Finite Group Databasebeta

← #6519681

Subgroups — #6519681

Order 512.

2392 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3C2×#2547C211yesmaximal, characteristic, complemented
4#34206C211yesmaximal, characteristic, complemented
5#28118C214yesmaximal, complemented
6#25367C214yesmaximal, complemented
7#25275C214yesmaximal, complemented
8#24727C214yesmaximal, complemented
9#24627C214yesmaximal, complemented
10#24215C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21804C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2212yes—
17C2×#520—21no—
18C22×C4×D4C2211yescharacteristic
19#3206—28no—
20C2×#492C2212yescomplemented
21C2×#487C2212yescomplemented
22#3100—28no—
23C2×#480C2212yes—
24C2×#474C2212yes—
25C2×#463C2218yes—
26C2×#461C2214yescomplemented
27C2×#459C2212yes—
28C2×#448C2214yes—
29C2×#447C2212yes—
30C2×C4×D8C2214yes—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2547—22no—
34#2547—22no—
35#2547C2214yes—
36#2547C2214yes—
37C2×#397—22no—
38#2367—22no—
39#2366C2212yescomplemented
40#2365C2212yescomplemented
41#2354C2212yescomplemented
42#2352—22no—
43D4×C4⋊C4C2212yescomplemented
44#2325—22no—
45#2311—21no—
46C4×#121C2214yes—
47C2×#391C2212yes—
48#2285C2211yescharacteristic
49C2×#380—21no—
50C2×#380—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2068C2218yescomplemented
53#2055C2218yescomplemented
54#2042C2218yes—
55#2012C2218yes—
56#1983C2218yes—
57#1981C2214yescomplemented
58#1956C2212yescomplemented
59#1956C2212yescomplemented
60#1939C2214yescomplemented
61#1939C2214yescomplemented
62#1928C2218yescomplemented
63#1877C2218yescomplemented
64#1823C2212yes—
65#1802C2212yes—
66#1802C2212yes—
67#1797C2214yes—
68#1787C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C22×#41—21no—
72C23×Q8C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×SD16C2318yes—
76C22×D8C2314yes—
77#532—24no—
78#532D418yes—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87C2×#124—21no—
88C2×#124—24no—
89C2×#124—48no—
90C2×#124C2318yes—
91C2×#121—21no—
92C2×#121C2312yes—
93#520—48no—
94C2×C4×Q8—22no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C2×#117—21no—
107C2×#117—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#492—28no—
120#487—28no—
121#480—28no—
122#474—28no—
123#463—416no—
124#461—216no—
125#459—28no—
126#448—216no—
127#447—28no—
128C4×D8—216no—
129#431—24no—
130#431—24no—
131C2×#107C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2318yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#103C2318yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146#397—28no—
147#397—28no—
148#391—22no—
149#391—24no—
150#391C2318yes—
151#390—22no—
152#390C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#387—24no—
157#386—24no—
158#386—24no—
159#385—24no—
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#382—21no—
163#380—21no—
164#380—22no—
165#380—22no—
166#380—24no—
167#380—24no—
168#380—24no—
169#380—24no—
170#380C2312yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C22⋊C4—24no—
179C2×#95—22no—
180C2×#95—22no—
181C2×#95D412yes—
182#340C2314yes—
183#340C2314yes—
184#336C2318yes—
185C25—21no—
186C2×#41—28no—
187C2×#41—44no—
188C2×#41—44no—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8C2414yes—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—42no—
204C22×D4C2412yes—
205C22×D4C2412yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C2×SD16—216no—
221C2×SD16—48no—
222C2×SD16—48no—
223C2×D8—28no—
224C2×D8—28no—
225C2×D8—44no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—22no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235C4⋊Q8C2×D414yes—
236#127—22no—
237#127C2×D414yes—
238#124—216no—
239#124—28no—
240#124—48no—
241#124—416no—
242#121—28no—
243#121—24no—
244#121C2×D418yes—
245C4×Q8—48no—
246C4×D4—28no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—48no—
253C4×D4—48no—
254C4×D4—416no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4C2×D418yes—
258#117—48no—
259#117—44no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—44no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D414yes—
306C2×C42#4114yes—
307C2×C42C2414yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310#107—28no—
311C4⋊C8—24no—
312C4⋊C8—24no—
313C4⋊C8—48no—
314#103—216no—
315#103—48no—
316#103—48no—
317#102—28no—
318#102—28no—
319#102—28no—
320#102—44no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C4×C8—24no—
324C4×C8—24no—
325#95—28no—
326#95—28no—
327#95—22no—
328#95—24no—
329#95—22no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24C22×D411yescharacteristic
343#41—432no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—88no—
355C2×Q8—88no—
356C2×D4—24no—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4C22×D414yes—
379C2×D4H(5,𝔽2)14yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—816no—
414C22×C4#12114yes—
415C22×C4#12114yes—
416C22×C4#13514yes—
417C22×C4AGL(1,ℤ/8)14yes—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4112yes—
427C22×C4C2511yescharacteristic
428SD16—816no—
429D8—416no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4C22×D414yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—28no—
468C42—44no—
469C42—48no—
470C42—44no—
471C42#12114yes—
472C42#12114yes—
473C42C22×D412yes—
474C42C22×D412yes—
475C42C2×#4112yes—
476C42C2×#4112yes—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—24no—
482C23—24no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23C2×#12112yes—
503C23C2×#13512yes—
504C23C2×AGL(1,ℤ/8)12yes—
505C23C23×D411yescharacteristic
506Q8—48no—
507Q8—48no—
508Q8—48no—
509Q8—88no—
510D4—24no—
511D4—28no—
512D4—24no—
513D4—48no—
514D4—416no—
515D4—42no—
516D4—44no—
517D4—42no—
518D4—48no—
519D4—84no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—416no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—88no—
554C2×C4—84no—
555C2×C4—88no—
556C2×C4—88no—
557C2×C4—168no—
558C2×C4#463116yescomplemented
559C2×C4C2×#12112yes—
560C2×C4C2×#12112yes—
561C2×C4C2×#12112yes—
562C2×C4C2×#12112yes—
563C2×C4C2×#12112yes—
564C2×C4C2×#13512yes—
565C2×C4C2×#13512yes—
566C2×C4C2×AGL(1,ℤ/8)12yes—
567C2×C4C2×AGL(1,ℤ/8)12yes—
568C2×C4C23×D411yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C2×H(5,𝔽2)12yes—
571C2×C4#58512yes—
572C8—44no—
573C8—44no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—816no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C2×#46314yes—
596C22C22×#12111yescharacteristic
597C22#348411yescharacteristic
598C22#348711yescharacteristic
599C22C22×#13511yescharacteristic
600C22C22×AGL(1,ℤ/8)11yescharacteristic
601C22#358611yescharacteristic
602C22#358611yescharacteristic
603C4—22no—
604C4—22no—
605C4—22no—
606C4—44no—
607C4—44no—
608C4—44no—
609C4—82no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4C2×#46314yes—
614C4C2×#46314yes—
615C4#348812yes—
616C4#349412yes—
617C2—24no—
618C2—84no—
619C2—84no—
620C2#3556311yescharacteristic
621C2C22×#46311yescharacteristic
622C2#6042711yescharacteristic
623C2#6156511yescharacteristic
624C2#6158511yescharacteristic
625C2#6188211yescharacteristic
626C2#6188811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3206C2semidirect
3. C2×#2547C2semidirect
4. #34206C2semidirect
5. #28118C2semidirect
6. #25367C2semidirect
7. #25275C2semidirect
8. #24727C2semidirect
9. #24627C2semidirect
10. #24215C2semidirect
11. #21951C2semidirect
12. #21804C2semidirect
13. #20732C2semidirect
20. C2×#492C22semidirect
21. C2×#487C22semidirect
26. C2×#461C22semidirect
39. #2366C22semidirect
40. #2365C22semidirect
41. #2354C22semidirect
43. D4×C4⋊C4C22semidirect
52. #2068C22semidirect
53. #2055C22semidirect
57. #1981C22semidirect
58. #1956C22semidirect
59. #1956C22semidirect
60. #1939C22semidirect
61. #1939C22semidirect
62. #1928C22semidirect
63. #1877C22semidirect
558. C2×C4#463semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
C2×#31002
C2×#25472
#342062
#281182
#253672
#252752
#247272
#246272
#242152
#219512
#218042
#207322