Finite Group Databasebeta

← #6512684

Subgroups — #6512684

Order 512.

3130 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3538C211yesmaximal, characteristic
2C2×#3127C212yesmaximal, complemented
3#35661C211yesmaximal, characteristic
4#24255C214yesmaximal, complemented
5#24243C214yesmaximal, complemented
6#24139C214yesmaximal, complemented
7#24127C214yesmaximal, complemented
8#24016C212yesmaximal, complemented
9#24010C212yesmaximal, complemented
10#23928C212yesmaximal, complemented
11#23921C212yesmaximal, complemented
12#21873C212yesmaximal, complemented
13#21792C211yesmaximal, characteristic
14#3538—216no—
15C2×#563—22no—
16C2×#552C2214yes—
17C2×#551—22no—
18C2×D4×Q8C2214yes—
19C2×#544—22no—
20C2×#541C2212yes—
21C2×#539—22no—
22C2×#536C2212yes—
23C2×#536C2211yescharacteristic
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C2×#519C2211yescharacteristic
28#3127—216no—
29C2×#485C2214yes—
30C2×#479C2214yescomplemented
31C2×#462C2214yescomplemented
32C2×#451C2214yes—
33C2×#451C2214yes—
34C2×#451C2212yescomplemented
35C2×#451C2212yescomplemented
36C2×#441C2214yescomplemented
37C2×C4×Q16C2214yescomplemented
38C2×#433C2212yes—
39#2538C2214yes—
40#2536C2214yes—
41#2429—22no—
42#2426—22no—
43#2413—22no—
44#2411—22no—
45#2317C2211yescharacteristic
46#2311C2211yescharacteristic
47#2280C2211yescharacteristic
48#2036C2214yes—
49#2036C2214yescomplemented
50#2035C2214yes—
51#2035C2214yescomplemented
52#2001C2214yes—
53#2001C2214yes—
54#2001C2214yescomplemented
55#2001C2214yescomplemented
56#1992C2212yescomplemented
57#1991C2212yescomplemented
58#1960C2212yescomplemented
59#1956C2212yescomplemented
60#1955C2214yes—
61#1954C2214yes—
62#1873C2214yes—
63#1873C2212yescomplemented
64#1873C2212yescomplemented
65#1872C2214yes—
66#1872C2214yes—
67#1854C2214yescomplemented
68#1853C2212yes—
69#1852C2212yes—
70#1805C2214yescomplemented
71#1803C2214yescomplemented
72#1787C2212yes—
73#1787C2212yes—
74#1787C2214yescomplemented
75#1780C2212yes—
76#1779C2212yes—
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C22×Q16C2314yes—
83C22×Q16C2314yes—
84#563—232no—
85#552—216no—
86#552—216no—
87#551—232no—
88D4×Q8—216no—
89D4×Q8—216no—
90#544—232no—
91#541—28no—
92#541D4116yes—
93#539—232no—
94#536—216no—
95#536—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2312yes—
100C2×#126—24no—
101C2×#126—24no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125C2312yes—
105C2×#124—22no—
106C2×#124—21no—
107#525—28no—
108C2×#123—22no—
109C2×#123—22no—
110C2×#123—24no—
111C2×#123—22no—
112C2×#123D412yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—216no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—44no—
120C2×C22⋊Q8D412yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C2×C22⋊Q8C2314yes—
124C2×C22⋊Q8C2312yes—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2314yes—
127C2×#121D412yes—
128C2×#120D412yes—
129#519—28no—
130C2×C4×Q8—22no—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2314yes—
133C2×C4×D4—24no—
134C2×C4×D4C2314yes—
135C2×#117—24no—
136C2×#117—22no—
137C2×#117C2311yescharacteristic
138C2×#117C2312yes—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—22no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2312yes—
144C22×C4⋊C4C2311yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—21no—
147#485—216no—
148#479—216no—
149#462—216no—
150#451—216no—
151#451—48no—
152#451—44no—
153#451—44no—
154#441—216no—
155C4×Q16—216no—
156#433—28no—
157C2×#107C2314yes—
158C2×#106C2314yes—
159C2×C4⋊C8C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2312yes—
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#398—24no—
178#396—24no—
179#396—24no—
180#394—24no—
181#391—21no—
182#390—21no—
183#390—24no—
184#389—22no—
185#389—24no—
186#388—24no—
187#388—24no—
188#386—22no—
189#385—22no—
190#384C2312yes—
191#382C2312yes—
192#381—21no—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#379—21no—
197C4×C22⋊C4—22no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202C25C2×D411yescharacteristic
203C2×#41—24no—
204C2×#41—42no—
205C2×#41—42no—
206C22×Q8—24no—
207C22×Q8—28no—
208C22×Q8—28no—
209C22×Q8—21no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—24no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8C2×D411yescharacteristic
221C22×Q8C2414yes—
222C22×Q8C2412yes—
223C22×D4C2×D414yes—
224C22×D4C2×D412yes—
225C22×D4C2×D411yescharacteristic
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C2×Q16—28no—
240C2×Q16—28no—
241C2×Q16—28no—
242C2×Q16—28no—
243C2×Q16—44no—
244C2×Q16—44no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—216no—
248C4⋊Q8—232no—
249C4⋊Q8—24no—
250C4⋊Q8—44no—
251C4⋊Q8C2×D418yes—
252#126—232no—
253#126—232no—
254#125—232no—
255#125—232no—
256#125—24no—
257#125C2×D418yes—
258#124—216no—
259#124—44no—
260#123—216no—
261#123—216no—
262#123—232no—
263#123—28no—
264#123—48no—
265C22⋊Q8—216no—
266C22⋊Q8—216no—
267C22⋊Q8—232no—
268C22⋊Q8—232no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—44no—
279C22⋊Q8—42no—
280C22⋊Q8—42no—
281C22⋊Q8—48no—
282C22⋊Q8C2×D4116yes—
283#121—28no—
284#120—28no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—48no—
289C4×Q8C2×D4116yes—
290C4×D4—232no—
291C4×D4—28no—
292C4×D4—28no—
293#117—232no—
294#117—28no—
295#117—24no—
296#117—42no—
297#117—48no—
298#117C2×D418yes—
299C2×C4⋊C4—216no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C22⋊C4—216no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C2412yes—
346#107—28no—
347#106—28no—
348C4⋊C8—28no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357C22⋊C8—28no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360C22⋊C8—44no—
361C22⋊C8—44no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364#97—24no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370C24—22no—
371C24—24no—
372C24—41no—
373C24—42no—
374C24—41no—
375C24C22×D412yes—
376C24C22×D411yescharacteristic
377C24C22×D412yes—
378#41—88no—
379C2×Q8—24no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—416no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—88no—
401C2×Q8—88no—
402C2×Q8—88no—
403C2×Q8—88no—
404C2×Q8C22×D418yes—
405C2×Q8H(5,𝔽2)14yes—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—42no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—88no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D414yes—
458C22×C4C2511yescharacteristic
459Q16—416no—
460Q16—416no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4#12018yes—
487C4⋊C4C22×D418yes—
488C22⋊C4—216no—
489C22⋊C4—28no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—82no—
501C22⋊C4—84no—
502C22⋊C4#12018yes—
503C42—28no—
504C42—216no—
505C42—42no—
506C42—42no—
507C42—48no—
508C42C22×D414yes—
509C42C22×D414yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—44no—
526C23—48no—
527C23—82no—
528C23—84no—
529C23—82no—
530C23C2×#12012yes—
531C23#57814yes—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—28no—
535Q8—28no—
536Q8—48no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—82no—
541Q8—88no—
542Q8—88no—
543D4—416no—
544D4—82no—
545D4—84no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—416no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—88no—
584C2×C4—88no—
585C2×C4—84no—
586C2×C4C2×#12018yes—
587C2×C4C2×#12012yes—
588C2×C4C23×D412yes—
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4#58512yes—
591C8—44no—
592C8—44no—
593C22—24no—
594C22—22no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—88no—
611C22C22×#12011yescharacteristic
612C22#348411yescharacteristic
613C22#348511yescharacteristic
614C22C2×#57812yes—
615C22#358612yes—
616C4—22no—
617C4—28no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—48no—
622C4—48no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—82no—
627C4#348814yes—
628C2—44no—
629C2—42no—
630C2—82no—
631C2#3551211yescharacteristic
632C2#6037512yes—
633C2#6163411yescharacteristic
634C2#6165211yescharacteristic
635C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3127C2semidirect
4. #24255C2semidirect
5. #24243C2semidirect
6. #24139C2semidirect
7. #24127C2semidirect
8. #24016C2semidirect
9. #24010C2semidirect
10. #23928C2semidirect
11. #23921C2semidirect
12. #21873C2semidirect
30. C2×#479C22semidirect
31. C2×#462C22semidirect
34. C2×#451C22semidirect
35. C2×#451C22semidirect
36. C2×#441C22semidirect
37. C2×C4×Q16C22semidirect
49. #2036C22semidirect
51. #2035C22semidirect
54. #2001C22semidirect
55. #2001C22semidirect
56. #1992C22semidirect
57. #1991C22semidirect
58. #1960C22semidirect
59. #1956C22semidirect
63. #1873C22semidirect
64. #1873C22semidirect
67. #1854C22semidirect
70. #1805C22semidirect
71. #1803C22semidirect
74. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35382
C2×#31272
#356612
#242552
#242432
#241392
#241272
#240162
#240102
#239282
#239212
#218732
#217922