Finite Group Databasebeta

← #4381119

Subgroups — #4381119

Order 512.

2282 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic
5#16675C214yesmaximal
6#15874C214yesmaximal, complemented
7#15867C214yesmaximal, complemented
8#15530C214yesmaximal, complemented
9#15517C214yesmaximal, complemented
10#14837C214yesmaximal
11#14660C211yesmaximal, characteristic
12#14297C211yesmaximal, characteristic
13#13939C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2625—28no—
18#2583—28no—
19#2484—28no—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#395C2212yes—
27#2313—28no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1446C2212yes—
45#1445C2212yes—
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C23×Q8C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4D418yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C42C2312yes—
104C22×C42C2314yes—
105C22×C42C2312yes—
106#400—28no—
107#400—28no—
108#398—28no—
109#398—28no—
110#398—28no—
111#397—216no—
112#395—28no—
113#391—28no—
114#389—216no—
115#389—28no—
116#387—216no—
117#387—216no—
118#387—216no—
119#385—28no—
120#385—28no—
121#385—28no—
122#381—24no—
123#381—28no—
124#381—28no—
125#380—24no—
126#380D418yes—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—28no—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C25C2411yescharacteristic
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C2×D414yes—
173C23×C4C2×D414yes—
174C23×C4C2×D414yes—
175C23×C4C2×D414yes—
176C23×C4#4114yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C22⋊Q8—416no—
190#117—416no—
191C2×C4⋊C4—232no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×D418yes—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4C2×D418yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C2×D418yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—44no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95C2×D418yes—
284C24—24no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24C22×D412yes—
290C24C22×D412yes—
291C24C2×#4112yes—
292C24C2511yescharacteristic
293C2×Q8—48no—
294C2×Q8—44no—
295C2×Q8—84no—
296C2×Q8—82no—
297C2×Q8—82no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
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—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—48no—
369C22×C4—84no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4#121116yes—
373C22×C4C22×D418yes—
374C22×C4C22×D414yes—
375C22×C4C22×D414yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C42−1+418yes—
383C4⋊C4—216no—
384C4⋊C4—216no—
385C4⋊C4—48no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C22⋊C4—216no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C42—48no—
403C42—84no—
404C42—88no—
405C23—24no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23C2×#12114yes—
421C23C23×D411yescharacteristic
422C23C23×D411yescharacteristic
423C23C22×#4111yescharacteristic
424C23C2×2−1+414yes—
425C23#58512yes—
426C23#58512yes—
427C23#58512yes—
428C23#58512yes—
429Q8—88no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—416no—
450C2×C4—416no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
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—44no—
499C2×C4—44no—
500C2×C4—88no—
501C2×C4C2×#12114yes—
502C2×C4C2×#12114yes—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—44no—
507C22—44no—
508C22—48no—
509C22—44no—
510C22—44no—
511C22C22×#12111yescharacteristic
512C22#344912yes—
513C22#348511yescharacteristic
514C22#348512yes—
515C22#348512yes—
516C22#348612yes—
517C22#348711yescharacteristic
518C22#349111yescharacteristic
519C22#349111yescharacteristic
520C22#349312yes—
521C22#349912yes—
522C22#350912yes—
523C22#355314yes—
524C22#355714yes—
525C22#356414yes—
526C22#356514yes—
527C4—24no—
528C4—24no—
529C4—44no—
530C4—44no—
531C4—48no—
532C4—48no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—84no—
538C4—84no—
539C2—44no—
540C2#3515011yescharacteristic
541C2#3544711yescharacteristic
542C2#3566711yescharacteristic
543C2#4141714yes—
544C2#4681812yes—
545C2#4681912yes—
546C2#4879912yes—
547C2#4880012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #15874C2semidirect
7. #15867C2semidirect
8. #15530C2semidirect
9. #15517C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24842
C2×#26252
C2×#23132
#166752
#158742
#158672
#155302
#155172
#148372
#146602
#142972
#139392