Finite Group Databasebeta

← #4930649

Subgroups — #4930649

Order 512.

4874 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2432C211yesmaximal, characteristic
4C2×#2625C211yesmaximal, characteristic, complemented
5#14906C214yesmaximal, complemented
6#14786C214yesmaximal, complemented
7#14731C218yesmaximal, complemented
8#14027C212yesmaximal, complemented
9#14024C212yesmaximal, complemented
10#13977C212yesmaximal, complemented
11#13976C212yesmaximal, complemented
12#13937C211yesmaximal, characteristic
13#13904C211yesmaximal, characteristic, complemented
14#13853C211yesmaximal, characteristic, complemented
15C23×#41C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C2×#520—28no—
20C2×#520—28no—
21C2×#520—28no—
22C22×C4×Q8C2212yes—
23C22×C4×D4C2214yes—
24C22×C4×D4C2212yes—
25C22×#117C2212yes—
26C22×#117C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29#2625—28no—
30#2502—28no—
31#2432—28no—
32C2×#400C2212yes—
33C2×#398C2212yes—
34C2×#398C2214yes—
35C2×#397C2214yes—
36C2×#394C2214yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic
43C2×#389C2212yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#385C2218yes—
47C2×#384C2212yes—
48C2×#381C2214yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1444C2218yes—
56#1441C2218yes—
57#1440C2214yes—
58#1440C2218yes—
59#1439C2218yes—
60#1438C2214yes—
61#1436C2218yes—
62#1436C2218yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2218yes—
67C4×#95C2212yes—
68C4×#95C2212yes—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41D418yes—
74C23×Q8C2311yescharacteristic
75C23×Q8C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—28no—
84C2×#124—24no—
85C2×#124—28no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89#520—2128no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×D4—216no—
94C2×C4×D4—28no—
95C2×C4×D4—216no—
96C2×C4×D4—28no—
97C2×C4×D4—216no—
98C2×C4×D4—28no—
99C2×#117—28no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117D418yes—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—28no—
110C22×C4⋊C4D418yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2318yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2318yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136#400—28no—
137#398—28no—
138#398—216no—
139#397—216no—
140#394—216no—
141#391—24no—
142#391—28no—
143#391—28no—
144#391—28no—
145#390—24no—
146#390—28no—
147#389—28no—
148#385—216no—
149#385—216no—
150#385—416no—
151#384—28no—
152#381—48no—
153#380—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C25C2411yescharacteristic
172C25C2412yes—
173C2×#41—264no—
174C2×#41—264no—
175C2×#41—232no—
176C2×#41—232no—
177C2×#41—232no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8C2×D414yes—
187C22×Q8#4118yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4C2×D418yes—
196C22×D4C2×D414yes—
197C22×D4#4118yes—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—48no—
212C23×C4—48no—
213C23×C4C2×D418yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D418yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C4⋊Q8—416no—
231#124—416no—
232C22⋊Q8—416no—
233C4×Q8—264no—
234C4×D4—2128no—
235C4×D4—264no—
236#117—264no—
237#117—232no—
238C2×C4⋊C4—264no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—216no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C2×D418yes—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42C2×D418yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—216no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—216no—
317#95—216no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—48no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—22no—
333C24—24no—
334C24C22×D412yes—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C22×D414yes—
338C24C2×#4118yes—
339C24C2511yescharacteristic
340#41—2256no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—216no—
344C2×Q8—216no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8#123116yes—
351C2×D4—232no—
352C2×D4—232no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—28no—
356C2×D4—216no—
357C2×D4—28no—
358C2×D4—216no—
359C2×D4—216no—
360C2×D4#123116yes—
361C22×C4—232no—
362C22×C4—232no—
363C22×C4—216no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—232no—
398C22×C4—216no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—416no—
420C22×C4—816no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—816no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4#120116yes—
428C22×C4C22×D414yes—
429C22×C4C22×D418yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C42−1+418yes—
443C4⋊C4—232no—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—816no—
453C4⋊C4—88no—
454C22⋊C4—216no—
455C22⋊C4—232no—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—432no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—84no—
463C42—232no—
464C42—216no—
465C42—44no—
466C42—44no—
467C42—48no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—22no—
485C23—216no—
486C23—216no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—48no—
495C23—44no—
496C23—44no—
497C23—48no—
498C23C2×#12014yes—
499C23C2×#12318yes—
500C23C23×D411yescharacteristic
501C23C23×D411yescharacteristic
502C23C23×D411yescharacteristic
503C23C22×#4112yes—
504C23C22×#4112yes—
505C23C2×H(5,𝔽2)12yes—
506C23C2×2−1+414yes—
507C23#58512yes—
508Q8—232no—
509Q8—88no—
510D4—264no—
511D4—232no—
512C2×C4—264no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—232no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—216no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—816no—
570C2×C4—832no—
571C2×C4C2×#12014yes—
572C2×C4C2×#12014yes—
573C2×C4C2×#12318yes—
574C2×C4C2×#12314yes—
575C2×C4C2×#12314yes—
576C2×C4#52518yes—
577C2×C4#52518yes—
578C22—24no—
579C22—28no—
580C22—24no—
581C22—216no—
582C22—24no—
583C22—44no—
584C22—48no—
585C22—44no—
586C22—48no—
587C22—416no—
588C22—44no—
589C22—48no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22C22×#12011yescharacteristic
595C22C22×#12311yescharacteristic
596C22C22×#12311yescharacteristic
597C22C2×#52512yes—
598C22C2×#52512yes—
599C22#348612yes—
600C22#348812yes—
601C22#349612yes—
602C22#349811yescharacteristic
603C22#349912yes—
604C22#349912yes—
605C22#350011yescharacteristic
606C22#351514yes—
607C22#351614yes—
608C22#352014yes—
609C22#352214yes—
610C4—24no—
611C4—28no—
612C4—44no—
613C4—48no—
614C4—44no—
615C4—48no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4#284918yes—
622C2—44no—
623C2—48no—
624C2#3551311yescharacteristic
625C2#3553512yes—
626C2#3562212yes—
627C2#3566611yescharacteristic
628C2#3572414yes—
629C2#3599612yes—
630C2#3599812yes—
631C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2502C2semidirect
4. C2×#2625C2semidirect
5. #14906C2semidirect
6. #14786C2semidirect
7. #14731C2semidirect
8. #14027C2semidirect
9. #14024C2semidirect
10. #13977C2semidirect
11. #13976C2semidirect
13. #13904C2semidirect
14. #13853C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#25022
C2×#24322
C2×#26252
#149062
#147862
#147312
#140272
#140242
#139772
#139762
#139372
#139042
#138532