Finite Group Databasebeta

← #366787

Subgroups — #366787

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1752C211yesmaximal, characteristic
6C22×#340C211yesmaximal, characteristic
7#13880C211yesmaximal, characteristic
8#11372C212yesmaximal, complemented
9#11050C212yesmaximal, complemented
10#10928C212yesmaximal, complemented
11#10843C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21#2503—24no—
22#2503C418yescomplemented
23C2×#400—22no—
24C2×#391C2212yes—
25C2×#385—22no—
26C2×#384C2212yes—
27C2×#382C2212yes—
28C2×#381—22no—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1892—24no—
32#1872—24no—
33#1823—24no—
34#1752—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—22no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C24×C4—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8D414yes—
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42C2312yes—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103—42no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8D414yes—
97C2×#97—24no—
98C2×#97—24no—
99C2×#97—42no—
100#400—216no—
101#391—24no—
102#391C2×C418yescomplemented
103#385—48no—
104#384—24no—
105#384—42no—
106#382—24no—
107#382C2×C418yescomplemented
108#381—216no—
109#380—22no—
110#380—22no—
111C4×C22⋊C4C2×C414yes—
112C4×C22⋊C4D414yes—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—21no—
117C2×#95D412yes—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—24no—
125#336—24no—
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C22×C412yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—82no—
156C23×C4C22×C412yes—
157C23×C4C2×D414yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—28no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—28no—
180C22×C8—44no—
181C22×C8—48no—
182C22×C8—48no—
183C22×C8C2×D412yes—
184C22×C8C2×D414yes—
185C22×C8C2×D412yes—
186C22×C8#4114yes—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—84no—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4C22⋊C418yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C22×C414yes—
248C2×C42C2×D414yes—
249#103—48no—
250#103—44no—
251#103—44no—
252C22⋊C8—28no—
253#97—48no—
254#95—216no—
255#95—216no—
256#95—48no—
257#95—42no—
258#95C22⋊C418yes—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24C2×C22⋊C412yes—
264C24#12412yes—
265C24#12712yes—
266C24AGL(1,ℤ/8)14yes—
267C24#13714yes—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
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—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—416no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—816no—
341C22×C4—84no—
342C22×C4—88no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4C2×C22⋊C418yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C4×D414yes—
349C22×C4C4×D414yes—
350C22×C4#12114yes—
351C22×C4#12114yes—
352C22×C4#12412yes—
353C22×C4#12712yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+412yes—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—416no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8#12314yes—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—816no—
404C22⋊C4—216no—
405C22⋊C4—48no—
406C22⋊C4—84no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—24no—
411C23—28no—
412C23—28no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—28no—
422C23—28no—
423C23#39014yes—
424C23#41718yes—
425C23#48918yes—
426C23#49614yes—
427C23#49714yes—
428C23C22×C22⋊C411yescharacteristic
429C23C2×#12411yescharacteristic
430C23C2×#12711yescharacteristic
431C23#54612yes—
432C23#54712yes—
433C23C2×AGL(1,ℤ/8)12yes—
434C23C2×#13712yes—
435C23#57712yes—
436C23#57812yes—
437Q8—42no—
438Q8—44no—
439Q8—42no—
440Q8—81no—
441Q8—81no—
442Q8—82no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—816no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—816no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—84no—
490C2×C4#39014yes—
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C4×#4112yes—
494C2×C4#52012yes—
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52912yes—
500C2×C4#53411yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53912yes—
503C2×C4#54612yes—
504C2×C4#54612yes—
505C2×C4#54712yes—
506C2×C4#54712yes—
507C2×C4#55112yes—
508C2×C4#55212yes—
509C8—24no—
510C8—42no—
511C8—44no—
512C22—24no—
513C22—24no—
514C22—216no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—216no—
520C22#195918yes—
521C22C2×#39011yescharacteristic
522C22#233912yes—
523C22#248812yes—
524C22#250111yescharacteristic
525C22#253911yescharacteristic
526C22C2×#41712yes—
527C22#289812yes—
528C22C2×#48912yes—
529C22#313712yes—
530C22C2×#49611yescharacteristic
531C22C2×#49711yescharacteristic
532C22#315411yescharacteristic
533C22#315611yescharacteristic
534C22#331014yes—
535C22#331214yes—
536C22#334314yes—
537C22#334414yes—
538C4—24no—
539C4—48no—
540C4—82no—
541C4—84no—
542C4—84no—
543C4—82no—
544C4—161no—
545C4—161no—
546C4C4×#12311yescharacteristic
547C4#232211yescharacteristic
548C4#235912yes—
549C4#241211yescharacteristic
550C4#241311yescharacteristic
551C4#242512yes—
552C2—28no—
553C2#1399611yescharacteristic
554C2C2×#195911yescharacteristic
555C2#2120511yescharacteristic
556C2#2249114yes—
557C2#2771314yes—
558C2#2793311yescharacteristic
559C2#2793911yescharacteristic
560C2#2834411yescharacteristic
561C2#2834511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
8. #11372C2semidirect
9. #11050C2semidirect
10. #10928C2semidirect
11. #10843C2semidirect
22. #2503C4semidirect
102. #391C2×C4semidirect
107. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#18922
C2×#18722
C2×#18232
C2×#17522
C22×#3402
#138802
#113722
#110502
#109282
#108432