Finite Group Databasebeta

← #6472783

Subgroups — #6472783

Order 512.

3338 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic, complemented
3#32981C214yesmaximal, complemented
4#29892C212yesmaximal, complemented
5#29890C212yesmaximal, complemented
6#27604C214yesmaximal, complemented
7#24560C218yesmaximal, complemented
8#24009C214yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21764C212yesmaximal, complemented
11#20579C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#121C2212yes—
15C22×#120—21no—
16C22×#117C2212yescomplemented
17C2×#482C2218yescomplemented
18C2×#407C2212yes—
19C2×#407C2214yescomplemented
20#2847—21no—
21#2847—21no—
22#2847—22no—
23#2847C2212yes—
24#2847C2212yes—
25#2847C2214yes—
26C2×#386—21no—
27C2×#386—22no—
28#2280—22no—
29#2280—22no—
30#2280C2214yes—
31#2280C2214yescomplemented
32C2×#381—21no—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic, complemented
37C2×#380C2211yescharacteristic, complemented
38C2×#379—21no—
39C2×C4×C22⋊C4—21no—
40C2×C4×C22⋊C4—21no—
41#2077C2218yescomplemented
42#2075C2218yescomplemented
43#2034C2218yescomplemented
44#2033C2218yescomplemented
45#1895C2218yescomplemented
46#1875C2218yescomplemented
47#1874C2214yescomplemented
48#1874C2214yescomplemented
49#1854C2218yescomplemented
50#1853C2214yescomplemented
51#1853C2214yescomplemented
52#1852C2214yescomplemented
53#1852C2214yescomplemented
54#1825C2218yescomplemented
55#1824C2214yescomplemented
56#1809C2218yescomplemented
57#1786C2214yescomplemented
58#1779C2214yes—
59#1779C2212yescomplemented
60#1779C2212yescomplemented
61C26D411yescharacteristic
62C23×D4—22no—
63C23×D4—21no—
64C23×D4C2311yescharacteristic
65C24×C4—28no—
66C24×C4—48no—
67C24×C4D411yescharacteristic
68C24×C4D411yescharacteristic
69C24×C4D412yes—
70C24×C4D411yescharacteristic
71C24×C4D412yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C22×M16C2314yes—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121—28no—
83C2×#121—48no—
84C2×#121C2314yes—
85C2×#121C2312yes—
86C2×#121C2312yes—
87C2×#121C2314yes—
88C2×#120—22no—
89C2×#120—24no—
90C2×#120—22no—
91C2×#120—48no—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—48no—
95C2×#117C2314yes—
96C2×#117C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—21no—
109C22×C42—22no—
110#482—416no—
111C2×#107C2318yescomplemented
112C2×#106C2318yescomplemented
113C2×#102C2318yescomplemented
114C2×#102C2318yescomplemented
115#407—28no—
116#407—48no—
117C2×C22⋊C8C2318yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2318yes—
121#386—24no—
122#386—22no—
123#386—24no—
124#386—24no—
125#386—24no—
126#386—22no—
127#386—24no—
128#381—24no—
129#381—24no—
130#380—22no—
131#380—22no—
132#380—21no—
133#380—21no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380C2312yes—
139#380C2314yes—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143#380C2314yes—
144#380C2312yes—
145#380C2314yes—
146#379—24no—
147#379—24no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C2×#95—22no—
153C2×#95—21no—
154C2×#95—21no—
155C2×#95—22no—
156C2×#95—21no—
157#340C2318yescomplemented
158#340C2318yescomplemented
159#340C2318yescomplemented
160#340C2318yescomplemented
161C25—24no—
162C25—24no—
163C25—21no—
164C25—44no—
165C25—42no—
166C25—44no—
167C25C2×D411yescharacteristic
168C25C2×D412yes—
169C25C2×D411yescharacteristic
170C25C2×D412yes—
171C25C2×D411yescharacteristic
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—24no—
178C22×D4—44no—
179C22×D4—48no—
180C22×D4—44no—
181C22×D4—48no—
182C22×D4—44no—
183C22×D4—48no—
184C22×D4—44no—
185C22×D4C2412yes—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—464no—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4#4118yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C2×M16—28no—
234C2×M16—44no—
235C2×M16—44no—
236C22×C8C2418yes—
237#121—28no—
238#121—28no—
239#121—216no—
240#121—48no—
241#121—48no—
242#121—48no—
243#121—44no—
244#121—44no—
245#120—416no—
246#120—48no—
247#120—48no—
248#117—216no—
249#117—48no—
250#117—48no—
251#117—48no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—48no—
289#107—48no—
290#106—48no—
291#102—48no—
292#102—48no—
293C22⋊C8—28no—
294C22⋊C8—28no—
295C22⋊C8—216no—
296C22⋊C8—48no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—24no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—416no—
334C24—416no—
335C24—432no—
336C24—416no—
337C24—44no—
338C24—42no—
339C24—42no—
340C24—432no—
341C24—42no—
342C24—44no—
343C24#12314yes—
344C24AGL(1,ℤ/8)14yes—
345C24C22×D411yescharacteristic
346C24C22×D412yes—
347C24C22×D412yes—
348C24C22×D411yescharacteristic
349C24C22×D411yescharacteristic
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—88no—
367C2×D4—832no—
368C2×D4—816no—
369C22×C4—232no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—416no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—432no—
415C22×C4—432no—
416C22×C4—416no—
417C22×C4—432no—
418C22×C4—432no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4#12314yes—
435C22×C4#12314yes—
436C22×C4#12314yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)18yes—
446C22×C4C2511yescharacteristic
447M16—88no—
448C2×C8—28no—
449C2×C8—28no—
450C2×C8—28no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C42—48no—
474C42—48no—
475C23—22no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—28no—
481C23—216no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—44no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—4128no—
523C23—48no—
524C23—48no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—88no—
532C23—84no—
533C23—84no—
534C23—88no—
535C23—84no—
536C23C2×#12312yes—
537C23C2×#12312yes—
538C23C2×#12312yes—
539C23#53414yes—
540C23C2×AGL(1,ℤ/8)14yes—
541C23C2×AGL(1,ℤ/8)12yes—
542C23#57514yes—
543C23#57512yes—
544C23C23×D411yescharacteristic
545D4—48no—
546D4—84no—
547D4—816no—
548D4—816no—
549D4—88no—
550D4—88no—
551C2×C4—216no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—216no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4#53414yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)14yes—
599C8—48no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—22no—
614C22—24no—
615C22—24no—
616C22—22no—
617C22—416no—
618C22—432no—
619C22—416no—
620C22—416no—
621C22—432no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—84no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—168no—
638C22#309314yes—
639C22#309414yes—
640C22C22×#12311yescharacteristic
641C22C2×#53412yes—
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57512yes—
645C4—22no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—84no—
651C4—84no—
652C4—84no—
653C4—82no—
654C4#349014yes—
655C2—24no—
656C2—24no—
657C2—44no—
658C2—44no—
659C2—42no—
660C2—162no—
661C2#3499211yescharacteristic
662C2C2×#309311yescharacteristic
663C2C2×#309411yescharacteristic
664C2#6074112yes—
665C2#6075012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2280C2semidirect
3. #32981C2semidirect
4. #29892C2semidirect
5. #29890C2semidirect
6. #27604C2semidirect
7. #24560C2semidirect
8. #24009C2semidirect
9. #21942C2semidirect
10. #21764C2semidirect
11. #20579C2semidirect
16. C22×#117C22semidirect
17. C2×#482C22semidirect
19. C2×#407C22semidirect
31. #2280C22semidirect
36. C2×#380C22semidirect
37. C2×#380C22semidirect
41. #2077C22semidirect
42. #2075C22semidirect
43. #2034C22semidirect
44. #2033C22semidirect
45. #1895C22semidirect
46. #1875C22semidirect
47. #1874C22semidirect
48. #1874C22semidirect
49. #1854C22semidirect
50. #1853C22semidirect
51. #1853C22semidirect
52. #1852C22semidirect
53. #1852C22semidirect
54. #1825C22semidirect
55. #1824C22semidirect
56. #1809C22semidirect
57. #1786C22semidirect
59. #1779C22semidirect
60. #1779C22semidirect
111. C2×#107C23semidirect
112. C2×#106C23semidirect
113. C2×#102C23semidirect
114. C2×#102C23semidirect
157. #340C23semidirect
158. #340C23semidirect
159. #340C23semidirect
160. #340C23semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C2×#22802
#329812
#298922
#298902
#276042
#245602
#240092
#219422
#217642
#205792