Finite Group Databasebeta

← #618454

Subgroups — #618454

Order 512.

3706 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3C2×#2484C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#16104C218yesmaximal, complemented
6#15881C214yesmaximal, complemented
7#15850C214yesmaximal, complemented
8#15092C212yesmaximal
9#15073C212yesmaximal
10#14705C212yesmaximal
11#14342C211yesmaximal, characteristic
12#14263C212yesmaximal
13#13870C211yesmaximal, characteristic
14#13870C211yesmaximal, characteristic
15C22×#125—28no—
16C22×#125—28no—
17C22×#125—28no—
18C22×#125C2214yes—
19C22×#125C2214yes—
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2484—28no—
30#2484—28no—
31C2×#400C2212yes—
32C2×#400C2212yes—
33C2×#398C2212yes—
34C2×#398C2212yes—
35#2317—28no—
36C2×#389C2214yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#384C2212yes—
44C2×#382C2212yes—
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#381C2212yes—
50C2×#381C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1444C2218yes—
54#1444C2218yes—
55#1442C2218yes—
56#1442C2218yes—
57#1441C2218yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2218yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2218yes—
64#1436C2218yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#125—2128no—
77C2×#125—216no—
78C2×#125—216no—
79C2×#125—216no—
80C2×#125—216no—
81C2×#125—216no—
82C2×#125—216no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4D418yes—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42—24no—
127C22×C42—24no—
128C22×C42—24no—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400—28no—
133#400—28no—
134#398—28no—
135#398—28no—
136#389—216no—
137#387—216no—
138#387—216no—
139#385—28no—
140#385—28no—
141#385—28no—
142#385—28no—
143#384—28no—
144#382—28no—
145#381—24no—
146#381—24no—
147#381—28no—
148#381—28no—
149#381—28no—
150#381—28no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2314yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C25C2411yescharacteristic
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214#125—2128no—
215#125—2128no—
216#117—232no—
217#117—416no—
218C2×C4⋊C4—232no—
219C2×C4⋊C4—232no—
220C2×C4⋊C4—232no—
221C2×C4⋊C4—232no—
222C2×C4⋊C4—232no—
223C2×C4⋊C4—232no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
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—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4#4118yes—
279C2×C42—232no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—216no—
301#95—216no—
302#95—216no—
303#95—48no—
304#95—48no—
305#95—48no—
306C24—24no—
307C24—24no—
308C24C22×D412yes—
309C24C22×D412yes—
310C24C22×D412yes—
311C24C2×#4114yes—
312C242−1+414yes—
313C24C2511yescharacteristic
314C22×C4—216no—
315C22×C4—216no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—216no—
320C22×C4—216no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—216no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
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—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4#120116yes—
392C22×C4C22×D414yes—
393C22×C4C22×D414yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D414yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D414yes—
404C22×C4C2×#4118yes—
405C22×C42−1+418yes—
406C4⋊C4—232no—
407C4⋊C4—232no—
408C4⋊C4—232no—
409C4⋊C4—216no—
410C4⋊C4—216no—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—416no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—416no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C22⋊C4—216no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C42—232no—
427C42—216no—
428C42—48no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—216no—
436C23—216no—
437C23C2×#12014yes—
438C23#52514yes—
439C23#53614yes—
440C23#54718yes—
441C23#54718yes—
442C23C23×D411yescharacteristic
443C23C23×D411yescharacteristic
444C23C23×D411yescharacteristic
445C23C22×#4112yes—
446C23C2×2−1+412yes—
447C23C2×2−1+414yes—
448C23#58514yes—
449C2×C4—232no—
450C2×C4—232no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—216no—
476C2×C4—416no—
477C2×C4—416no—
478C2×C4—416no—
479C2×C4—416no—
480C2×C4—416no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4C2×#12014yes—
515C2×C4C2×#12014yes—
516C2×C4#53618yes—
517C22—216no—
518C22—24no—
519C22—28no—
520C22—28no—
521C22—216no—
522C22—24no—
523C22#240618yes—
524C22C22×#12011yescharacteristic
525C22C2×#52511yescharacteristic
526C22C2×#53612yes—
527C22C2×#53611yescharacteristic
528C22#344912yes—
529C22C2×#54712yes—
530C22C2×#54712yes—
531C22#348512yes—
532C22#348612yes—
533C22#348612yes—
534C22#348712yes—
535C22#351518yes—
536C22#356318yes—
537C4—28no—
538C4—24no—
539C4—24no—
540C4—44no—
541C4—44no—
542C4—44no—
543C4—48no—
544C4—44no—
545C4—44no—
546C4—84no—
547C4—82no—
548C4—84no—
549C4—82no—
550C2—28no—
551C2C2×#240611yescharacteristic
552C2#3514911yescharacteristic
553C2#3524211yescharacteristic
554C2#3545312yes—
555C2#3695912yes—
556C2#3804514yes—
557C2#3891914yes—
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
5. #16104C2semidirect
6. #15881C2semidirect
7. #15850C2semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#24842
C2×#23172
#161042
#158812
#158502
#150922
#150732
#147052
#143422
#142632
#138702