Finite Group Databasebeta

← #4930363

Subgroups — #4930363

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#14965C214yesmaximal
4#14965C214yesmaximal
5#14850C214yesmaximal, complemented
6#14794C214yesmaximal, complemented
7#14258C211yesmaximal, characteristic
8#14142C212yesmaximal, complemented
9#14138C211yesmaximal, characteristic
10#14129C211yesmaximal, characteristic
11#14021C212yesmaximal, complemented
12#13981C212yesmaximal, complemented
13#13904C211yesmaximal, characteristic
14#13884C212yesmaximal, complemented
15#13844C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2502—28no—
21C2×#398C2214yes—
22C2×#391C2212yes—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#387C2214yes—
29C2×#385C2214yes—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#381C2214yes—
33C2×#380C2211yescharacteristic
34#2274—28no—
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43#1445C2212yes—
44#1445C2212yes—
45#1444C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2212yes—
58#1440C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—28no—
86C2×#117—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2314yes—
117#398—216no—
118#391—24no—
119#391—24no—
120#391—28no—
121#391—28no—
122#389—28no—
123#387—48no—
124#387—48no—
125#385—216no—
126#384—24no—
127#384—24no—
128#381—48no—
129#380—24no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—216no—
133C4×C4⋊C4—28no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—24no—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25C2411yescharacteristic
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×D414yes—
180C23×C4C2×D418yes—
181C23×C4#4114yes—
182C23×C4#4114yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C22⋊Q8—416no—
198#117—232no—
199#117—232no—
200C2×C4⋊C4—232no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4#4118yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42#4118yes—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—28no—
279#95—28no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24C22×D412yes—
311C24C2×#4112yes—
312C24C2×#4112yes—
313C24C2511yescharacteristic
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—48no—
318C2×Q8—48no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—216no—
324C22×C4—216no—
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—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
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—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—416no—
395C22×C4—88no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4#123116yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C2×#4118yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C42−1+414yes—
414C22×C42−1+414yes—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—216no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—216no—
421C4⋊C4—48no—
422C4⋊C4—416no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C22⋊C4—216no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—216no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—84no—
435C42—216no—
436C42—216no—
437C42—44no—
438C42—44no—
439C42—48no—
440C42—88no—
441C42—88no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23C2×#12314yes—
463C23C23×D411yescharacteristic
464C23C22×#4111yescharacteristic
465C23C22×#4111yescharacteristic
466C23C22×#4112yes—
467C23C22×#4112yes—
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+412yes—
470C23C2×2−1+412yes—
471C23#58512yes—
472Q8—88no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—232no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
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—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—816no—
555C2×C4C2×#12314yes—
556C2×C4C2×#12314yes—
557C2×C4#52514yes—
558C2×C4#52514yes—
559C2×C4#52918yes—
560C2×C4#52918yes—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22C22×#12311yescharacteristic
571C22C2×#52511yescharacteristic
572C22C2×#52511yescharacteristic
573C22C2×#52912yes—
574C22C2×#52912yes—
575C22#348612yes—
576C22#349212yes—
577C22#349412yes—
578C22#349512yes—
579C22#349612yes—
580C22#350712yes—
581C22#350811yescharacteristic
582C22#350911yescharacteristic
583C22#350911yescharacteristic
584C22#350912yes—
585C22#351011yescharacteristic
586C22#351112yes—
587C22#351512yes—
588C22#351612yes—
589C22#352012yes—
590C22#352212yes—
591C4—24no—
592C4—44no—
593C4—44no—
594C4—44no—
595C4—44no—
596C4—48no—
597C4—44no—
598C4—44no—
599C4—44no—
600C4—82no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4#285318yes—
605C2—44no—
606C2#3554111yescharacteristic
607C2#3559912yes—
608C2#3561312yes—
609C2#3567411yescharacteristic
610C2#3568812yes—
611C2#3572412yes—
612C2#3574512yes—
613C2#3581812yes—
614C2C2×#285311yescharacteristic
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. #14850C2semidirect
6. #14794C2semidirect
8. #14142C2semidirect
11. #14021C2semidirect
12. #13981C2semidirect
14. #13884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#22742
#149652
#148502
#147942
#142582
#141422
#141382
#141292
#140212
#139812
#139042
#138842
#138442