Finite Group Databasebeta

← #6472774

Subgroups — #6472774

Order 512.

3628 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic, complemented
3#32979C214yesmaximal, complemented
4#29892C214yesmaximal, complemented
5#27601C214yesmaximal, complemented
6#24553C218yesmaximal, complemented
7#24005C214yesmaximal, complemented
8#21938C212yesmaximal, complemented
9#21758C212yesmaximal, complemented
10C2×#1779C211yesmaximal, characteristic, complemented
11C25×C4C2211yescharacteristic
12C22×#121C2211yescharacteristic
13C22×#121C2212yes—
14C22×#120—21no—
15C23×C4⋊C4C2212yescomplemented
16C23×C22⋊C4—21no—
17C2×#481C2218yescomplemented
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2214yescomplemented
20#2847—21no—
21#2847—22no—
22#2847—21no—
23#2847C2212yes—
24#2847C2212yes—
25#2847C2214yes—
26C2×#386—22no—
27C2×#386—21no—
28C2×#380—22no—
29C2×#380—21no—
30C2×#380—22no—
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×#380C2212yescomplemented
37C2×#380C2214yescomplemented
38C22×#95—21no—
39#2077C22116yescomplemented
40#2034C22116yescomplemented
41#1893C2218yescomplemented
42#1874C2218yescomplemented
43#1874C2218yescomplemented
44#1853C2218yescomplemented
45#1853C2218yescomplemented
46#1853C2218yescomplemented
47#1824C2218yescomplemented
48#1823C2214yescomplemented
49#1807C2218yescomplemented
50#1786C2214yescomplemented
51#1779C2214yes—
52#1779C2212yescomplemented
53#1779C2212yescomplemented
54C26D411yescharacteristic
55C23×D4—22no—
56C23×D4—21no—
57C23×D4C2311yescharacteristic
58C24×C4—28no—
59C24×C4—22no—
60C24×C4—48no—
61C24×C4D411yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4D412yes—
64C24×C4D411yescharacteristic
65C24×C4D412yes—
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C23×C8C2314yes—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—28no—
77C2×#121—48no—
78C2×#121C2314yes—
79C2×#121C2312yes—
80C2×#121C2312yes—
81C2×#121C2314yes—
82C2×#120—22no—
83C2×#120—24no—
84C2×#120—22no—
85C2×#120—48no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—48no—
104#481—416no—
105C2×#106C23116yescomplemented
106C2×#102C23116yescomplemented
107C2×C22⋊C8—28no—
108C2×C22⋊C8—48no—
109C2×C22⋊C8C2318yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2318yes—
113#386—24no—
114#386—22no—
115#386—24no—
116#386—24no—
117#386—24no—
118#386—22no—
119#386—24no—
120#380—22no—
121#380—22no—
122#380—21no—
123#380—21no—
124#380—24no—
125#380—24no—
126#380—22no—
127#380—22no—
128#380—24no—
129#380—24no—
130#380—24no—
131#380C2312yes—
132#380C2314yes—
133#380C2314yes—
134#380C2318yes—
135#380C2314yes—
136#380C2314yes—
137#380C2312yes—
138C2×#95—24no—
139C2×#95—22no—
140C2×#95—24no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144#340C23116yescomplemented
145#340C23116yescomplemented
146C25—24no—
147C25—24no—
148C25—21no—
149C25—44no—
150C25—44no—
151C25SD1618yes—
152C25C2×D411yescharacteristic
153C25C2×D412yes—
154C25C2×D412yes—
155C25C2×D411yescharacteristic
156C25C2×D411yescharacteristic
157C22×D4—24no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—24no—
161C22×D4—21no—
162C22×D4—24no—
163C22×D4—44no—
164C22×D4—48no—
165C22×D4—48no—
166C22×D4—44no—
167C22×D4—48no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4C2412yes—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—464no—
200C23×C4—48no—
201C23×C4—48no—
202C23×C4—48no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4#4118yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C22×C8—28no—
223C22×C8—48no—
224C22×C8C2418yes—
225#121—28no—
226#121—28no—
227#121—216no—
228#121—48no—
229#121—48no—
230#121—48no—
231#121—44no—
232#121—44no—
233#120—48no—
234#120—416no—
235#120—48no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4C2418yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—416no—
266C2×C22⋊C4—416no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—44no—
271#106—416no—
272#102—416no—
273C22⋊C8—28no—
274C22⋊C8—28no—
275C22⋊C8—216no—
276C22⋊C8—48no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—24no—
284C24—22no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—232no—
299C24—22no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—432no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—416no—
310C24—432no—
311C24—416no—
312C24—44no—
313C24—44no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24#12314yes—
318C24C2×SD1614yes—
319C24C2×SD1618yes—
320C24C2×SD1618yes—
321C24C2×SD1614yes—
322C24C2×SD1614yes—
323C24AGL(1,ℤ/8)14yes—
324C24C22×D412yes—
325C24C22×D411yescharacteristic
326C24C22×D411yescharacteristic
327C24C22×D411yescharacteristic
328C24C22×D412yes—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—88no—
346C2×D4—832no—
347C2×D4—816no—
348C22×C4—232no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—416no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—432no—
393C22×C4—432no—
394C22×C4—416no—
395C22×C4—432no—
396C22×C4—432no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—832no—
416C22×C4#12314yes—
417C22×C4#12314yes—
418C22×C4#12314yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)18yes—
428C22×C4C2511yescharacteristic
429C2×C8—28no—
430C2×C8—216no—
431C2×C8—88no—
432C4⋊C4—24no—
433C4⋊C4—216no—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C23—28no—
452C23—216no—
453C23—28no—
454C23—28no—
455C23—22no—
456C23—216no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—216no—
467C23—216no—
468C23—28no—
469C23—22no—
470C23—216no—
471C23—216no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—44no—
478C23—44no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—44no—
493C23—4128no—
494C23—48no—
495C23—44no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—88no—
502C23—84no—
503C23—88no—
504C23—84no—
505C23—84no—
506C23#481116yescomplemented
507C23C2×#12312yes—
508C23C2×#12312yes—
509C23C2×#12312yes—
510C23#53414yes—
511C23C22×SD1614yes—
512C23C22×SD1612yes—
513C23C22×SD1612yes—
514C23C22×SD1612yes—
515C23C22×SD1614yes—
516C23C2×AGL(1,ℤ/8)12yes—
517C23C2×AGL(1,ℤ/8)12yes—
518C23C2×AGL(1,ℤ/8)12yes—
519C23#57514yes—
520C23C23×D411yescharacteristic
521D4—48no—
522D4—84no—
523D4—816no—
524D4—88no—
525D4—88no—
526D4—816no—
527C2×C4—216no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—216no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—816no—
563C2×C4—816no—
564C2×C4—816no—
565C2×C4—88no—
566C2×C4—84no—
567C2×C4C2×#12312yes—
568C2×C4C2×#12312yes—
569C2×C4C2×#12312yes—
570C2×C4#53414yes—
571C2×C4C22×#4112yes—
572C2×C4C2×H(5,𝔽2)14yes—
573C8—48no—
574C22—232no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—22no—
592C22—22no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—416no—
597C22—432no—
598C22—432no—
599C22—416no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—168no—
610C22C2×#48114yes—
611C22C2×#48114yes—
612C22C2×#48114yes—
613C22#318718yes—
614C22C22×#12311yescharacteristic
615C22C2×#53412yes—
616C22C23×SD1611yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C2×#57512yes—
619C4—22no—
620C4—24no—
621C4—22no—
622C4—22no—
623C4—42no—
624C4—84no—
625C4—88no—
626C4—82no—
627C4#349014yes—
628C2—24no—
629C2—28no—
630C2—24no—
631C2—24no—
632C2—44no—
633C2—162no—
634C2#3499211yescharacteristic
635C2C22×#48111yescharacteristic
636C2#6051111yescharacteristic
637C2C2×#318712yes—
638C2#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. C22×#380C2semidirect
3. #32979C2semidirect
4. #29892C2semidirect
5. #27601C2semidirect
6. #24553C2semidirect
7. #24005C2semidirect
8. #21938C2semidirect
9. #21758C2semidirect
10. C2×#1779C2semidirect
15. C23×C4⋊C4C22semidirect
17. C2×#481C22semidirect
19. C22×C22⋊C8C22semidirect
36. C2×#380C22semidirect
37. C2×#380C22semidirect
39. #2077C22semidirect
40. #2034C22semidirect
41. #1893C22semidirect
42. #1874C22semidirect
43. #1874C22semidirect
44. #1853C22semidirect
45. #1853C22semidirect
46. #1853C22semidirect
47. #1824C22semidirect
48. #1823C22semidirect
49. #1807C22semidirect
50. #1786C22semidirect
52. #1779C22semidirect
53. #1779C22semidirect
105. C2×#106C23semidirect
106. C2×#102C23semidirect
144. #340C23semidirect
145. #340C23semidirect
506. C23#481semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C22×#3802
#329792
#298922
#276012
#245532
#240052
#219382
#217582
C2×#17792