Finite Group Databasebeta

← #712177

Subgroups — #712177

Order 512.

2486 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3C2×#2345C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#16396C214yesmaximal
6#15885C214yesmaximal, complemented
7#15789C214yesmaximal, complemented
8#15225C212yesmaximal, complemented
9#15189C212yesmaximal, complemented
10#14837C214yesmaximal
11#14142C212yesmaximal, complemented
12#14138C211yesmaximal, characteristic
13#14129C211yesmaximal, characteristic
14#14006C212yesmaximal, complemented
15#13853C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2625—28no—
21#2503—28no—
22C2×#400C2214yes—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26#2345—28no—
27#2317—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#384C2212yes—
40C2×#382C2212yes—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#381C2214yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2212yes—
65#1440C2212yes—
66#1440C2212yes—
67#1440C2212yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—28no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4D418yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120#400—216no—
121#400—28no—
122#398—28no—
123#397—216no—
124#391—28no—
125#391—24no—
126#391—24no—
127#391—28no—
128#389—28no—
129#389—28no—
130#389—28no—
131#389—28no—
132#389—28no—
133#387—216no—
134#385—216no—
135#384—28no—
136#382—28no—
137#382—24no—
138#382—24no—
139#381—216no—
140#380—28no—
141#380D418yes—
142C4×C4⋊C4—28no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—28no—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C25C2411yescharacteristic
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D418yes—
190C23×C4#4114yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C22⋊Q8—416no—
206#117—232no—
207#117—416no—
208C2×C4⋊C4—232no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
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×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C2×D418yes—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—44no—
282C2×C42—44no—
283#95—24no—
284#95—28no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
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—24no—
301#95—24no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D412yes—
319C24C22×D412yes—
320C24C2×#4112yes—
321C24C2511yescharacteristic
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—48no—
326C2×Q8—48no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—216no—
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—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—416no—
406C22×C4—88no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4#121116yes—
410C22×C4C22×D414yes—
411C22×C4C22×D414yes—
412C22×C4C22×D418yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C2×#4118yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4114yes—
421C22×C42−1+414yes—
422C22×C42−1+414yes—
423C4⋊C4—216no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C22⋊C4—28no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C42—216no—
445C42—48no—
446C42—44no—
447C42—44no—
448C42—48no—
449C42—88no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23C2×#12114yes—
471C23C23×D411yescharacteristic
472C23C23×D411yescharacteristic
473C23C22×#4111yescharacteristic
474C23C22×#4112yes—
475C23C2×2−1+412yes—
476C23C2×2−1+412yes—
477C23#58512yes—
478C23#58512yes—
479C23#58512yes—
480Q8—88no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—232no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
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—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—816no—
563C2×C4C2×#12114yes—
564C2×C4C2×#12114yes—
565C2×C4#52514yes—
566C2×C4#53614yes—
567C2×C4#54718yes—
568C2×C4#55218yes—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22C22×#12111yescharacteristic
579C22C2×#52511yescharacteristic
580C22C2×#53611yescharacteristic
581C22#346112yes—
582C22C2×#54712yes—
583C22C2×#55212yes—
584C22#348512yes—
585C22#348511yescharacteristic
586C22#348611yescharacteristic
587C22#349712yes—
588C22#350012yes—
589C22#351011yescharacteristic
590C22#351011yescharacteristic
591C22#351512yes—
592C22#351612yes—
593C22#352112yes—
594C22#352412yes—
595C22#355512yes—
596C22#356312yes—
597C22#356512yes—
598C22#356512yes—
599C4—24no—
600C4—44no—
601C4—44no—
602C4—48no—
603C4—44no—
604C4—48no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4#241318yes—
613C2—44no—
614C2C2×#241311yescharacteristic
615C2#3515311yescharacteristic
616C2#3561711yescharacteristic
617C2#3580212yes—
618C2#3826512yes—
619C2#3908112yes—
620C2#4264412yes—
621C2#4509612yes—
622C2#4517712yes—
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
6. #15885C2semidirect
7. #15789C2semidirect
8. #15225C2semidirect
9. #15189C2semidirect
11. #14142C2semidirect
14. #14006C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#26252
C2×#23452
C2×#23172
#163962
#158852
#157892
#152252
#151892
#148372
#141422
#141382
#141292
#140062
#138532