Finite Group Databasebeta

← #581621

Subgroups — #581621

Order 512.

3746 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2308C211yesmaximal, characteristic
2#14674C211yesmaximal, characteristic, complemented
3#14428C214yesmaximal, complemented
4#14397C214yesmaximal, complemented
5#14365C214yesmaximal, complemented
6#14165C211yesmaximal, characteristic, complemented
7#14076C214yesmaximal, complemented
8#14012C214yesmaximal, complemented
9#13902C212yesmaximal
10#13873C214yesmaximal, complemented
11#13780C212yesmaximal, complemented
12C24×D4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14#2308C4116yescomplemented
15C2×#390C2212yes—
16C2×#390C2212yes—
17C2×#390C2212yes—
18C2×#390C2212yes—
19C2×#388C2214yescomplemented
20C2×#388C2214yescomplemented
21C2×#388C2214yescomplemented
22C2×#388C2214yescomplemented
23C2×#386C2214yes—
24C2×#386C2214yes—
25C2×#381C2211yescharacteristic
26C2×#381C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2214yescomplemented
29C2×#379C2214yes—
30C2×#379C2214yes—
31C2×#379C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2214yescomplemented
34#1444C2214yescomplemented
35#1442C2214yescomplemented
36#1440C2214yescomplemented
37#1440C2214yescomplemented
38#1440C2214yescomplemented
39#1440C2214yescomplemented
40#1439C2214yes—
41#1439C2214yes—
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1438C2214yes—
45#1438C2214yes—
46#1437C2214yes—
47#1437C2214yes—
48#1437C2214yescomplemented
49#1437C2214yescomplemented
50#1436C2214yescomplemented
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53#1436C2214yescomplemented
54#1434C2214yescomplemented
55C22⋊#95C2214yescomplemented
56C22⋊#95C2214yescomplemented
57C4×#95C2214yescomplemented
58C26C2311yescharacteristic
59C26C2311yescharacteristic
60C23×D4—24no—
61C23×D4—24no—
62C23×D4—44no—
63C23×D4—44no—
64C23×D4C2×C418yes—
65C23×D4C2312yes—
66C23×D4C2312yes—
67C24×C4C2314yes—
68C24×C4C2311yescharacteristic
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C22⋊C4—44no—
74C22×C22⋊C4—44no—
75C22×C22⋊C4C2×C4116yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C42C2312yes—
92C22×C42C2314yes—
93C22×C42C2312yes—
94C22×C42C2314yes—
95#390—44no—
96#390—44no—
97#390—44no—
98#390—44no—
99#388—48no—
100#388—48no—
101#388—48no—
102#388—48no—
103#386C2×C4132yes—
104#386C2×C4132yes—
105#381C2×C418yes—
106#381C2×C418yes—
107#380—28no—
108#380—48no—
109#379—48no—
110#379—48no—
111#379C2×C4116yes—
112C4×C22⋊C4—28no—
113C4×C22⋊C4—48no—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C25—22no—
131C25—22no—
132C25—24no—
133C25—24no—
134C25—22no—
135C25—22no—
136C25—42no—
137C25—42no—
138C25—44no—
139C25—44no—
140C25—44no—
141C25—44no—
142C25C22×C414yes—
143C25C22×C414yes—
144C25C2411yescharacteristic
145C25C2411yescharacteristic
146C25C2411yescharacteristic
147C25C2412yes—
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—232no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—432no—
162C22×D4—432no—
163C22×D4—416no—
164C22×D4—416no—
165C22×D4C22×C418yes—
166C22×D4C22×C418yes—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C22×C4116yes—
176C23×C4C22×C414yes—
177C23×C4#4114yes—
178C23×C4#4114yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—432no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4C22×C4116yes—
240C2×C22⋊C4C22×C418yes—
241C2×C22⋊C4C22×C418yes—
242C2×C22⋊C4C22×C4116yes—
243C2×C22⋊C4C22×C4116yes—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95C22×C4116yes—
271#95C22×C4116yes—
272C24—216no—
273C24—28no—
274C24—28no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—28no—
288C24—216no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—24no—
294C24—24no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—24no—
301C24—28no—
302C24—48no—
303C24—416no—
304C24—416no—
305C24—48no—
306C24—44no—
307C24—44no—
308C24—88no—
309C24—88no—
310C24—84no—
311C24—84no—
312C24—84no—
313C24—84no—
314C24—88no—
315C24—88no—
316C24—88no—
317C24—88no—
318C24—88no—
319C24—88no—
320C24—88no—
321C24—88no—
322C24C23×C412yes—
323C24C23×C412yes—
324C24C23×C412yes—
325C24C23×C414yes—
326C24C2×#4112yes—
327C24C2×#4112yes—
328C24H(5,𝔽2)14yes—
329C24H(5,𝔽2)14yes—
330C24C2511yescharacteristic
331C2×D4—216no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—216no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—4128no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4#117116yes—
401C22×C4C23×C418yes—
402C22×C4C23×C418yes—
403C22×C4C23×C414yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4H(5,𝔽2)18yes—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—88no—
410C4⋊C4—88no—
411C22⋊C4—232no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—416no—
415C22⋊C4—416no—
416C22⋊C4—416no—
417C22⋊C4—416no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C42—84no—
427C42—84no—
428C42—88no—
429C42—88no—
430C23—232no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—28no—
435C23—28no—
436C23—28no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—28no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—416no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—832no—
493C23—816no—
494C23—816no—
495C23—832no—
496C23—816no—
497C23—88no—
498C23—88no—
499C23—832no—
500C23—832no—
501C23—816no—
502C23—88no—
503C23—88no—
504C23C2×#11714yes—
505C23#51818yes—
506C23C24×C411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)14yes—
512C23#58514yes—
513D4—432no—
514D4—432no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—416no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—832no—
565C2×C4C2×#11718yes—
566C22—24no—
567C22—28no—
568C22—216no—
569C22—28no—
570C22—48no—
571C22—48no—
572C22—48no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—48no—
581C22—48no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—44no—
590C22—832no—
591C22—832no—
592C22—84no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C22×#11711yescharacteristic
601C22C2×#51812yes—
602C22C4×H(5,𝔽2)14yes—
603C22#349012yes—
604C22#349011yescharacteristic
605C22#349011yescharacteristic
606C22#349414yes—
607C22#350812yes—
608C22#351012yes—
609C22#353414yes—
610C22#353714yes—
611C22#355814yes—
612C22#356014yes—
613C4—28no—
614C4—48no—
615C4—48no—
616C4—48no—
617C4—84no—
618C4—84no—
619C4—84no—
620C2—48no—
621C2—44no—
622C2—82no—
623C2—82no—
624C2#3417611yescharacteristic
625C2#3425512yes—
626C2#3443714yes—
627C2#4639314yes—
628C2#4645212yes—
629C2#4645411yescharacteristic
630C2#4645711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14674C2semidirect
3. #14428C2semidirect
4. #14397C2semidirect
5. #14365C2semidirect
6. #14165C2semidirect
7. #14076C2semidirect
8. #14012C2semidirect
10. #13873C2semidirect
11. #13780C2semidirect
14. #2308C4semidirect
19. C2×#388C22semidirect
20. C2×#388C22semidirect
21. C2×#388C22semidirect
22. C2×#388C22semidirect
28. C2×#380C22semidirect
33. C2×C4×C22⋊C4C22semidirect
34. #1444C22semidirect
35. #1442C22semidirect
36. #1440C22semidirect
37. #1440C22semidirect
38. #1440C22semidirect
39. #1440C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
48. #1437C22semidirect
49. #1437C22semidirect
50. #1436C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect
53. #1436C22semidirect
54. #1434C22semidirect
55. C22⋊#95C22semidirect
56. C22⋊#95C22semidirect
57. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23082
#146742
#144282
#143972
#143652
#141652
#140762
#140122
#139022
#138732
#137802