Finite Group Databasebeta

← #6481752

Subgroups — #6481752

Order 512.

2200 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3#35180C211yesmaximal, characteristic, complemented
4C2×#2354C211yesmaximal, characteristic, complemented
5#33111C214yesmaximal, complemented
6#29718C214yesmaximal, complemented
7#24681C214yesmaximal, complemented
8#24677C214yesmaximal, complemented
9#24631C214yesmaximal, complemented
10#24627C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#21859C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18#3101—28no—
19#3100—28no—
20C2×#483C2212yes—
21C2×#480C2212yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#461C2212yes—
25C2×#459C2212yes—
26C2×#450C2212yes—
27C2×#447C2212yes—
28C2×C4×SD16C2214yes—
29C2×C4×D8C2214yes—
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2850C2214yes—
33#2639—22no—
34#2592—22no—
35#2568C2212yescomplemented
36#2563C2212yescomplemented
37#2562—22no—
38#2548C2212yescomplemented
39#2547C2212yescomplemented
40#2354—22no—
41#2354—22no—
42#2354C2214yes—
43#2354C2214yes—
44#2311—21no—
45#2311—21no—
46C2×#390C2212yes—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2073C2218yes—
53#2055C2218yes—
54#2054C2218yes—
55#2012C2218yes—
56#1990C2212yescomplemented
57#1990C2212yescomplemented
58#1983C2218yescomplemented
59#1957C2212yescomplemented
60#1956C2212yescomplemented
61#1949C2218yescomplemented
62#1939C2214yescomplemented
63#1939C2214yescomplemented
64#1928C2218yescomplemented
65#1926C2218yescomplemented
66#1874C2214yes—
67#1872C2214yes—
68#1803C2214yescomplemented
69#1802C2214yescomplemented
70#1797C2214yes—
71C22×#41—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×SD16C2314yes—
75C22×D8C2314yes—
76#533—24no—
77#533—24no—
78#532—24no—
79#532—24no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#125—21no—
85C2×#124—21no—
86C2×#124C2314yes—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8C2312yes—
89C2×#121—21no—
90C2×#121C2312yes—
91C2×C4×Q8—21no—
92C2×C4×Q8C2312yes—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—24no—
97C2×C4×D4D418yes—
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×#117—21no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117—48no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C42—22no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#483—28no—
118#480—28no—
119#476—28no—
120#474—28no—
121#461—28no—
122#459—28no—
123#450—28no—
124#447—28no—
125C4×SD16—216no—
126C4×D8—216no—
127#431—24no—
128#431—24no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2318yes—
134C2×#103C2314yes—
135C2×#103C2318yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2318yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—24no—
145#397—24no—
146#394—24no—
147#393—24no—
148#391—22no—
149#391—22no—
150#391C2314yes—
151#390—22no—
152#390—22no—
153#390—24no—
154#390C2314yes—
155#390C2318yes—
156#388—24no—
157#388—24no—
158#388—28no—
159#388—28no—
160#386—24no—
161#386—24no—
162#384—21no—
163#384—21no—
164#384C2312yes—
165#384C2312yes—
166#382C2314yes—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—24no—
172#380—24no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2318yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C2×#95—22no—
180C2×#95D412yes—
181#340C2318yes—
182#340C2318yes—
183#336C2318yes—
184C25—21no—
185C2×#41—44no—
186C2×#41—44no—
187C2×#41—44no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8C2412yes—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4C2×D414yes—
199C22×D4C2412yes—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2411yescharacteristic
214C2×SD16—28no—
215C2×SD16—44no—
216C2×SD16—44no—
217C2×D8—28no—
218C2×D8—44no—
219C2×D8—44no—
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228#127—22no—
229#127—22no—
230#125—44no—
231#124—48no—
232#124—44no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—44no—
236#121—24no—
237#121—24no—
238#121—44no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—44no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4C2×D418yes—
250C4×D4C2×D418yes—
251#117—416no—
252#117—416no—
253#117—44no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—84no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—44no—
298C2×C42—48no—
299C2×C42C2×D414yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D414yes—
303C2×C42#4114yes—
304C2×C42C2414yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—28no—
308#106—28no—
309C4⋊C8—24no—
310C4⋊C8—24no—
311C4⋊C8—48no—
312#103—28no—
313#103—48no—
314#102—28no—
315#102—28no—
316#102—28no—
317#102—48no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C4×C8—24no—
321C4×C8—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24C22×D411yescharacteristic
339#41—816no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—48no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4C22×D414yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—432no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4#12114yes—
406C22×C4#12114yes—
407C22×C4#13514yes—
408C22×C4AGL(1,ℤ/8)14yes—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4112yes—
418C22×C4C2511yescharacteristic
419SD16—416no—
420D8—416no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—28no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4C22×D414yes—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—416no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—88no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—28no—
462C42—44no—
463C42—416no—
464C42—44no—
465C42#12114yes—
466C42#12114yes—
467C42C22×D412yes—
468C42C22×D412yes—
469C42C2×#4112yes—
470C42C2×#4112yes—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—24no—
476C23—48no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—48no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23C2×#12112yes—
497C23C2×#13512yes—
498C23C2×AGL(1,ℤ/8)12yes—
499C23C23×D411yescharacteristic
500Q8—44no—
501Q8—44no—
502Q8—84no—
503Q8—84no—
504D4—24no—
505D4—24no—
506D4—48no—
507D4—44no—
508D4—42no—
509D4—42no—
510D4—44no—
511D4—84no—
512D4—88no—
513D4—84no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—416no—
535C2×C4—416no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4#463116yes—
552C2×C4C2×#12112yes—
553C2×C4C2×#12112yes—
554C2×C4C2×#12112yes—
555C2×C4C2×#12112yes—
556C2×C4C2×#12112yes—
557C2×C4C2×#13512yes—
558C2×C4C2×#13512yes—
559C2×C4C2×AGL(1,ℤ/8)12yes—
560C2×C4C2×AGL(1,ℤ/8)12yes—
561C2×C4C23×D411yescharacteristic
562C2×C4C22×#4111yescharacteristic
563C2×C4#58512yes—
564C2×C4#58512yes—
565C8—44no—
566C8—44no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—816no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22C2×#46314yes—
589C22C22×#12111yescharacteristic
590C22#348511yescharacteristic
591C22#348711yescharacteristic
592C22C22×#13511yescharacteristic
593C22C22×AGL(1,ℤ/8)11yescharacteristic
594C22#358511yescharacteristic
595C22#358611yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—44no—
600C4—44no—
601C4—44no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4C2×#46314yes—
607C4C2×#46314yes—
608C4#344912yes—
609C4#349312yes—
610C2—24no—
611C2—84no—
612C2—162no—
613C2#3515011yescharacteristic
614C2C22×#46311yescharacteristic
615C2#6042511yescharacteristic
616C2#6170011yescharacteristic
617C2#6171711yescharacteristic
618C2#6188211yescharacteristic
619C2#6189011yescharacteristic
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×#3101C2semidirect
3. #35180C2semidirect
4. C2×#2354C2semidirect
5. #33111C2semidirect
6. #29718C2semidirect
7. #24681C2semidirect
8. #24677C2semidirect
9. #24631C2semidirect
10. #24627C2semidirect
11. #21951C2semidirect
12. #21860C2semidirect
13. #21859C2semidirect
35. #2568C22semidirect
36. #2563C22semidirect
38. #2548C22semidirect
39. #2547C22semidirect
56. #1990C22semidirect
57. #1990C22semidirect
58. #1983C22semidirect
59. #1957C22semidirect
60. #1956C22semidirect
61. #1949C22semidirect
62. #1939C22semidirect
63. #1939C22semidirect
64. #1928C22semidirect
65. #1926C22semidirect
68. #1803C22semidirect
69. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#31002
#351802
C2×#23542
#331112
#297182
#246812
#246772
#246312
#246272
#219512
#218602
#218592