Finite Group Databasebeta

← #4452948

Subgroups — #4452948

Order 512.

2525 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic, complemented
3#16883C214yesmaximal, complemented
4#16854C214yesmaximal, complemented
5#16779C214yesmaximal, complemented
6#16716C214yesmaximal, complemented
7#15073C212yesmaximal, complemented
8#15073C211yesmaximal, characteristic, complemented
9#15069C212yesmaximal
10#15069C211yesmaximal, characteristic
11#14955C211yesmaximal, characteristic, complemented
12#14945C211yesmaximal, characteristic
13#13996C212yesmaximal
14#13972C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C22×#123C2211yescharacteristic
17C22×#123C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2623—28no—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#398—28no—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#397C2212yes—
29C2×#386C2211yescharacteristic
30C2×#386C2211yescharacteristic
31C2×#386C2212yes—
32C2×#385C2211yescharacteristic
33C2×#385C2211yescharacteristic
34C2×#385C2212yes—
35C2×#381C2211yescharacteristic
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43#1445C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#123—24no—
80C2×#123—28no—
81C2×#123—44no—
82C2×#123—44no—
83C2×#123—44no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4D418yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2314yes—
114#400—28no—
115#399—28no—
116#398—28no—
117#398—216no—
118#398—216no—
119#397—28no—
120#386—24no—
121#386—24no—
122#386—44no—
123#385—24no—
124#385—24no—
125#385—44no—
126#381—24no—
127#381—28no—
128#381—28no—
129#381—44no—
130#381—44no—
131C4×C22⋊C4—44no—
132C2×#95—24no—
133C2×#95—28no—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C22×D4—22no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203#123—432no—
204#123—416no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—416no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—432no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4C2×D418yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4#4118yes—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—44no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—24no—
294C24—24no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—24no—
301C24—24no—
302C24C22×D412yes—
303C24C2×#4114yes—
304C24H(5,𝔽2)14yes—
305C24H(5,𝔽2)12yes—
306C24H(5,𝔽2)12yes—
307C24C2511yescharacteristic
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C22×C4—216no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
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—24no—
381C22×C4—24no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—416no—
385C22×C4—416no—
386C22×C4—48no—
387C22×C4—416no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—88no—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C2×#4118yes—
404C22×C42−1+418yes—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—416no—
408C4⋊C4—416no—
409C4⋊C4—416no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—416no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—88no—
416C22⋊C4—216no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—416no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—88no—
427C42—88no—
428C23—24no—
429C23—24no—
430C23—28no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—28no—
435C23—28no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23#53214yes—
450C23#53414yes—
451C23#53514yes—
452C23#53818yes—
453C23#54018yes—
454C23C23×D411yescharacteristic
455C23C22×#4112yes—
456C23C2×H(5,𝔽2)12yes—
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)11yescharacteristic
459C23C2×2−1+414yes—
460C23#58514yes—
461D4—88no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—416no—
481C2×C4—432no—
482C2×C4—416no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
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—88no—
535C2×C4—816no—
536C22—216no—
537C22—216no—
538C22—24no—
539C22—24no—
540C22—28no—
541C22—28no—
542C22—81no—
543C22—84no—
544C22—84no—
545C22—81no—
546C22—81no—
547C22—82no—
548C22—82no—
549C22#266118yes—
550C22C2×#53211yescharacteristic
551C22C2×#53411yescharacteristic
552C22C2×#53511yescharacteristic
553C22C2×#53812yes—
554C22C2×#54012yes—
555C22#348512yes—
556C22#348612yes—
557C22#351514yes—
558C22#351514yes—
559C22#353314yes—
560C22#353514yes—
561C22#353514yes—
562C22#354414yes—
563C4—42no—
564C4—42no—
565C4—48no—
566C4—44no—
567C4—44no—
568C4—48no—
569C4—84no—
570C4—81no—
571C4—84no—
572C4—81no—
573C4—81no—
574C4—82no—
575C4—82no—
576C2—28no—
577C2—161no—
578C2C2×#266111yescharacteristic
579C2#3546812yes—
580C2#3695511yescharacteristic
581C2#3711812yes—
582C2#3725811yescharacteristic
583C2#4153214yes—
584C2#4153314yes—
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
3. #16883C2semidirect
4. #16854C2semidirect
5. #16779C2semidirect
6. #16716C2semidirect
7. #15073C2semidirect
8. #15073C2semidirect
11. #14955C2semidirect
14. #13972C2semidirect
15. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C22×#3982
#168832
#168542
#167792
#167162
#150732
#150692
#149552
#149452
#139962
#139722
#137762