Finite Group Databasebeta

← #6508742

Subgroups — #6508742

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C22×#485C211yesmaximal, characteristic
3#35311C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic
5#28132C214yesmaximal, complemented
6#27657C214yesmaximal, complemented
7#25226C214yesmaximal, complemented
8#25220C214yesmaximal, complemented
9#24580C212yesmaximal, complemented
10#24577C212yesmaximal, complemented
11#24532C212yesmaximal, complemented
12#24529C212yesmaximal, complemented
13#21933C211yesmaximal, characteristic
14#21759C211yesmaximal, characteristic
15#20702C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3195—28no—
20C2×#485—28no—
21C2×#485C2214yes—
22C2×#485C2214yes—
23C2×#483C2212yes—
24C2×#483C2212yes—
25C2×#470C2212yes—
26C2×#470C2212yes—
27C2×#451C2212yes—
28C2×#451C2212yes—
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2855—21no—
36#2855—21no—
37#2625—22no—
38#2503C2212yes—
39#2502C2212yes—
40#2499C2214yes—
41#2484—22no—
42#2483—22no—
43#2378—22no—
44#2360C2214yes—
45#2344C2212yes—
46#2342—22no—
47Q8×C22⋊C4C2212yes—
48#2329—22no—
49#2323C2214yes—
50#2317C2212yes—
51#2316—21no—
52#2313—21no—
53#2310C2212yes—
54#2280C2211yescharacteristic
55C2×#380C2211yescharacteristic
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2074C2214yescomplemented
59#2072C2214yescomplemented
60#2068C2214yescomplemented
61#2066C2214yescomplemented
62#2056C2214yescomplemented
63#2056C2214yescomplemented
64#1949C2214yescomplemented
65#1948C2214yescomplemented
66#1928C2214yescomplemented
67#1927C2214yescomplemented
68#1892C2212yes—
69#1892C2212yes—
70#1878C2214yescomplemented
71#1878C2214yescomplemented
72#1872C2214yes—
73#1851C2212yes—
74#1851C2212yes—
75#1850C2212yes—
76#1850C2212yes—
77#1825C2214yes—
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1805C2214yescomplemented
82#1803C2214yescomplemented
83#1798C2214yes—
84#1787C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×Q16C2312yes—
91C22×Q16C2312yes—
92C22×M16C2311yescharacteristic
93C22×M16C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8D418yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8C2314yes—
109C2×#117—21no—
110C2×#117—21no—
111C2×#117—22no—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129#485—216no—
130#485—216no—
131#483—28no—
132#483—28no—
133#470—28no—
134#470—28no—
135#451—28no—
136#451—28no—
137C2×#107—24no—
138C2×#107C2314yes—
139C2×#106—24no—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155#407—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#400—24no—
162#398—24no—
163#397—24no—
164#397—24no—
165#391—22no—
166#391—22no—
167#391—24no—
168#391C2314yes—
169#391C2314yes—
170#391C2314yes—
171#389—24no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389C2314yes—
176#388—24no—
177#388—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#385—24no—
184#385—24no—
185#385—22no—
186#385—22no—
187#384C2314yes—
188#382C2314yes—
189#381—24no—
190#381—22no—
191#381—22no—
192#381—22no—
193#381D412yes—
194#380—21no—
195#380—21no—
196#380—22no—
197#380D411yescharacteristic
198#380D411yescharacteristic
199#380D412yes—
200#380C2312yes—
201#380C2312yes—
202#379D412yes—
203C4×C4⋊C4C2314yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4C2312yes—
207C4×C22⋊C4C2312yes—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8C2×D414yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4#4114yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—42no—
254C2×M16—42no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—42no—
258C22×C8—42no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8C2×D418yes—
274C22⋊Q8C2×D418yes—
275C4×Q8—44no—
276C4×Q8—44no—
277#117—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—42no—
282#117—42no—
283#117—44no—
284#117—44no—
285#117—84no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42C2414yes—
340#107—28no—
341#106—28no—
342#106—24no—
343#106—24no—
344C4⋊C8—44no—
345C4⋊C8—44no—
346#103—28no—
347#103—28no—
348#103—24no—
349#103—24no—
350#103—24no—
351#103—24no—
352#103—44no—
353#103—44no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D414yes—
373C24—22no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24#12114yes—
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×Q8—84no—
397C2×Q8—84no—
398C2×Q8C22×D414yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4#12114yes—
436C22×C4#12114yes—
437C22×C4#12114yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450Q16—84no—
451Q16—84no—
452M16—82no—
453M16—82no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—44no—
463C2×C8—44no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4C22×D414yes—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—82no—
501C42—24no—
502C42—24no—
503C42—44no—
504C42—44no—
505C42—84no—
506C42—84no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—28no—
518C23—44no—
519C23—44no—
520C23—84no—
521C23—84no—
522C23C2×#12112yes—
523C23C2×#12112yes—
524C23C2×#12112yes—
525C23#57712yes—
526C23#57812yes—
527C23C23×D411yescharacteristic
528Q8—24no—
529Q8—24no—
530Q8—44no—
531Q8—42no—
532Q8—42no—
533Q8—44no—
534Q8—84no—
535Q8—84no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—164no—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4C2×#12112yes—
585C2×C4C23×D411yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4#58512yes—
588C2×C4#58512yes—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—42no—
593C22—22no—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—88no—
608C22#304614yes—
609C22C22×#12111yescharacteristic
610C22#348512yes—
611C22C2×#57711yescharacteristic
612C22C2×#57811yescharacteristic
613C22#358612yes—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4#348511yescharacteristic
627C4#348511yescharacteristic
628C4#349112yes—
629C2—24no—
630C2—42no—
631C2—42no—
632C2#3544711yescharacteristic
633C2C2×#304611yescharacteristic
634C2#6042411yescharacteristic
635C2#6175512yes—
636C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28132C2semidirect
6. #27657C2semidirect
7. #25226C2semidirect
8. #25220C2semidirect
9. #24580C2semidirect
10. #24577C2semidirect
11. #24532C2semidirect
12. #24529C2semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2074C22semidirect
59. #2072C22semidirect
60. #2068C22semidirect
61. #2066C22semidirect
62. #2056C22semidirect
63. #2056C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1928C22semidirect
67. #1927C22semidirect
70. #1878C22semidirect
71. #1878C22semidirect
81. #1805C22semidirect
82. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C22×#4852
#353112
#343802
#281322
#276572
#252262
#252202
#245802
#245772
#245322
#245292
#219332
#217592
#207022