Finite Group Databasebeta

← #4433059

Subgroups — #4433059

Order 512.

2302 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic
3C2×#2633C211yesmaximal, characteristic
4C2×#2357C211yesmaximal, characteristic
5#16596C214yesmaximal
6#16394C212yesmaximal, complemented
7#16393C212yesmaximal, complemented
8#15758C214yesmaximal, complemented
9#15751C214yesmaximal, complemented
10#14844C214yesmaximal, complemented
11#14841C212yesmaximal, complemented
12#14840C212yesmaximal, complemented
13#14296C211yesmaximal, characteristic, complemented
14#14289C211yesmaximal, characteristic, complemented
15#14160C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#2822—28no—
20#2633—28no—
21#2562—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2214yes—
30C2×#397C2212yes—
31C2×#394C2212yes—
32C2×#394C2212yes—
33#2357—28no—
34C2×#390C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#383C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#381C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1445C2211yescharacteristic
49#1445C2214yes—
50#1445C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2211yescharacteristic
53#1444C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78#1434C2214yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#125—24no—
82C2×#125—24no—
83C2×#125—24no—
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#400—28no—
120#400—28no—
121#400—28no—
122#399—28no—
123#398—216no—
124#398—28no—
125#398—28no—
126#398—28no—
127#397—28no—
128#394—28no—
129#394—28no—
130#390—28no—
131#388—216no—
132#388—216no—
133#385—28no—
134#385—28no—
135#383—24no—
136#382D418yes—
137#381—28no—
138#380—28no—
139#380—28no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—24no—
143C4×C22⋊C4—28no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C25C2412yes—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C2×D414yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4C2×D418yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197#125—416no—
198#125—416no—
199C4×D4—232no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
243C2×C4⋊C4C2×D418yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×D418yes—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—24no—
272C2×C42—44no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42C2×D414yes—
278C2×C42C2×D414yes—
279C2×C42#4114yes—
280#95—22no—
281#95—22no—
282#95—28no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95C2×D418yes—
307#95C2×D418yes—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D414yes—
319C24C2511yescharacteristic
320C2×D4—28no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
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—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—216no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4C22×D414yes—
394C22×C4C22×D418yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C42−1+414yes—
405C4⋊C4—28no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—88no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C22⋊C4—216no—
422C22⋊C4—416no—
423C22⋊C4—416no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—88no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—88no—
430C42—48no—
431C42—84no—
432C42—88no—
433C42#121116yes—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23C23×D411yescharacteristic
458C23C23×D411yescharacteristic
459C23C22×#4111yescharacteristic
460C23C2×2−1+412yes—
461C23#58512yes—
462C23#58512yes—
463C23#58512yes—
464C23#58514yes—
465D4—416no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
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—816no—
539C2×C4C2×#12114yes—
540C2×C4C2×#12114yes—
541C2×C4C2×#12114yes—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22C22×#12111yescharacteristic
554C22#344912yes—
555C22#346112yes—
556C22#348511yescharacteristic
557C22#348512yes—
558C22#348711yescharacteristic
559C22#348912yes—
560C22#349312yes—
561C22#349911yescharacteristic
562C22#350012yes—
563C22#350011yescharacteristic
564C22#351012yes—
565C22#355314yes—
566C22#355714yes—
567C22#356614yes—
568C22#356714yes—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—44no—
573C4—48no—
574C4—44no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4—82no—
583C2—84no—
584C2#3515011yescharacteristic
585C2#3530811yescharacteristic
586C2#3561711yescharacteristic
587C2#4138212yes—
588C2#4138312yes—
589C2#4174614yes—
590C2#5121112yes—
591C2#5121212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2822C2semidirect
6. #16394C2semidirect
7. #16393C2semidirect
8. #15758C2semidirect
9. #15751C2semidirect
10. #14844C2semidirect
11. #14841C2semidirect
12. #14840C2semidirect
13. #14296C2semidirect
14. #14289C2semidirect
15. #14160C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25622
C2×#26332
C2×#23572
#165962
#163942
#163932
#157582
#157512
#148442
#148412
#148402
#142962
#142892
#141602