Finite Group Databasebeta

← #6495669

Subgroups — #6495669

Order 512.

1684 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#33117C212yesmaximal, complemented
4#32914C212yesmaximal
5#29982C212yesmaximal, complemented
6#29798C212yesmaximal, complemented
7#29560C212yesmaximal, complemented
8#29554C212yesmaximal, complemented
9#29378C214yesmaximal, complemented
10#29344C212yesmaximal, complemented
11#28320C212yesmaximal
12#24065C212yesmaximal, complemented
13#23933C212yesmaximal
14#22162C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#518C2211yescharacteristic
18C2×#485C2214yes—
19C2×#461C2214yes—
20C2×#450C2214yes—
21#2868C2211yescharacteristic, complemented
22#2862C2212yescomplemented
23#2862C2211yescharacteristic, complemented
24#2621C2212yes—
25#2621C2212yes—
26#2565—22no—
27#2565—22no—
28#2499—22no—
29#2499—22no—
30#2357—22no—
31#2354—22no—
32#2342—22no—
33#2342—22no—
34#2327C2212yescomplemented
35#2327C2212yescomplemented
36#2319C2212yes—
37#2319C2212yes—
38#2317C2211yescharacteristic
39#2309C2211yescharacteristic
40#2296C2211yescharacteristic
41#2293C2212yes—
42#2293C2212yes—
43C2×#390C2211yescharacteristic
44#2274C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yescomplemented
47#2092C2214yes—
48#2058C2212yescomplemented
49#2057C2212yescomplemented
50#2043C2212yescomplemented
51#2043C2212yescomplemented
52#2043C2214yescomplemented
53#2023C2212yes—
54#2023C2212yes—
55#2013C2212yes—
56#2013C2212yes—
57#2013C2214yes—
58#1981C2214yescomplemented
59#1973C2214yes—
60#1968C2214yes—
61#1949C2214yescomplemented
62#1935C2214yescomplemented
63#1933C2214yescomplemented
64#1928C2214yescomplemented
65#1927C2214yescomplemented
66#1880C2214yes—
67#1880C2214yes—
68#1877C2214yes—
69#1877C2214yes—
70#1873C2214yes—
71#1872C2214yes—
72#1865C2214yescomplemented
73#1863C2214yescomplemented
74#1840C2212yescomplemented
75#1831C2212yescomplemented
76#1817C2212yes—
77#1804C2214yes—
78#1798C2212yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86#533—48no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#125—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8C2314yes—
94#518—28no—
95C2×C4×Q8—22no—
96C2×C4×Q8C2314yes—
97C2×C4×D4—22no—
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C42C2311yescharacteristic
111#485—48no—
112#461—216no—
113#450—48no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#398—24no—
129#398—22no—
130#398—22no—
131#397—24no—
132#397—22no—
133#397—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391C2314yes—
139#391C2314yes—
140#390—21no—
141#390—21no—
142#390—22no—
143#390C2312yes—
144#390C2312yes—
145#390C2312yes—
146#390C2312yes—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#387—24no—
160#387—24no—
161#387—24no—
162#387—24no—
163#385—22no—
164#385—21no—
165#385—21no—
166#384—21no—
167#384—22no—
168#384—22no—
169#382—21no—
170#382—22no—
171#382—22no—
172#382C2312yes—
173#382C2312yes—
174#381—22no—
175#381—22no—
176#380—22no—
177#380—22no—
178#379—21no—
179#379—21no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
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#336C2312yes—
202#336C2312yes—
203#336C2314yes—
204#336C2314yes—
205C25—21no—
206C22×Q8—22no—
207C22×Q8C2414yes—
208C22×D4—24no—
209C22×D4—21no—
210C22×D4—24no—
211C22×D4—42no—
212C22×D4C2412yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218C2×SD16—28no—
219C2×SD16—28no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C22×C8C2414yes—
225C22×C8C2414yes—
226C4⋊Q8—28no—
227C4⋊Q8—44no—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—44no—
231#125—48no—
232C22⋊Q8—28no—
233C22⋊Q8—28no—
234C22⋊Q8—48no—
235C4×Q8—44no—
236C4×Q8—44no—
237C4×Q8—48no—
238C4×D4—216no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4#4118yes—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—42no—
247#117—48no—
248#117—84no—
249#117—84no—
250C2×C4⋊C4—21no—
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—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—84no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C22⋊C4C2×D411yescharacteristic
289C2×C42—21no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#107—44no—
307#106—44no—
308C4⋊C8—28no—
309#103—28no—
310#103—28no—
311#103—44no—
312#103—44no—
313#103—44no—
314#103—44no—
315#102—44no—
316#102—44no—
317C22⋊C8—28no—
318#97—42no—
319C4×C8—42no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337C24—24no—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—82no—
342C24—82no—
343C24C22×D411yescharacteristic
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—416no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4C2×#4114yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4H(5,𝔽2)14yes—
389C22×C4C2511yescharacteristic
390SD16—88no—
391SD16—88no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4C2×#4114yes—
420C22⋊C4—24no—
421C22⋊C4—28no—
422C22⋊C4—24no—
423C22⋊C4—28no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C42—24no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—42no—
446C42—42no—
447C42—48no—
448C42—42no—
449C42—82no—
450C42—82no—
451C42C2×#4114yes—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—24no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—88no—
465C23—164no—
466C23—162no—
467C23—162no—
468C23#53512yes—
469C23C23×D411yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—88no—
475D4—28no—
476D4—48no—
477D4—44no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—88no—
516C2×C4—88no—
517C2×C4—162no—
518C2×C4#52514yes—
519C2×C4#53512yes—
520C2×C4#57714yes—
521C2×C4C22×#4112yes—
522C2×C4C2×H(5,𝔽2)12yes—
523C2×C4#58512yes—
524C8—44no—
525C8—44no—
526C22—24no—
527C22—22no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—168no—
539C22C2×#52511yescharacteristic
540C22C2×#53511yescharacteristic
541C22#344911yescharacteristic
542C22C2×#57712yes—
543C22#358612yes—
544C4—24no—
545C4—24no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4—84no—
558C4#349414yes—
559C2—42no—
560C2—84no—
561C2#3530111yescharacteristic
562C2#6059112yes—
563C2#6081312yes—
564C2#6101211yescharacteristic
565C2#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. #35322C2semidirect
3. #33117C2semidirect
5. #29982C2semidirect
6. #29798C2semidirect
7. #29560C2semidirect
8. #29554C2semidirect
9. #29378C2semidirect
10. #29344C2semidirect
12. #24065C2semidirect
14. #22162C2semidirect
15. #20780C2semidirect
21. #2868C22semidirect
22. #2862C22semidirect
23. #2862C22semidirect
34. #2327C22semidirect
35. #2327C22semidirect
46. #2101C22semidirect
48. #2058C22semidirect
49. #2057C22semidirect
50. #2043C22semidirect
51. #2043C22semidirect
52. #2043C22semidirect
58. #1981C22semidirect
61. #1949C22semidirect
62. #1935C22semidirect
63. #1933C22semidirect
64. #1928C22semidirect
65. #1927C22semidirect
72. #1865C22semidirect
73. #1863C22semidirect
74. #1840C22semidirect
75. #1831C22semidirect
79. #1768C22semidirect
81. #1745C22semidirect
82. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353222
#342552
#331172
#329142
#299822
#297982
#295602
#295542
#293782
#293442
#283202
#240652
#239332
#221622
#207802