Finite Group Databasebeta

← #4453284

Subgroups — #4453284

Order 512.

4010 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
1C22×#394C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic, complemented
3#16271C218yesmaximal, complemented
4#16267C218yesmaximal, complemented
5#15015C212yesmaximal, complemented
6#14821C212yesmaximal, complemented
7#14820C212yesmaximal, complemented
8#14734C212yesmaximal, complemented
9#14341C211yesmaximal, characteristic, complemented
10#14127C212yesmaximal, complemented
11#13867C212yesmaximal
12C24×D4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C2×#399C2214yescomplemented
20C2×#398C2214yescomplemented
21C2×#397C2214yescomplemented
22C2×#394—28no—
23C2×#394C2218yes—
24#2315—28no—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×#385C2214yescomplemented
29C2×#385C2214yescomplemented
30C2×#385C2214yescomplemented
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2212yes—
34C2×#379C2214yes—
35C22×#95C2211yescharacteristic
36#1445C2214yescomplemented
37#1443C2218yescomplemented
38#1442C2214yescomplemented
39#1441C2214yescomplemented
40#1441C2218yescomplemented
41#1440C2218yescomplemented
42#1440C2218yescomplemented
43#1439C2218yescomplemented
44#1439C2214yescomplemented
45#1439C2218yescomplemented
46#1439C2214yescomplemented
47#1438C2218yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2214yescomplemented
53C26C2311yescharacteristic
54C26C2311yescharacteristic
55C23×D4—24no—
56C23×D4—28no—
57C23×D4—44no—
58C23×D4—44no—
59C23×D4C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2312yes—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2318yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—28no—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4—44no—
81C22×C22⋊C4—44no—
82C22×C22⋊C4—48no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4D418yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2318yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2318yes—
96#399—48no—
97#398—48no—
98#397—48no—
99#394—232no—
100#386—48no—
101#386—48no—
102#385—216no—
103#385—216no—
104#385—48no—
105#385—48no—
106#380—28no—
107#380—28no—
108#379—28no—
109#379—48no—
110C2×#95—24no—
111C2×#95C2314yes—
112C2×#95C2318yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2318yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2318yes—
122C2×#95C2318yes—
123C25—24no—
124C25—24no—
125C25—22no—
126C25—24no—
127C25—24no—
128C25—24no—
129C25—24no—
130C25—42no—
131C25—42no—
132C25—42no—
133C25—42no—
134C25—42no—
135C25—42no—
136C25—42no—
137C25C2×D414yes—
138C25#4118yes—
139C25C2412yes—
140C25C2412yes—
141C25C2411yescharacteristic
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—28no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—416no—
150C22×D4—464no—
151C22×D4—416no—
152C22×D4—416no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4C2418yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
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—216no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—416no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—44no—
204C2×C22⋊C4—232no—
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—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—216no—
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—416no—
231C2×C22⋊C4—416no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—432no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4C2×D4116yes—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—216no—
246#95—44no—
247#95—48no—
248#95—48no—
249#95—48no—
250#95—44no—
251#95—48no—
252#95—44no—
253#95—44no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—28no—
265C24—28no—
266C24—232no—
267C24—28no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—24no—
273C24—22no—
274C24—24no—
275C24—24no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—416no—
280C24—44no—
281C24—416no—
282C24—416no—
283C24—416no—
284C24—416no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—416no—
292C24—88no—
293C24—88no—
294C24—88no—
295C24—88no—
296C24—88no—
297C24—88no—
298C24—88no—
299C24#123116yes—
300C24C22×D414yes—
301C24C22×D412yes—
302C24C2×#4114yes—
303C24C2×#4114yes—
304C24C2×#4114yes—
305C24H(5,𝔽2)14yes—
306C24H(5,𝔽2)14yes—
307C242−1+414yes—
308C24C2511yescharacteristic
309C2×D4—232no—
310C2×D4—416no—
311C2×D4—416no—
312C2×D4—416no—
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—4128no—
316C2×D4—416no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—88no—
378C22×C4—816no—
379C22×C4—816no—
380C22×C4C22×D414yes—
381C22×C4C22×D414yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4H(5,𝔽2)14yes—
384C4⋊C4—416no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—816no—
390C22⋊C4—232no—
391C22⋊C4—416no—
392C22⋊C4—416no—
393C22⋊C4—416no—
394C22⋊C4—416no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C22⋊C4—88no—
399C22⋊C4—816no—
400C22⋊C4—816no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—28no—
406C23—28no—
407C23—28no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—24no—
412C23—24no—
413C23—28no—
414C23—28no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—28no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—232no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—416no—
429C23—416no—
430C23—416no—
431C23—416no—
432C23—416no—
433C23—48no—
434C23—48no—
435C23—48no—
436C23—48no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—48no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23—48no—
449C23—48no—
450C23—864no—
451C23—88no—
452C23—88no—
453C23—88no—
454C23—88no—
455C23—864no—
456C23—88no—
457C23—88no—
458C23—88no—
459C23—88no—
460C23C2×#12318yes—
461C23C2×#12314yes—
462C23#53514yes—
463C23#53614yes—
464C23#539116yes—
465C23C23×D411yescharacteristic
466C23C22×#4112yes—
467C23C2×H(5,𝔽2)12yes—
468C23C2×H(5,𝔽2)12yes—
469C23C2×H(5,𝔽2)12yes—
470C23C2×H(5,𝔽2)12yes—
471C23C2×2−1+412yes—
472C23#58512yes—
473D4—464no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—416no—
485C2×C4—416no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—88no—
516C2×C4—88no—
517C2×C4—88no—
518C2×C4—816no—
519C2×C4—816no—
520C22—24no—
521C22—24no—
522C22—232no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—28no—
528C22—28no—
529C22—28no—
530C22—416no—
531C22—416no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—416no—
538C22—44no—
539C22—44no—
540C22—832no—
541C22—832no—
542C22—82no—
543C22—82no—
544C22—88no—
545C22—82no—
546C22—82no—
547C22#262318yes—
548C22C22×#12311yescharacteristic
549C22C2×#53511yescharacteristic
550C22C2×#53611yescharacteristic
551C22C2×#53914yes—
552C22#348411yescharacteristic
553C22#348411yescharacteristic
554C22#348411yescharacteristic
555C22#348511yescharacteristic
556C22#349014yes—
557C22#350514yes—
558C22#353414yes—
559C22#353614yes—
560C22#353714yes—
561C22#356114yes—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—88no—
567C4—84no—
568C4—82no—
569C4—84no—
570C2—28no—
571C2—48no—
572C2—84no—
573C2C2×#262311yescharacteristic
574C2#3535011yescharacteristic
575C2#3535111yescharacteristic
576C2#3671711yescharacteristic
577C2#3671811yescharacteristic
578C2#3672111yescharacteristic
579C2#3696611yescharacteristic
580C2#4053818yes—
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
2. C2×#2315C2semidirect
3. #16271C2semidirect
4. #16267C2semidirect
5. #15015C2semidirect
6. #14821C2semidirect
7. #14820C2semidirect
8. #14734C2semidirect
9. #14341C2semidirect
10. #14127C2semidirect
19. C2×#399C22semidirect
20. C2×#398C22semidirect
21. C2×#397C22semidirect
28. C2×#385C22semidirect
29. C2×#385C22semidirect
30. C2×#385C22semidirect
36. #1445C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
41. #1440C22semidirect
42. #1440C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
52. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3942
C2×#23152
#162712
#162672
#150152
#148212
#148202
#147342
#143412
#141272
#138672