Finite Group Databasebeta

← #4607239

Subgroups — #4607239

Order 512.

2224 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×#2804C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#17098C214yesmaximal, complemented
4#16694C212yesmaximal, complemented
5#16570C212yesmaximal, complemented
6#16447C214yesmaximal, complemented
7#16269C214yesmaximal, complemented
8#15984C212yesmaximal
9#15398C214yesmaximal, complemented
10#15342C212yesmaximal, complemented
11#15050C212yesmaximal, complemented
12#14893C212yesmaximal, complemented
13#14667C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2804—28no—
17#2500—28no—
18C2×#400C2212yes—
19C2×#399C2212yes—
20C2×#397C2214yes—
21C2×#397C2212yes—
22C2×#394C2214yes—
23C2×#393C2214yes—
24C2×#393C2212yes—
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#381C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1446C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1440C2214yes—
53#1440C2212yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1438C2212yes—
64#1436C2214yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#121—24no—
75C2×#121—42no—
76C2×#121—42no—
77C2×#117—24no—
78C2×#117—42no—
79C2×#117—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#400—28no—
105#399—44no—
106#397—44no—
107#397—48no—
108#394—216no—
109#393—216no—
110#393—44no—
111#391—28no—
112#390—24no—
113#390D418yes—
114#389—44no—
115#388—216no—
116#386—28no—
117#386—44no—
118#385—216no—
119#385—48no—
120#384—24no—
121#382—24no—
122#381—44no—
123C4×C22⋊C4—24no—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2412yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4C2×D414yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186#121—416no—
187#117—416no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
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—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4C2×D4116yes—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D414yes—
254#95—24no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—42no—
276#95—44no—
277#95—42no—
278C24—21no—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—22no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24C22×D414yes—
294C24H(5,𝔽2)12yes—
295C24C2511yescharacteristic
296C2×Q8—48no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×D4—28no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4—88no—
309C2×D4—84no—
310C2×D4—88no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
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—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
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—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4—84no—
377C22×C4—88no—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×D412yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+414yes—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—84no—
389C4⋊C4—88no—
390C4⋊C4—84no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C22⋊C4—416no—
394C22⋊C4—48no—
395C22⋊C4—416no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—416no—
399C22⋊C4—84no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—84no—
403C22⋊C4—88no—
404C22⋊C4—84no—
405C42—44no—
406C42—48no—
407C42—44no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—28no—
412C23—24no—
413C23—28no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—41no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23C23×D411yescharacteristic
446C23C2×H(5,𝔽2)11yescharacteristic
447C23C2×H(5,𝔽2)11yescharacteristic
448C23C2×H(5,𝔽2)12yes—
449C23C2×2−1+412yes—
450C23#58514yes—
451C23#58512yes—
452C23#58512yes—
453Q8—816no—
454D4—88no—
455D4—816no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—416no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—816no—
520C2×C4#53614yes—
521C22—48no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—42no—
528C22—48no—
529C22—48no—
530C22—84no—
531C22—84no—
532C22—82no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22C2×#53611yescharacteristic
539C22#348411yescharacteristic
540C22#348512yes—
541C22#348511yescharacteristic
542C22#348812yes—
543C22#352911yescharacteristic
544C22#353312yes—
545C22#353512yes—
546C22#353512yes—
547C22#354014yes—
548C22#354012yes—
549C22#354911yescharacteristic
550C22#355812yes—
551C22#355914yes—
552C22#356514yes—
553C22#356614yes—
554C4—24no—
555C4—44no—
556C4—42no—
557C4—44no—
558C4—44no—
559C4—84no—
560C4—84no—
561C4—82no—
562C4—82no—
563C4—84no—
564C4—84no—
565C2—82no—
566C2—84no—
567C2#3672011yescharacteristic
568C2#4268212yes—
569C2#4622511yescharacteristic
570C2#5017112yes—
571C2#5205411yescharacteristic
572C2#5290112yes—
573C2#5538112yes—
574C2#5814314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2804C2semidirect
3. #17098C2semidirect
4. #16694C2semidirect
5. #16570C2semidirect
6. #16447C2semidirect
7. #16269C2semidirect
9. #15398C2semidirect
10. #15342C2semidirect
11. #15050C2semidirect
12. #14893C2semidirect
13. #14667C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#25002
#170982
#166942
#165702
#164472
#162692
#159842
#153982
#153422
#150502
#148932
#146672