Finite Group Databasebeta

← #6508709

Subgroups — #6508709

Order 512.

2570 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3186C211yesmaximal, characteristic
2C22×#480C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic, complemented
4#34205C211yesmaximal, characteristic, complemented
5#28114C214yesmaximal, complemented
6#27644C214yesmaximal, complemented
7#25200C218yesmaximal, complemented
8#24576C214yesmaximal, complemented
9#24526C214yesmaximal, complemented
10#21930C211yesmaximal, characteristic, complemented
11#21758C211yesmaximal, characteristic, complemented
12C2×#1823C211yesmaximal, characteristic, complemented
13C22×#123—21no—
14C22×#123—21no—
15C22×#121C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C2×#518—21no—
18C23×C4⋊C4C2211yescharacteristic
19#3186—28no—
20C2×#480—28no—
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#480C2214yes—
24C2×#480C2214yescomplemented
25C2×#466C2214yes—
26C2×#447C2212yes—
27C2×#447C2212yes—
28C22×#107C2212yescomplemented
29C22×#102C2212yescomplemented
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2501—22no—
33#2501—22no—
34#2501C2214yes—
35#2501C2214yescomplemented
36C2×#399—22no—
37#2364—22no—
38#2361C2214yescomplemented
39#2344C2212yescomplemented
40#2342—22no—
41D4×C22⋊C4C2212yescomplemented
42#2325—22no—
43#2308—21no—
44C4×#121C2214yes—
45C2×#390C2212yescomplemented
46C2×#384C2212yescomplemented
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2083C2218yescomplemented
50#2071C2218yescomplemented
51#2067C2218yescomplemented
52#2054C2218yescomplemented
53#1947C2218yescomplemented
54#1925C2218yescomplemented
55#1893C2214yes—
56#1879C2218yescomplemented
57#1874C2214yescomplemented
58#1850C2214yescomplemented
59#1850C2214yescomplemented
60#1823C2214yescomplemented
61#1823C2214yescomplemented
62#1823C2212yescomplemented
63#1802C2218yescomplemented
64#1797C2214yescomplemented
65#1786C2212yescomplemented
66C2×#340C2214yescomplemented
67C23×D4—21no—
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4—21no—
72C24×C4C2311yescharacteristic
73C22×D8C2314yes—
74C23×C8C2312yes—
75C23×C8C2312yes—
76C2×#123—24no—
77C2×#123—24no—
78C2×#123—24no—
79C2×#123—24no—
80C2×#123—48no—
81C2×#123—48no—
82C2×#121—21no—
83C2×#121—21no—
84C2×#121—24no—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121D418yes—
90C2×#121C2312yes—
91C2×#121C2312yes—
92C2×#121C2312yes—
93C2×#121C2312yes—
94#518—48no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4C2314yes—
98C2×#117—22no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42C2312yes—
121#480—216no—
122#480—216no—
123#480—28no—
124#480—28no—
125#466—216no—
126#447—28no—
127#447—28no—
128C2×#107—28no—
129C2×#107C2314yes—
130C2×#107C2318yes—
131C2×C4⋊C8C2318yes—
132C2×#102—28no—
133C2×#102C2318yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2318yes—
137C2×C22⋊C8—24no—
138C2×C22⋊C8—24no—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143#399—28no—
144#399—28no—
145#390—24no—
146#390C2314yes—
147#390C2318yes—
148#389C2314yes—
149#388—24no—
150#388—24no—
151#387—24no—
152#386—24no—
153#386—24no—
154#386—24no—
155#385—24no—
156#384—24no—
157#384C2318yes—
158#381—22no—
159#380—21no—
160#380—21no—
161#380C2312yes—
162#380C2312yes—
163#379—22no—
164C4×C4⋊C4C2314yes—
165C4×C22⋊C4—24no—
166C4×C22⋊C4D412yes—
167C4×C22⋊C4D412yes—
168C4×C22⋊C4C2312yes—
169C4×C22⋊C4C2312yes—
170C2×#95—21no—
171C2×#95D412yes—
172#340C2318yes—
173#340C2318yes—
174#340C2318yes—
175#336C2318yes—
176C25—21no—
177C25—21no—
178C25C2×D411yescharacteristic
179C22×D4—28no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—21no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4C2×D414yes—
197C22×D4C2412yes—
198C22×D4C2412yes—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D414yes—
215C23×C4C2×D412yes—
216C23×C4#4114yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C2×D8—28no—
221C2×D8—48no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228#123—416no—
229#123—416no—
230#123—416no—
231#123—416no—
232#121—28no—
233#121—28no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—24no—
238#121—24no—
239#121—24no—
240#121—44no—
241#121—44no—
242#121C2×D418yes—
243#121C2×D418yes—
244C4×D4—48no—
245C4×D4—416no—
246C4×D4—416no—
247#117—48no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4D818yescomplemented
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2414yes—
304#107—216no—
305#107—28no—
306C4⋊C8—48no—
307#102—216no—
308#102—28no—
309#102—28no—
310#102—48no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95C2×D414yes—
320#95C2×D414yes—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—42no—
325C24—42no—
326C24—44no—
327C24—44no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—82no—
334C24—82no—
335C24#12114yes—
336C24C2×D814yes—
337C24AGL(1,ℤ/8)14yes—
338C24C22×D411yescharacteristic
339C24C22×D411yescharacteristic
340C24C22×D411yescharacteristic
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—432no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—42no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—82no—
367C2×D4—816no—
368C2×D4C22×D414yes—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—48no—
379C22×C4—48no—
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—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—816no—
408C22×C4#12114yes—
409C22×C4#12114yes—
410C22×C4#12114yes—
411C22×C4C2×D818yes—
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2511yescharacteristic
424D8—88no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—28no—
428C2×C8—28no—
429C2×C8—48no—
430C2×C8—84no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—28no—
438C4⋊C4—416no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—416no—
444C4⋊C4—48no—
445C4⋊C4—82no—
446C4⋊C4C22×D414yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—416no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4C2×D818yes—
461C22⋊C4C2×D818yes—
462C42—28no—
463C42—44no—
464C42—48no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—44no—
473C23—44no—
474C23—48no—
475C23—48no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—48no—
481C23—44no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—88no—
490C23—84no—
491C23—88no—
492C23—84no—
493C23—164no—
494C23—164no—
495C23#459116yes—
496C23C2×#12112yes—
497C23C2×#12112yes—
498C23C2×#12112yes—
499C23C22×D812yes—
500C23C22×D812yes—
501C23C22×D812yes—
502C23C2×AGL(1,ℤ/8)12yes—
503C23C2×AGL(1,ℤ/8)12yes—
504C23C2×AGL(1,ℤ/8)12yes—
505C23C23×D411yescharacteristic
506D4—28no—
507D4—416no—
508D4—416no—
509D4—44no—
510D4—44no—
511D4—44no—
512D4—84no—
513D4—84no—
514D4—88no—
515D4—88no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4—84no—
552C2×C4—168no—
553C2×C4—164no—
554C2×C4C2×#12112yes—
555C2×C4C2×#12112yes—
556C2×C4C2×#12112yes—
557C2×C4C22×D818yes—
558C2×C4C23×D411yescharacteristic
559C2×C4C22×#4111yescharacteristic
560C2×C4#58514yes—
561C8—44no—
562C8—44no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—416no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—84no—
586C22—168no—
587C22—168no—
588C22C2×#45914yes—
589C22C2×#45914yes—
590C22C2×#45914yes—
591C22C22×#12111yescharacteristic
592C22#348512yes—
593C22C23×D811yescharacteristic
594C22C22×AGL(1,ℤ/8)11yescharacteristic
595C22#358612yes—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—28no—
600C4—44no—
601C4—44no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4#348512yes—
606C4#349112yes—
607C2—24no—
608C2—24no—
609C2—24no—
610C2—84no—
611C2—162no—
612C2#3544711yescharacteristic
613C2C22×#45911yescharacteristic
614C2#6042711yescharacteristic
615C2#6169412yes—
616C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3186C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#480C2semidirect
3. C2×#2501C2semidirect
4. #34205C2semidirect
5. #28114C2semidirect
6. #27644C2semidirect
7. #25200C2semidirect
8. #24576C2semidirect
9. #24526C2semidirect
10. #21930C2semidirect
11. #21758C2semidirect
12. C2×#1823C2semidirect
24. C2×#480C22semidirect
28. C22×#107C22semidirect
29. C22×#102C22semidirect
35. #2501C22semidirect
38. #2361C22semidirect
39. #2344C22semidirect
41. D4×C22⋊C4C22semidirect
45. C2×#390C22semidirect
46. C2×#384C22semidirect
49. #2083C22semidirect
50. #2071C22semidirect
51. #2067C22semidirect
52. #2054C22semidirect
53. #1947C22semidirect
54. #1925C22semidirect
56. #1879C22semidirect
57. #1874C22semidirect
58. #1850C22semidirect
59. #1850C22semidirect
60. #1823C22semidirect
61. #1823C22semidirect
62. #1823C22semidirect
63. #1802C22semidirect
64. #1797C22semidirect
65. #1786C22semidirect
66. C2×#340C22semidirect
295. C2×C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
C22×#4802
C2×#25012
#342052
#281142
#276442
#252002
#245762
#245262
#219302
#217582
C2×#18232