Finite Group Databasebeta

← #6480634

Subgroups — #6480634

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#33072C214yesmaximal, complemented
4#30094C212yesmaximal, complemented
5#30075C212yesmaximal, complemented
6#29578C214yesmaximal, complemented
7#29320C212yesmaximal, complemented
8#29130C212yesmaximal, complemented
9#27677C212yesmaximal, complemented
10#27645C212yesmaximal, complemented
11#27614C212yesmaximal, complemented
12#24061C212yesmaximal
13#22082C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×#125—21no—
17C2×C4×#41—21no—
18C2×#481C2214yes—
19#2859—21no—
20#2859—21no—
21#2859—21no—
22#2859—21no—
23#2859C2212yes—
24#2859C2212yes—
25#2859C2212yes—
26#2859C2212yes—
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2211yescharacteristic
30C4×#127C2211yescharacteristic
31C4×#124—21no—
32C4×#124—21no—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121C2212yes—
36C4×#121C2212yes—
37#2297—21no—
38#2293—21no—
39#2293C2212yes—
40C2×#389—21no—
41C2×#389—21no—
42C2×#389C2212yes—
43C2×#389C2212yes—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#382—21no—
48C42×D4—21no—
49C42×D4C2212yes—
50C2×C43C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2093C2214yes—
54#2084C2214yes—
55#2083C2214yes—
56#2071C2214yes—
57#2058C2214yes—
58#2041C2214yes—
59#1948C2214yes—
60#1947C2214yes—
61#1944C2218yes—
62#1936C2218yes—
63#1921C2218yes—
64#1919C2218yes—
65#1874C2214yes—
66#1864C2218yes—
67#1862C2218yes—
68#1852C2214yes—
69#1839C2212yes—
70#1831C2212yes—
71#1824C2214yes—
72#1817C2212yes—
73#1802C2214yes—
74#1802C2214yes—
75#1797C2212yes—
76#1767C2212yes—
77#1749C2212yes—
78#1745C2212yes—
79C22×#41—21no—
80#532—28no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#125—22no—
84C2×#125—22no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—48no—
88C2×#124—21no—
89C2×#121—21no—
90C2×#121C2312yes—
91C4×#41—48no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C2×#117—21no—
105C2×#117—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#481—216no—
123C2×#107C2318yes—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×#102C2318yes—
127C2×#102C2318yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130#391—21no—
131#391—21no—
132#390—21no—
133#390—21no—
134#390—21no—
135#390—21no—
136#390C2312yes—
137#390C2312yes—
138#389—21no—
139#389—21no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2312yes—
153#389C2312yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#386—22no—
163#386—22no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#382—22no—
175#382—22no—
176#382—22no—
177#382—22no—
178#382—21no—
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C2×#95—22no—
194C2×#95—22no—
195C43D411yescharacteristic
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43C2311yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211C2×#41—28no—
212C2×#41—44no—
213C2×#41—44no—
214C22×Q8—21no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8C2418yes—
232C4⋊Q8—24no—
233#127—24no—
234#125—48no—
235#125—48no—
236#125—48no—
237#125—48no—
238#124—28no—
239#121—28no—
240#121—24no—
241#121#4118yes—
242C4×Q8—44no—
243C4×Q8—44no—
244C4×D4—28no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—28no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4—48no—
254C4×D4—44no—
255#117—48no—
256#117—44no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—48no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—48no—
340#106—216no—
341C4⋊C8—28no—
342#102—216no—
343#102—48no—
344C22⋊C8—28no—
345#97—44no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356C24—22no—
357C24C22×D411yescharacteristic
358#41—432no—
359C2×Q8—24no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4C2×#4114yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4AGL(1,ℤ/8)14yes—
414C22×C4#13714yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4114yes—
429C22×C4C2511yescharacteristic
430C2×C8—28no—
431C2×C8—28no—
432C2×C8—28no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4C2×#4114yes—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—44no—
497C42—44no—
498C42—44no—
499C42AGL(1,ℤ/8)14yes—
500C422−1+414yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23#52512yes—
513C23C2×AGL(1,ℤ/8)12yes—
514C23C2×#13712yes—
515C23C23×D411yescharacteristic
516Q8—48no—
517D4—28no—
518D4—48no—
519D4—48no—
520D4—416no—
521D4—44no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—416no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4#52512yes—
578C2×C4#53612yes—
579C2×C4#53612yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C2×AGL(1,ℤ/8)12yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×#13712yes—
586C2×C4C2×#13712yes—
587C2×C4#57512yes—
588C2×C4C22×#4112yes—
589C2×C4C2×2−1+412yes—
590C2×C4#58512yes—
591C8—48no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22#311014yes—
606C22C2×#52511yescharacteristic
607C22C2×#53611yescharacteristic
608C22#344911yescharacteristic
609C22C22×AGL(1,ℤ/8)11yescharacteristic
610C22C22×AGL(1,ℤ/8)11yescharacteristic
611C22C22×#13711yescharacteristic
612C22C2×#57511yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—44no—
625C4—41no—
626C4—82no—
627C4—84no—
628C4—84no—
629C4—82no—
630C4#323414yes—
631C4#351514yes—
632C2—24no—
633C2—84no—
634C2#3513811yescharacteristic
635C2C2×#311011yescharacteristic
636C2#6056111yescharacteristic
637C2C2×#323411yescharacteristic
638C2#6086011yescharacteristic
639C2#6095211yescharacteristic
640C2#6096811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. C2×#2859C2semidirect
3. #33072C2semidirect
4. #30094C2semidirect
5. #30075C2semidirect
6. #29578C2semidirect
7. #29320C2semidirect
8. #29130C2semidirect
9. #27677C2semidirect
10. #27645C2semidirect
11. #27614C2semidirect
13. #22082C2semidirect
14. #20778C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C4×#5322
#330722
#300942
#300752
#295782
#293202
#291302
#276772
#276452
#276142
#240612
#220822
#207782