Finite Group Databasebeta

← #6526688

Subgroups — #6526688

Order 512.

2228 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3C2×#2563C211yesmaximal, characteristic, complemented
4#34206C211yesmaximal, characteristic, complemented
5#28117C214yesmaximal, complemented
6#27651C214yesmaximal, complemented
7#25452C214yesmaximal, complemented
8#25321C214yesmaximal, complemented
9#24812C214yesmaximal, complemented
10#24677C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21834C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#520—21no—
17C22×C4×D4C2211yescharacteristic
18#3227—28no—
19C2×#498C2212yescomplemented
20C2×#486C2212yescomplemented
21#3100—28no—
22C2×#481C2214yes—
23C2×#480C2212yes—
24C2×#476C2214yescomplemented
25C2×#474C2212yes—
26C2×#459C2212yes—
27C2×#447C2212yes—
28C2×C4×D8C2214yes—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2563—22no—
32#2563—22no—
33#2563C2214yes—
34#2563C2214yes—
35C2×#393—22no—
36#2367—22no—
37#2366C2212yescomplemented
38#2365C2212yescomplemented
39#2354C2212yescomplemented
40#2352—22no—
41D4×C4⋊C4C2212yescomplemented
42#2325—22no—
43#2311—21no—
44C4×#121C2214yes—
45C2×#390C2212yes—
46#2285C2211yescharacteristic
47C2×#382C2212yes—
48C2×#380—21no—
49C2×#380—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2085C2218yescomplemented
52#2073C2218yescomplemented
53#2065C2218yes—
54#2054C2218yes—
55#1990C2212yescomplemented
56#1990C2212yescomplemented
57#1987C2214yescomplemented
58#1949C2218yescomplemented
59#1939C2214yescomplemented
60#1939C2214yescomplemented
61#1926C2218yes—
62#1879C2218yes—
63#1824C2214yes—
64#1823C2212yes—
65#1804C2218yescomplemented
66#1802C2212yes—
67#1802C2212yes—
68#1797C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C22×#41—21no—
72C23×Q8—21no—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×D8C2314yes—
76#532—24no—
77#532D418yes—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#124—21no—
87C2×#121—21no—
88C2×#121C2312yes—
89#520—48no—
90C2×C4×Q8—22no—
91C2×C4×D4—24no—
92C2×C4×D4—22no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2314yes—
102C2×#117—21no—
103C2×#117—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#498—28no—
116#486—28no—
117#481—216no—
118#480—28no—
119#476—216no—
120#474—28no—
121#459—28no—
122#447—28no—
123C4×D8—216no—
124#431—24no—
125#431—24no—
126C2×#107C2314yes—
127C2×#106C2318yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2318yes—
131C2×#102C2314yes—
132C2×#102C2318yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#393—28no—
141#393—28no—
142#391—22no—
143#390—22no—
144#390—24no—
145#390C2318yes—
146#390C2314yes—
147#389C2314yes—
148#388—24no—
149#388—24no—
150#387—24no—
151#386—24no—
152#386—24no—
153#385—24no—
154#384C2311yescharacteristic
155#384C2311yescharacteristic
156#382—24no—
157#382—21no—
158#382C2318yes—
159#380—21no—
160#380—24no—
161#380—24no—
162#380—24no—
163#380—24no—
164#380—22no—
165#380—22no—
166#380C2312yes—
167C4×C4⋊C4—21no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C22⋊C4—24no—
175C2×#95—22no—
176C2×#95—22no—
177C2×#95D412yes—
178#340C2318yes—
179#340C2314yes—
180#340C2314yes—
181#340C2318yes—
182#336C2318yes—
183C25—21no—
184C2×#41—28no—
185C2×#41—44no—
186C2×#41—44no—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—44no—
191C22×Q8—82no—
192C22×Q8—82no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—42no—
201C22×D4C2412yes—
202C22×D4C2412yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×D8—28no—
218C2×D8—28no—
219C2×D8—44no—
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—24no—
223C4⋊Q8—24no—
224C4⋊Q8—22no—
225C4⋊Q8—24no—
226C4⋊Q8—24no—
227C4⋊Q8—44no—
228C4⋊Q8—44no—
229C4⋊Q8C2×D414yes—
230#127—22no—
231#127C2×D414yes—
232#124—28no—
233#121—28no—
234#121—24no—
235#121C2×D418yes—
236C4×Q8—48no—
237C4×D4—28no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—48no—
244C4×D4—48no—
245C4×D4—416no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4C2×D418yes—
249#117—48no—
250#117—44no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—84no—
290C2×C22⋊C4—84no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—44no—
294C2×C42C2×D414yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D414yes—
298C2×C42#4114yes—
299C2×C42C2414yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—28no—
303#106—216no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306C4⋊C8—48no—
307#102—28no—
308#102—216no—
309#102—28no—
310#102—28no—
311#102—44no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C4×C8—24no—
315C4×C8—24no—
316#95—28no—
317#95—28no—
318#95—22no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24C22×D411yescharacteristic
334#41—432no—
335C2×Q8—24no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8—816no—
345C2×D4—24no—
346C2×D4—28no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4C22×D414yes—
368C2×D42−1+414yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—816no—
403C22×C4#12114yes—
404C22×C4#12114yes—
405C22×C4#13514yes—
406C22×C4AGL(1,ℤ/8)14yes—
407C22×C4C22×D414yes—
408C22×C4C22×D414yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2511yescharacteristic
417D8—416no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4C22×D414yes—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—84no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—84no—
456C42—28no—
457C42—44no—
458C42—48no—
459C42—44no—
460C42#12114yes—
461C42#12114yes—
462C42C22×D412yes—
463C42C22×D412yes—
464C42C2×#4112yes—
465C42C2×#4112yes—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—24no—
471C23—24no—
472C23—48no—
473C23—48no—
474C23—48no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23C2×#12112yes—
492C23C2×#13512yes—
493C23C2×AGL(1,ℤ/8)12yes—
494C23C23×D411yescharacteristic
495Q8—48no—
496Q8—88no—
497Q8—88no—
498D4—24no—
499D4—28no—
500D4—24no—
501D4—48no—
502D4—416no—
503D4—42no—
504D4—44no—
505D4—42no—
506D4—48no—
507D4—84no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—88no—
542C2×C4—84no—
543C2×C4—88no—
544C2×C4—88no—
545C2×C4—168no—
546C2×C4#463116yes—
547C2×C4C2×#12112yes—
548C2×C4C2×#12112yes—
549C2×C4C2×#12112yes—
550C2×C4C2×#12112yes—
551C2×C4C2×#12112yes—
552C2×C4C2×#13512yes—
553C2×C4C2×#13512yes—
554C2×C4C2×AGL(1,ℤ/8)12yes—
555C2×C4C2×AGL(1,ℤ/8)12yes—
556C2×C4C23×D411yescharacteristic
557C2×C4C22×#4111yescharacteristic
558C2×C4C2×2−1+412yes—
559C2×C4#58512yes—
560C8—44no—
561C8—44no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—816no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22C2×#46314yes—
584C22C22×#12111yescharacteristic
585C22#348511yescharacteristic
586C22#348611yescharacteristic
587C22C22×#13511yescharacteristic
588C22C22×AGL(1,ℤ/8)11yescharacteristic
589C22#358511yescharacteristic
590C22#358511yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—84no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4C2×#46314yes—
602C4C2×#46314yes—
603C4#348912yes—
604C4#349512yes—
605C2—24no—
606C2—84no—
607C2—84no—
608C2#3563311yescharacteristic
609C2C22×#46311yescharacteristic
610C2#6042611yescharacteristic
611C2#6170011yescharacteristic
612C2#6172011yescharacteristic
613C2#6181911yescharacteristic
614C2#6182511yescharacteristic
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×#3227C2semidirect
3. C2×#2563C2semidirect
4. #34206C2semidirect
5. #28117C2semidirect
6. #27651C2semidirect
7. #25452C2semidirect
8. #25321C2semidirect
9. #24812C2semidirect
10. #24677C2semidirect
11. #21951C2semidirect
12. #21834C2semidirect
13. #20732C2semidirect
19. C2×#498C22semidirect
20. C2×#486C22semidirect
24. C2×#476C22semidirect
37. #2366C22semidirect
38. #2365C22semidirect
39. #2354C22semidirect
41. D4×C4⋊C4C22semidirect
51. #2085C22semidirect
52. #2073C22semidirect
55. #1990C22semidirect
56. #1990C22semidirect
57. #1987C22semidirect
58. #1949C22semidirect
59. #1939C22semidirect
60. #1939C22semidirect
65. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C2×#31002
C2×#25632
#342062
#281172
#276512
#254522
#253212
#248122
#246772
#219512
#218342
#207322