Finite Group Databasebeta

← #6752307

Subgroups — #6752307

Order 512.

1511 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36181C211yesmaximal, characteristic, complemented
3#33048C211yesmaximal, characteristic, complemented
4#32213C214yesmaximal, complemented
5#32020C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#29842C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#27973C211yesmaximal, characteristic, complemented
11#27890C211yesmaximal, characteristic, complemented
12#27876C211yesmaximal, characteristic, complemented
13#27875C211yesmaximal, characteristic, complemented
14#27386C214yesmaximal, complemented
15#27344C214yesmaximal, complemented
16#22219C211yesmaximal, characteristic, complemented
17#21981C211yesmaximal, characteristic, complemented
18#21102C211yesmaximal, characteristic, complemented
19#20824C211yesmaximal, characteristic, complemented
20C2×#552—21no—
21C2×#486C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2856C2211yescharacteristic
30#2565—21no—
31#2535C2212yes—
32#2513—21no—
33#2513—21no—
34#2506—21no—
35#2506C2212yes—
36#2503—21no—
37#2499—21no—
38#2496—21no—
39#2496C2212yes—
40#2490C2212yes—
41#2490C2212yes—
42#2486—21no—
43#2486C2212yes—
44#2420—21no—
45#2420—21no—
46#2414—21no—
47#2414—21no—
48#2363—21no—
49#2363—21no—
50#2357C2212yes—
51#2344—21no—
52#2344C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C42×D4C2211yescharacteristic
60#2101C2214yes—
61#2085C2214yes—
62#2077C2214yes—
63#2077C2214yes—
64#2071C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2059C2214yes—
68#2034C2214yes—
69#2033C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1948C2214yes—
78#1936C2214yes—
79#1926C2214yes—
80#1926C2214yes—
81#1924C2212yes—
82#1920C2214yes—
83#1919C2214yes—
84#1917C2212yes—
85#1895C2214yes—
86#1874C2214yes—
87#1870C2212yes—
88#1864C2214yes—
89#1864C2214yes—
90#1853C2214yes—
91#1851C2212yes—
92#1845C2211yescharacteristic
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1834C2211yescharacteristic
101#1809C2214yes—
102#1776C2214yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×M16C2312yes—
108#552—48no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#126—22no—
112C2×#125—22no—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×#121—21no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—21no—
134C2×#117—22no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#486—24no—
141#431—24no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C4×M16—24no—
156C2×#97C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161#400—22no—
162#400—22no—
163#398—22no—
164#398—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#397—22no—
169#397—22no—
170#394—22no—
171#394—22no—
172#392—22no—
173#391—21no—
174#391—21no—
175#391—21no—
176#390—22no—
177#390—22no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#386—22no—
210#386—22no—
211#384—21no—
212#384—21no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2312yes—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382C2312yes—
225#381—22no—
226#381—22no—
227#381—22no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380C2312yes—
234#380C2312yes—
235#380C2312yes—
236#380C2312yes—
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4—22no—
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2×D411yescharacteristic
276C22×D4C2412yes—
277C23×C4—22no—
278C23×C4—22no—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C23×C4C2411yescharacteristic
285C2×M16—24no—
286C2×M16—42no—
287C2×M16—42no—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294#127—22no—
295#127—22no—
296#126—48no—
297#125—44no—
298#125—44no—
299#125—48no—
300#123—48no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303#121—44no—
304#121—44no—
305#121—44no—
306#121—44no—
307#121—82no—
308#121—82no—
309C4×Q8—44no—
310C4×D4—24no—
311C4×D4—44no—
312C4×D4—42no—
313C4×D4—42no—
314C4×D4—82no—
315C4×D4—82no—
316#117—48no—
317#117—44no—
318#117—82no—
319#117—82no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397#107—44no—
398#106—44no—
399#106—44no—
400#106—44no—
401C4⋊C8—24no—
402C4⋊C8—42no—
403#102—28no—
404#102—44no—
405#102—44no—
406#102—44no—
407C22⋊C8—24no—
408C22⋊C8—42no—
409C22⋊C8—42no—
410#97—24no—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8#4114yes—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24—22no—
438C24C22×D411yescharacteristic
439C24C22×D411yescharacteristic
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—82no—
459C2×D4—82no—
460C2×D4—82no—
461C2×D4—82no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4AGL(1,ℤ/8)14yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2511yescharacteristic
508M16—48no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8C2×#4114yes—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—21no—
584C42—21no—
585C42—21no—
586C42—21no—
587C42—22no—
588C42—22no—
589C42—44no—
590C42—44no—
591C42—82no—
592C42—82no—
593C42C2×#4112yes—
594C422−1+412yes—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23C2×AGL(1,ℤ/8)12yes—
616C23C23×D411yescharacteristic
617Q8—84no—
618D4—44no—
619D4—44no—
620D4—44no—
621D4—84no—
622D4—84no—
623D4—164no—
624D4—164no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—28no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—88no—
684C2×C4—164no—
685C2×C4#54712yes—
686C2×C4#54712yes—
687C2×C4C2×AGL(1,ℤ/8)12yes—
688C2×C4C2×AGL(1,ℤ/8)12yes—
689C2×C4#57512yes—
690C2×C4#57712yes—
691C2×C4#57712yes—
692C2×C4C22×#4111yescharacteristic
693C2×C4C2×2−1+411yescharacteristic
694C2×C4#58512yes—
695C2×C4#58512yes—
696C8—24no—
697C8—42no—
698C8—42no—
699C8—42no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22#344911yescharacteristic
722C22C2×#54711yescharacteristic
723C22#350011yescharacteristic
724C22C22×AGL(1,ℤ/8)11yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57711yescharacteristic
727C22C2×#57711yescharacteristic
728C4—22no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#334514yes—
746C4#335114yes—
747C4#349312yes—
748C4#356512yes—
749C2—42no—
750C2—81no—
751C2—162no—
752C2#3787111yescharacteristic
753C2#6096511yescharacteristic
754C2#6101211yescharacteristic
755C2C2×#334511yescharacteristic
756C2C2×#335111yescharacteristic
757C2#6254111yescharacteristic
758C2#6254411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36181C2semidirect
3. #33048C2semidirect
4. #32213C2semidirect
5. #32020C2semidirect
6. #30307C2semidirect
7. #29842C2semidirect
8. #29220C2semidirect
9. #29190C2semidirect
10. #27973C2semidirect
11. #27890C2semidirect
12. #27876C2semidirect
13. #27875C2semidirect
14. #27386C2semidirect
15. #27344C2semidirect
16. #22219C2semidirect
17. #21981C2semidirect
18. #21102C2semidirect
19. #20824C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361812
#330482
#322132
#320202
#303072
#298422
#292202
#291902
#279732
#278902
#278762
#278752
#273862
#273442
#222192
#219812
#211022
#208242