Finite Group Databasebeta

← #416898

Subgroups — #416898

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic, complemented
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#13955C211yesmaximal, characteristic, complemented
7#11268C212yesmaximal, complemented
8#11267C212yesmaximal, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10834C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2566—24no—
19#2566—24no—
20C2×#396—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#385—22no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1991—24no—
30#1989—24no—
31#1804—24no—
32C4×#102—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125D414yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4D414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#396—216no—
90#390—22no—
91#390—22no—
92#390—22no—
93#390—22no—
94#385—216no—
95#384—22no—
96#384—22no—
97#382—22no—
98#382—22no—
99#382—22no—
100#382—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#336—24no—
121#336—24no—
122C25—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C2×D412yes—
132C22×D4#4114yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8#4114yes—
162#125—28no—
163#125—28no—
164#125—44no—
165#125—44no—
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—44no—
169C4×D4—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C4⋊C8—44no—
238#102—216no—
239#102—48no—
240C4×C8—44no—
241#95—28no—
242#95—24no—
243#95—24no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—44no—
248#95—82no—
249#95—82no—
250C24—22no—
251C24—24no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24C22×D411yescharacteristic
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4#12414yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4#12012yes—
330C22×C4#12012yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12412yes—
334C22×C4#12412yes—
335C22×C4#13514yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C42−1+412yes—
345C22×C42−1+412yes—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—88no—
384C22⋊C4—416no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—84no—
394C42#12114yes—
395C42#12114yes—
396C42#12314yes—
397C42#12314yes—
398C23—24no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—44no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—81no—
414C23—81no—
415C23—82no—
416C23—82no—
417C23—81no—
418C23—81no—
419C23C2×#12011yescharacteristic
420C23C2×#12111yescharacteristic
421C23C2×#12411yescharacteristic
422C23#54712yes—
423C23#54812yes—
424C23C2×#13512yes—
425C23#57512yes—
426C23#57612yes—
427C23#57712yes—
428D4—24no—
429D4—24no—
430D4—42no—
431D4—44no—
432D4—42no—
433D4—44no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—816no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#39414yes—
487C2×C4#39414yes—
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4#52512yes—
493C2×C4C2×#12411yescharacteristic
494C2×C4#52912yes—
495C2×C4#53211yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53612yes—
499C2×C4#54112yes—
500C2×C4#54812yes—
501C2×C4#54812yes—
502C2×C4#55212yes—
503C2×C4#55212yes—
504C2×C4#55212yes—
505C2×C4#55612yes—
506C8—44no—
507C8—82no—
508C22—48no—
509C22—44no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—82no—
516C22—82no—
517C22C2×#39411yescharacteristic
518C22#241512yes—
519C22#243211yescharacteristic
520C22#243611yescharacteristic
521C22#248912yes—
522C22#300212yes—
523C22#301212yes—
524C22#303512yes—
525C22#304111yescharacteristic
526C22#304311yescharacteristic
527C22#313111yescharacteristic
528C22#313511yescharacteristic
529C22#313612yes—
530C22#333712yes—
531C22#333912yes—
532C22#335112yes—
533C22#335512yes—
534C22#336012yes—
535C22#336112yes—
536C22#337012yes—
537C22#337612yes—
538C4—24no—
539C4—24no—
540C4—44no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—162no—
547C4#242112yes—
548C4#249911yescharacteristic
549C4#250311yescharacteristic
550C4#253612yes—
551C4#262111yescharacteristic
552C4#285011yescharacteristic
553C2—84no—
554C2#1523611yescharacteristic
555C2#2350411yescharacteristic
556C2#2351111yescharacteristic
557C2#2468412yes—
558C2#2468512yes—
559C2#2570712yes—
560C2#2593812yes—
561C2#2773112yes—
562C2#2773312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1991C2semidirect
3. C2×#1989C2semidirect
6. #13955C2semidirect
7. #11268C2semidirect
8. #11267C2semidirect
11. #10834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#19912
C2×#19892
C2×#18042
C2×C4×#1022
#139552
#112682
#112672
#111892
#108342