Finite Group Databasebeta

← #7503937

Subgroups — #7503937

Order 512.

2064 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3084C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic, complemented
3#35639C211yesmaximal, characteristic, complemented
4#35174C211yesmaximal, characteristic, complemented
5#31245C214yesmaximal, complemented
6#31244C214yesmaximal, complemented
7#31243C214yesmaximal, complemented
8#31241C214yesmaximal, complemented
9#23614C211yesmaximal, characteristic, complemented
10#23612C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic, complemented
12#23609C211yesmaximal, characteristic, complemented
13#23555C211yesmaximal, characteristic, complemented
14#23553C211yesmaximal, characteristic, complemented
15#21235C211yesmaximal, characteristic
16#21157C211yesmaximal, characteristic, complemented
17#21147C211yesmaximal, characteristic, complemented
18#20932C211yesmaximal, characteristic
19#20925C211yesmaximal, characteristic
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#3084—28no—
26C2×#479C2212yescomplemented
27C2×#475C2212yescomplemented
28C2×#478—28no—
29C2×#478C2214yes—
30C2×#478C2214yes—
31C2×#478C2212yescomplemented
32C2×#476C2212yescomplemented
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#424C2211yescharacteristic
40C22×C4⋊C8C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2848—22no—
49#2626—22no—
50#2579—22no—
51#2487—22no—
52#2454—22no—
53#2454C2214yes—
54#2454C2214yescomplemented
55#2453C2214yescomplemented
56#2452—22no—
57#2452C2214yes—
58#2442—22no—
59#2441—22no—
60#2317C2211yescharacteristic
61#2316C2211yescharacteristic
62#2316C2211yescharacteristic
63#2311C2211yescharacteristic
64#2309C2211yescharacteristic
65#2309C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68#2296C2211yescharacteristic
69#2068C2214yescomplemented
70#2068C2212yescomplemented
71#2067C2214yes—
72#2067C2214yes—
73#2067C2214yescomplemented
74#2067C2212yescomplemented
75#2066C2212yescomplemented
76#2065C2214yes—
77#2065C2214yes—
78#2065C2212yescomplemented
79#2059C2214yescomplemented
80#2059C2214yescomplemented
81#2059C2214yescomplemented
82#2057C2214yescomplemented
83#1973C2212yes—
84#1972C2212yes—
85#1970C2212yes—
86#1968C2212yes—
87#1944C2214yescomplemented
88#1941C2214yescomplemented
89#1940C2214yescomplemented
90#1939C2214yescomplemented
91#1880C2214yes—
92#1879C2214yes—
93#1879C2214yes—
94#1877C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C2×AGL(1,ℤ/8)—42no—
105C2×AGL(1,ℤ/8)—42no—
106C2×AGL(1,ℤ/8)Q818yescomplemented
107C22×SD16C2312yes—
108C22×D8C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—24no—
114#544—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2312yes—
117C2×#125—21no—
118C2×#125—24no—
119C2×#125—22no—
120C2×#125—42no—
121C2×#125C2314yes—
122C2×#125C2312yes—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—22no—
125#520—24no—
126#520—42no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—22no—
129C2×C4×D4—24no—
130C2×C4×D4—22no—
131C2×C4×D4—42no—
132C2×C4×D4C2314yes—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C22⋊C4—21no—
147C22×C42C2311yescharacteristic
148#479—28no—
149#478—216no—
150#478—216no—
151#478—28no—
152#476—28no—
153#475—28no—
154#428—24no—
155#428—42no—
156#427—42no—
157C2×#107—24no—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#106D418yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168#424—24no—
169#424—42no—
170C2×C4⋊C8—24no—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2314yes—
176#418—42no—
177#417—24no—
178#416—24no—
179#416—42no—
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2311yescharacteristic
183C2×#103C2312yes—
184C2×#102—24no—
185C2×#102—24no—
186C2×#102C2312yes—
187C2×#102C2314yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×#102C2314yes—
193C2×#102C2312yes—
194#405—42no—
195C2×#97C2312yes—
196C2×C4×C8C2312yes—
197#400—24no—
198#399—24no—
199#399—24no—
200#398—24no—
201#396—24no—
202#396—24no—
203#396—24no—
204#393—24no—
205#391C2312yes—
206#391C2312yes—
207#390—21no—
208#390—22no—
209#390—21no—
210#390C2312yes—
211#390C2314yes—
212#390C2312yes—
213#390C2312yes—
214#390C2312yes—
215#389—22no—
216#389—22no—
217#389—24no—
218#389D412yes—
219#389D412yes—
220#389C2314yes—
221#389C2312yes—
222#389C2312yes—
223#389C2314yes—
224#389C2314yes—
225#387—24no—
226#386—21no—
227#386—21no—
228#386—22no—
229#385—21no—
230#385—21no—
231#385—24no—
232#385—24no—
233#385—22no—
234#385—24no—
235#385D412yes—
236#385D412yes—
237#384—21no—
238#382—21no—
239#380—22no—
240#380—22no—
241#380—22no—
242#380—24no—
243#380—24no—
244#379—22no—
245#379—22no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—22no—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256C25—21no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41C2×Q814yes—
263C22×Q8—22no—
264C22×Q8C2411yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4—22no—
272C22×D4C2×Q814yes—
273C22×D4C2411yescharacteristic
274C22×D4C2412yes—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—24no—
280C23×C4—22no—
281C23×C4—44no—
282C23×C4—84no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286AGL(1,ℤ/8)—416no—
287C2×SD16—42no—
288C2×SD16—42no—
289C2×SD16C2×Q818yes—
290C2×D8—42no—
291C2×D8—42no—
292C2×D8C2×Q818yes—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16C2×Q814yes—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8C2×D414yes—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C22×C8C2411yescharacteristic
308C22×C8C2411yescharacteristic
309C22×C8C2412yes—
310C4⋊Q8—24no—
311C4⋊Q8—42no—
312C4⋊Q8—44no—
313#125—28no—
314#125—24no—
315#125—44no—
316#125—42no—
317#125—48no—
318#125—44no—
319C22⋊Q8—48no—
320C22⋊Q8—88no—
321C4×Q8—24no—
322C4×Q8—42no—
323C4×D4—28no—
324C4×D4—24no—
325C4×D4—44no—
326C4×D4—42no—
327C4×D4—48no—
328C4×D4—48no—
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—22no—
339#117—44no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2×D414yes—
367C2×C4⋊C4C2×D414yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—44no—
388C2×C22⋊C4C2×D412yes—
389C2×C22⋊C4C2×D412yes—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42C2×D412yes—
397C2×C42C2412yes—
398C2×C42C2412yes—
399C2×C42C2412yes—
400#107—28no—
401#107—24no—
402#107—42no—
403#107—42no—
404#106—24no—
405#106—42no—
406#106—42no—
407#106C2×D418yes—
408#106C2×D418yes—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413C4⋊C8—44no—
414#103—22no—
415#103—22no—
416#103—24no—
417#103—42no—
418#102—24no—
419#102—24no—
420#102—22no—
421#102—22no—
422#102—28no—
423#102—24no—
424#102—42no—
425#102—44no—
426#97—42no—
427C4×C8—42no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—24no—
433#95—24no—
434#95—24no—
435#95—24no—
436#95C2×D414yes—
437#95C2×D414yes—
438C24—21no—
439C24—22no—
440C24—24no—
441C24—42no—
442C24—42no—
443C24—82no—
444C24—82no—
445C24C22×D411yescharacteristic
446#41—28no—
447#41—28no—
448#41—44no—
449#41—44no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—44no—
456C2×Q8—84no—
457C2×Q8C22×Q812yes—
458C2×D4—24no—
459C2×D4—28no—
460C2×D4—28no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—44no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—44no—
478C2×D4C22×Q812yes—
479C2×D4C22×Q814yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—48no—
496C22×C4—44no—
497C22×C4—48no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4—816no—
507C22×C4C22×D414yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×Q812yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C42−1+412yes—
518C22×C42−1+412yes—
519C22×C4C2511yescharacteristic
520SD16—48no—
521D8—48no—
522M16—28no—
523M16—44no—
524C2×C8—28no—
525C2×C8—21no—
526C2×C8—22no—
527C2×C8—21no—
528C2×C8—22no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—44no—
538C2×C8C22×D412yes—
539C2×C8C22×D412yes—
540C2×C8C22×Q814yes—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—48no—
562C4⋊C4—48no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—42no—
568C4⋊C4—84no—
569C4⋊C4—82no—
570C4⋊C4—84no—
571C4⋊C4—84no—
572C4⋊C4C22×D414yes—
573C4⋊C4C22×D414yes—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—48no—
577C22⋊C4—44no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—84no—
582C22⋊C4—84no—
583C22⋊C4—82no—
584C22⋊C4—84no—
585C22⋊C4—82no—
586C22⋊C4—84no—
587C22⋊C4—84no—
588C42—24no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—44no—
596C42—42no—
597C42—42no—
598C42—42no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—24no—
603C23—44no—
604C23—42no—
605C23—42no—
606C23—44no—
607C23—41no—
608C23—42no—
609C23—48no—
610C23—41no—
611C23—42no—
612C23—88no—
613C23—84no—
614C23—88no—
615C23—84no—
616C23#53712yes—
617C23D4212yes—
618C23D4×Q812yes—
619C23C23×D411yescharacteristic
620Q8—22no—
621Q8—22no—
622Q8—41no—
623Q8—41no—
624Q8—164no—
625D4—22no—
626D4—24no—
627D4—24no—
628D4—22no—
629D4—48no—
630D4—42no—
631D4—41no—
632D4—48no—
633D4—42no—
634D4—41no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—816no—
673C2×C4—84no—
674C2×C4—164no—
675C2×C4#53714yes—
676C2×C4#53712yes—
677C2×C4#54314yes—
678C2×C4D4212yes—
679C2×C4D4×Q812yes—
680C2×C4D4×Q814yes—
681C2×C4#56314yes—
682C2×C4C23×D411yescharacteristic
683C2×C4C23×Q811yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×2−1+411yescharacteristic
687C2×C4C2×2−1+411yescharacteristic
688C8—22no—
689C8—22no—
690C8—24no—
691C8—42no—
692C8—42no—
693C22—24no—
694C22—22no—
695C22—22no—
696C22—24no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22—48no—
701C22—816no—
702C22—82no—
703C22—84no—
704C22—81no—
705C22—82no—
706C22—82no—
707C22—81no—
708C22#268314yes—
709C22C2×#53711yescharacteristic
710C22C2×D4211yescharacteristic
711C22C2×D4×Q811yescharacteristic
712C22#358412yes—
713C22#358512yes—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—44no—
721C4—42no—
722C4—82no—
723C4—82no—
724C4—81no—
725C4—82no—
726C4—81no—
727C4#268314yes—
728C4C2×#53711yescharacteristic
729C4C2×#54311yescharacteristic
730C4C2×D4×Q811yescharacteristic
731C4C2×#56311yescharacteristic
732C2—24no—
733C2—82no—
734C2—84no—
735C2C2×#268311yescharacteristic
736C2#6093312yes—
737C2#6105211yescharacteristic
738C2#6105911yescharacteristic
739C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3084C2semidirect
2. C22×#478C2semidirect
3. #35639C2semidirect
4. #35174C2semidirect
5. #31245C2semidirect
6. #31244C2semidirect
7. #31243C2semidirect
8. #31241C2semidirect
9. #23614C2semidirect
10. #23612C2semidirect
11. #23611C2semidirect
12. #23609C2semidirect
13. #23555C2semidirect
14. #23553C2semidirect
16. #21157C2semidirect
17. #21147C2semidirect
26. C2×#479C22semidirect
27. C2×#475C22semidirect
31. C2×#478C22semidirect
32. C2×#476C22semidirect
54. #2454C22semidirect
55. #2453C22semidirect
69. #2068C22semidirect
70. #2068C22semidirect
73. #2067C22semidirect
74. #2067C22semidirect
75. #2066C22semidirect
78. #2065C22semidirect
79. #2059C22semidirect
80. #2059C22semidirect
81. #2059C22semidirect
82. #2057C22semidirect
87. #1944C22semidirect
88. #1941C22semidirect
89. #1940C22semidirect
90. #1939C22semidirect
106. C2×AGL(1,ℤ/8)Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30842
C22×#4782
#356392
#351742
#312452
#312442
#312432
#312412
#236142
#236122
#236112
#236092
#235552
#235532
#212352
#211572
#211472
#209322
#209252