Finite Group Databasebeta

← #4457341

Subgroups — #4457341

Order 512.

4077 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2613C211yesmaximal, characteristic
2C2×#2309C211yesmaximal, characteristic, complemented
3#16573C214yesmaximal, complemented
4#16530C214yesmaximal, complemented
5#16527C214yesmaximal, complemented
6#16526C214yesmaximal, complemented
7#14820C214yesmaximal, complemented
8#14725C214yesmaximal, complemented
9#14135C212yesmaximal, complemented
10#13878C212yesmaximal, complemented
11C2×#1438C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C22×#123C2211yescharacteristic
14C22×#120C2212yes—
15C22×#117C2212yes—
16C23×C22⋊C4C2214yes—
17#2613—28no—
18C2×#397C2214yescomplemented
19C2×#397C2214yescomplemented
20C2×#394C2214yes—
21C2×#394C2214yescomplemented
22C2×#394C2214yescomplemented
23C2×#392C2214yes—
24#2309—28no—
25C2×#390C2214yes—
26C2×#390C2214yes—
27C2×#386C2214yes—
28C2×#381C2211yescharacteristic
29C2×#380C2214yes—
30C2×#379C2214yes—
31C2×#379C2212yes—
32C22×#95C2212yes—
33#1444C2214yescomplemented
34#1442C2214yescomplemented
35#1441C2214yescomplemented
36#1440C2214yescomplemented
37#1440C2214yescomplemented
38#1440C2214yescomplemented
39#1440C2214yescomplemented
40#1440C2214yescomplemented
41#1440C2214yescomplemented
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yescomplemented
50#1438C2214yescomplemented
51#1437C2212yescomplemented
52#1437C2212yescomplemented
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55#1436C2212yescomplemented
56#1436C2212yescomplemented
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C26C2312yes—
60C23×D4—28no—
61C23×D4—48no—
62C23×D4D418yes—
63C23×D4C2311yescharacteristic
64C23×D4C2314yes—
65C23×D4C2314yes—
66C24×C4C2314yes—
67C24×C4C2311yescharacteristic
68C2×#123—24no—
69C2×#123—28no—
70C2×#120—28no—
71C2×#120—48no—
72C2×#117—28no—
73C2×#117—48no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C22⋊C4—48no—
79C22×C22⋊C4—48no—
80C22×C22⋊C4—48no—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2311yescharacteristic
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C42C2312yes—
99C22×C42C2312yes—
100#397—48no—
101#397—48no—
102#394—216no—
103#394—48no—
104#394—48no—
105#392—216no—
106#390—216no—
107#390—216no—
108#386—48no—
109#381—24no—
110#380—48no—
111#379—216no—
112#379—28no—
113C2×#95—28no—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C25—28no—
131C25—24no—
132C25—24no—
133C25—28no—
134C25—44no—
135C25—44no—
136C25—44no—
137C25—44no—
138C25C2×D418yes—
139C25C2411yescharacteristic
140C25C2414yes—
141C25C2411yescharacteristic
142C22×D4—22no—
143C22×D4—24no—
144C22×D4—28no—
145C22×D4—28no—
146C22×D4—232no—
147C22×D4—28no—
148C22×D4—28no—
149C22×D4—432no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—464no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4C2×D4116yes—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×D414yes—
171C23×C4#4118yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2411yescharacteristic
178C23×C4C2414yes—
179C23×C4C2412yes—
180#123—416no—
181#120—432no—
182#117—432no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
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—464no—
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⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—44no—
248C2×C42—44no—
249#95—28no—
250#95—24no—
251#95—28no—
252#95—24no—
253#95—28no—
254#95—28no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265C24—24no—
266C24—24no—
267C24—216no—
268C24—24no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—216no—
283C24—216no—
284C24—21no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—416no—
290C24—416no—
291C24—416no—
292C24—416no—
293C24—416no—
294C24—416no—
295C24—416no—
296C24—416no—
297C24—88no—
298C24—88no—
299C24—84no—
300C24—84no—
301C24—84no—
302C24—84no—
303C24—88no—
304C24—88no—
305C24—88no—
306C24—88no—
307C24C22×D418yes—
308C24C22×D412yes—
309C24C2×#4114yes—
310C24H(5,𝔽2)18yes—
311C24C2511yescharacteristic
312C2×D4—232no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—4128no—
316C2×D4—416no—
317C2×D4—416no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—44no—
322C2×D4—48no—
323C2×D4—416no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—88no—
330C22×C4—216no—
331C22×C4—232no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
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—416no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—416no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4#123116yes—
384C22×C4C22×D414yes—
385C22×C4C2×#4114yes—
386C22×C4C2×#4114yes—
387C22×C4H(5,𝔽2)18yes—
388C4⋊C4—416no—
389C4⋊C4—48no—
390C4⋊C4—88no—
391C4⋊C4—88no—
392C22⋊C4—416no—
393C22⋊C4—416no—
394C22⋊C4—416no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—44no—
398C22⋊C4—416no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C42—48no—
410C42—48no—
411C23—28no—
412C23—28no—
413C23—24no—
414C23—24no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—28no—
425C23—28no—
426C23—464no—
427C23—432no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—41no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—832no—
468C23—816no—
469C23—816no—
470C23—832no—
471C23—832no—
472C23—832no—
473C23—832no—
474C23C2×#12314yes—
475C23#53418yes—
476C23#538116yes—
477C23C23×D411yescharacteristic
478C23C22×#4112yes—
479C23C2×H(5,𝔽2)14yes—
480C23C2×H(5,𝔽2)14yes—
481C23#58514yes—
482D4—464no—
483D4—832no—
484D4—88no—
485C2×C4—232no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—832no—
534C2×C4C2×#12318yes—
535C22—216no—
536C22—28no—
537C22—216no—
538C22—216no—
539C22—48no—
540C22—48no—
541C22—48no—
542C22—48no—
543C22—48no—
544C22—48no—
545C22—48no—
546C22—864no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—81no—
560C22#262218yes—
561C22C22×#12311yescharacteristic
562C22C2×#53412yes—
563C22C2×#53814yes—
564C22#349414yes—
565C22#349614yes—
566C22#350812yes—
567C22#351012yes—
568C22#353214yes—
569C22#353314yes—
570C22#354214yes—
571C22#354214yes—
572C4—28no—
573C4—48no—
574C4—84no—
575C4—84no—
576C4—84no—
577C4—82no—
578C4—84no—
579C4—81no—
580C2—28no—
581C2—88no—
582C2—161no—
583C2C2×#262211yescharacteristic
584C2#3554112yes—
585C2#3673714yes—
586C2#4196214yes—
587C2#4196314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2309C2semidirect
3. #16573C2semidirect
4. #16530C2semidirect
5. #16527C2semidirect
6. #16526C2semidirect
7. #14820C2semidirect
8. #14725C2semidirect
9. #14135C2semidirect
10. #13878C2semidirect
11. C2×#1438C2semidirect
18. C2×#397C22semidirect
19. C2×#397C22semidirect
21. C2×#394C22semidirect
22. C2×#394C22semidirect
33. #1444C22semidirect
34. #1442C22semidirect
35. #1441C22semidirect
36. #1440C22semidirect
37. #1440C22semidirect
38. #1440C22semidirect
39. #1440C22semidirect
40. #1440C22semidirect
41. #1440C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
49. #1438C22semidirect
50. #1438C22semidirect
51. #1437C22semidirect
52. #1437C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26132
C2×#23092
#165732
#165302
#165272
#165262
#148202
#147252
#141352
#138782
C2×#14382