Finite Group Databasebeta

← #6508835

Subgroups — #6508835

Order 512.

2376 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic, complemented
2C2×#2973C211yesmaximal, characteristic, complemented
3#35000C211yesmaximal, characteristic
4C2×#2486C211yesmaximal, characteristic, complemented
5#28107C214yesmaximal, complemented
6#27667C214yesmaximal, complemented
7#25250C218yesmaximal, complemented
8#24608C214yesmaximal, complemented
9#24558C214yesmaximal, complemented
10#21938C211yesmaximal, characteristic, complemented
11#21758C211yesmaximal, characteristic, complemented
12C2×#1823C211yesmaximal, characteristic, complemented
13C2×#534—21no—
14C22×#123—21no—
15C22×#121C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#480—28no—
19C2×#480C2214yes—
20C2×#480C2214yescomplemented
21#2973—28no—
22C2×#435C2214yes—
23C2×#435C2214yescomplemented
24C2×#432C2212yescomplemented
25C22×#107C2212yescomplemented
26C22×#102C2212yescomplemented
27C2×#407C2212yescomplemented
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30#2854C2212yes—
31#2847C2212yes—
32#2618—21no—
33#2617—21no—
34#2543—22no—
35#2534—22no—
36#2486—22no—
37#2486—22no—
38#2486C2212yes—
39#2486C2214yes—
40#2486C2214yescomplemented
41#2416C2214yes—
42#2408C2212yes—
43C2×#398—21no—
44C2×#397—21no—
45#2308—21no—
46C2×#389C2212yescomplemented
47C2×#381—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2212yescomplemented
51#2087C2218yescomplemented
52#2076C2218yescomplemented
53#2065C2218yescomplemented
54#2034C2218yescomplemented
55#1944C2218yescomplemented
56#1936C2218yescomplemented
57#1895C2218yescomplemented
58#1893C2214yescomplemented
59#1874C2214yes—
60#1853C2214yescomplemented
61#1853C2214yescomplemented
62#1823C2212yescomplemented
63#1823C2214yescomplemented
64#1823C2214yescomplemented
65#1809C2218yescomplemented
66#1807C2214yescomplemented
67#1786C2212yescomplemented
68C2×#340C2214yescomplemented
69C23×D4—21no—
70C23×D4D411yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4—21no—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C22×M16C2314yes—
76C23×C8C2312yes—
77C23×C8C2312yes—
78#534—48no—
79C2×#123—24no—
80C2×#123—24no—
81C2×#123—22no—
82C2×#123—22no—
83C2×#123—48no—
84C2×#121—24no—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121C2312yes—
90C2×#121C2312yes—
91C2×#121C2314yes—
92C2×#120—22no—
93C2×#120—22no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4D412yes—
97C2×C4×D4D412yes—
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42C2311yescharacteristic
120#480—216no—
121#480—216no—
122#435—216no—
123#435—216no—
124#432—28no—
125C2×#107—28no—
126C2×#107C2318yes—
127C2×#106C2318yes—
128C2×C4⋊C8C2314yes—
129C2×#102—28no—
130C2×#102C2318yes—
131C2×#102C2318yes—
132#407—28no—
133C2×C22⋊C8—24no—
134C2×C22⋊C8D418yescomplemented
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141#398—22no—
142#398—24no—
143#398—24no—
144#397—22no—
145#397—24no—
146#397—24no—
147#396—24no—
148#394—22no—
149#394—24no—
150#392—22no—
151#390C2314yes—
152#389—24no—
153#389C2318yes—
154#389C2314yes—
155#388—24no—
156#386—22no—
157#386—22no—
158#386—24no—
159#386—24no—
160#381—22no—
161#381—24no—
162#381—24no—
163#380—21no—
164#380—21no—
165#380—21no—
166#380—24no—
167#380—21no—
168#380C2312yes—
169#380C2312yes—
170#380C2314yes—
171#380C2318yes—
172#380C2312yes—
173#380C2312yes—
174#379—22no—
175C2×#95—21no—
176C2×#95—22no—
177C2×#95—22no—
178C2×#95—21no—
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182#340C2318yes—
183C25—21no—
184C25C2×D411yescharacteristic
185C25C2×D411yescharacteristic
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4C2×D412yes—
200C22×D4C2×D412yes—
201C22×D4C2412yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D414yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C2×M16—28no—
229C2×M16—48no—
230C22×C8—24no—
231C22×C8—44no—
232C22×C8—44no—
233C22×C8C2×D418yes—
234C22×C8C2414yes—
235C22×C8C2414yes—
236#123—48no—
237#123—416no—
238#123—416no—
239#123—48no—
240#121—24no—
241#121—24no—
242#121—24no—
243#121—24no—
244#121—416no—
245#121—44no—
246#121—44no—
247#121—48no—
248#120—48no—
249#120—48no—
250C4×D4—28no—
251C4×D4—28no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2414yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42C2×D412yes—
309C2×C42C2412yes—
310#107—216no—
311#106—48no—
312C4⋊C8—28no—
313#102—216no—
314#102—48no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—28no—
318C22⋊C8C2×D418yes—
319C22⋊C8C2×D418yes—
320#97—28no—
321#95—28no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—28no—
327#95—24no—
328#95—24no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—82no—
341C24—82no—
342C24#12114yes—
343C24AGL(1,ℤ/8)14yes—
344C24AGL(1,ℤ/8)14yes—
345C24C22×D412yes—
346C24C22×D411yescharacteristic
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—28no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4—82no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—816no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—816no—
418C22×C4#12114yes—
419C22×C4#12114yes—
420C22×C4#12114yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2511yescharacteristic
434M16—88no—
435C2×C8—28no—
436C2×C8—24no—
437C2×C8—28no—
438C2×C8—48no—
439C2×C8—84no—
440C2×C8C22×D418yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—44no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—82no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—48no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C42—24no—
473C42—22no—
474C42—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—48no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—82no—
498C23—84no—
499C23—82no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—164no—
504C23—164no—
505C23#469116yes—
506C23C2×#12112yes—
507C23C2×#12112yes—
508C23C2×#12112yes—
509C23C2×AGL(1,ℤ/8)12yes—
510C23C2×AGL(1,ℤ/8)12yes—
511C23C2×AGL(1,ℤ/8)12yes—
512C23C2×AGL(1,ℤ/8)12yes—
513C23C2×AGL(1,ℤ/8)12yes—
514C23C2×AGL(1,ℤ/8)12yes—
515C23C23×D411yescharacteristic
516D4—48no—
517D4—48no—
518D4—44no—
519D4—44no—
520D4—84no—
521D4—88no—
522D4—88no—
523D4—168no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—84no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—88no—
562C2×C4—88no—
563C2×C4—88no—
564C2×C4—88no—
565C2×C4—84no—
566C2×C4—164no—
567C2×C4C2×#12112yes—
568C2×C4C2×#12112yes—
569C2×C4C2×#12112yes—
570C2×C4C23×D411yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4#58514yes—
573C8—28no—
574C8—44no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—416no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—88no—
594C22—84no—
595C22—84no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—168no—
600C22C2×#46914yes—
601C22C2×#46914yes—
602C22C2×#46914yes—
603C22C22×#12111yescharacteristic
604C22#348512yes—
605C22C22×AGL(1,ℤ/8)11yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22#358612yes—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—84no—
615C4—84no—
616C4—84no—
617C4—82no—
618C4#348512yes—
619C4#349112yes—
620C2—24no—
621C2—24no—
622C2—24no—
623C2—82no—
624C2—162no—
625C2#3544711yescharacteristic
626C2C22×#46911yescharacteristic
627C2#6045511yescharacteristic
628C2#6170412yes—
629C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#480C2semidirect
2. C2×#2973C2semidirect
4. C2×#2486C2semidirect
5. #28107C2semidirect
6. #27667C2semidirect
7. #25250C2semidirect
8. #24608C2semidirect
9. #24558C2semidirect
10. #21938C2semidirect
11. #21758C2semidirect
12. C2×#1823C2semidirect
20. C2×#480C22semidirect
23. C2×#435C22semidirect
24. C2×#432C22semidirect
25. C22×#107C22semidirect
26. C22×#102C22semidirect
27. C2×#407C22semidirect
40. #2486C22semidirect
46. C2×#389C22semidirect
50. C2×#380C22semidirect
51. #2087C22semidirect
52. #2076C22semidirect
53. #2065C22semidirect
54. #2034C22semidirect
55. #1944C22semidirect
56. #1936C22semidirect
57. #1895C22semidirect
58. #1893C22semidirect
60. #1853C22semidirect
61. #1853C22semidirect
62. #1823C22semidirect
63. #1823C22semidirect
64. #1823C22semidirect
65. #1809C22semidirect
66. #1807C22semidirect
67. #1786C22semidirect
68. C2×#340C22semidirect
134. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#29732
#350002
C2×#24862
#281072
#276672
#252502
#246082
#245582
#219382
#217582
C2×#18232