Finite Group Databasebeta

← #5025774

Subgroups — #5025774

Order 512.

3689 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic, complemented
2C2×#2613C211yesmaximal, characteristic
3#15640C214yesmaximal, complemented
4#15617C214yesmaximal, complemented
5#15448C214yesmaximal, complemented
6#15348C214yesmaximal, complemented
7#15297C214yesmaximal, complemented
8#15268C214yesmaximal, complemented
9#14287C212yesmaximal, complemented
10#13902C212yesmaximal, complemented
11#13828C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C22×#123C2211yescharacteristic
14C22×#120C2212yes—
15#2613—28no—
16#2608—28no—
17C2×#397C2214yescomplemented
18C2×#396C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yescomplemented
21C2×#394C2214yescomplemented
22C2×#392C2214yes—
23C2×#390C2214yes—
24C2×#390C2214yes—
25C2×#390C2214yescomplemented
26C2×#389C2214yescomplemented
27C2×#388C2214yescomplemented
28C2×#388C2214yescomplemented
29C2×#387C2214yescomplemented
30C2×#381C2211yescharacteristic
31C2×#380C2212yes—
32C2×#379C2212yes—
33C2×#379C2214yes—
34C2×#379C2214yes—
35C2×C4×C22⋊C4C2212yes—
36#1441C2214yescomplemented
37#1440C2214yes—
38#1440C2214yescomplemented
39#1440C2214yescomplemented
40#1439C2214yescomplemented
41#1439C2214yescomplemented
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1438C2214yescomplemented
51#1437C2214yes—
52#1437C2212yescomplemented
53#1437C2212yescomplemented
54#1437C2214yescomplemented
55#1436C2214yescomplemented
56#1436C2214yescomplemented
57#1436C2212yescomplemented
58#1436C2212yescomplemented
59C26C2312yes—
60C23×D4—24no—
61C23×D4—28no—
62C23×D4—48no—
63C23×D4C2314yes—
64C23×D4C2311yescharacteristic
65C23×D4C2314yes—
66C24×C4C2311yescharacteristic
67C2×#123—24no—
68C2×#123—28no—
69C2×#120—28no—
70C2×#120—48no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2311yescharacteristic
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⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C42C2312yes—
95C22×C42C2312yes—
96C22×C42C2314yes—
97#397—48no—
98#396—216no—
99#394—216no—
100#394—48no—
101#394—48no—
102#392—216no—
103#390—48no—
104#390—48no—
105#390—48no—
106#389—216no—
107#388—48no—
108#388—48no—
109#387—216no—
110#381D418yes—
111#380—28no—
112#379—28no—
113#379—48no—
114#379—48no—
115C4×C22⋊C4—28no—
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—24no—
131C25—28no—
132C25—24no—
133C25—24no—
134C25—44no—
135C25—44no—
136C25—44no—
137C25—44no—
138C25—44no—
139C25C2411yescharacteristic
140C25C2414yes—
141C25C2411yescharacteristic
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—22no—
145C22×D4—28no—
146C22×D4—24no—
147C22×D4—28no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—464no—
160C22×D4—432no—
161C22×D4—416no—
162C23×C4—44no—
163C23×C4C2×D414yes—
164C23×C4#4118yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2411yescharacteristic
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173#123—416no—
174#120—432no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
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—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
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⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4C2×D418yes—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249#95—28no—
250#95—28no—
251#95—24no—
252#95—24no—
253#95—28no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95C2×D4116yes—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—216no—
267C24—24no—
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—21no—
282C24—22no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—416no—
287C24—416no—
288C24—48no—
289C24—416no—
290C24—416no—
291C24—48no—
292C24—88no—
293C24—88no—
294C24—84no—
295C24—84no—
296C24—84no—
297C24—84no—
298C24—88no—
299C24—88no—
300C24—88no—
301C24—88no—
302C24—88no—
303C24—88no—
304C24—88no—
305C24—88no—
306C24C22×D412yes—
307C24C2×#4114yes—
308C24H(5,𝔽2)18yes—
309C24C2511yescharacteristic
310C2×D4—232no—
311C2×D4—416no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—416no—
316C2×D4—416no—
317C2×D4—4128no—
318C2×D4—48no—
319C2×D4—416no—
320C2×D4—44no—
321C2×D4—416no—
322C2×D4—88no—
323C2×D4—88no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—88no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
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—24no—
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—48no—
372C22×C4—416no—
373C22×C4#124116yes—
374C22×C4C22×D418yes—
375C22×C4C22×D414yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4114yes—
378C22×C4H(5,𝔽2)18yes—
379C4⋊C4—416no—
380C4⋊C4—416no—
381C4⋊C4—48no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C22⋊C4—48no—
385C22⋊C4—416no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—44no—
390C22⋊C4—48no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—84no—
403C42—84no—
404C42—88no—
405C23—24no—
406C23—28no—
407C23—28no—
408C23—28no—
409C23—24no—
410C23—24no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—432no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
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—41no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—816no—
463C23—816no—
464C23—832no—
465C23—832no—
466C23—832no—
467C23—832no—
468C23—832no—
469C23—832no—
470C23C2×#12414yes—
471C23C23×D411yescharacteristic
472C23C22×#4112yes—
473C23C2×H(5,𝔽2)14yes—
474C23C2×H(5,𝔽2)14yes—
475C23#58514yes—
476D4—464no—
477D4—832no—
478D4—88no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—416no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
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—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
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×C4C2×#12418yes—
528C22—24no—
529C22—28no—
530C22—48no—
531C22—48no—
532C22—48no—
533C22—48no—
534C22—48no—
535C22—48no—
536C22—48no—
537C22—48no—
538C22—48no—
539C22—48no—
540C22—864no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—81no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—82no—
554C22C22×#12411yescharacteristic
555C22#348412yes—
556C22#348814yes—
557C22#349014yes—
558C22#350614yes—
559C22#350812yes—
560C22#351012yes—
561C22#353414yes—
562C22#353714yes—
563C22#354214yes—
564C22#356014yes—
565C4—28no—
566C4—48no—
567C4—48no—
568C4—81no—
569C4—84no—
570C4—84no—
571C4—84no—
572C4—82no—
573C2—44no—
574C2—88no—
575C2—161no—
576C2#3536711yescharacteristic
577C2#3555612yes—
578C2#4800714yes—
579C2#4800914yes—
580C2#5063414yes—
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
1. C2×#2608C2semidirect
3. #15640C2semidirect
4. #15617C2semidirect
5. #15448C2semidirect
6. #15348C2semidirect
7. #15297C2semidirect
8. #15268C2semidirect
9. #14287C2semidirect
10. #13902C2semidirect
11. #13828C2semidirect
17. C2×#397C22semidirect
20. C2×#394C22semidirect
21. C2×#394C22semidirect
25. C2×#390C22semidirect
26. C2×#389C22semidirect
27. C2×#388C22semidirect
28. C2×#388C22semidirect
29. C2×#387C22semidirect
36. #1441C22semidirect
38. #1440C22semidirect
39. #1440C22semidirect
40. #1439C22semidirect
41. #1439C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
50. #1438C22semidirect
52. #1437C22semidirect
53. #1437C22semidirect
54. #1437C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
C2×#26132
#156402
#156172
#154482
#153482
#152972
#152682
#142872
#139022
#138282