Finite Group Databasebeta

← #407065

Subgroups — #407065

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1892C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#13880C211yesmaximal, characteristic
7#11378C212yesmaximal, complemented
8#11373C212yesmaximal, complemented
9#11122C212yesmaximal, complemented
10#11050C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#391C2212yes—
22C2×#385—22no—
23C2×#384C2212yes—
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1892—24no—
27#1872—24no—
28#1853—24no—
29C2×#340—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—42no—
50C23×C8—42no—
51C23×C8D414yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8Q814yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42C2312yes—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8D414yes—
95#391—24no—
96#391—42no—
97#385—48no—
98#384—24no—
99#384—42no—
100#380—22no—
101#380—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95—24no—
105C2×#95—24no—
106C2×#95—21no—
107C2×#95—24no—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—24no—
115#336—42no—
116#336—42no—
117C25C2×D411yescharacteristic
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8C2×Q812yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C2×D414yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×Q812yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C22×C8—28no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—44no—
171C22×C8—44no—
172C22×C8—48no—
173C22×C8—48no—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C22×C8C2×D414yes—
177C22⋊Q8—28no—
178C22⋊Q8—28no—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—84no—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4114yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4SD1618yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237#106—48no—
238#103—48no—
239#103—44no—
240#103—44no—
241C22⋊C8—28no—
242#95—24no—
243#95—48no—
244#95—42no—
245#95—48no—
246#95—84no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24C22⋊Q812yes—
252C24#12312yes—
253C24#12712yes—
254C24C2×SD1614yes—
255C24#13714yes—
256C24C22×D411yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—816no—
328C22×C4—816no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4C22⋊Q814yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22⋊Q812yes—
336C22×C4#12314yes—
337C22×C4#12314yes—
338C22×C4#12312yes—
339C22×C4#12712yes—
340C22×C4C2×SD1618yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×Q811yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)12yes—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—416no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—48no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—816no—
391C22⋊C4—216no—
392C22⋊C4—48no—
393C22⋊C4—84no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—28no—
405C23—28no—
406C23—22no—
407C23—22no—
408C23—28no—
409C23—28no—
410C23#39914yes—
411C23#47518yes—
412C23#48318yes—
413C23#49214yes—
414C23#49714yes—
415C23C2×C22⋊Q811yescharacteristic
416C23C2×#12311yescharacteristic
417C23C2×#12711yescharacteristic
418C23#54612yes—
419C23#54612yes—
420C23C22×SD1612yes—
421C23C2×#13712yes—
422C23#57612yes—
423C23#57812yes—
424Q8—42no—
425Q8—42no—
426Q8—44no—
427Q8—81no—
428Q8—81no—
429Q8—82no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—816no—
456C2×C4—816no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—84no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4#39914yes—
478C2×C4C2×C22⋊Q811yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4#52912yes—
485C2×C4#52912yes—
486C2×C4#53411yescharacteristic
487C2×C4#53411yescharacteristic
488C2×C4#53812yes—
489C2×C4#53812yes—
490C2×C4#54612yes—
491C2×C4#54612yes—
492C2×C4#55312yes—
493C2×C4#55312yes—
494C2×C4#55412yes—
495C2×C4#55412yes—
496C8—42no—
497C8—42no—
498C8—44no—
499C22—216no—
500C22—216no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22#209618yes—
508C22C2×#39911yescharacteristic
509C22#244412yes—
510C22#245012yes—
511C22#253911yescharacteristic
512C22#253911yescharacteristic
513C22C2×#47512yes—
514C22#308312yes—
515C22C2×#48312yes—
516C22#309812yes—
517C22C2×#49211yescharacteristic
518C22C2×#49711yescharacteristic
519C22#315511yescharacteristic
520C22#315611yescharacteristic
521C22#329514yes—
522C22#329814yes—
523C22#331214yes—
524C22#331414yes—
525C4—24no—
526C4—48no—
527C4—82no—
528C4—84no—
529C4—84no—
530C4—82no—
531C4—161no—
532C4—161no—
533C4#244412yes—
534C4#245012yes—
535C4#257911yescharacteristic
536C4#257911yescharacteristic
537C4#262211yescharacteristic
538C4#285311yescharacteristic
539C2—28no—
540C2#1494611yescharacteristic
541C2C2×#209611yescharacteristic
542C2#2373711yescharacteristic
543C2#2620014yes—
544C2#2641214yes—
545C2#2833211yescharacteristic
546C2#2833611yescharacteristic
547C2#2834211yescharacteristic
548C2#2834511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11378C2semidirect
8. #11373C2semidirect
9. #11122C2semidirect
10. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18922
C2×#18722
C2×#18532
C22×#3402
#141392
#138802
#113782
#113732
#111222
#110502
#107142