Finite Group Databasebeta

← #6498089

Subgroups — #6498089

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#33147C211yesmaximal, characteristic, complemented
5#32956C211yesmaximal, characteristic
6#32953C211yesmaximal, characteristic
7#29986C212yesmaximal, complemented
8#29865C212yesmaximal, complemented
9#29576C214yesmaximal, complemented
10#29396C214yesmaximal, complemented
11#29298C212yesmaximal, complemented
12#29104C212yesmaximal
13#27617C212yesmaximal, complemented
14#27588C212yesmaximal
15#22111C212yesmaximal, complemented
16#22110C212yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#476C2212yes—
21C2×#474C2212yes—
22#2864C2212yescomplemented
23#2863—21no—
24#2863—21no—
25#2862C2211yescharacteristic, complemented
26#2859C2211yescharacteristic, complemented
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2855—21no—
30#2855—21no—
31#2851C2212yes—
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×C4⋊Q8C2211yescharacteristic, complemented
34C4×#125C2212yescomplemented
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37#2295—21no—
38#2295—21no—
39#2293—21no—
40#2293—21no—
41#2293—21no—
42#2293—21no—
43#2293C2212yes—
44#2293C2212yes—
45#2293C2212yes—
46#2293C2212yes—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#388—21no—
52C2×#388—21no—
53#2283C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4—21no—
56C2×C43C2211yescharacteristic
57#2101C2214yescomplemented
58#2092C2214yes—
59#2085C2214yescomplemented
60#2085C2214yescomplemented
61#2073C2214yes—
62#2073C2214yes—
63#2059C2214yescomplemented
64#2025C2214yes—
65#1988C2212yescomplemented
66#1987C2212yescomplemented
67#1948C2212yescomplemented
68#1947C2212yescomplemented
69#1942C2214yescomplemented
70#1941C2214yescomplemented
71#1934C2214yes—
72#1934C2214yes—
73#1926C2214yes—
74#1925C2214yes—
75#1921C2214yescomplemented
76#1919C2214yescomplemented
77#1919C2214yescomplemented
78#1918C2214yescomplemented
79#1864C2214yes—
80#1862C2214yes—
81#1853C2212yes—
82#1853C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1839C2212yescomplemented
86#1831C2212yescomplemented
87#1824C2212yes—
88#1824C2212yes—
89#1817C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92#1797C2211yescharacteristic
93#1797C2211yescharacteristic
94#1786C2212yes—
95#1786C2212yes—
96C4×C4⋊C8C2212yescomplemented
97C4×C22⋊C8C2212yes—
98#1744C2212yescomplemented
99C23×D4C2311yescharacteristic
100C24×C4—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—28no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—22no—
117C2×#117—48no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#476—28no—
137#474—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2314yes—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
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#389—21no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#385—22no—
204#385—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#383—22no—
208#382C2312yes—
209#382C2312yes—
210#382C2312yes—
211#381—22no—
212#380—21no—
213#380—21no—
214#380C2312yes—
215#380C2312yes—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231#376—22no—
232C2×#95—21no—
233C2×#95—21no—
234C2×#95—21no—
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256C25—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×D4—24no—
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—42no—
264C22×D4C2412yes—
265C23×C4—28no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—44no—
272C23×C4—44no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C22×C8C2414yes—
281C22×C8C2414yes—
282C4⋊Q8—24no—
283C4⋊Q8—24no—
284#125—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4#4118yes—
295#117—48no—
296#117—48no—
297#117—48no—
298#117—48no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
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—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—28no—
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×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42—42no—
360C2×C42—44no—
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×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#107—44no—
391#106—28no—
392#106—44no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395#102—28no—
396#102—28no—
397#102—44no—
398#102—44no—
399C22⋊C8—24no—
400C22⋊C8—24no—
401C4×C8—44no—
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—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24C22×D411yescharacteristic
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4C2×#4114yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—28no—
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—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—44no—
478C22×C4—432no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4#13514yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
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×C42−1+412yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4C2×#4114yes—
542C22⋊C4—28no—
543C22⋊C4—28no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—22no—
573C42—22no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42AGL(1,ℤ/8)14yes—
587C42C2×#4114yes—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—24no—
593C23—48no—
594C23—48no—
595C23—48no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—84no—
603C23—84no—
604C23—84no—
605C23—84no—
606C23#53612yes—
607C23C2×#13512yes—
608C23#57512yes—
609C23C23×D411yescharacteristic
610Q8—84no—
611Q8—84no—
612D4—24no—
613D4—24no—
614D4—48no—
615D4—42no—
616D4—42no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—24no—
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—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—416no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—416no—
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—84no—
680C2×C4—84no—
681C2×C4#52512yes—
682C2×C4#52512yes—
683C2×C4#53612yes—
684C2×C4C2×#13512yes—
685C2×C4C2×#13512yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4C2×AGL(1,ℤ/8)12yes—
688C2×C4C2×AGL(1,ℤ/8)12yes—
689C2×C4#57512yes—
690C2×C4C22×#4112yes—
691C2×C4C2×2−1+411yescharacteristic
692C2×C4C2×2−1+411yescharacteristic
693C2×C4#58512yes—
694C8—44no—
695C8—44no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—816no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22#323314yes—
711C22C2×#52511yescharacteristic
712C22C2×#53611yescharacteristic
713C22#344911yescharacteristic
714C22C22×#13511yescharacteristic
715C22C22×AGL(1,ℤ/8)11yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57511yescharacteristic
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—41no—
725C4—41no—
726C4—44no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#310514yes—
738C4#349512yes—
739C4#349512yes—
740C2—24no—
741C2—84no—
742C2#3532511yescharacteristic
743C2C2×#310511yescharacteristic
744C2#6056311yescharacteristic
745C2C2×#323311yescharacteristic
746C2#6086411yescharacteristic
747C2#6094211yescharacteristic
748C2#6096211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #33149C2semidirect
4. #33147C2semidirect
7. #29986C2semidirect
8. #29865C2semidirect
9. #29576C2semidirect
10. #29396C2semidirect
11. #29298C2semidirect
13. #27617C2semidirect
15. #22111C2semidirect
16. #22110C2semidirect
17. #20777C2semidirect
22. #2864C22semidirect
25. #2862C22semidirect
26. #2859C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×C4⋊Q8C22semidirect
34. C4×#125C22semidirect
57. #2101C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
63. #2059C22semidirect
65. #1988C22semidirect
66. #1987C22semidirect
67. #1948C22semidirect
68. #1947C22semidirect
69. #1942C22semidirect
70. #1941C22semidirect
75. #1921C22semidirect
76. #1919C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
85. #1839C22semidirect
86. #1831C22semidirect
96. C4×C4⋊C8C22semidirect
98. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#22932
#331492
#331472
#329562
#329532
#299862
#298652
#295762
#293962
#292982
#291042
#276172
#275882
#221112
#221102
#207772