Finite Group Databasebeta

← #6510791

Subgroups — #6510791

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3196C212yesmaximal, complemented
2C2×#3092C212yesmaximal, complemented
3#35032C211yesmaximal, characteristic
4C2×#2503C211yesmaximal, characteristic, complemented
5#28366C214yesmaximal, complemented
6#28356C214yesmaximal, complemented
7#28314C211yesmaximal, characteristic, complemented
8#28313C211yesmaximal, characteristic, complemented
9#28306C212yesmaximal, complemented
10#27946C214yesmaximal, complemented
11#27943C214yesmaximal, complemented
12#27919C211yesmaximal, characteristic, complemented
13#27918C211yesmaximal, characteristic, complemented
14#27912C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C2×#525C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3196—216no—
20#3092—216no—
21C2×#484C2212yes—
22C2×#484C2214yes—
23C2×#481C2212yes—
24C2×#481C2214yes—
25C2×#485C2214yes—
26C2×#483C2212yes—
27C2×#483C2212yes—
28C2×#480C2212yes—
29C2×#480C2212yes—
30C2×#470C2214yes—
31C2×#468C2214yes—
32C2×#452C2214yes—
33C2×#451C2212yes—
34C2×#448C2212yes—
35C2×#428C2214yes—
36C2×#417C2214yes—
37C2×#407C2212yes—
38#2857C2211yescharacteristic
39#2856C2211yescharacteristic
40#2850C2212yes—
41#2617—22no—
42#2602—24no—
43#2503—24no—
44#2503C2214yes—
45#2503C2214yes—
46#2500C2212yes—
47#2499C2212yes—
48#2432C2212yes—
49#2431C2212yes—
50C2×#400—22no—
51#2325—22no—
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#381—22no—
57C2×C4×C22⋊C4C2211yescharacteristic
58#2097C2214yescomplemented
59#2097C2214yescomplemented
60#2097C2214yescomplemented
61#2096C2212yescomplemented
62#2095C2212yescomplemented
63#2092C2214yescomplemented
64#2092C2214yescomplemented
65#2016C2212yescomplemented
66#2015C2214yescomplemented
67#2014C2212yescomplemented
68#1973C2214yescomplemented
69#1971C2214yescomplemented
70#1969C2214yescomplemented
71#1968C2214yescomplemented
72#1961C2212yescomplemented
73#1960C2212yescomplemented
74#1958C2214yescomplemented
75#1847C2212yescomplemented
76#1847C2212yescomplemented
77#1846C2214yescomplemented
78#1753C2214yes—
79#1753C2214yes—
80#1749C2211yescharacteristic
81#1749C2211yescharacteristic
82C4×C22⋊C8C2212yes—
83C22×#41C2312yes—
84C23×Q8C2311yescharacteristic
85C24×C4—21no—
86C24×C4C2311yescharacteristic
87C22×Q16C2314yes—
88C22×SD16C2314yes—
89C22×M16C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#124C2312yes—
93#525—28no—
94#525—28no—
95C2×#123—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102C2×#121C2312yes—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×#117C2312yes—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D412yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#485—216no—
121#484—216no—
122#484—28no—
123#483—28no—
124#483—28no—
125#481—216no—
126#481—28no—
127#480—28no—
128#480—28no—
129#470—216no—
130#468—216no—
131#452—216no—
132#451—28no—
133#448—28no—
134#428—216no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141#417—216no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#407D4116yescomplemented
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×C4×C8C2314yes—
157#400—28no—
158#400—28no—
159#397—24no—
160#394—24no—
161#394—24no—
162#392—24no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2314yes—
170#390—22no—
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#388—24no—
175#388—24no—
176#387—24no—
177#386—24no—
178#386—24no—
179#385—24no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#382—22no—
183#382—22no—
184#382C2312yes—
185#382C2312yes—
186#382C2314yes—
187#382C2312yes—
188#381—28no—
189#381—28no—
190#380—22no—
191C4×C22⋊C4D414yes—
192C4×C22⋊C4C2312yes—
193C4×C22⋊C4C2312yes—
194C2×#95—22no—
195C2×#95—22no—
196C2×#95—22no—
197C2×#95—22no—
198C2×#95D412yes—
199#336C2312yes—
200#336C2314yes—
201#336C2312yes—
202C25C2×D411yescharacteristic
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—44no—
206C2×#41—44no—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8C2412yes—
212C22×Q8C2412yes—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4C2412yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—48no—
227C23×C4—44no—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D412yes—
232C23×C4C2×D418yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C2×Q16—28no—
236C2×Q16—44no—
237C2×Q16—44no—
238C2×SD16—28no—
239C2×SD16—44no—
240C2×SD16—44no—
241C2×M16—44no—
242C2×M16—44no—
243C2×M16C2×D4116yes—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—42no—
247#127—42no—
248#124—44no—
249#123—48no—
250C22⋊Q8—28no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—48no—
257#121—24no—
258#121—24no—
259#121—24no—
260#121—24no—
261#121—48no—
262#120—48no—
263C4×D4—48no—
264C4×D4—48no—
265#117—24no—
266#117—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2412yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—28no—
312#107—28no—
313#107—28no—
314#106—28no—
315#106—28no—
316#106—28no—
317#103—28no—
318#103—28no—
319#103—28no—
320#103—28no—
321#103—28no—
322#102—28no—
323#102—28no—
324#102—28no—
325C22⋊C8C2×D418yes—
326C22⋊C8C2×D418yes—
327C22⋊C8C2×D4116yes—
328#97—42no—
329#97—42no—
330C4×C8—44no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95C2×D414yes—
341#95C2×D414yes—
342C24—24no—
343C24—22no—
344C24—41no—
345C24—41no—
346C24—42no—
347C24#12714yes—
348C24C22×D412yes—
349C24C22×D411yescharacteristic
350#41—416no—
351#41—88no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—88no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—84no—
376C2×D4—82no—
377C2×D4—82no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—416no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—416no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—816no—
415C22×C4#12714yes—
416C22×C4#12718yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C2511yescharacteristic
426Q16—88no—
427SD16—88no—
428M16—416no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8C22×D418yes—
434C2×C8C22×D418yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—42no—
467C42—42no—
468C42—44no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—28no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—44no—
478C23—44no—
479C23—48no—
480C23—48no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23C2×#12714yes—
487C23C2×#12712yes—
488C23#57514yes—
489C23C23×D411yescharacteristic
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—84no—
495D4—44no—
496D4—44no—
497D4—88no—
498D4—88no—
499D4—88no—
500D4—164no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—88no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—816no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4C2×#12714yes—
544C2×C4C2×#12712yes—
545C2×C4C23×D412yes—
546C2×C4#58512yes—
547C2×C4#58512yes—
548C8—28no—
549C8—28no—
550C22—24no—
551C22—24no—
552C22—22no—
553C22—22no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—416no—
558C22—44no—
559C22—44no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22#315314yes—
566C22C22×#12711yescharacteristic
567C22#348511yescharacteristic
568C22#348511yescharacteristic
569C22C2×#57512yes—
570C22#358612yes—
571C4—24no—
572C4—22no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4#348514yes—
584C2—24no—
585C2—44no—
586C2—162no—
587C2#3546311yescharacteristic
588C2C2×#315311yescharacteristic
589C2#6064511yescharacteristic
590C2#6171112yes—
591C2#6171612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3196C2semidirect
2. C2×#3092C2semidirect
4. C2×#2503C2semidirect
5. #28366C2semidirect
6. #28356C2semidirect
7. #28314C2semidirect
8. #28313C2semidirect
9. #28306C2semidirect
10. #27946C2semidirect
11. #27943C2semidirect
12. #27919C2semidirect
13. #27918C2semidirect
14. #27912C2semidirect
15. #20524C2semidirect
58. #2097C22semidirect
59. #2097C22semidirect
60. #2097C22semidirect
61. #2096C22semidirect
62. #2095C22semidirect
63. #2092C22semidirect
64. #2092C22semidirect
65. #2016C22semidirect
66. #2015C22semidirect
67. #2014C22semidirect
68. #1973C22semidirect
69. #1971C22semidirect
70. #1969C22semidirect
71. #1968C22semidirect
72. #1961C22semidirect
73. #1960C22semidirect
74. #1958C22semidirect
75. #1847C22semidirect
76. #1847C22semidirect
77. #1846C22semidirect
150. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31962
C2×#30922
#350322
C2×#25032
#283662
#283562
#283142
#283132
#283062
#279462
#279432
#279192
#279182
#279122
#205242