Finite Group Databasebeta

← #4381120

Subgroups — #4381120

Order 512.

2456 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2624C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic, complemented
5#16661C214yesmaximal, complemented
6#15902C214yesmaximal, complemented
7#15895C214yesmaximal, complemented
8#15584C214yesmaximal, complemented
9#15583C214yesmaximal, complemented
10#14838C214yesmaximal, complemented
11#14659C211yesmaximal, characteristic
12#14297C211yesmaximal, characteristic, complemented
13#13939C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2624—28no—
18#2591—28no—
19#2484—28no—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#394C2214yes—
26C2×#394C2212yes—
27#2313—28no—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1445C2214yes—
45#1445C2212yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2214yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#121—24no—
77C2×#121—24no—
78C2×#121—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4D418yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2312yes—
110#400—28no—
111#400—28no—
112#399—28no—
113#398—28no—
114#397—28no—
115#394—28no—
116#394—216no—
117#390—28no—
118#389—28no—
119#388—216no—
120#387—216no—
121#387—216no—
122#387—216no—
123#386—28no—
124#386—28no—
125#385—28no—
126#381—24no—
127#381—28no—
128#381—28no—
129#380—24no—
130#380D418yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—28no—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—44no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D414yes—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4C2×D414yes—
182C23×C4#4114yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194#121—416no—
195#117—416no—
196C2×C4⋊C4—232no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C2×D418yes—
225C2×C4⋊C4C2×D418yes—
226C2×C4⋊C4C2×D418yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4C2×D418yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—44no—
270C2×C42—44no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95C2×D418yes—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24C22×D412yes—
309C24C22×D412yes—
310C24C2×#4112yes—
311C24C2511yescharacteristic
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—84no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—88no—
321C2×D4—88no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
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—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—84no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4#121116yes—
393C22×C4C22×D414yes—
394C22×C4C22×D418yes—
395C22×C4C22×D414yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C4⋊C4—216no—
403C4⋊C4—216no—
404C4⋊C4—48no—
405C4⋊C4—416no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C22⋊C4—48no—
413C22⋊C4—416no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C42—48no—
425C42—84no—
426C42—88no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—44no—
444C23—42no—
445C23—44no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23C2×#12114yes—
464C23C23×D411yescharacteristic
465C23C23×D411yescharacteristic
466C23C22×#4111yescharacteristic
467C23#58512yes—
468C23#58512yes—
469C23#58512yes—
470C23#58512yes—
471C23#58514yes—
472D4—88no—
473D4—816no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—416no—
491C2×C4—416no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
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—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—88no—
540C2×C4C2×#12114yes—
541C2×C4C2×#12114yes—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—44no—
546C22—44no—
547C22—48no—
548C22—44no—
549C22—44no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—84no—
558C22—84no—
559C22C22×#12111yescharacteristic
560C22#344912yes—
561C22#348511yescharacteristic
562C22#348512yes—
563C22#348512yes—
564C22#348711yescharacteristic
565C22#348712yes—
566C22#349111yescharacteristic
567C22#349111yescharacteristic
568C22#349312yes—
569C22#350012yes—
570C22#351012yes—
571C22#356614yes—
572C22#356614yes—
573C22#356614yes—
574C22#356714yes—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—44no—
579C4—48no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—84no—
585C4—84no—
586C2—44no—
587C2—162no—
588C2#3515011yescharacteristic
589C2#3544711yescharacteristic
590C2#3567711yescharacteristic
591C2#4141614yes—
592C2#4681712yes—
593C2#4682012yes—
594C2#4879812yes—
595C2#4880112yes—
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
2. C2×#2484C2semidirect
4. C2×#2313C2semidirect
5. #16661C2semidirect
6. #15902C2semidirect
7. #15895C2semidirect
8. #15584C2semidirect
9. #15583C2semidirect
10. #14838C2semidirect
12. #14297C2semidirect
13. #13939C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#24842
C2×#26242
C2×#23132
#166612
#159022
#158952
#155842
#155832
#148382
#146592
#142972
#139392