Finite Group Databasebeta

← #624969

Subgroups — #624969

Order 512.

2248 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2642C211yesmaximal, characteristic, complemented
2C2×#2298C211yesmaximal, characteristic
3#15723C214yesmaximal, complemented
4#15718C214yesmaximal
5#15580C214yesmaximal
6#15574C214yesmaximal, complemented
7#15218C211yesmaximal, characteristic, complemented
8#15216C211yesmaximal, characteristic
9#15180C211yesmaximal, characteristic, complemented
10#15178C211yesmaximal, characteristic
11#14703C212yesmaximal, complemented
12#14258C212yesmaximal
13#14021C212yesmaximal, complemented
14#13997C212yesmaximal
15#13797C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×#117C2211yescharacteristic
20#2642—28no—
21C2×#400C2214yes—
22C2×#400C2214yes—
23#2298—28no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1445C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C23×Q8C2312yes—
74C24×C4C2311yescharacteristic
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—44no—
77C2×#125—24no—
78C2×#125—44no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—44no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111#400—216no—
112#400—216no—
113#391—28no—
114#391—28no—
115#391—28no—
116#391—28no—
117#391—44no—
118#389—44no—
119#388—28no—
120#388—28no—
121#387—28no—
122#387—28no—
123#380—44no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—24no—
129#376D418yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C43—24no—
158C25C2411yescharacteristic
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4#4114yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C4⋊Q8—416no—
186#125—416no—
187C22⋊Q8—432no—
188#117—232no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4#4118yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42C2×D414yes—
260C2×C42C2×D414yes—
261C2×C42C2×D414yes—
262C2×C42#4114yes—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95C2×D418yes—
289#95C2×D418yes—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C2×#4112yes—
299C24C2511yescharacteristic
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
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—24no—
379C22×C4—24no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4C22×D414yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4118yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4114yes—
395C22×C4H(5,𝔽2)14yes—
396C22×C42−1+414yes—
397C22×C42−1+414yes—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—216no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—416no—
404C4⋊C4—416no—
405C4⋊C4—416no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—88no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—416no—
414C22⋊C4—88no—
415C42—28no—
416C42—28no—
417C42—28no—
418C42—28no—
419C42—48no—
420C42—48no—
421C42—44no—
422C42—44no—
423C42—88no—
424C23—22no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23C23×D411yescharacteristic
446C23C22×#4111yescharacteristic
447C23C22×#4111yescharacteristic
448C23C22×#4112yes—
449C23C2×H(5,𝔽2)12yes—
450C23C2×2−1+412yes—
451C23C2×2−1+412yes—
452C23#58512yes—
453C23#58512yes—
454Q8—816no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—416no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4#52514yes—
538C2×C4#53214yes—
539C2×C4#53514yes—
540C2×C4#55118yes—
541C2×C4#55518yes—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×#52511yescharacteristic
552C22C2×#53211yescharacteristic
553C22C2×#53511yescharacteristic
554C22#346112yes—
555C22#346112yes—
556C22C2×#55112yes—
557C22C2×#55512yes—
558C22#348912yes—
559C22#348912yes—
560C22#350911yescharacteristic
561C22#350911yescharacteristic
562C22#351211yescharacteristic
563C22#351211yescharacteristic
564C22#351512yes—
565C22#351612yes—
566C22#352112yes—
567C22#352312yes—
568C22#355212yes—
569C22#355412yes—
570C22#356112yes—
571C22#356212yes—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—48no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—84no—
583C4—84no—
584C4—84no—
585C4#261018yes—
586C2—82no—
587C2C2×#261011yescharacteristic
588C2#3530911yescharacteristic
589C2#3530911yescharacteristic
590C2#3563512yes—
591C2#3737912yes—
592C2#3828712yes—
593C2#3828912yes—
594C2#3906712yes—
595C2#3906812yes—
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×#2642C2semidirect
3. #15723C2semidirect
6. #15574C2semidirect
7. #15218C2semidirect
9. #15180C2semidirect
11. #14703C2semidirect
13. #14021C2semidirect
15. #13797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26422
C2×#22982
#157232
#157182
#155802
#155742
#152182
#152162
#151802
#151782
#147032
#142582
#140212
#139972
#137972