Finite Group Databasebeta

← #6498206

Subgroups — #6498206

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33212C211yesmaximal, characteristic, complemented
4#33210C211yesmaximal, characteristic, complemented
5#33048C211yesmaximal, characteristic
6#33048C211yesmaximal, characteristic
7#30307C211yesmaximal, characteristic, complemented
8#30278C211yesmaximal, characteristic, complemented
9#30229C211yesmaximal, characteristic, complemented
10#30204C211yesmaximal, characteristic, complemented
11#29582C214yesmaximal, complemented
12#29523C214yesmaximal, complemented
13#29167C214yesmaximal, complemented
14#27865C214yesmaximal, complemented
15#22247C211yesmaximal, characteristic, complemented
16#22246C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic, complemented
18#22219C211yesmaximal, characteristic, complemented
19#20364C211yesmaximal, characteristic, complemented
20C2×#529—21no—
21C2×#500C2211yescharacteristic
22C2×#498C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#2864C2212yes—
28#2863—21no—
29#2863—21no—
30#2862C2211yescharacteristic
31#2861—21no—
32#2860—21no—
33#2860C2212yes—
34#2859C2211yescharacteristic
35#2859C2211yescharacteristic
36#2858C2211yescharacteristic
37#2857C2211yescharacteristic
38#2857C2211yescharacteristic
39#2857C2211yescharacteristic
40#2856C2211yescharacteristic
41#2855—21no—
42#2855—21no—
43#2854—21no—
44#2854C2212yes—
45#2851C2212yes—
46#2850—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#125C2212yes—
52C4×#124—21no—
53C4×C22⋊Q8—21no—
54C4×C22⋊Q8—21no—
55C4×#121—21no—
56C4×#121C2212yes—
57#2295—21no—
58#2295—21no—
59#2293—21no—
60#2293C2212yes—
61#2283C2212yes—
62C2×C43C2211yescharacteristic
63#2101C2214yes—
64#2101C2214yes—
65#2092C2214yes—
66#2092C2214yes—
67#2085C2214yes—
68#2073C2214yes—
69#2059C2214yes—
70#2025C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1944C2214yes—
80#1943C2214yes—
81#1936C2214yes—
82#1936C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1920C2214yes—
86#1919C2214yes—
87#1918C2214yes—
88#1918C2214yes—
89#1895C2214yes—
90#1895C2214yes—
91#1864C2214yes—
92#1862C2214yes—
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1809C2214yes—
98#1809C2214yes—
99#1745C2212yes—
100#1745C2212yes—
101#1744C2211yescharacteristic
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105#1728C2212yes—
106#1728C2212yes—
107C22×M16C2312yes—
108C22×M16C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2311yescharacteristic
113#529—48no—
114C2×#126—22no—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#124—21no—
118C2×#124—21no—
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×#121—21no—
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×D4—22no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×#117—22no—
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#500—24no—
142#498—24no—
143#486—24no—
144#486—24no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C4×M16—24no—
154C4×M16—24no—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#388—22no—
208#388—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#386—22no—
218#386—22no—
219#385—22no—
220#385—22no—
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#383—22no—
226#382—21no—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#381—22no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380C2312yes—
235#380C2312yes—
236#380C2312yes—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248#376—22no—
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43D411yescharacteristic
252C43D411yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
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—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×D4—22no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2412yes—
280C23×C4—22no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C2×M16—24no—
289C2×M16—24no—
290C2×M16—24no—
291C2×M16—24no—
292C2×M16—42no—
293C2×M16—42no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304#127—22no—
305#127—22no—
306#126—48no—
307#125—44no—
308#125—44no—
309#124—44no—
310#124—44no—
311#123—48no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315#121—28no—
316#121—44no—
317#121—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×D4—44no—
322C4×D4—44no—
323C4×D4—48no—
324C4×D4—44no—
325#117—48no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
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—21no—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
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×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42—42no—
377C2×C42—42no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—28no—
407#107—44no—
408#106—28no—
409#106—44no—
410#102—28no—
411#102—28no—
412#102—44no—
413#102—44no—
414#97—24no—
415#97—24no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8#4114yes—
419C4×C8#4114yes—
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#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440C24—22no—
441C24C22×D411yescharacteristic
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
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×C4C22×D411yescharacteristic
508C22×C42−1+414yes—
509C22×C4C2511yescharacteristic
510M16—48no—
511M16—48no—
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×C8—22no—
521C2×C8—22no—
522C2×C8C2×#4114yes—
523C2×C8C2×#4114yes—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—21no—
582C42—21no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—44no—
596C42—44no—
597C42—44no—
598C42AGL(1,ℤ/8)14yes—
599C42C2×#4112yes—
600C42C2×#4112yes—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23#53612yes—
615C23C23×D411yescharacteristic
616Q8—84no—
617Q8—84no—
618Q8—84no—
619D4—44no—
620D4—44no—
621D4—88no—
622D4—84no—
623D4—84no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—24no—
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—21no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
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#53212yes—
685C2×C4#53212yes—
686C2×C4#53612yes—
687C2×C4C2×AGL(1,ℤ/8)12yes—
688C2×C4C2×AGL(1,ℤ/8)12yes—
689C2×C4C2×AGL(1,ℤ/8)12yes—
690C2×C4#57512yes—
691C2×C4#57512yes—
692C2×C4#57512yes—
693C2×C4C22×#4111yescharacteristic
694C2×C4C22×#4111yescharacteristic
695C2×C4C2×2−1+412yes—
696C2×C4#58512yes—
697C8—24no—
698C8—24no—
699C8—42no—
700C8—42no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22C2×#53211yescharacteristic
716C22C2×#53611yescharacteristic
717C22#344911yescharacteristic
718C22C22×AGL(1,ℤ/8)11yescharacteristic
719C22C2×#57511yescharacteristic
720C22C2×#57511yescharacteristic
721C22C2×#57511yescharacteristic
722C4—22no—
723C4—22no—
724C4—22no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4#317114yes—
741C4#318314yes—
742C4#349512yes—
743C4#349512yes—
744C2—42no—
745C2—162no—
746C2#3532611yescharacteristic
747C2C2×#317111yescharacteristic
748C2C2×#318311yescharacteristic
749C2#6086211yescharacteristic
750C2#6086411yescharacteristic
751C2#6096511yescharacteristic
752C2#6097211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #33212C2semidirect
4. #33210C2semidirect
7. #30307C2semidirect
8. #30278C2semidirect
9. #30229C2semidirect
10. #30204C2semidirect
11. #29582C2semidirect
12. #29523C2semidirect
13. #29167C2semidirect
14. #27865C2semidirect
15. #22247C2semidirect
16. #22246C2semidirect
17. #22220C2semidirect
18. #22219C2semidirect
19. #20364C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#332122
#332102
#330482
#303072
#302782
#302292
#302042
#295822
#295232
#291672
#278652
#222472
#222462
#222202
#222192
#203642