Finite Group Databasebeta

← #4452771

Subgroups — #4452771

Order 512.

3862 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic, complemented
2C22×#394C211yesmaximal, characteristic
3#16374C218yesmaximal, complemented
4#16371C218yesmaximal, complemented
5#14820C212yesmaximal, complemented
6#14820C212yesmaximal, complemented
7#14717C212yesmaximal, complemented
8#14717C212yesmaximal, complemented
9#13923C212yesmaximal, complemented
10#13902C212yesmaximal
11#13796C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2659—28no—
20C2×#398C2214yescomplemented
21C2×#397C2214yescomplemented
22C2×#397C2214yescomplemented
23C2×#395C2214yescomplemented
24C2×#394—28no—
25C2×#394C2218yes—
26C2×#393C2214yescomplemented
27C2×#393C2214yescomplemented
28C2×#390C2214yes—
29C2×#390C2214yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#379C2214yes—
33C2×#379C2214yes—
34C2×C4×C22⋊C4C2212yes—
35C22×#95C2211yescharacteristic
36#1441C2214yescomplemented
37#1441C2214yescomplemented
38#1440C2218yescomplemented
39#1440C2218yescomplemented
40#1439C2214yescomplemented
41#1439C2214yescomplemented
42#1439C2218yescomplemented
43#1439C2218yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1438C2218yes—
47#1436C2218yescomplemented
48#1434C2218yescomplemented
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2212yes—
52C22⋊#95C2212yes—
53C26C2311yescharacteristic
54C26C2311yescharacteristic
55C23×Q8C2312yes—
56C23×D4—24no—
57C23×D4—28no—
58C23×D4—44no—
59C23×D4—44no—
60C23×D4C2314yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—48no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—28no—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4—44no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4—44no—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2318yes—
92C22×C22⋊C4C2318yes—
93C22×C22⋊C4C2318yes—
94C22×C22⋊C4C2318yes—
95C22×C42C2314yes—
96#398—216no—
97#397—48no—
98#397—48no—
99#395—216no—
100#394—232no—
101#393—48no—
102#393—48no—
103#390—48no—
104#390—48no—
105#380—28no—
106#380—28no—
107#379—48no—
108#379—48no—
109C4×C22⋊C4—28no—
110C2×#95D418yes—
111C2×#95C2318yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C25—22no—
124C25—22no—
125C25—24no—
126C25—24no—
127C25—24no—
128C25—24no—
129C25—24no—
130C25—24no—
131C25—24no—
132C25—24no—
133C25—42no—
134C25—42no—
135C25—42no—
136C25—42no—
137C25—42no—
138C25—42no—
139C25C2412yes—
140C25C2412yes—
141C25C2411yescharacteristic
142C22×Q8—24no—
143C22×Q8—44no—
144C22×Q8—44no—
145C22×Q8—44no—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—464no—
154C22×D4—416no—
155C22×D4—416no—
156C22×D4—416no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C2×D414yes—
174C23×C4C2×D414yes—
175C23×C4C2×D414yes—
176C23×C4#4118yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2418yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C22⋊Q8—432no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—416no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
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—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—216no—
218C2×C22⋊C4—216no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—416no—
224C2×C22⋊C4—416no—
225C2×C22⋊C4—416no—
226C2×C22⋊C4—416no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—48no—
237C2×C42—28no—
238C2×C42—44no—
239C2×C42—44no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—216no—
244#95—48no—
245#95—44no—
246#95—44no—
247#95—48no—
248#95—44no—
249#95—44no—
250#95—48no—
251#95C2×D4116yes—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—28no—
257C24—28no—
258C24—22no—
259C24—22no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—416no—
275C24—416no—
276C24—416no—
277C24—416no—
278C24—416no—
279C24—416no—
280C24—44no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—416no—
293C24—416no—
294C24—88no—
295C24—88no—
296C24—88no—
297C24—88no—
298C24—88no—
299C24—88no—
300C24C22×D412yes—
301C24C2×#4114yes—
302C24H(5,𝔽2)14yes—
303C24H(5,𝔽2)14yes—
304C24H(5,𝔽2)14yes—
305C242−1+414yes—
306C24C2511yescharacteristic
307C2×Q8—48no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—816no—
311C2×D4—232no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—416no—
316C2×D4—416no—
317C2×D4—4128no—
318C2×D4—416no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—416no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—416no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4#124116yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D414yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4114yes—
377C22×C4H(5,𝔽2)14yes—
378C4⋊C4—48no—
379C4⋊C4—416no—
380C4⋊C4—416no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C22⋊C4—416no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—416no—
390C22⋊C4—416no—
391C22⋊C4—816no—
392C22⋊C4—816no—
393C22⋊C4—816no—
394C22⋊C4—816no—
395C42—88no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—28no—
401C23—28no—
402C23—28no—
403C23—28no—
404C23—28no—
405C23—28no—
406C23—24no—
407C23—24no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—28no—
414C23—416no—
415C23—416no—
416C23—416no—
417C23—416no—
418C23—416no—
419C23—416no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—48no—
427C23—48no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—48no—
437C23—48no—
438C23—48no—
439C23—44no—
440C23—44no—
441C23—48no—
442C23—44no—
443C23—48no—
444C23—48no—
445C23—44no—
446C23—864no—
447C23—864no—
448C23—88no—
449C23—88no—
450C23—88no—
451C23—88no—
452C23—88no—
453C23—88no—
454C23—88no—
455C23—88no—
456C23C2×#12414yes—
457C23#53514yes—
458C23#53614yes—
459C23#542116yes—
460C23C23×D411yescharacteristic
461C23C22×#4112yes—
462C23C2×H(5,𝔽2)12yes—
463C23C2×H(5,𝔽2)12yes—
464C23C2×H(5,𝔽2)12yes—
465C23C2×H(5,𝔽2)12yes—
466C23C2×2−1+412yes—
467C23#58512yes—
468Q8—816no—
469D4—464no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—416no—
483C2×C4—416no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—88no—
511C2×C4—88no—
512C2×C4—88no—
513C2×C4C2×#12418yes—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—232no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—416no—
528C22—416no—
529C22—832no—
530C22—832no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—88no—
540C22—88no—
541C22#264018yes—
542C22C22×#12411yescharacteristic
543C22C2×#53511yescharacteristic
544C22C2×#53611yescharacteristic
545C22C2×#54214yes—
546C22#348411yescharacteristic
547C22#348411yescharacteristic
548C22#348411yescharacteristic
549C22#348711yescharacteristic
550C22#349014yes—
551C22#349414yes—
552C22#353414yes—
553C22#353514yes—
554C22#353614yes—
555C22#353714yes—
556C4—28no—
557C4—44no—
558C4—44no—
559C4—44no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4—88no—
564C2—28no—
565C2—84no—
566C2—84no—
567C2C2×#264011yescharacteristic
568C2#3536711yescharacteristic
569C2#3558411yescharacteristic
570C2#3671011yescharacteristic
571C2#3671111yescharacteristic
572C2#3671711yescharacteristic
573C2#3671811yescharacteristic
574C2#4064418yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2659C2semidirect
3. #16374C2semidirect
4. #16371C2semidirect
5. #14820C2semidirect
6. #14820C2semidirect
7. #14717C2semidirect
8. #14717C2semidirect
9. #13923C2semidirect
11. #13796C2semidirect
20. C2×#398C22semidirect
21. C2×#397C22semidirect
22. C2×#397C22semidirect
23. C2×#395C22semidirect
26. C2×#393C22semidirect
27. C2×#393C22semidirect
36. #1441C22semidirect
37. #1441C22semidirect
38. #1440C22semidirect
39. #1440C22semidirect
40. #1439C22semidirect
41. #1439C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
47. #1436C22semidirect
48. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
C22×#3942
#163742
#163712
#148202
#147172
#139232
#139022
#137962