Finite Group Databasebeta

← #6479679

Subgroups — #6479679

Order 512.

1886 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35321C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#32966C212yesmaximal, complemented
4#32914C212yesmaximal, complemented
5#29911C212yesmaximal, complemented
6#29625C212yesmaximal
7#29498C212yesmaximal, complemented
8#29492C212yesmaximal, complemented
9#29377C214yesmaximal, complemented
10#29136C212yesmaximal, complemented
11#28388C212yesmaximal, complemented
12#28207C212yesmaximal, complemented
13#28157C212yesmaximal, complemented
14#22052C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#518C2212yes—
18C2×#478C2214yescomplemented
19C2×#461C2214yescomplemented
20C2×#450C2214yes—
21#2863C2212yescomplemented
22#2855C2212yes—
23#2647C2211yescharacteristic, complemented
24#2642C2211yescharacteristic, complemented
25#2629C2211yescharacteristic
26#2621C2211yescharacteristic
27#2585—22no—
28#2573—22no—
29#2524—22no—
30#2515—22no—
31#2366—22no—
32#2366—22no—
33D4×C22⋊C4—22no—
34D4×C22⋊C4—22no—
35#2328C2212yes—
36#2328C2212yes—
37#2323C2212yes—
38#2320C2212yescomplemented
39#2309C2212yes—
40C4×C4⋊Q8C2211yescharacteristic, complemented
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×#125C2211yescharacteristic, complemented
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yescomplemented
47#2097C2214yescomplemented
48#2084C2212yescomplemented
49#2083C2212yescomplemented
50#2072C2212yescomplemented
51#2072C2212yescomplemented
52#2041C2212yes—
53#2041C2212yes—
54#2041C2214yes—
55#2014C2212yes—
56#2014C2212yes—
57#2014C2214yes—
58#1973C2214yescomplemented
59#1968C2214yescomplemented
60#1936C2214yescomplemented
61#1934C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1880C2214yescomplemented
65#1880C2214yescomplemented
66#1877C2214yescomplemented
67#1877C2214yescomplemented
68#1873C2214yes—
69#1864C2214yescomplemented
70#1862C2214yescomplemented
71#1854C2214yes—
72#1840C2212yescomplemented
73#1831C2212yescomplemented
74#1817C2212yes—
75#1804C2214yescomplemented
76#1802C2214yescomplemented
77#1798C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127—28no—
89C2×#127—22no—
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#125C2311yescharacteristic
93C2×C22⋊Q8C2312yes—
94#518—48no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4C2314yes—
102C2×C4×D4C2314yes—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C22⋊C4—24no—
107C22×C42C2311yescharacteristic
108#478—48no—
109#461—48no—
110#450—216no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125#400—22no—
126#400—22no—
127#400—22no—
128#398—22no—
129#398—22no—
130#398—22no—
131#397—22no—
132#396—22no—
133#391C2312yes—
134#391C2312yes—
135#391C2312yes—
136#391C2312yes—
137#390—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#388—22no—
152#388—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#386—24no—
163#386—24no—
164#386—24no—
165#386—24no—
166#385—22no—
167#384C2312yes—
168#384C2311yescharacteristic
169#383—22no—
170#383—22no—
171#382—22no—
172#382C2311yescharacteristic
173#382C2312yes—
174#381—22no—
175#380—22no—
176#380C2312yes—
177#379—22no—
178#379—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C25—22no—
202C22×Q8C2412yes—
203C22×D4—216no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—84no—
212C22×D4—84no—
213C22×D4#4118yes—
214C22×D4C2414yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4C2×D412yes—
218C23×C4C2411yescharacteristic
219C2×SD16—28no—
220C2×SD16—28no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—24no—
228#127—216no—
229#127—28no—
230#127—28no—
231#125—24no—
232C22⋊Q8—28no—
233C4×Q8—44no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—44no—
237C4×D4—44no—
238C4×D4—416no—
239C4×D4—84no—
240C4×D4—84no—
241C4×D4—84no—
242C4×D4—84no—
243#117—24no—
244#117—42no—
245#117—44no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
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—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—416no—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#107—44no—
293#106—44no—
294C4⋊C8—28no—
295#103—44no—
296#103—44no—
297#102—28no—
298#102—28no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303C22⋊C8—28no—
304#97—42no—
305C4×C8—42no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326C24—24no—
327C24—28no—
328C24—44no—
329C24—44no—
330C24—84no—
331C24—84no—
332C24C22×D411yescharacteristic
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—216no—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—416no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—416no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—832no—
357C2×D4C2×#4118yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C2×#4114yes—
384C22×C4C2511yescharacteristic
385SD16—88no—
386SD16—88no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C22⋊C4—24no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—48no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—82no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—42no—
434C42—42no—
435C42—44no—
436C42—44no—
437C42—82no—
438C42H(5,𝔽2)14yes—
439C23—22no—
440C23—21no—
441C23—21no—
442C23—22no—
443C23—28no—
444C23—416no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—816no—
452C23—164no—
453C23—164no—
454C23—168no—
455C23#52512yes—
456C23C23×D411yescharacteristic
457Q8—44no—
458Q8—44no—
459D4—28no—
460D4—28no—
461D4—416no—
462D4—416no—
463D4—416no—
464D4—44no—
465D4—44no—
466D4—816no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—168no—
505C2×C4—168no—
506C2×C4#52512yes—
507C2×C4#53514yes—
508C2×C4#57714yes—
509C2×C4C22×#4112yes—
510C2×C4C2×H(5,𝔽2)12yes—
511C2×C4#58512yes—
512C8—44no—
513C8—44no—
514C22—22no—
515C22—24no—
516C22—41no—
517C22—41no—
518C22—42no—
519C22—48no—
520C22—48no—
521C22—48no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—1616no—
527C22C2×#52511yescharacteristic
528C22C2×#53511yescharacteristic
529C22#344911yescharacteristic
530C22C2×#57712yes—
531C22#358512yes—
532C4—22no—
533C4—24no—
534C4—24no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—42no—
540C4—42no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—82no—
546C4#350514yes—
547C2—42no—
548C2—88no—
549C2#3510211yescharacteristic
550C2#6057912yes—
551C2#6083312yes—
552C2#6101211yescharacteristic
553C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #35321C2semidirect
3. #32966C2semidirect
4. #32914C2semidirect
5. #29911C2semidirect
7. #29498C2semidirect
8. #29492C2semidirect
9. #29377C2semidirect
10. #29136C2semidirect
11. #28388C2semidirect
12. #28207C2semidirect
13. #28157C2semidirect
14. #22052C2semidirect
15. #20780C2semidirect
18. C2×#478C22semidirect
19. C2×#461C22semidirect
21. #2863C22semidirect
23. #2647C22semidirect
24. #2642C22semidirect
38. #2320C22semidirect
40. C4×C4⋊Q8C22semidirect
43. C4×#125C22semidirect
46. #2101C22semidirect
47. #2097C22semidirect
48. #2084C22semidirect
49. #2083C22semidirect
50. #2072C22semidirect
51. #2072C22semidirect
58. #1973C22semidirect
59. #1968C22semidirect
60. #1936C22semidirect
61. #1934C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1880C22semidirect
65. #1880C22semidirect
66. #1877C22semidirect
67. #1877C22semidirect
69. #1864C22semidirect
70. #1862C22semidirect
72. #1840C22semidirect
73. #1831C22semidirect
75. #1804C22semidirect
76. #1802C22semidirect
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
#353212
#342102
#329662
#329142
#299112
#296252
#294982
#294922
#293772
#291362
#283882
#282072
#281572
#220522
#207802