Finite Group Databasebeta

← #6470493

Subgroups — #6470493

Order 512.

2470 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37307C211yesmaximal, characteristic, complemented
2#34256C211yesmaximal, characteristic
3#32966C212yesmaximal, complemented
4#32956C212yesmaximal, complemented
5#29965C212yesmaximal, complemented
6#29955C212yesmaximal, complemented
7#25978C214yesmaximal, complemented
8#25968C214yesmaximal, complemented
9#25964C214yesmaximal, complemented
10#25958C214yesmaximal, complemented
11#21946C212yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#20602C212yesmaximal
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#518C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#478C2214yescomplemented
18C2×#476C2214yescomplemented
19C2×#417C2212yes—
20C2×#418C2212yes—
21C2×#416C2212yes—
22C22×#102C2212yes—
23C2×#408C2212yes—
24#2857C2212yescomplemented
25#2806—22no—
26#2748—22no—
27#2732—22no—
28#2683—22no—
29#2629C2212yescomplemented
30#2441C2214yescomplemented
31#2440C2214yescomplemented
32#2339—22no—
33#2339C2214yes—
34D4×C22⋊C4—22no—
35D4×C22⋊C4C2214yes—
36#2311C2211yescharacteristic
37#2311C2212yes—
38#2311C2211yescharacteristic
39#2308—22no—
40#2297C2211yescharacteristic
41#2297C2211yescharacteristic
42C2×#386—22no—
43#2096C2212yescomplemented
44#2096C2212yescomplemented
45#2094C2212yescomplemented
46#2093C2212yescomplemented
47#2068C2214yescomplemented
48#2066C2214yescomplemented
49#2065C2214yescomplemented
50#2065C2214yescomplemented
51#2026C2212yescomplemented
52#2025C2212yescomplemented
53#2024C2212yescomplemented
54#2023C2212yescomplemented
55#1864C2214yescomplemented
56#1864C2214yescomplemented
57#1864C2214yescomplemented
58#1862C2214yescomplemented
59#1854C2214yescomplemented
60#1854C2214yescomplemented
61#1853C2214yescomplemented
62#1853C2214yescomplemented
63#1851C2214yescomplemented
64#1850C2214yescomplemented
65#1809C2214yescomplemented
66#1807C2214yescomplemented
67#1802C2214yescomplemented
68#1802C2214yescomplemented
69#1798C2212yescomplemented
70#1797C2212yescomplemented
71#1788C2214yes—
72#1788C2214yes—
73#1786C2214yes—
74#1786C2214yes—
75C2×H(5,𝔽2)—24no—
76C2×H(5,𝔽2)—44no—
77C2×H(5,𝔽2)—44no—
78C2×H(5,𝔽2)Q818yescomplemented
79C22×#41—21no—
80C22×#41C2312yes—
81C22×#41C2311yescharacteristic
82C22×#41C2311yescharacteristic
83C23×D4—22no—
84C23×D4—21no—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4—21no—
88C22×M16C2312yes—
89C23×C8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#125C2312yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94#518—44no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—44no—
98C2×C4×D4C2314yes—
99C2×C4×D4C2314yes—
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—24no—
108C22×C42C2311yescharacteristic
109#478—48no—
110#476—48no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2314yes—
116C2×C4⋊C8C2314yes—
117#418—44no—
118#417—44no—
119#416—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102—44no—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#408—28no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132#399—24no—
133#398—24no—
134#398—24no—
135#396—24no—
136#396—24no—
137#395—24no—
138#391—21no—
139#391—22no—
140#391—22no—
141#391D412yes—
142#391C2312yes—
143#391C2312yes—
144#390—21no—
145#390—22no—
146#390D412yes—
147#390C2312yes—
148#390C2312yes—
149#389—22no—
150#389—24no—
151#389C2314yes—
152#389C2314yes—
153#387—24no—
154#386—22no—
155#386—22no—
156#386—22no—
157#386—21no—
158#386—21no—
159#386—22no—
160#386—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#386D412yes—
165#386D412yes—
166#385—24no—
167#384C2312yes—
168#384C2312yes—
169#381—24no—
170#380—22no—
171#380—22no—
172#380C2314yes—
173#380C2314yes—
174#380C2314yes—
175#379—24no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4C2314yes—
178C2×#95—24no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2312yes—
184#336C2312yes—
185#336C2312yes—
186#336C2312yes—
187C25—21no—
188C25—22no—
189H(5,𝔽2)—432no—
190H(5,𝔽2)—432no—
191C2×#41—28no—
192C2×#41—22no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—42no—
199C2×#41—42no—
200C2×#41—42no—
201C2×#41—42no—
202C2×#41—44no—
203C2×#41—44no—
204C2×#41C2×Q818yes—
205C2×#41C2×Q814yes—
206C2×#41C2×Q814yes—
207C2×#41#4114yes—
208C22×Q8—22no—
209C22×Q8C2×D411yescharacteristic
210C22×Q8C2411yescharacteristic
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—28no—
218C22×D4—216no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—42no—
224C22×D4—42no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4—48no—
228C22×D4—48no—
229C22×D4C2×D412yes—
230C22×D4C2×D411yescharacteristic
231C22×D4C2×Q814yes—
232C22×D4C2×Q818yes—
233C22×D4C2411yescharacteristic
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—44no—
242C23×C4—84no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C2×M16—24no—
248C2×M16—42no—
249C2×M16—42no—
250C22×C8—24no—
251C22×C8—42no—
252C22×C8—42no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—44no—
256#125—44no—
257C22⋊Q8—88no—
258C22⋊Q8—84no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—48no—
264C4×D4—48no—
265#117—42no—
266#117—42no—
267#117—42no—
268#117—42no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—88no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2412yes—
307C2×C42C2412yes—
308#107—44no—
309#106—44no—
310#106—44no—
311#106—44no—
312C4⋊C8—44no—
313C4⋊C8—44no—
314#103—44no—
315#103—44no—
316#102—28no—
317#102—28no—
318#102—28no—
319#102—28no—
320#102—44no—
321#102—44no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—21no—
339C24—28no—
340C24—44no—
341C24—44no—
342C24—82no—
343C24—82no—
344C24—84no—
345C24—84no—
346C24C22×D412yes—
347C24C22×D411yescharacteristic
348#41—216no—
349#41—48no—
350#41—48no—
351#41—48no—
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356#41—432no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—44no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8C22⋊Q814yes—
366C2×Q8C22×Q812yes—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—28no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—432no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—464no—
397C2×D4C22⋊Q818yes—
398C2×D4C22⋊Q814yes—
399C2×D4C22×Q812yes—
400C2×D4C22×Q814yes—
401C2×D4H(5,𝔽2)14yes—
402C2×D4H(5,𝔽2)14yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—48no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—84no—
434C22×C4—816no—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q814yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4114yes—
444C22×C4C2511yescharacteristic
445M16—84no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—84no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—82no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C22⋊C4—28no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—82no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—82no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C42—24no—
483C42—24no—
484C42—42no—
485C42—42no—
486C42—82no—
487C23—216no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—21no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—21no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—416no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—88no—
508C23—816no—
509C23—84no—
510C23—84no—
511C23—88no—
512C23—164no—
513C23—164no—
514C23—168no—
515C23C2×C22⋊Q812yes—
516C23C2×C22⋊Q814yes—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—44no—
520Q8—42no—
521Q8—164no—
522D4—216no—
523D4—24no—
524D4—24no—
525D4—24no—
526D4—24no—
527D4—44no—
528D4—48no—
529D4—416no—
530D4—416no—
531D4—48no—
532D4—42no—
533D4#54318yes—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—44no—
554C2×C4—416no—
555C2×C4—48no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—816no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—168no—
573C2×C4—164no—
574C2×C4C2×C22⋊Q812yes—
575C2×C4C2×C22⋊Q814yes—
576C2×C4#57714yes—
577C2×C4C23×Q811yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×H(5,𝔽2)12yes—
580C2×C4C2×H(5,𝔽2)12yes—
581C8—44no—
582C8—44no—
583C22—24no—
584C22—28no—
585C22—22no—
586C22—24no—
587C22—28no—
588C22—48no—
589C22—41no—
590C22—44no—
591C22—42no—
592C22—41no—
593C22—42no—
594C22—44no—
595C22—48no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—816no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22—1616no—
605C22C22×C22⋊Q811yescharacteristic
606C22#348411yescharacteristic
607C22#348411yescharacteristic
608C22C2×#57712yes—
609C22#358512yes—
610C4—22no—
611C4—24no—
612C4—24no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—82no—
617C4—84no—
618C4—84no—
619C4—82no—
620C4C2×#54312yes—
621C4#349012yes—
622C2—42no—
623C2—44no—
624C2—48no—
625C2—82no—
626C2#3498211yescharacteristic
627C2#6049512yes—
628C2#6163111yescharacteristic
629C2#6164811yescharacteristic
630C2#6168512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37307C2semidirect
3. #32966C2semidirect
4. #32956C2semidirect
5. #29965C2semidirect
6. #29955C2semidirect
7. #25978C2semidirect
8. #25968C2semidirect
9. #25964C2semidirect
10. #25958C2semidirect
11. #21946C2semidirect
12. #21937C2semidirect
17. C2×#478C22semidirect
18. C2×#476C22semidirect
24. #2857C22semidirect
29. #2629C22semidirect
30. #2441C22semidirect
31. #2440C22semidirect
43. #2096C22semidirect
44. #2096C22semidirect
45. #2094C22semidirect
46. #2093C22semidirect
47. #2068C22semidirect
48. #2066C22semidirect
49. #2065C22semidirect
50. #2065C22semidirect
51. #2026C22semidirect
52. #2025C22semidirect
53. #2024C22semidirect
54. #2023C22semidirect
55. #1864C22semidirect
56. #1864C22semidirect
57. #1864C22semidirect
58. #1862C22semidirect
59. #1854C22semidirect
60. #1854C22semidirect
61. #1853C22semidirect
62. #1853C22semidirect
63. #1851C22semidirect
64. #1850C22semidirect
65. #1809C22semidirect
66. #1807C22semidirect
67. #1802C22semidirect
68. #1802C22semidirect
69. #1798C22semidirect
70. #1797C22semidirect
78. C2×H(5,𝔽2)Q8semidirect

Maximal subgroups

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

subgroupindex
#373072
#342562
#329662
#329562
#299652
#299552
#259782
#259682
#259642
#259582
#219462
#219372
#206022