Finite Group Databasebeta

← #6498232

Subgroups — #6498232

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33218C212yesmaximal, complemented
4#33058C212yesmaximal
5#30310C211yesmaximal, characteristic, complemented
6#30283C211yesmaximal, characteristic, complemented
7#30235C211yesmaximal, characteristic, complemented
8#30213C211yesmaximal, characteristic, complemented
9#29576C214yesmaximal, complemented
10#29514C212yesmaximal, complemented
11#29513C212yesmaximal, complemented
12#29163C212yesmaximal, complemented
13#29162C212yesmaximal, complemented
14#27865C214yesmaximal, complemented
15#22250C212yesmaximal, complemented
16#22226C212yesmaximal, complemented
17#20364C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#501C2212yes—
20C2×#490C2212yes—
21C2×C4×M16C2212yes—
22#2864C2212yes—
23#2863—21no—
24#2863—21no—
25#2862C2211yescharacteristic
26#2861—21no—
27#2860—21no—
28#2860C2212yes—
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2856C2211yescharacteristic
36#2855—21no—
37#2855—21no—
38#2854—21no—
39#2854C2212yes—
40#2851C2212yes—
41#2850—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2212yes—
47C4×#124—21no—
48C4×C22⋊Q8—21no—
49C4×C22⋊Q8—21no—
50C4×#121—21no—
51C4×#121C2212yes—
52#2295—21no—
53#2295—21no—
54#2293—21no—
55#2293C2212yes—
56#2283C2212yes—
57C2×C43C2211yescharacteristic
58#2101C2214yes—
59#2100C2212yes—
60#2099C2212yes—
61#2094C2212yes—
62#2093C2212yes—
63#2092C2214yes—
64#2085C2214yes—
65#2073C2214yes—
66#2058C2212yes—
67#2057C2212yes—
68#2023C2212yes—
69#2023C2212yes—
70#1993C2212yes—
71#1992C2212yes—
72#1992C2212yes—
73#1961C2212yes—
74#1942C2214yes—
75#1941C2214yes—
76#1934C2214yes—
77#1934C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1921C2214yes—
81#1919C2214yes—
82#1919C2214yes—
83#1918C2214yes—
84#1895C2214yes—
85#1895C2214yes—
86#1864C2214yes—
87#1862C2214yes—
88#1847C2212yes—
89#1836C2212yes—
90#1809C2214yes—
91#1809C2214yes—
92#1745C2211yescharacteristic
93#1745C2211yescharacteristic
94#1745C2212yes—
95#1744C2212yes—
96#1744C2212yes—
97#1728C2212yes—
98#1728C2212yes—
99C22×M16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104#529—48no—
105C2×#126—22no—
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#124—21no—
109C2×#124—21no—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#501—28no—
133#490—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C4×M16—28no—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2314yes—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#383—22no—
211#382—21no—
212#382C2312yes—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#380—21no—
217#380—21no—
218#380—21no—
219#380C2312yes—
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233#376—22no—
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2312yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2314yes—
255#336C2314yes—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2412yes—
263C23×C4—22no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×M16—28no—
272C2×M16—28no—
273C2×M16—44no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279#127—24no—
280#126—48no—
281#125—24no—
282#125—24no—
283#125—44no—
284#124—44no—
285#124—44no—
286#123—48no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290#121—28no—
291#121—44no—
292#121—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—48no—
300#117—48no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—44no—
382#106—28no—
383#106—44no—
384#102—28no—
385#102—28no—
386#102—44no—
387#102—44no—
388#97—24no—
389#97#4118yes—
390C4×C8—28no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411C24—22no—
412C24C22×D411yescharacteristic
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C42−1+414yes—
478C22×C4C2511yescharacteristic
479M16—416no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8C2×#4118yes—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42AGL(1,ℤ/8)14yes—
559C42C2×#4114yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23#53612yes—
574C23C23×D411yescharacteristic
575Q8—84no—
576Q8—84no—
577Q8—84no—
578D4—48no—
579D4—84no—
580D4—84no—
581D4—88no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—88no—
641C2×C4#53212yes—
642C2×C4#53212yes—
643C2×C4#53612yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4#57512yes—
648C2×C4#57512yes—
649C2×C4#57512yes—
650C2×C4C22×#4112yes—
651C2×C4C2×2−1+412yes—
652C2×C4#58512yes—
653C8—28no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22C2×#53211yescharacteristic
670C22C2×#53611yescharacteristic
671C22#344911yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57511yescharacteristic
675C22C2×#57511yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#349514yes—
697C2—42no—
698C2—162no—
699C2#3532611yescharacteristic
700C2#6070011yescharacteristic
701C2#6070211yescharacteristic
702C2#6086311yescharacteristic
703C2#6086411yescharacteristic
704C2#6094911yescharacteristic
705C2#6095411yescharacteristic
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. #33218C2semidirect
5. #30310C2semidirect
6. #30283C2semidirect
7. #30235C2semidirect
8. #30213C2semidirect
9. #29576C2semidirect
10. #29514C2semidirect
11. #29513C2semidirect
12. #29163C2semidirect
13. #29162C2semidirect
14. #27865C2semidirect
15. #22250C2semidirect
16. #22226C2semidirect
17. #20364C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#332182
#330582
#303102
#302832
#302352
#302132
#295762
#295142
#295132
#291632
#291622
#278652
#222502
#222262
#203642