Finite Group Databasebeta

← #6644684

Subgroups — #6644684

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35082C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3#33004C212yesmaximal, complemented
4#32446C214yesmaximal, complemented
5#32344C214yesmaximal, complemented
6#31634C214yesmaximal, complemented
7#31538C214yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#29104C211yesmaximal, characteristic, complemented
10#28310C211yesmaximal, characteristic
11#28117C212yesmaximal, complemented
12#27965C211yesmaximal, characteristic, complemented
13#27906C211yesmaximal, characteristic, complemented
14#27645C212yesmaximal, complemented
15#21914C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—22no—
18C22×C4×D4C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#481C2212yes—
22C2×#447C2212yes—
23#2856C2211yescharacteristic
24#2854C2212yes—
25#2851C2211yescharacteristic
26#2651—22no—
27#2542—22no—
28#2542—22no—
29#2542C2214yes—
30#2542C2214yes—
31#2513C2212yescomplemented
32#2507C2212yescomplemented
33#2454C2212yescomplemented
34#2445C2212yescomplemented
35#2443—22no—
36#2425—22no—
37C2×#399—22no—
38#2335—22no—
39C4×#121C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2099C2214yescomplemented
46#2094C2212yes—
47#2094C2214yescomplemented
48#2084C2214yescomplemented
49#2073C2212yes—
50#2071C2214yescomplemented
51#2065C2214yes—
52#2058C2214yescomplemented
53#2058C2214yescomplemented
54#2054C2214yes—
55#2025C2212yes—
56#2023C2214yescomplemented
57#2023C2214yescomplemented
58#2011C2212yes—
59#1992C2212yescomplemented
60#1972C2214yes—
61#1957C2212yescomplemented
62#1948C2214yescomplemented
63#1934C2214yes—
64#1926C2214yes—
65#1917C2214yescomplemented
66#1916C2214yescomplemented
67#1879C2214yes—
68#1879C2214yes—
69#1879C2214yes—
70#1864C2214yes—
71#1864C2214yes—
72#1844C2212yescomplemented
73#1836C2212yescomplemented
74#1824C2212yes—
75#1824C2212yes—
76#1824C2212yes—
77#1817C2212yes—
78#1802C2214yescomplemented
79#1797C2212yes—
80#1786C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C22×D8C2314yes—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8C2314yes—
90C2×#127C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#123—22no—
93C2×#123—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—416no—
99C2×#121—22no—
100C2×#121C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—44no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4D412yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#481—44no—
120#481—44no—
121#481—44no—
122#447—44no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#399—22no—
138#399—24no—
139#399—24no—
140#399—24no—
141#399—24no—
142#398—22no—
143#398—22no—
144#397—22no—
145#397—22no—
146#396—22no—
147#394—22no—
148#394—22no—
149#390—22no—
150#390—21no—
151#390—22no—
152#390—21no—
153#390C2312yes—
154#390C2314yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#387—22no—
171#387—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—22no—
179#384—21no—
180#384C2314yes—
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382C2311yescharacteristic
185#380—22no—
186#380C2312yes—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C2×#95D412yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203#336C2312yes—
204#336C2312yes—
205C25—21no—
206C22×Q8—24no—
207C22×Q8—48no—
208C22×Q8—44no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—42no—
215C22×D4C2412yes—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—44no—
223C23×C4—48no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2411yescharacteristic
229C2×D8—28no—
230C2×D8—44no—
231C2×D8—44no—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—48no—
235C4⋊Q8—48no—
236#127—42no—
237#125—42no—
238#123—48no—
239#123—84no—
240C22⋊Q8—416no—
241C22⋊Q8—416no—
242C22⋊Q8—416no—
243C22⋊Q8—416no—
244#121—24no—
245#121—24no—
246#121—84no—
247C4×D4—48no—
248C4×D4—48no—
249C4×D4—44no—
250C4×D4—84no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
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⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2412yes—
310C2×C42C2412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#106—44no—
314#106—44no—
315#106—44no—
316C4⋊C8—44no—
317#102—28no—
318#102—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325#97—42no—
326C4×C8—42no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—44no—
342C24—42no—
343C24—42no—
344C24—44no—
345C24C22×D411yescharacteristic
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—816no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—416no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D42−1+414yes—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—816no—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4H(5,𝔽2)14yes—
407C22×C4C2511yescharacteristic
408D8—88no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4AGL(1,ℤ/8)18yes—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C42—24no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—24no—
461C42—41no—
462C42—42no—
463C42—41no—
464C42—82no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—24no—
469C23—48no—
470C23—416no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—84no—
483C23—88no—
484C23—84no—
485C23#53512yes—
486C23#57714yes—
487C23C23×D411yescharacteristic
488Q8—816no—
489D4—28no—
490D4—48no—
491D4—44no—
492D4—44no—
493D4—44no—
494D4—88no—
495D4—164no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—816no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—164no—
538C2×C4#53512yes—
539C2×C4C2×AGL(1,ℤ/8)14yes—
540C2×C4C2×AGL(1,ℤ/8)14yes—
541C2×C4C2×AGL(1,ℤ/8)14yes—
542C2×C4C2×H(5,𝔽2)12yes—
543C2×C4C2×2−1+412yes—
544C2×C4#58512yes—
545C8—44no—
546C8—44no—
547C22—24no—
548C22—22no—
549C22—24no—
550C22—22no—
551C22—48no—
552C22—44no—
553C22—816no—
554C22—84no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22#322114yes—
565C22C2×#53511yescharacteristic
566C22#348511yescharacteristic
567C22#348611yescharacteristic
568C22C22×AGL(1,ℤ/8)12yes—
569C22C2×#57712yes—
570C4—24no—
571C4—24no—
572C4—24no—
573C4—22no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4#353514yes—
584C2—24no—
585C2—84no—
586C2—162no—
587C2#3711811yescharacteristic
588C2C2×#322111yescharacteristic
589C2#6079811yescharacteristic
590C2#6172812yes—
591C2#6184312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28310C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35082C2semidirect
2. C2×#2542C2semidirect
3. #33004C2semidirect
4. #32446C2semidirect
5. #32344C2semidirect
6. #31634C2semidirect
7. #31538C2semidirect
8. #29639C2semidirect
9. #29104C2semidirect
11. #28117C2semidirect
12. #27965C2semidirect
13. #27906C2semidirect
14. #27645C2semidirect
15. #21914C2semidirect
31. #2513C22semidirect
32. #2507C22semidirect
33. #2454C22semidirect
34. #2445C22semidirect
45. #2099C22semidirect
47. #2094C22semidirect
48. #2084C22semidirect
50. #2071C22semidirect
52. #2058C22semidirect
53. #2058C22semidirect
56. #2023C22semidirect
57. #2023C22semidirect
59. #1992C22semidirect
61. #1957C22semidirect
62. #1948C22semidirect
65. #1917C22semidirect
66. #1916C22semidirect
72. #1844C22semidirect
73. #1836C22semidirect
78. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#350822
C2×#25422
#330042
#324462
#323442
#316342
#315382
#296392
#291042
#283102
#281172
#279652
#279062
#276452
#219142