Finite Group Databasebeta

← #6630470

Subgroups — #6630470

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35261C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic, complemented
3#33156C212yesmaximal, complemented
4#33151C212yesmaximal, complemented
5#33116C212yesmaximal, complemented
6#33106C212yesmaximal, complemented
7#33103C214yesmaximal, complemented
8#32981C212yesmaximal, complemented
9#32978C212yesmaximal, complemented
10#21590C214yesmaximal, complemented
11#21578C212yesmaximal, complemented
12C4×#480C212yesmaximal, complemented
13#21399C212yesmaximal, complemented
14#21368C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#482C2214yes—
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#433C2212yes—
24#2856C2211yescharacteristic
25#2851C2211yescharacteristic
26#2847C2212yes—
27#2629C2211yescharacteristic
28#2624C2212yes—
29#2619C2211yescharacteristic
30#2476—22no—
31#2471—22no—
32#2443—22no—
33#2424—22no—
34#2324C2212yes—
35#2314C2211yescharacteristic
36C4×#121C2212yes—
37#2274—22no—
38#2274—22no—
39#2274C2212yes—
40#2274C2212yes—
41#2274C2212yes—
42#2274C2212yes—
43C2×C4×C4⋊C4—22no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4—22no—
47#1989C2214yescomplemented
48#1982C2214yescomplemented
49#1949C2214yescomplemented
50#1947C2214yescomplemented
51#1926C2214yescomplemented
52#1925C2214yescomplemented
53#1925C2214yescomplemented
54#1925C2214yescomplemented
55#1921C2214yescomplemented
56#1921C2214yescomplemented
57#1919C2214yescomplemented
58#1918C2214yescomplemented
59#1875C2214yes—
60#1874C2212yes—
61#1874C2212yes—
62#1874C2214yes—
63#1874C2214yes—
64#1854C2214yes—
65#1853C2212yes—
66#1852C2212yes—
67#1852C2214yes—
68#1852C2214yes—
69#1825C2214yes—
70#1823C2212yes—
71#1823C2212yes—
72#1818C2212yes—
73#1798C2212yes—
74#1797C2212yes—
75#1779C2212yes—
76#1776C2214yescomplemented
77C4×#107C2214yescomplemented
78#1762C2214yescomplemented
79C4×#102C2214yescomplemented
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85#540—28no—
86C2×#127C2311yescharacteristic
87C2×#125C2311yescharacteristic
88C2×#123—22no—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121C2312yes—
92C2×#121C2312yes—
93C2×#121C2312yes—
94C2×#120—22no—
95C2×C4×D4—22no—
96C2×C4×D4C2312yes—
97C2×#117—21no—
98C2×#117—21no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117D412yes—
102C2×#117D412yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4D412yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4—21no—
119C22×C22⋊C4D412yes—
120C22×C42—21no—
121C22×C42—22no—
122C22×C42—22no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#482—216no—
126#480—28no—
127#480—28no—
128#433—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144#399—22no—
145#398—22no—
146#397—22no—
147#396—22no—
148#394—22no—
149#394—22no—
150#394—22no—
151#392—22no—
152#390—22no—
153#390—22no—
154#390C2312yes—
155#390C2312yes—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#387—22no—
162#387—22no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#385—22no—
169#385—22no—
170#384C2311yescharacteristic
171#382C2311yescharacteristic
172#380C2312yes—
173#380C2312yes—
174#380C2312yes—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—24no—
189C4×C22⋊C4—24no—
190C4×C22⋊C4—24no—
191C4×C22⋊C4—24no—
192C4×C22⋊C4—22no—
193C2×#95—22no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206C25C2×D411yescharacteristic
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2412yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—28no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—84no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C22×C8C2414yes—
231C22×C8C2414yes—
232#127—24no—
233#125—24no—
234#123—48no—
235#121—28no—
236#121—24no—
237#121—24no—
238#121—28no—
239#121—48no—
240#121—84no—
241#120—84no—
242C4×D4—28no—
243C4×D4—84no—
244#117—28no—
245#117—28no—
246#117—24no—
247#117—24no—
248#117—28no—
249#117—28no—
250#117—28no—
251#117—24no—
252#117—24no—
253#117#4118yes—
254#117#4118yes—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4#4118yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2414yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C42—28no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—84no—
311C2×C42—84no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318#107—28no—
319#107—28no—
320#107—28no—
321#106—28no—
322C4⋊C8—28no—
323#102—28no—
324#102—28no—
325#102—28no—
326#102—28no—
327C22⋊C8—28no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330C22⋊C8—28no—
331#97—24no—
332C4×C8—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—42no—
350C24C22×D411yescharacteristic
351C24C22×D411yescharacteristic
352C24C22×D411yescharacteristic
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C22×C4—232no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—816no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4114yes—
420C22×C4C2511yescharacteristic
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—44no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4#12318yes—
445C4⋊C4C2×#4118yes—
446C4⋊C4C2×#4118yes—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4#12318yes—
464C42—28no—
465C42—24no—
466C42—24no—
467C42—28no—
468C42—22no—
469C42—22no—
470C42—84no—
471C42—84no—
472C42—84no—
473C42—84no—
474C42—82no—
475C42C2×#4114yes—
476C42C2×#4114yes—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—28no—
481C23—22no—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—48no—
490C23—48no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C2×#12312yes—
501C23#52512yes—
502C23#52512yes—
503C23#57614yes—
504C23C23×D411yescharacteristic
505D4—44no—
506D4—44no—
507D4—88no—
508D4—164no—
509D4—164no—
510D4—164no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—416no—
538C2×C4—44no—
539C2×C4—416no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—88no—
553C2×C4—88no—
554C2×C4—164no—
555C2×C4C2×#12314yes—
556C2×C4C2×#12314yes—
557C2×C4C2×#12312yes—
558C2×C4#52512yes—
559C2×C4#52512yes—
560C2×C4C22×#4112yes—
561C2×C4C22×#4112yes—
562C2×C4C22×#4112yes—
563C8—44no—
564C8—44no—
565C22—22no—
566C22—24no—
567C22—22no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—416no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22#312214yes—
588C22C22×#12311yescharacteristic
589C22C2×#52511yescharacteristic
590C22C2×#52511yescharacteristic
591C22C2×#57612yes—
592C22#358612yes—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—24no—
597C4—24no—
598C4—42no—
599C4—42no—
600C4—48no—
601C4—48no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4#344414yes—
606C2—24no—
607C2—42no—
608C2—42no—
609C2—162no—
610C2#3662611yescharacteristic
611C2#6052112yes—
612C2C2×#312211yescharacteristic
613C2#6056611yescharacteristic
614C2#6058412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35261C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2274C2semidirect
3. #33156C2semidirect
4. #33151C2semidirect
5. #33116C2semidirect
6. #33106C2semidirect
7. #33103C2semidirect
8. #32981C2semidirect
9. #32978C2semidirect
10. #21590C2semidirect
11. #21578C2semidirect
12. C4×#480C2semidirect
13. #21399C2semidirect
14. #21368C2semidirect
15. #20716C2semidirect
47. #1989C22semidirect
48. #1982C22semidirect
49. #1949C22semidirect
50. #1947C22semidirect
51. #1926C22semidirect
52. #1925C22semidirect
53. #1925C22semidirect
54. #1925C22semidirect
55. #1921C22semidirect
56. #1921C22semidirect
57. #1919C22semidirect
58. #1918C22semidirect
76. #1776C22semidirect
77. C4×#107C22semidirect
78. #1762C22semidirect
79. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#352612
C2×#22742
#331562
#331512
#331162
#331062
#331032
#329812
#329782
#215902
#215782
C4×#4802
#213992
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