Finite Group Databasebeta

← #6513528

Subgroups — #6513528

Order 512.

2182 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic
4C2×#2538C211yesmaximal, characteristic
5#28232C214yesmaximal, complemented
6#28151C214yesmaximal, complemented
7#25394C214yesmaximal, complemented
8#25349C214yesmaximal, complemented
9#24556C214yesmaximal, complemented
10#24386C214yesmaximal, complemented
11#21981C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#20734C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3104—28no—
19C2×#484C2212yes—
20C2×#477C2212yescomplemented
21C2×#470C2218yescomplemented
22C2×#462C2212yescomplemented
23C2×#451C2212yes—
24#2975—28no—
25C2×C4×Q16C2214yescomplemented
26C2×#436C2214yes—
27C2×#436C2214yes—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#407C2214yes—
31#2848C2214yes—
32#2626—22no—
33#2580—22no—
34#2566—22no—
35#2563C2212yes—
36#2542C2212yes—
37#2538—22no—
38#2538—22no—
39#2538C2214yes—
40#2538C2212yes—
41#2538C2214yes—
42#2538C2212yes—
43C2×#396—22no—
44#2317—21no—
45#2309—21no—
46C2×#390—21no—
47C2×#389—21no—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#380C2212yes—
51#2075C2218yescomplemented
52#2035C2218yescomplemented
53#2035C2218yescomplemented
54#2001C2218yescomplemented
55#1990C2212yescomplemented
56#1956C2212yescomplemented
57#1948C2214yescomplemented
58#1894C2218yescomplemented
59#1894C2218yescomplemented
60#1872C2214yes—
61#1853C2214yes—
62#1844C2212yescomplemented
63#1834C2212yescomplemented
64#1824C2212yes—
65#1809C2218yescomplemented
66#1803C2212yescomplemented
67#1803C2214yescomplemented
68#1803C2212yescomplemented
69#1787C2212yes—
70C4×#103C2214yescomplemented
71C23×Q8C2311yescharacteristic
72C23×D4D411yescharacteristic
73C24×C4C2312yes—
74C22×Q16C2314yes—
75C22×M16C2318yes—
76#533—24no—
77#533—24no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×#125—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—48no—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2318yes—
86C2×C22⋊Q8C2314yes—
87C2×C22⋊Q8C2314yes—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C2×#117—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4D412yes—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116#484—28no—
117#477—28no—
118#470—416no—
119#462—28no—
120#451—28no—
121C4×Q16—216no—
122#436—216no—
123#436—216no—
124#431—24no—
125#431D418yes—
126C2×#107C2314yes—
127C2×#106C2318yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2318yes—
134C2×#103C2314yes—
135#407—216no—
136C2×C22⋊C8C2318yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#399—24no—
142#396—24no—
143#396—24no—
144#396—28no—
145#396—28no—
146#393—24no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—24no—
152#390—24no—
153#389—22no—
154#389—22no—
155#389—24no—
156#389—24no—
157#385—24no—
158#385—24no—
159#385—24no—
160#384—22no—
161#384—21no—
162#384—21no—
163#384C2311yescharacteristic
164#384C2312yes—
165#384C2311yescharacteristic
166#384C2312yes—
167#382—22no—
168#382—21no—
169#380—21no—
170#380—24no—
171#380C2314yes—
172#380C2314yes—
173#380C2318yes—
174#380C2312yes—
175#379—24no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2312yes—
178C2×#95—22no—
179#340C2314yes—
180#340C2318yes—
181#340C2314yes—
182#340C2318yes—
183C25C2×D412yes—
184C22×Q8—24no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8C2412yes—
191C22×Q8C2412yes—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4C2×D412yes—
196C22×D4C2×D412yes—
197C23×C4—21no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D412yes—
205C23×C4C2×D414yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C2×Q16—28no—
211C2×Q16—28no—
212C2×Q16—44no—
213C2×M16—216no—
214C2×M16—48no—
215C2×M16—48no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222#125—44no—
223C22⋊Q8—216no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—48no—
227C22⋊Q8—416no—
228C22⋊Q8—44no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—44no—
238C4×Q8—44no—
239C4×Q8—44no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246#117—44no—
247#117—44no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—84no—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—44no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D414yes—
304C2×C42#4114yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—28no—
308#106—48no—
309C4⋊C8—24no—
310C4⋊C8C2×D418yes—
311#103—28no—
312#103—28no—
313#103—28no—
314#103—44no—
315#103—48no—
316C22⋊C8—24no—
317C22⋊C8—216no—
318C22⋊C8C2×D418yes—
319C4×C8—24no—
320C4×C8C2×D418yes—
321#95—28no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24C22×D414yes—
334C24C22×D411yescharacteristic
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8H(5,𝔽2)14yes—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—416no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4#12114yes—
393C22×C4#12114yes—
394C22×C4#13714yes—
395C22×C4#13714yes—
396C22×C4C22×D414yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2511yescharacteristic
405Q16—416no—
406M16—816no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8C22×D418yes—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—416no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—416no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—88no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—44no—
456C42#12114yes—
457C42#12114yes—
458C42C22×D412yes—
459C42C22×D412yes—
460C42C2×#4112yes—
461C42C2×#4112yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—48no—
474C23—48no—
475C23—48no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23—84no—
480C23C2×#12112yes—
481C23C2×#13712yes—
482C23C2×#13712yes—
483C23C23×D411yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—48no—
487Q8—42no—
488Q8—42no—
489Q8—44no—
490Q8—44no—
491Q8—84no—
492D4—48no—
493D4—48no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—88no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—88no—
534C2×C4#470116yescomplemented
535C2×C4C2×#12112yes—
536C2×C4C2×#12112yes—
537C2×C4C2×#12112yes—
538C2×C4C2×#12112yes—
539C2×C4C2×#12112yes—
540C2×C4C2×#13712yes—
541C2×C4C2×#13712yes—
542C2×C4C2×#13712yes—
543C2×C4C2×#13712yes—
544C2×C4C23×D411yescharacteristic
545C2×C4C22×#4111yescharacteristic
546C2×C4C2×H(5,𝔽2)12yes—
547C2×C4#58512yes—
548C8—28no—
549C8—44no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—816no—
564C22C2×#47014yes—
565C22C22×#12111yescharacteristic
566C22#348411yescharacteristic
567C22#348511yescharacteristic
568C22C22×#13711yescharacteristic
569C22C22×#13711yescharacteristic
570C22#358511yescharacteristic
571C22#358611yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—44no—
576C4—44no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—84no—
582C4—82no—
583C4C2×#47014yes—
584C4C2×#47014yes—
585C4#348812yes—
586C4#349412yes—
587C2—24no—
588C2—84no—
589C2#3551411yescharacteristic
590C2C22×#47011yescharacteristic
591C2#6045611yescharacteristic
592C2#6158411yescharacteristic
593C2#6158811yescharacteristic
594C2#6171911yescharacteristic
595C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3104C2semidirect
5. #28232C2semidirect
6. #28151C2semidirect
7. #25394C2semidirect
8. #25349C2semidirect
9. #24556C2semidirect
10. #24386C2semidirect
11. #21981C2semidirect
12. #21860C2semidirect
13. #20734C2semidirect
20. C2×#477C22semidirect
21. C2×#470C22semidirect
22. C2×#462C22semidirect
25. C2×C4×Q16C22semidirect
51. #2075C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
54. #2001C22semidirect
55. #1990C22semidirect
56. #1956C22semidirect
57. #1948C22semidirect
58. #1894C22semidirect
59. #1894C22semidirect
62. #1844C22semidirect
63. #1834C22semidirect
65. #1809C22semidirect
66. #1803C22semidirect
67. #1803C22semidirect
68. #1803C22semidirect
70. C4×#103C22semidirect
534. C2×C4#470semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#29752
#351722
C2×#25382
#282322
#281512
#253942
#253492
#245562
#243862
#219812
#218602
#207342