Finite Group Databasebeta

← #4456846

Subgroups — #4456846

Order 512.

2290 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic
5#17061C212yesmaximal, complemented
6#16806C212yesmaximal, complemented
7#16804C212yesmaximal, complemented
8#16705C212yesmaximal, complemented
9#16675C214yesmaximal
10#16396C214yesmaximal
11#15005C214yesmaximal, complemented
12#14967C214yesmaximal, complemented
13#14292C211yesmaximal, characteristic
14#13953C211yesmaximal, characteristic
15#13853C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2583—28no—
20#2499—28no—
21#2484—28no—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#395C2212yes—
27#2313—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2214yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1436C2212yes—
73#1436C2214yes—
74#1436C2212yes—
75C4⋊#95C2212yes—
76#1434C2212yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#400—28no—
118#398—28no—
119#398—28no—
120#397—216no—
121#395—28no—
122#391—24no—
123#391—24no—
124#389—216no—
125#387—216no—
126#387—28no—
127#387—28no—
128#387—28no—
129#387—28no—
130#387—216no—
131#385—28no—
132#385—28no—
133#385—28no—
134#384—24no—
135#382—24no—
136#381—24no—
137#381—28no—
138#381—28no—
139#380—28no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4D418yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—42no—
178C22×Q8—42no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4C2×D414yes—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4#4114yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C22⋊Q8—416no—
209#117—416no—
210C2×C4⋊C4—232no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D418yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C2×D418yes—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C2×D414yes—
278C2×C42C2×D414yes—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95C2×D418yes—
309C24—24no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24C22×D412yes—
315C24C22×D412yes—
316C24C2×#4112yes—
317C24C2511yescharacteristic
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—84no—
322C2×Q8—84no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—84no—
402C22×C4#121116yes—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C42−1+418yes—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—416no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C42—44no—
434C42—44no—
435C42—48no—
436C42—84no—
437C42—82no—
438C42—82no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23C2×#12114yes—
455C23C23×D411yescharacteristic
456C23C23×D411yescharacteristic
457C23C22×#4111yescharacteristic
458C23C2×2−1+414yes—
459C23#58512yes—
460C23#58512yes—
461C23#58512yes—
462C23#58512yes—
463Q8—88no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—416no—
487C2×C4—416no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
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—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—88no—
541C2×C4C2×#12114yes—
542C2×C4C2×#12114yes—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—48no—
551C22C22×#12111yescharacteristic
552C22#344912yes—
553C22#344912yes—
554C22#345511yescharacteristic
555C22#345511yescharacteristic
556C22#348511yescharacteristic
557C22#348511yescharacteristic
558C22#348512yes—
559C22#348612yes—
560C22#349112yes—
561C22#349912yes—
562C22#350912yes—
563C22#355314yes—
564C22#355714yes—
565C22#356414yes—
566C22#356514yes—
567C4—24no—
568C4—24no—
569C4—44no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4—48no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—84no—
581C2—44no—
582C2#3463311yescharacteristic
583C2#3544711yescharacteristic
584C2#3566711yescharacteristic
585C2#3701112yes—
586C2#3701712yes—
587C2#4138312yes—
588C2#4138412yes—
589C2#4141714yes—
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
5. #17061C2semidirect
6. #16806C2semidirect
7. #16804C2semidirect
8. #16705C2semidirect
11. #15005C2semidirect
12. #14967C2semidirect

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×#24992
C2×#23132
#170612
#168062
#168042
#167052
#166752
#163962
#150052
#149672
#142922
#139532
#138532