Finite Group Databasebeta

← #6510790

Subgroups — #6510790

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 598 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×#2502C211yesmaximal, characteristic, complemented
5#28362C214yesmaximal, complemented
6#28360C214yesmaximal, complemented
7#28315C212yesmaximal, complemented
8#28305C211yesmaximal, characteristic, complemented
9#28304C211yesmaximal, characteristic, complemented
10#27945C214yesmaximal, complemented
11#27940C214yesmaximal, complemented
12#27913C212yesmaximal, complemented
13#27910C211yesmaximal, characteristic, complemented
14#27906C211yesmaximal, characteristic, complemented
15#20524C211yesmaximal, characteristic, complemented
16C2×#525C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3196—216no—
20#3092—216no—
21C2×#484C2212yes—
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#481C2212yes—
25C2×#481C2212yes—
26C2×#481C2212yes—
27C2×#485C2214yes—
28C2×#483C2214yes—
29C2×#480C2214yes—
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#2502—24no—
44#2502C2214yes—
45#2502C2214yes—
46#2500C2212yes—
47#2499C2212yes—
48#2432C2212yes—
49#2431C2212yes—
50C2×#398—22no—
51#2325—22no—
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#385—22no—
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58#2097C2214yescomplemented
59#2096C2214yescomplemented
60#2096C2212yescomplemented
61#2095C2214yescomplemented
62#2095C2212yescomplemented
63#2094C2214yescomplemented
64#2093C2214yescomplemented
65#2016C2212yescomplemented
66#2015C2214yescomplemented
67#2014C2212yescomplemented
68#1973C2214yescomplemented
69#1971C2214yescomplemented
70#1969C2214yescomplemented
71#1968C2214yescomplemented
72#1959C2214yescomplemented
73#1957C2212yescomplemented
74#1956C2212yescomplemented
75#1847C2214yescomplemented
76#1845C2212yescomplemented
77#1844C2212yescomplemented
78#1753C2214yes—
79#1753C2214yes—
80#1749C2212yes—
81C4×C22⋊C8C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
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⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4D412yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#485—216no—
123#484—28no—
124#484—28no—
125#484—28no—
126#483—216no—
127#481—28no—
128#481—28no—
129#481—28no—
130#480—216no—
131#470—216no—
132#468—216no—
133#452—216no—
134#451—28no—
135#448—28no—
136#428—216no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143#417—216no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#407D4116yescomplemented
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#398—28no—
160#398—28no—
161#397—24no—
162#394—24no—
163#394—24no—
164#392—24no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2314yes—
172#390—22no—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#388—24no—
177#388—24no—
178#387—24no—
179#386—24no—
180#386—24no—
181#385—28no—
182#385—28no—
183#385—24no—
184#384—22no—
185#384—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2314yes—
189#384C2312yes—
190#384C2312yes—
191#382C2312yes—
192#380—22no—
193C4×C22⋊C4D414yes—
194C4×C22⋊C4C2312yes—
195C4×C22⋊C4C2312yes—
196C2×#95—22no—
197C2×#95—22no—
198C2×#95D412yes—
199#336C2314yes—
200#336C2312yes—
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—44no—
227C23×C4—84no—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D418yes—
232C23×C4C2×D412yes—
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—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
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—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—84no—
289C2×C4⋊C4—84no—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—22no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#107—28no—
317#107—28no—
318#107—28no—
319#106—28no—
320#106—28no—
321#106—28no—
322#103—28no—
323#103—28no—
324#103—28no—
325#103—28no—
326#103—28no—
327#102—28no—
328#102—28no—
329#102—28no—
330C22⋊C8C2×D418yes—
331C22⋊C8C2×D418yes—
332C22⋊C8C2×D4116yes—
333#97—44no—
334C4×C8—42no—
335C4×C8—42no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95C2×D414yes—
343#95C2×D414yes—
344C24—24no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—42no—
349C24#12714yes—
350C24C22×D412yes—
351C24C22×D411yescharacteristic
352#41—416no—
353#41—88no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—88no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—84no—
378C2×D4—82no—
379C2×D4—82no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—416no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—88no—
415C22×C4—816no—
416C22×C4#12714yes—
417C22×C4#12718yes—
418C22×C4#13718yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C2511yescharacteristic
428Q16—88no—
429SD16—88no—
430M16—416no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8C22×D418yes—
436C2×C8C22×D418yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—82no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C42—22no—
469C42—22no—
470C42—24no—
471C42—42no—
472C42—42no—
473C42—44no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—24no—
478C23—28no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—44no—
483C23—44no—
484C23—48no—
485C23—48no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23C2×#12714yes—
492C23C2×#12712yes—
493C23C2×#13714yes—
494C23C23×D411yescharacteristic
495Q8—44no—
496Q8—44no—
497Q8—44no—
498Q8—44no—
499Q8—84no—
500D4—44no—
501D4—44no—
502D4—88no—
503D4—88no—
504D4—88no—
505D4—164no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—88no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—816no—
546C2×C4C2×#12714yes—
547C2×C4C2×#12712yes—
548C2×C4C2×#13714yes—
549C2×C4C2×#13714yes—
550C2×C4C23×D412yes—
551C2×C4#58512yes—
552C2×C4#58512yes—
553C8—28no—
554C8—28no—
555C22—24no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—416no—
563C22—44no—
564C22—44no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22#315214yes—
571C22C22×#12711yescharacteristic
572C22#348511yescharacteristic
573C22#348511yescharacteristic
574C22C22×#13712yes—
575C22#358512yes—
576C4—24no—
577C4—24no—
578C4—24no—
579C4—22no—
580C4—44no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4#348514yes—
589C2—24no—
590C2—44no—
591C2—162no—
592C2#3546311yescharacteristic
593C2C2×#315211yescharacteristic
594C2#6064411yescharacteristic
595C2#6170912yes—
596C2#6171912yes—
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×#2502C2semidirect
5. #28362C2semidirect
6. #28360C2semidirect
7. #28315C2semidirect
8. #28305C2semidirect
9. #28304C2semidirect
10. #27945C2semidirect
11. #27940C2semidirect
12. #27913C2semidirect
13. #27910C2semidirect
14. #27906C2semidirect
15. #20524C2semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2096C22semidirect
61. #2095C22semidirect
62. #2095C22semidirect
63. #2094C22semidirect
64. #2093C22semidirect
65. #2016C22semidirect
66. #2015C22semidirect
67. #2014C22semidirect
68. #1973C22semidirect
69. #1971C22semidirect
70. #1969C22semidirect
71. #1968C22semidirect
72. #1959C22semidirect
73. #1957C22semidirect
74. #1956C22semidirect
75. #1847C22semidirect
76. #1845C22semidirect
77. #1844C22semidirect
152. #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×#25022
#283622
#283602
#283152
#283052
#283042
#279452
#279402
#279132
#279102
#279062
#205242