Finite Group Databasebeta

← #765948

Subgroups — #765948

Order 512.

3012 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2413C212yesmaximal
2C22×#398C211yesmaximal, characteristic, complemented
3C2×#2322C211yesmaximal, characteristic
4#15735C214yesmaximal, complemented
5#15676C214yesmaximal, complemented
6#15590C214yesmaximal, complemented
7#15518C214yesmaximal, complemented
8#15073C211yesmaximal, characteristic, complemented
9#15069C211yesmaximal, characteristic
10#14955C211yesmaximal, characteristic, complemented
11#14945C211yesmaximal, characteristic
12#13992C212yesmaximal, complemented
13#13968C212yesmaximal
14#13870C212yesmaximal, complemented
15#13795C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2413—216no—
24C2×#398—28no—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#396C2214yes—
29#2322—28no—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#387C2212yes—
35C2×#386C2211yescharacteristic
36C2×#386C2212yes—
37C2×#385C2211yescharacteristic
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2212yes—
47#1444C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67#1434C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#123—24no—
82C2×#123—24no—
83C2×#123—28no—
84C2×#123—28no—
85C2×#123—44no—
86C2×#123D418yes—
87C2×C4×D4—24no—
88C2×C4×D4—44no—
89C2×#117—28no—
90C2×#117—28no—
91C2×#117—44no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#398—216no—
124#398—216no—
125#398—216no—
126#396—216no—
127#389—28no—
128#389—216no—
129#388—28no—
130#388—216no—
131#387—28no—
132#386—28no—
133#386—24no—
134#385—24no—
135#385—28no—
136#385—28no—
137#381—24no—
138#381—28no—
139#381—28no—
140#381—44no—
141C4×C22⋊C4—24no—
142C2×#95—24no—
143C2×#95—28no—
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—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4C2×D414yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4#4114yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214#123—232no—
215#123—432no—
216C4×D4—416no—
217#117—432no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—432no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D418yes—
277C2×C22⋊C4#4118yes—
278C2×C42—28no—
279C2×C42—22no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—44no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309C24—24no—
310C24—24no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C22×D412yes—
320C24C2×#4112yes—
321C24C2×#4114yes—
322C24H(5,𝔽2)14yes—
323C24C2511yescharacteristic
324C2×D4—28no—
325C2×D4—216no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—48no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—216no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
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—28no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—416no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—84no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4#121116yes—
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C22×D412yes—
417C22×C4C22×D414yes—
418C22×C4C22×D414yes—
419C22×C4C2×#4118yes—
420C22×C4C2×#4114yes—
421C22×C42−1+418yes—
422C4⋊C4—216no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—44no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—84no—
433C22⋊C4—28no—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—84no—
443C42—416no—
444C42—44no—
445C42—84no—
446C23—24no—
447C23—24no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—22no—
455C23—24no—
456C23—28no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23C2×#12114yes—
468C23#53214yes—
469C23#53414yes—
470C23#54618yes—
471C23#55318yes—
472C23C23×D411yescharacteristic
473C23C23×D411yescharacteristic
474C23C22×#4111yescharacteristic
475C23C22×#4112yes—
476C23C2×H(5,𝔽2)12yes—
477C23C2×2−1+414yes—
478C23#58514yes—
479D4—416no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—232no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—432no—
505C2×C4—416no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
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—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4—816no—
552C2×C4C2×#12118yes—
553C22—28no—
554C22—24no—
555C22—24no—
556C22—28no—
557C22—216no—
558C22—216no—
559C22—42no—
560C22—44no—
561C22—48no—
562C22—81no—
563C22—81no—
564C22—82no—
565C22—84no—
566C22#254318yes—
567C22C22×#12111yescharacteristic
568C22C2×#53211yescharacteristic
569C22C2×#53411yescharacteristic
570C22C2×#54612yes—
571C22C2×#55312yes—
572C22#348512yes—
573C22#348612yes—
574C22#348914yes—
575C22#349514yes—
576C22#351514yes—
577C22#351514yes—
578C22#353314yes—
579C22#354414yes—
580C4—28no—
581C4—42no—
582C4—44no—
583C4—48no—
584C4—44no—
585C4—42no—
586C4—42no—
587C4—48no—
588C4—84no—
589C4—81no—
590C4—81no—
591C4—82no—
592C4—84no—
593C2—28no—
594C2—82no—
595C2C2×#254311yescharacteristic
596C2#3546812yes—
597C2#3563312yes—
598C2#3696511yescharacteristic
599C2#3725811yescharacteristic
600C2#4449414yes—
601C2#4449514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#398C2semidirect
4. #15735C2semidirect
5. #15676C2semidirect
6. #15590C2semidirect
7. #15518C2semidirect
8. #15073C2semidirect
10. #14955C2semidirect
12. #13992C2semidirect
14. #13870C2semidirect
15. #13795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24132
C22×#3982
C2×#23222
#157352
#156762
#155902
#155182
#150732
#150692
#149552
#149452
#139922
#139682
#138702
#137952