Finite Group Databasebeta

← #366749

Subgroups — #366749

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4C2×C8×C22⋊C4C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#14139C211yesmaximal, characteristic
7#13904C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11122C212yesmaximal, complemented
10#10926C212yesmaximal, complemented
11#10861C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#483—24no—
17C2×#483—24no—
18C2×#483C418yescomplemented
19C22×#106C2212yes—
20C22×#106C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2212yes—
26C2×#391C2212yes—
27C2×#381—22no—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1892—24no—
31#1853—24no—
32C8×C22⋊C4—24no—
33C2×#340—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#336C2212yes—
40#1445—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
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×C8C2311yescharacteristic
56C23×C8C2312yes—
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—42no—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D412yes—
85C22×C42C2312yes—
86#483—232no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×#106C2×C418yescomplemented
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103C2×C418yescomplemented
100C2×C22⋊C8C2×C414yes—
101C2×C22⋊C8D414yes—
102C2×C22⋊C8D414yes—
103C2×C4×C8—24no—
104C2×C4×C8—24no—
105C2×C4×C8—42no—
106#391—24no—
107#391—42no—
108#381—48no—
109#380—22no—
110#380—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—21no—
117C2×#95—24no—
118C2×#95D412yes—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#336—24no—
125C25C2×D411yescharacteristic
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8C22×C412yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—82no—
155C23×C4C22×C412yes—
156C23×C4C2×D414yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—28no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—28no—
177C22×C8—44no—
178C22×C8—48no—
179C22×C8—48no—
180C22×C8C22×C414yes—
181C22×C8C2×D412yes—
182C22×C8C2×D414yes—
183C22×C8C2×D414yes—
184C22×C8C2×D412yes—
185C22×C8#4114yes—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
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—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4#4114yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4SD1618yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245#106—216no—
246#106—48no—
247#103—216no—
248#103—44no—
249#103—44no—
250C22⋊C8C22⋊C4116yes—
251C4×C8—48no—
252#95—24no—
253#95—48no—
254#95—42no—
255#95—48no—
256#95—48no—
257#95—84no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24C4×D412yes—
263C24#12112yes—
264C24#12712yes—
265C24C2×SD1614yes—
266C24#13514yes—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
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—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—48no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—416no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—816no—
340C22×C4—84no—
341C22×C4—88no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4C2×C22⋊C414yes—
346C22×C4C4×D414yes—
347C22×C4C4×D412yes—
348C22×C4#12114yes—
349C22×C4#12112yes—
350C22×C4#12414yes—
351C22×C4#12712yes—
352C22×C4C2×SD1618yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—416no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8C2×C22⋊C418yes—
380C2×C8#12314yes—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C22⋊C4—216no—
403C22⋊C4—48no—
404C22⋊C4—84no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C23—28no—
410C23—28no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—28no—
418C23—28no—
419C23—22no—
420C23—24no—
421C23#39014yes—
422C23C4×SD1618yes—
423C23#46518yes—
424C23#49214yes—
425C23#49514yes—
426C23C2×C4×D411yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×#12711yescharacteristic
429C23#54612yes—
430C23#54712yes—
431C23C22×SD1612yes—
432C23C2×#13512yes—
433C23#57712yes—
434C23#57812yes—
435Q8—42no—
436Q8—44no—
437Q8—42no—
438Q8—81no—
439Q8—81no—
440Q8—82no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—48no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—816no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—816no—
475C2×C4—81no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—84no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—84no—
487C2×C4—81no—
488C2×C4#39014yes—
489C2×C4C22×C22⋊C411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C4×#4112yes—
492C2×C4#52012yes—
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4C2×#12411yescharacteristic
497C2×C4#52912yes—
498C2×C4#53411yescharacteristic
499C2×C4#53611yescharacteristic
500C2×C4#53912yes—
501C2×C4#54612yes—
502C2×C4#54612yes—
503C2×C4#54712yes—
504C2×C4#54712yes—
505C2×C4#55112yes—
506C2×C4#55212yes—
507C8—28no—
508C8—24no—
509C8—42no—
510C22—24no—
511C22—24no—
512C22—216no—
513C22—216no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22#196818yes—
519C22C2×#39011yescharacteristic
520C22#235912yes—
521C22#242512yes—
522C22#250111yescharacteristic
523C22#253911yescharacteristic
524C22C2×C4×SD1612yes—
525C22#294712yes—
526C22C2×#46512yes—
527C22#306312yes—
528C22C2×#49211yescharacteristic
529C22C2×#49511yescharacteristic
530C22#315411yescharacteristic
531C22#315611yescharacteristic
532C22#329614yes—
533C22#329814yes—
534C22#333514yes—
535C22#333614yes—
536C4—24no—
537C4—48no—
538C4—84no—
539C4—82no—
540C4—84no—
541C4—82no—
542C4—161no—
543C4—161no—
544C4C4×#12311yescharacteristic
545C4#232211yescharacteristic
546C4#233912yes—
547C4#241211yescharacteristic
548C4#241311yescharacteristic
549C4#248812yes—
550C2—28no—
551C2#1399611yescharacteristic
552C2C2×#196811yescharacteristic
553C2#2123411yescharacteristic
554C2#2292014yes—
555C2#2523614yes—
556C2#2792411yescharacteristic
557C2#2793111yescharacteristic
558C2#2833011yescharacteristic
559C2#2833611yescharacteristic
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. #11378C2semidirect
9. #11122C2semidirect
10. #10926C2semidirect
11. #10861C2semidirect
18. C2×#483C4semidirect
92. C2×#106C2×C4semidirect
99. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#18922
C2×#18532
C2×C8×C22⋊C42
C22×#3402
#141392
#139042
#113782
#111222
#109262
#108612