—
598C42C2×#4112yes—
599C23—22no—
600C23—22no—
601C23—24no—
602C23—22no—
603C23—42no—
604C23—44no—
605C23—44no—
606C23—42no—
607C23—41no—
608C23—48no—
609C23—41no—
610C23—42no—
611C23—42no—
612C23—84no—
613C23—84no—
614C23—88no—
615C23—88no—
616C23C2×#12112yes—
617C23#54712yes—
618C23#54812yes—
619C23C23×D411yescharacteristic
620Q8—22no—
621Q8—22no—
622Q8—41no—
623Q8—41no—
624D4—22no—
625D4—24no—
626D4—24no—
627D4—22no—
628D4—48no—
629D4—48no—
630D4—41no—
631D4—42no—
632D4—42no—
633D4—41no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—416no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—88no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4C2×#12112yes—
674C2×C4C2×#12112yes—
675C2×C4C2×#12112yes—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4#52514yes—
679C2×C4#53614yes—
680C2×C4#54712yes—
681C2×C4#54812yes—
682C2×C4#55214yes—
683C2×C4#55614yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×2−1+411yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C8—24no—
691C8—42no—
692C8—42no—
693C8—42no—
694C22—22no—
695C22—22no—
696C22—24no—
697C22—24no—
698C22—44no—
699C22—44no—
700C22—42no—
701C22—44no—
702C22—44no—
703C22—816no—
704C22—81no—
705C22—81no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22#241514yes—
710C22C22×#12111yescharacteristic
711C22C2×#54711yescharacteristic
712C22C2×#54811yescharacteristic
713C22#358412yes—
714C22#358512yes—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—42no—
723C4—82no—
724C4—81no—
725C4—82no—
726C4—81no—
727C4—82no—
728C4—82no—
729C4#241514yes—
730C4C2×#52511yescharacteristic
731C4C2×#53611yescharacteristic
732C4C2×#55211yescharacteristic
733C4C2×#55611yescharacteristic
734C2—24no—
735C2—82no—
736C2—84no—
737C2C2×#241511yescharacteristic
738C2#6042211yescharacteristic
739C2#6042311yescharacteristic
740C2#6128312yes—
741C2#6134912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#478C2semidirect
3. #35333C2semidirect
4. #34186C2semidirect
5. #24690C2semidirect
6. #24688C2semidirect
7. #24686C2semidirect
8. #24684C2semidirect
9. #23632C2semidirect
10. #23632C2semidirect
13. #21325C2semidirect
14. #21174C2semidirect
15. #21173C2semidirect
16. #21145C2semidirect
17. #21068C2semidirect
18. #21067C2semidirect
19. #20645C2semidirect
49. #2566C22semidirect
50. #2566C22semidirect
51. #2565C22semidirect
52. #2565C22semidirect
77. #1992C22semidirect
78. #1991C22semidirect
79. #1949C22semidirect
80. #1949C22semidirect
81. #1948C22semidirect
82. #1947C22semidirect
83. #1940C22semidirect
84. #1939C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
87. #1926C22semidirect
88. #1925C22semidirect
89. #1805C22semidirect
90. #1804C22semidirect
91. #1803C22semidirect
92. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×C4×AGL(1,ℤ/8)2
#353332
#341862
#246902
#246882
#246862
#246842
#236322
#235442
#213252
#211742
#211732
#211452
#210682
#210672
#206452