Finite Group Databasebeta

← #4418517

Subgroups — #4418517

Order 512.

2385 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic, complemented
2C2×#2637C211yesmaximal, characteristic
3C2×#2664C211yesmaximal, characteristic
4C2×#2351C211yesmaximal, characteristic, complemented
5#17040C212yesmaximal
6#16992C212yesmaximal, complemented
7#16564C212yesmaximal
8#16517C212yesmaximal, complemented
9#15676C214yesmaximal, complemented
10#15620C214yesmaximal
11#15176C212yesmaximal
12#14880C212yesmaximal, complemented
13#14869C212yesmaximal, complemented
14#14808C212yesmaximal
15#14664C211yesmaximal, characteristic, complemented
16#14291C211yesmaximal, characteristic
17#13955C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2664—28no—
23#2637—28no—
24#2584—28no—
25C2×#400C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#398C2212yes—
28C2×#398C2211yescharacteristic
29C2×#398C2212yes—
30C2×#398C2211yescharacteristic
31C2×#398C2212yes—
32C2×#398C2214yes—
33C2×#397C2211yescharacteristic
34C2×#397C2211yescharacteristic
35C2×#396C2212yes—
36C2×#396C2212yes—
37C2×#395C2211yescharacteristic
38C2×#395C2212yes—
39C2×#393C2211yescharacteristic
40#2351—28no—
41C2×#389C2214yes—
42C2×#389C2214yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#388C2212yes—
48C2×#388C2212yes—
49C2×#388C2214yes—
50C2×#387C2212yes—
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#382C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1444C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2212yes—
79#1437C2212yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2212yes—
83C4⋊#95C2214yes—
84#1434C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C2×#126—24no—
89C2×#126—24no—
90C2×#126—24no—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—48no—
98C2×C4×Q8D418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#400—24no—
134#399—24no—
135#398—28no—
136#398—24no—
137#398—28no—
138#398—28no—
139#398—24no—
140#398—48no—
141#397—24no—
142#397—24no—
143#396—28no—
144#396—28no—
145#395—28no—
146#395—24no—
147#393—24no—
148#389—216no—
149#389—216no—
150#389—28no—
151#389—28no—
152#389—28no—
153#389—28no—
154#388—28no—
155#388—28no—
156#388—48no—
157#387—28no—
158#382—24no—
159#382—24no—
160#382—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4D418yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C25C2411yescharacteristic
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8C2×D414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216#126—416no—
217#125—416no—
218C4×Q8—232no—
219C4×Q8—432no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42C2×D418yes—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42#4114yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95C2×D414yes—
342#95C2×D414yes—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24C2511yescharacteristic
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—216no—
355C2×Q8—28no—
356C2×Q8—216no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C42−1+418yes—
442C22×C42−1+414yes—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—28no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—416no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C42—216no—
473C42—48no—
474C42—416no—
475C42—84no—
476C42#121116yes—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23C23×D411yescharacteristic
494C23C23×D411yescharacteristic
495C23C22×#4111yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×2−1+412yes—
498C23C2×2−1+414yes—
499C23#58512yes—
500C23#58512yes—
501Q8—232no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4C2×#12114yes—
585C2×C4C2×#12114yes—
586C2×C4C2×#12114yes—
587C2×C4#53718yes—
588C22—81no—
589C22—82no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22C22×#12111yescharacteristic
597C22C2×#53712yes—
598C22#344912yes—
599C22#348411yescharacteristic
600C22#348612yes—
601C22#348611yescharacteristic
602C22#349212yes—
603C22#349912yes—
604C22#350011yescharacteristic
605C22#350011yescharacteristic
606C22#350912yes—
607C22#351212yes—
608C22#354114yes—
609C22#355414yes—
610C22#355514yes—
611C22#356314yes—
612C4—24no—
613C4—24no—
614C4—24no—
615C4—28no—
616C4—44no—
617C4—48no—
618C4—44no—
619C4—44no—
620C4—81no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—81no—
626C4—81no—
627C4—82no—
628C2—161no—
629C2#3522311yescharacteristic
630C2#3532811yescharacteristic
631C2#3566711yescharacteristic
632C2#4050312yes—
633C2#4231312yes—
634C2#4279012yes—
635C2#4279512yes—
636C2#5124014yes—
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
1. C2×#2584C2semidirect
4. C2×#2351C2semidirect
6. #16992C2semidirect
8. #16517C2semidirect
9. #15676C2semidirect
12. #14880C2semidirect
13. #14869C2semidirect
15. #14664C2semidirect
17. #13955C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×#26372
C2×#26642
C2×#23512
#170402
#169922
#165642
#165172
#156762
#156202
#151762
#148802
#148692
#148082
#146642
#142912
#139552