Finite Group Databasebeta

← #6506433

Subgroups — #6506433

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic, complemented
2#35305C211yesmaximal, characteristic
3#29120C214yesmaximal, complemented
4#27281C214yesmaximal, complemented
5#27277C214yesmaximal, complemented
6#27265C214yesmaximal, complemented
7#27261C214yesmaximal, complemented
8#25866C214yesmaximal, complemented
9#21768C211yesmaximal, characteristic, complemented
10#21762C211yesmaximal, characteristic, complemented
11#20600C211yesmaximal, characteristic
12#20597C211yesmaximal, characteristic
13#20527C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C2×#536C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C2×#489C2214yescomplemented
18C2×#487C2214yescomplemented
19C2×#419C2212yes—
20C2×#418C2212yes—
21C2×#416C2212yes—
22C2×#416C2212yes—
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2848—22no—
26#2842—21no—
27#2840—21no—
28#2665—22no—
29#2665—22no—
30#2628C2214yescomplemented
31#2616—22no—
32#2607—22no—
33#2439C2214yes—
34#2439C2214yescomplemented
35#2435C2212yes—
36#2434C2212yes—
37#2432C2214yes—
38#2406—22no—
39#2405—21no—
40#2405—21no—
41#2311C2212yes—
42#2280C2211yescharacteristic
43#2280C2211yescharacteristic
44#2280C2211yescharacteristic
45#2074C2214yescomplemented
46#2073C2214yescomplemented
47#2043C2214yescomplemented
48#2043C2214yescomplemented
49#2042C2214yescomplemented
50#2042C2214yescomplemented
51#2024C2214yescomplemented
52#2023C2214yescomplemented
53#1960C2214yescomplemented
54#1958C2214yescomplemented
55#1936C2214yescomplemented
56#1935C2214yescomplemented
57#1934C2214yescomplemented
58#1933C2214yescomplemented
59#1894C2214yes—
60#1892C2212yescomplemented
61#1892C2212yescomplemented
62#1873C2214yes—
63#1873C2214yescomplemented
64#1824C2212yescomplemented
65#1823C2212yescomplemented
66#1788C2214yes—
67#1788C2214yes—
68#1788C2214yes—
69#1787C2212yes—
70#1787C2212yescomplemented
71#1787C2212yescomplemented
72#1786C2212yes—
73#1753C2214yes—
74#1752C2212yescomplemented
75C8×C22⋊C4C2212yescomplemented
76#1750C2214yescomplemented
77C22×#41—21no—
78C22×#41—21no—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—44no—
82C22×#41—44no—
83C22×#41C2312yes—
84C22×#41C2312yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94#536—44no—
95#536—44no—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2314yes—
98C2×#124C2314yes—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114#489—48no—
115#487—48no—
116#419—44no—
117#418—44no—
118#416—44no—
119#416—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132#407—24no—
133C2×C22⋊C8—24no—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2314yes—
140#400—24no—
141#399—24no—
142#396—24no—
143#396—24no—
144#395—24no—
145#395—24no—
146#394—24no—
147#394—24no—
148#394—24no—
149#393—24no—
150#393—24no—
151#393—24no—
152#391—24no—
153#391—24no—
154#391C2314yes—
155#390C2314yes—
156#389C2314yes—
157#389C2314yes—
158#387—24no—
159#387—24no—
160#386—22no—
161#386—22no—
162#385—22no—
163#385—22no—
164#385—24no—
165#385—24no—
166#382C2314yes—
167#381—21no—
168#381—21no—
169#381D412yes—
170#380—24no—
171#380D412yes—
172#380D412yes—
173#380C2312yes—
174#380C2312yes—
175#380C2312yes—
176#380C2312yes—
177#379—21no—
178#379—21no—
179#379—22no—
180#379—22no—
181#379D411yescharacteristic
182#379D411yescharacteristic
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4C2312yes—
188C4×C22⋊C4C2312yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—24no—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—42no—
209C2×#41—48no—
210C2×#41—416no—
211C2×#41—48no—
212C2×#41—416no—
213C2×#41—432no—
214C2×#41—44no—
215C2×#41—44no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4—24no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—42no—
236C22×D4—48no—
237C22×D4—44no—
238C22×D4C2412yes—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—48no—
247C23×C4—44no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4#4114yes—
252C23×C4#4114yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×M16—24no—
257C2×M16—42no—
258C2×M16—42no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—42no—
262C22×C8—42no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—28no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#124—28no—
271#124—44no—
272#123—48no—
273#123—48no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—416no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#117—42no—
282#117—42no—
283#117—42no—
284#117—42no—
285#117—42no—
286#117—42no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42C2414yes—
324#103—28no—
325#103—28no—
326#103—44no—
327#103—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#103—44no—
332#102—28no—
333#102—28no—
334#102—44no—
335#102—44no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340#97—44no—
341C4×C8—44no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—44no—
359C24—44no—
360C24—42no—
361C24—82no—
362C24#12414yes—
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366#41—216no—
367#41—48no—
368#41—48no—
369#41—48no—
370#41—48no—
371#41—48no—
372#41—48no—
373#41—48no—
374#41—432no—
375#41—432no—
376C2×Q8—22no—
377C2×Q8—28no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×Q8—84no—
395C2×Q8—82no—
396C2×Q8—84no—
397C2×Q8—82no—
398C2×Q8H(5,𝔽2)14yes—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—432no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4H(5,𝔽2)14yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—432no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4#12414yes—
448C22×C4#12414yes—
449C22×C4#12414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462M16—84no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—82no—
471C2×C8—82no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C42—24no—
506C42—24no—
507C42—82no—
508C42—82no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—48no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—88no—
530C23—816no—
531C23C2×#12412yes—
532C23C2×#12412yes—
533C23C2×#12412yes—
534C23C23×D411yescharacteristic
535Q8—24no—
536Q8—24no—
537Q8—48no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—44no—
542Q8—42no—
543Q8—84no—
544Q8—168no—
545D4—24no—
546D4—24no—
547D4—48no—
548D4—416no—
549D4—416no—
550D4—42no—
551D4—42no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4C2×#12412yes—
592C2×C4C2×#12412yes—
593C2×C4C2×#12412yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×H(5,𝔽2)12yes—
598C8—42no—
599C8—42no—
600C8—44no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22—88no—
616C22—816no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22C22×#12411yescharacteristic
621C22#348412yes—
622C22#358412yes—
623C22#358612yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4#349011yescharacteristic
635C4#349012yes—
636C4#349011yescharacteristic
637C2—42no—
638C2—42no—
639C2—42no—
640C2—84no—
641C2#3536711yescharacteristic
642C2#6061211yescharacteristic
643C2#6062011yescharacteristic
644C2#6162714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37480C2semidirect
3. #29120C2semidirect
4. #27281C2semidirect
5. #27277C2semidirect
6. #27265C2semidirect
7. #27261C2semidirect
8. #25866C2semidirect
9. #21768C2semidirect
10. #21762C2semidirect
13. #20527C2semidirect
17. C2×#489C22semidirect
18. C2×#487C22semidirect
30. #2628C22semidirect
34. #2439C22semidirect
45. #2074C22semidirect
46. #2073C22semidirect
47. #2043C22semidirect
48. #2043C22semidirect
49. #2042C22semidirect
50. #2042C22semidirect
51. #2024C22semidirect
52. #2023C22semidirect
53. #1960C22semidirect
54. #1958C22semidirect
55. #1936C22semidirect
56. #1935C22semidirect
57. #1934C22semidirect
58. #1933C22semidirect
60. #1892C22semidirect
61. #1892C22semidirect
63. #1873C22semidirect
64. #1824C22semidirect
65. #1823C22semidirect
70. #1787C22semidirect
71. #1787C22semidirect
74. #1752C22semidirect
75. C8×C22⋊C4C22semidirect
76. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#353052
#291202
#272812
#272772
#272652
#272612
#258662
#217682
#217622
#206002
#205972
#205272