Finite Group Databasebeta

← #407067

Subgroups — #407067

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 546 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#13881C211yesmaximal, characteristic
7#11378C212yesmaximal, complemented
8#11372C212yesmaximal, complemented
9#11122C212yesmaximal, complemented
10#11052C212yesmaximal, 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×#382C2212yes—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1892—24no—
26#1872—24no—
27#1853—24no—
28C2×#340—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1442—22no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—22no—
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—42no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42C2312yes—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#103—24no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8D414yes—
94#391—24no—
95#391—42no—
96#382—24no—
97#382—42no—
98#380—22no—
99#380—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C2×#95—24no—
103C2×#95—24no—
104C2×#95—21no—
105C2×#95—24no—
106C2×#95—24no—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—24no—
114#336—42no—
115#336—42no—
116C25C2×D411yescharacteristic
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8C2×Q812yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C2×D414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—28no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
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—44no—
170C22×C8—44no—
171C22×C8—48no—
172C22×C8—48no—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D414yes—
176C22⋊Q8—28no—
177C22⋊Q8—28no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4#4114yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233#106—48no—
234#103—48no—
235#103—44no—
236#103—44no—
237C22⋊C8—28no—
238#95—24no—
239#95—48no—
240#95—48no—
241#95—42no—
242#95—84no—
243#95—84no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24C22⋊Q812yes—
249C24#12312yes—
250C24#12712yes—
251C24#13514yes—
252C24#13714yes—
253C24C22×D411yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—48no—
300C22×C4—48no—
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—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—816no—
326C22×C4—816no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4C22⋊Q814yes—
332C22×C4C22⋊Q814yes—
333C22×C4C22⋊Q812yes—
334C22×C4#12314yes—
335C22×C4#12314yes—
336C22×C4#12312yes—
337C22×C4#12712yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4112yes—
346C22×C4H(5,𝔽2)12yes—
347C22×C4H(5,𝔽2)12yes—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—416no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—48no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—84no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C23—24no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—28no—
401C23—28no—
402C23—22no—
403C23—22no—
404C23—28no—
405C23—28no—
406C23#39914yes—
407C23#47918yes—
408C23#48518yes—
409C23#49514yes—
410C23#49714yes—
411C23C2×C22⋊Q811yescharacteristic
412C23C2×#12311yescharacteristic
413C23C2×#12711yescharacteristic
414C23#54612yes—
415C23#54612yes—
416C23C2×#13512yes—
417C23C2×#13712yes—
418C23#57712yes—
419C23#57712yes—
420Q8—42no—
421Q8—42no—
422Q8—44no—
423Q8—81no—
424Q8—81no—
425Q8—82no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—816no—
452C2×C4—816no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4#39914yes—
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12311yescharacteristic
478C2×C4#52511yescharacteristic
479C2×C4#52511yescharacteristic
480C2×C4#52912yes—
481C2×C4#52912yes—
482C2×C4#53411yescharacteristic
483C2×C4#53411yescharacteristic
484C2×C4#53812yes—
485C2×C4#53812yes—
486C2×C4#54612yes—
487C2×C4#54612yes—
488C2×C4#55312yes—
489C2×C4#55312yes—
490C2×C4#55412yes—
491C2×C4#55412yes—
492C8—42no—
493C8—42no—
494C8—44no—
495C22—216no—
496C22—216no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22#209718yes—
504C22C2×#39911yescharacteristic
505C22#244412yes—
506C22#245012yes—
507C22#253911yescharacteristic
508C22#253911yescharacteristic
509C22C2×#47912yes—
510C22#308412yes—
511C22C2×#48512yes—
512C22#309912yes—
513C22C2×#49511yescharacteristic
514C22C2×#49711yescharacteristic
515C22#315411yescharacteristic
516C22#315411yescharacteristic
517C22#329614yes—
518C22#329614yes—
519C22#331614yes—
520C22#332014yes—
521C4—24no—
522C4—84no—
523C4—82no—
524C4—84no—
525C4—84no—
526C4—82no—
527C4—161no—
528C4—161no—
529C4#244412yes—
530C4#245012yes—
531C4#257911yescharacteristic
532C4#257911yescharacteristic
533C4#262211yescharacteristic
534C4#285311yescharacteristic
535C2—28no—
536C2#1494611yescharacteristic
537C2C2×#209711yescharacteristic
538C2#2373811yescharacteristic
539C2#2620614yes—
540C2#2641614yes—
541C2#2833011yescharacteristic
542C2#2833011yescharacteristic
543C2#2834711yescharacteristic
544C2#2834811yescharacteristic
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. #11372C2semidirect
9. #11122C2semidirect
10. #11052C2semidirect

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
#138812
#113782
#113722
#111222
#110522
#107142