Finite Group Databasebeta

← #676394

Subgroups — #676394

Order 512.

2746 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#15555C214yesmaximal, complemented
4#15392C214yesmaximal, complemented
5#15221C212yesmaximal, complemented
6#15133C212yesmaximal, complemented
7#14840C214yesmaximal, complemented
8#14709C211yesmaximal, characteristic, complemented
9#14160C211yesmaximal, characteristic, complemented
10#14044C214yesmaximal, complemented
11#14018C212yesmaximal
12#14018C212yesmaximal
13#13954C211yesmaximal, characteristic, complemented
14C4×#384C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19#2566—28no—
20C2×#398C2214yes—
21C2×#396C2214yes—
22#2349—28no—
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#384C2211yescharacteristic
34C2×#384C2212yes—
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2214yes—
44C22×#95C2212yes—
45C2×C43C2211yescharacteristic
46#1445C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2212yes—
60#1437C2212yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×D4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125—24no—
83C2×C4×D4—28no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—44no—
87C2×C4×D4—44no—
88C2×C4×D4D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#398—48no—
122#396—216no—
123#390—24no—
124#390—24no—
125#390Q8116yes—
126#389—28no—
127#389—44no—
128#389—44no—
129#385—216no—
130#385—216no—
131#385—216no—
132#385—48no—
133#384—24no—
134#384—28no—
135#382—24no—
136#382—24no—
137#380—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—44no—
142C4×C4⋊C4—44no—
143C4×C22⋊C4—48no—
144C2×#95—28no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C43—24no—
170C25C2412yes—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4C2×D414yes—
175C22×D4C2×Q818yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D418yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202#125—416no—
203C4×D4—232no—
204C4×D4—432no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×Q818yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D418yes—
272C2×C22⋊C4C2×Q8116yes—
273C2×C22⋊C4C2×Q8116yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42C2×D414yes—
293C2×C42C2×Q818yes—
294C2×C42#4114yes—
295C2×C42#4114yes—
296#95—22no—
297#95—24no—
298#95—28no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24C22×D414yes—
328C24C22×Q818yes—
329C24C2511yescharacteristic
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4C4⋊Q8116yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×Q814yes—
419C22×C4C22×Q818yes—
420C22×C4C22×Q814yes—
421C22×C4C22×Q814yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)14yes—
433C4⋊C4—28no—
434C4⋊C4—216no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—216no—
451C22⋊C4—232no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—88no—
456C42—28no—
457C42—216no—
458C42—48no—
459C42—44no—
460C42—44no—
461C42—44no—
462C42—48no—
463C42—84no—
464C42#123116yes—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C2×C4⋊Q818yes—
486C23C23×D411yescharacteristic
487C23C23×Q812yes—
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)12yes—
493C23#58512yes—
494C23#58512yes—
495D4—232no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—816no—
575C2×C4—832no—
576C2×C4C2×#12314yes—
577C2×C4C2×#12314yes—
578C2×C4C2×#12314yes—
579C2×C4C2×C4⋊Q814yes—
580C2×C4#53514yes—
581C2×C4#55418yes—
582C2×C4#55418yes—
583C2×C4#55418yes—
584C22—216no—
585C22—28no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—48no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22C22×#12311yescharacteristic
594C22C22×C4⋊Q811yescharacteristic
595C22#344411yescharacteristic
596C22C2×#53511yescharacteristic
597C22#345511yescharacteristic
598C22D4×#4112yes—
599C22C2×#55412yes—
600C22C2×#55412yes—
601C22C2×#55412yes—
602C22Q8×#4112yes—
603C22#348012yes—
604C22#349412yes—
605C22#349612yes—
606C22#350112yes—
607C22#350412yes—
608C22#350512yes—
609C22#350811yescharacteristic
610C22#351011yescharacteristic
611C22#351912yes—
612C22#354612yes—
613C22#355112yes—
614C4—24no—
615C4—24no—
616C4—28no—
617C4—44no—
618C4—44no—
619C4—48no—
620C4—44no—
621C4—44no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4—82no—
627C4#250218yes—
628C2—48no—
629C2#3474411yescharacteristic
630C2C2×#250211yescharacteristic
631C2#3554111yescharacteristic
632C2#3626112yes—
633C2#3657012yes—
634C2#3960212yes—
635C2#3994812yes—
636C2#5945712yes—
637C2#5990912yes—
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
3. #15555C2semidirect
4. #15392C2semidirect
5. #15221C2semidirect
6. #15133C2semidirect
7. #14840C2semidirect
8. #14709C2semidirect
9. #14160C2semidirect
10. #14044C2semidirect
13. #13954C2semidirect
14. C4×#384C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#23492
#155552
#153922
#152212
#151332
#148402
#147092
#141602
#140442
#140182
#139542
C4×#3842
C4×#3902