Finite Group Databasebeta

← #4452857

Subgroups — #4452857

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3#16829C214yesmaximal, complemented
4#16828C214yesmaximal, complemented
5#16525C214yesmaximal, complemented
6#16523C214yesmaximal, complemented
7#15092C212yesmaximal
8#15092C212yesmaximal
9#14955C212yesmaximal
10#14786C212yesmaximal
11#13966C211yesmaximal, characteristic
12#13965C211yesmaximal, characteristic
13#13881C211yesmaximal, characteristic
14#13880C211yesmaximal, characteristic
15#13796C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2646—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#398—28no—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#384C2212yes—
36C2×#382C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1445C2214yes—
45#1444C2214yes—
46#1444C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115#400—28no—
116#400—28no—
117#398—28no—
118#398—28no—
119#398—216no—
120#398—216no—
121#398—48no—
122#385—28no—
123#385—28no—
124#385—44no—
125#385—44no—
126#385—44no—
127#385—44no—
128#384—44no—
129#382—44no—
130#380—24no—
131#380—24no—
132C4×C22⋊C4Q818yes—
133C4×C22⋊C4Q818yes—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95D418yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
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—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2411yescharacteristic
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×Q814yes—
195C23×C4C2×Q814yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—416no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
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⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4C2×Q818yes—
265C2×C22⋊C4C2×Q818yes—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42C2×Q818yes—
271C2×C42C2×Q818yes—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
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—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95C2×D418yes—
298#95C2×D418yes—
299#95C2×Q818yes—
300#95C2×Q818yes—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24C22×D412yes—
306C24C22×Q812yes—
307C24C22×Q812yes—
308C24H(5,𝔽2)14yes—
309C242−1+414yes—
310C24C2511yescharacteristic
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—416no—
382C22×C4—44no—
383C22×C4—416no—
384C22×C4—416no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—88no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4C4⋊Q8116yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×Q818yes—
404C22×C4C22×Q818yes—
405C22×C4C22×Q812yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4H(5,𝔽2)14yes—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—416no—
413C4⋊C4—416no—
414C4⋊C4—416no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C22⋊C4—216no—
428C22⋊C4—216no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C42—48no—
432C42—48no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23C2×C4⋊Q814yes—
446C23#53514yes—
447C23#53614yes—
448C23#54418yes—
449C23#54418yes—
450C23C23×D411yescharacteristic
451C23C23×Q811yescharacteristic
452C23C23×Q811yescharacteristic
453C23C2×H(5,𝔽2)12yes—
454C23C2×H(5,𝔽2)12yes—
455C23C2×2−1+412yes—
456C23#58512yes—
457C23#58512yes—
458C23#58512yes—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—416no—
482C2×C4—416no—
483C2×C4—416no—
484C2×C4—416no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
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—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
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—48no—
530C2×C4—48no—
531C2×C4—88no—
532C2×C4—88no—
533C2×C4—88no—
534C2×C4C2×C4⋊Q814yes—
535C2×C4C2×C4⋊Q814yes—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—216no—
543C22—216no—
544C22#266018yes—
545C22C22×C4⋊Q811yescharacteristic
546C22C2×#53511yescharacteristic
547C22C2×#53611yescharacteristic
548C22C2×#54412yes—
549C22C2×#54412yes—
550C22#348411yescharacteristic
551C22#348511yescharacteristic
552C22#348511yescharacteristic
553C22#348711yescharacteristic
554C22#350112yes—
555C22#350112yes—
556C22#350412yes—
557C22#350412yes—
558C22#353314yes—
559C22#353414yes—
560C22#354114yes—
561C22#355314yes—
562C4—24no—
563C4—24no—
564C4—44no—
565C4—44no—
566C4—44no—
567C4—48no—
568C4—48no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C2—28no—
577C2C2×#266011yescharacteristic
578C2#3539111yescharacteristic
579C2#3564011yescharacteristic
580C2#3670511yescharacteristic
581C2#3695611yescharacteristic
582C2#3712011yescharacteristic
583C2#3718511yescharacteristic
584C2#4061514yes—
585C2#4061714yes—
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
3. #16829C2semidirect
4. #16828C2semidirect
5. #16525C2semidirect
6. #16523C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3982
#168292
#168282
#165252
#165232
#150922
#149552
#147862
#139662
#139652
#138812
#138802
#137962