Finite Group Databasebeta

← #6644689

Subgroups — #6644689

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35087C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#33009C212yesmaximal
4#32448C214yesmaximal, complemented
5#32342C214yesmaximal, complemented
6#31636C214yesmaximal, complemented
7#31540C214yesmaximal, complemented
8#29642C212yesmaximal, complemented
9#29104C211yesmaximal, characteristic
10#28311C211yesmaximal, characteristic
11#28123C212yesmaximal
12#27965C211yesmaximal, characteristic, complemented
13#27910C211yesmaximal, characteristic, complemented
14#27648C212yesmaximal, complemented
15#21914C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—22no—
18C22×C4×D4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#484C2212yes—
22C2×#450C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2851C2211yescharacteristic
26#2655—22no—
27#2542—22no—
28#2542—22no—
29#2542C2214yes—
30#2542C2214yes—
31#2514C2212yescomplemented
32#2506C2212yescomplemented
33#2493—22no—
34#2473—22no—
35#2453C2212yescomplemented
36#2446C2212yescomplemented
37C2×#399—22no—
38#2336—22no—
39C4×C22⋊Q8C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2099C2214yescomplemented
46#2095C2214yescomplemented
47#2093C2212yes—
48#2083C2214yescomplemented
49#2073C2212yes—
50#2072C2214yescomplemented
51#2067C2214yes—
52#2057C2214yescomplemented
53#2057C2214yescomplemented
54#2055C2214yes—
55#2025C2212yes—
56#2024C2214yescomplemented
57#2024C2214yescomplemented
58#2013C2212yes—
59#1992C2212yescomplemented
60#1973C2214yes—
61#1956C2212yescomplemented
62#1947C2214yescomplemented
63#1934C2214yes—
64#1925C2214yes—
65#1917C2214yescomplemented
66#1916C2214yescomplemented
67#1880C2214yes—
68#1877C2214yes—
69#1877C2214yes—
70#1862C2214yes—
71#1862C2214yes—
72#1845C2212yescomplemented
73#1836C2212yescomplemented
74#1824C2212yes—
75#1824C2212yes—
76#1824C2212yes—
77#1817C2212yes—
78#1803C2214yescomplemented
79#1797C2212yes—
80#1786C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C22×SD16C2314yes—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2314yes—
91C2×#125—22no—
92C2×#125C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—416no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×D4—22no—
101C2×C4×D4—44no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4D412yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#484—44no—
117#484—44no—
118#484—44no—
119#450—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×#97C2312yes—
133C2×C4×C8C2312yes—
134#401—22no—
135#400—22no—
136#400—22no—
137#400—22no—
138#400—22no—
139#399—24no—
140#399—24no—
141#399—24no—
142#399—24no—
143#398—22no—
144#398—22no—
145#398—22no—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#390—21no—
151#390—21no—
152#390—22no—
153#390C2314yes—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#384—21no—
175#384—21no—
176#384—22no—
177#384C2314yes—
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#383—22no—
182#382C2311yescharacteristic
183#381—22no—
184#380—22no—
185#380C2312yes—
186C4×C4⋊C4—22no—
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×Q8C2412yes—
210C22×D4—24no—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4C2412yes—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—48no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2411yescharacteristic
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—48no—
232C4⋊Q8—48no—
233C4⋊Q8—42no—
234#125—42no—
235#125—48no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—416no—
239C22⋊Q8—416no—
240C22⋊Q8—416no—
241C22⋊Q8—416no—
242C4×Q8—44no—
243C4×D4—48no—
244C4×D4—48no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D412yes—
298C2×C42C2412yes—
299C2×C42C2412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—44no—
303#107—44no—
304#107—44no—
305C4⋊C8—44no—
306#103—44no—
307#103—44no—
308#103—44no—
309#102—28no—
310#102—44no—
311#102—44no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314#97—42no—
315C4×C8—42no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335C24—24no—
336C24—22no—
337C24—44no—
338C24—42no—
339C24—42no—
340C24—44no—
341C24C22×D411yescharacteristic
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—816no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—416no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D42−1+414yes—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—816no—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4H(5,𝔽2)14yes—
395C22×C4C2511yescharacteristic
396SD16—88no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4AGL(1,ℤ/8)18yes—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—24no—
443C42—41no—
444C42—42no—
445C42—41no—
446C42—44no—
447C42—82no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—24no—
452C23—48no—
453C23—416no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—84no—
459C23—88no—
460C23—84no—
461C23#53512yes—
462C23#57814yes—
463C23C23×D411yescharacteristic
464Q8—44no—
465Q8—44no—
466Q8—816no—
467D4—28no—
468D4—48no—
469D4—44no—
470C2×C4—28no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—816no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4#53512yes—
514C2×C4C2×AGL(1,ℤ/8)14yes—
515C2×C4C2×AGL(1,ℤ/8)14yes—
516C2×C4C2×AGL(1,ℤ/8)14yes—
517C2×C4C2×H(5,𝔽2)12yes—
518C2×C4C2×2−1+412yes—
519C2×C4#58512yes—
520C8—44no—
521C8—44no—
522C22—24no—
523C22—22no—
524C22—24no—
525C22—22no—
526C22—48no—
527C22—44no—
528C22—816no—
529C22—84no—
530C22—82no—
531C22—82no—
532C22#322314yes—
533C22C2×#53511yescharacteristic
534C22#348511yescharacteristic
535C22#348611yescharacteristic
536C22C22×AGL(1,ℤ/8)12yes—
537C22C2×#57812yes—
538C4—24no—
539C4—24no—
540C4—24no—
541C4—22no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4#353514yes—
553C2—24no—
554C2—84no—
555C2#3711811yescharacteristic
556C2C2×#322311yescharacteristic
557C2#6079811yescharacteristic
558C2#6173712yes—
559C2#6183712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35087C2semidirect
4. #32448C2semidirect
5. #32342C2semidirect
6. #31636C2semidirect
7. #31540C2semidirect
8. #29642C2semidirect
12. #27965C2semidirect
13. #27910C2semidirect
14. #27648C2semidirect
15. #21914C2semidirect
31. #2514C22semidirect
32. #2506C22semidirect
35. #2453C22semidirect
36. #2446C22semidirect
45. #2099C22semidirect
46. #2095C22semidirect
48. #2083C22semidirect
50. #2072C22semidirect
52. #2057C22semidirect
53. #2057C22semidirect
56. #2024C22semidirect
57. #2024C22semidirect
59. #1992C22semidirect
61. #1956C22semidirect
62. #1947C22semidirect
65. #1917C22semidirect
66. #1916C22semidirect
72. #1845C22semidirect
73. #1836C22semidirect
78. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#350872
C2×#25422
#330092
#324482
#323422
#316362
#315402
#296422
#291042
#283112
#281232
#279652
#279102
#276482
#219142