Finite Group Databasebeta

← #6468226

Subgroups — #6468226

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#29148C212yesmaximal, complemented
4#29137C214yesmaximal, complemented
5#29118C212yesmaximal, complemented
6#29116C212yesmaximal
7#28727C212yesmaximal, complemented
8#27968C212yesmaximal, complemented
9#25864C212yesmaximal
10#23827C214yesmaximal, complemented
11#21655C212yesmaximal, complemented
12C4×#489C212yesmaximal, complemented
13#20745C212yesmaximal, complemented
14#20744C212yesmaximal
15#20465C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C2×#519C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#489C2212yes—
20C2×#489C2212yes—
21C2×#433C2212yes—
22#2868C2211yescharacteristic, complemented
23#2862C2211yescharacteristic, complemented
24#2857C2212yes—
25#2647C2212yescomplemented
26#2628C2212yes—
27#2511—22no—
28#2492—22no—
29#2451—22no—
30#2423—22no—
31#2367—22no—
32#2367—22no—
33#2342—22no—
34#2342—22no—
35#2331C2212yescomplemented
36#2326C2212yes—
37#2326C2212yes—
38#2314C2212yes—
39#2310C2211yescharacteristic
40#2309C2211yescharacteristic
41C4×C4⋊Q8C2212yescomplemented
42C4×#124C2212yes—
43C4×#124C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2100C2214yescomplemented
47#2099C2214yescomplemented
48#2074C2214yescomplemented
49#2073C2214yes—
50#2042C2214yes—
51#2042C2214yes—
52#2024C2214yescomplemented
53#2023C2214yes—
54#1993C2212yescomplemented
55#1993C2212yescomplemented
56#1959C2212yes—
57#1959C2212yes—
58#1936C2214yes—
59#1935C2214yescomplemented
60#1934C2214yes—
61#1933C2214yescomplemented
62#1836C2212yescomplemented
63#1836C2212yescomplemented
64#1825C2214yes—
65#1825C2214yes—
66#1805C2214yescomplemented
67#1804C2214yes—
68#1803C2214yescomplemented
69#1802C2214yes—
70#1787C2214yes—
71#1786C2214yes—
72#1768C2212yescomplemented
73#1767C2212yescomplemented
74#1763C2214yescomplemented
75#1762C2214yes—
76C4×#103C2214yescomplemented
77C4×#102C2214yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yescomplemented
81C4×#97C2211yescharacteristic, complemented
82C4×#97C2211yescharacteristic, complemented
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#124—28no—
89C2×#124—22no—
90C2×#124C2312yes—
91C2×#124C2312yes—
92C2×C22⋊Q8—22no—
93#519—28no—
94#518—28no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C22⋊C4—22no—
108C22×C22⋊C4—22no—
109C22×C42C2311yescharacteristic
110#489—28no—
111#489—28no—
112#433—28no—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×#97C2312yes—
125C2×#97C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129#400—22no—
130#400—24no—
131#399—22no—
132#398—24no—
133#395—22no—
134#393—22no—
135#391—21no—
136#391—21no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—24no—
142#391C2312yes—
143#391C2312yes—
144#390—21no—
145#390—21no—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#388—24no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#387—24no—
162#387—24no—
163#385—22no—
164#385—22no—
165#385—22no—
166#385—22no—
167#384—22no—
168#384C2312yes—
169#384C2312yes—
170#384C2312yes—
171#382C2312yes—
172#382C2312yes—
173#382C2312yes—
174#381—21no—
175#381—21no—
176#380—22no—
177#380—22no—
178#380C2314yes—
179#379—21no—
180#379—21no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4D412yes—
191C4×C22⋊C4D412yes—
192#376C2311yescharacteristic
193#376C2311yescharacteristic
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C22×Q8—24no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—24no—
212C22×Q8—42no—
213C22×Q8C2412yes—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—42no—
218C22×D4C2412yes—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2×D412yes—
223C23×C4C2411yescharacteristic
224C22×C8C2414yes—
225C22×C8C2414yes—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—84no—
231#124—216no—
232#124—24no—
233#124—24no—
234#124—24no—
235#124#4118yes—
236C22⋊Q8—84no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—48no—
240C4×Q8—84no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—48no—
244#117—24no—
245#117—24no—
246#117—42no—
247#117—48no—
248#117#4118yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D412yes—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—24no—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C4⋊C8—28no—
303#103—28no—
304#103—28no—
305#103—44no—
306#103—44no—
307#102—28no—
308#102—28no—
309#102—44no—
310#102—44no—
311C22⋊C8—28no—
312#97—24no—
313#97—24no—
314#97—24no—
315#97—42no—
316C4×C8—24no—
317C4×C8—42no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335C24—24no—
336C24—22no—
337C24—42no—
338C24—42no—
339C24—82no—
340C24—82no—
341C24C22×D411yescharacteristic
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—416no—
347C2×Q8—42no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—42no—
353C2×Q8—84no—
354C2×Q8C2×#4114yes—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—416no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4C2×#4114yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C2×#4114yes—
392C22×C4C2511yescharacteristic
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4#12418yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—84no—
434C22⋊C4#12418yes—
435C42—24no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—44no—
446C42—44no—
447C42—84no—
448C42C2×#4114yes—
449C42C2×#4114yes—
450C42C2×#4114yes—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—24no—
456C23—48no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—88no—
464C23—164no—
465C23—162no—
466C23—162no—
467C23C2×#12412yes—
468C23C23×D411yescharacteristic
469Q8—28no—
470Q8—48no—
471Q8—44no—
472Q8—88no—
473Q8—164no—
474D4—28no—
475D4—48no—
476D4—44no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—84no—
514C2×C4—164no—
515C2×C4—164no—
516C2×C4#52514yes—
517C2×C4C2×#12412yes—
518C2×C4C2×#12414yes—
519C2×C4C2×#12414yes—
520C2×C4C22×#4112yes—
521C2×C4C22×#4112yes—
522C2×C4#58512yes—
523C8—44no—
524C8—44no—
525C22—22no—
526C22—24no—
527C22—41no—
528C22—41no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—168no—
538C22C2×#52511yescharacteristic
539C22C22×#12411yescharacteristic
540C22#344911yescharacteristic
541C22#358512yes—
542C22#358612yes—
543C4—22no—
544C4—24no—
545C4—24no—
546C4—24no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—44no—
552C4—42no—
553C4—84no—
554C4—84no—
555C4—82no—
556C4—82no—
557C4#347514yes—
558C2—42no—
559C2—84no—
560C2#3494611yescharacteristic
561C2#6056712yes—
562C2#6062512yes—
563C2#6101711yescharacteristic
564C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35283C2semidirect
3. #29148C2semidirect
4. #29137C2semidirect
5. #29118C2semidirect
7. #28727C2semidirect
8. #27968C2semidirect
10. #23827C2semidirect
11. #21655C2semidirect
12. C4×#489C2semidirect
13. #20745C2semidirect
15. #20465C2semidirect
22. #2868C22semidirect
23. #2862C22semidirect
25. #2647C22semidirect
35. #2331C22semidirect
41. C4×C4⋊Q8C22semidirect
46. #2100C22semidirect
47. #2099C22semidirect
48. #2074C22semidirect
52. #2024C22semidirect
54. #1993C22semidirect
55. #1993C22semidirect
59. #1935C22semidirect
61. #1933C22semidirect
62. #1836C22semidirect
63. #1836C22semidirect
66. #1805C22semidirect
68. #1803C22semidirect
72. #1768C22semidirect
73. #1767C22semidirect
74. #1763C22semidirect
76. C4×#103C22semidirect
80. #1746C22semidirect
81. C4×#97C22semidirect
82. C4×#97C22semidirect

Maximal subgroups

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

subgroupindex
#352832
#342112
#291482
#291372
#291182
#291162
#287272
#279682
#258642
#238272
#216552
C4×#4892
#207452
#207442
#204652