Finite Group Databasebeta

← #6489381

Subgroups — #6489381

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#32967C212yesmaximal, complemented
4#32966C212yesmaximal
5#30088C214yesmaximal, complemented
6#28726C212yesmaximal, complemented
7#28517C212yesmaximal, complemented
8#28215C214yesmaximal, complemented
9#27955C212yesmaximal, complemented
10#27592C212yesmaximal, complemented
11#27591C212yesmaximal
12#25864C212yesmaximal
13#22224C212yesmaximal, complemented
14#22223C212yesmaximal, complemented
15#20783C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C2×#519C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#489C2212yes—
20C2×#489C2212yes—
21C2×#479C2214yescomplemented
22C2×#478C2214yes—
23#2868C2211yescharacteristic, complemented
24#2859C2211yescharacteristic, complemented
25#2857C2212yes—
26#2647C2212yescomplemented
27#2629C2212yes—
28#2507—22no—
29#2497—22no—
30#2446—22no—
31#2429—22no—
32#2367—22no—
33#2367—22no—
34#2342—22no—
35#2342—22no—
36#2330C2212yescomplemented
37#2326C2212yes—
38#2326C2212yes—
39#2314C2212yes—
40#2310C2211yescharacteristic
41#2309C2211yescharacteristic
42C4×C4⋊Q8C2212yescomplemented
43C4×#124C2212yes—
44C4×#124C2212yes—
45#2285C2211yescharacteristic
46C4×#117C2211yescharacteristic
47#2097C2214yescomplemented
48#2092C2214yes—
49#2084C2214yescomplemented
50#2083C2214yescomplemented
51#2072C2214yescomplemented
52#2071C2214yes—
53#2042C2214yes—
54#2042C2214yes—
55#1993C2212yescomplemented
56#1993C2212yescomplemented
57#1959C2212yes—
58#1959C2212yes—
59#1928C2214yescomplemented
60#1927C2214yescomplemented
61#1926C2214yes—
62#1925C2214yes—
63#1865C2214yescomplemented
64#1864C2214yes—
65#1863C2214yescomplemented
66#1862C2214yes—
67#1854C2214yes—
68#1854C2214yes—
69#1847C2212yescomplemented
70#1847C2212yescomplemented
71#1840C2212yescomplemented
72#1832C2212yescomplemented
73#1818C2212yes—
74#1803C2214yescomplemented
75#1802C2214yes—
76#1798C2212yes—
77#1787C2214yes—
78#1786C2214yes—
79#1767C2212yescomplemented
80#1749C2212yes—
81#1745C2211yescharacteristic, complemented
82#1745C2211yescharacteristic, complemented
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×#125C2312yes—
88C2×#124—28no—
89C2×#124—21no—
90C2×#124—21no—
91C2×#124C2312yes—
92C2×#124C2312yes—
93C2×C22⋊Q8—22no—
94#519—44no—
95#518—44no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×D4—22no—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×#117—22no—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—22no—
111C22×C42C2311yescharacteristic
112#489—28no—
113#489—28no—
114#479—48no—
115#478—48no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2312yes—
131C2×#97C2312yes—
132#400—24no—
133#398—24no—
134#398—24no—
135#396—24no—
136#391—22no—
137#391—22no—
138#391—22no—
139#391C2312yes—
140#391C2312yes—
141#390—22no—
142#390C2312yes—
143#390C2312yes—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#389C2314yes—
151#388—24no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—24no—
161#387—24no—
162#385—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#384C2312yes—
171#382C2312yes—
172#382C2312yes—
173#381—22no—
174#380—21no—
175#380—21no—
176#380—22no—
177#380C2314yes—
178#379—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2314yes—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C25—21no—
206C22×Q8—24no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8C2412yes—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4—42no—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C22×C8C2414yes—
224C22×C8C2414yes—
225C4⋊Q8—24no—
226C4⋊Q8—24no—
227C4⋊Q8—84no—
228#125—28no—
229#124—28no—
230#124—28no—
231#124—24no—
232#124—24no—
233#124—24no—
234#124#4118yes—
235C22⋊Q8—84no—
236C4×Q8—28no—
237C4×Q8—44no—
238C4×Q8—48no—
239C4×Q8—84no—
240C4×D4—28no—
241C4×D4—44no—
242C4×D4—48no—
243#117—24no—
244#117—24no—
245#117—48no—
246#117—42no—
247#117—42no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2414yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
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×D412yes—
283C2×C22⋊C4C2×D412yes—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—24no—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—44no—
303#107—44no—
304#106—44no—
305#106—44no—
306C4⋊C8—28no—
307#103—28no—
308#103—28no—
309#103—44no—
310#103—44no—
311#102—28no—
312#102—28no—
313#102—44no—
314#102—44no—
315C22⋊C8—28no—
316#97—24no—
317#97—24no—
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—24no—
329#95C2×D412yes—
330#95C2×D412yes—
331#95C2×D412yes—
332#95C2×D412yes—
333C24—24no—
334C24—22no—
335C24—42no—
336C24—42no—
337C24—82no—
338C24—82no—
339C24C22×D411yescharacteristic
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—416no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—42no—
350C2×Q8C2×#4114yes—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—416no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4C2×#4114yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C42−1+414yes—
389C22×C4C2511yescharacteristic
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C22⋊C4—84no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—44no—
444C42—44no—
445C42—84no—
446C42C2×#4114yes—
447C42H(5,𝔽2)14yes—
448C23—22no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—24no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—84no—
461C23—84no—
462C23—164no—
463C23—164no—
464C23#53612yes—
465C23C23×D411yescharacteristic
466Q8—28no—
467Q8—48no—
468Q8—44no—
469Q8—88no—
470D4—28no—
471D4—48no—
472D4—44no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
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—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—164no—
512C2×C4—164no—
513C2×C4#52512yes—
514C2×C4#52512yes—
515C2×C4#53514yes—
516C2×C4#53612yes—
517C2×C4C22×#4112yes—
518C2×C4C2×H(5,𝔽2)12yes—
519C2×C4C2×2−1+412yes—
520C8—44no—
521C8—44no—
522C22—22no—
523C22—24no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—168no—
535C22C2×#52511yescharacteristic
536C22C2×#53511yescharacteristic
537C22C2×#53611yescharacteristic
538C22#358512yes—
539C22#358612yes—
540C4—22no—
541C4—22no—
542C4—24no—
543C4—22no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—44no—
548C4—42no—
549C4—42no—
550C4—84no—
551C4—84no—
552C4—82no—
553C4—82no—
554C4#352014yes—
555C2—42no—
556C2—84no—
557C2#3524011yescharacteristic
558C2#6056811yescharacteristic
559C2#6056911yescharacteristic
560C2#6082912yes—
561C2#6088512yes—
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. #35284C2semidirect
3. #32967C2semidirect
5. #30088C2semidirect
6. #28726C2semidirect
7. #28517C2semidirect
8. #28215C2semidirect
9. #27955C2semidirect
10. #27592C2semidirect
13. #22224C2semidirect
14. #22223C2semidirect
15. #20783C2semidirect
21. C2×#479C22semidirect
23. #2868C22semidirect
24. #2859C22semidirect
26. #2647C22semidirect
36. #2330C22semidirect
42. C4×C4⋊Q8C22semidirect
47. #2097C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
51. #2072C22semidirect
55. #1993C22semidirect
56. #1993C22semidirect
59. #1928C22semidirect
60. #1927C22semidirect
63. #1865C22semidirect
65. #1863C22semidirect
69. #1847C22semidirect
70. #1847C22semidirect
71. #1840C22semidirect
72. #1832C22semidirect
74. #1803C22semidirect
79. #1767C22semidirect
81. #1745C22semidirect
82. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#352842
#342112
#329672
#329662
#300882
#287262
#285172
#282152
#279552
#275922
#275912
#258642
#222242
#222232
#207832