Finite Group Databasebeta

← #402300

Subgroups — #402300

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C211yesmaximal, characteristic, complemented
2C2×#2077C212yesmaximal
3C2×#2035C212yesmaximal
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#14686C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic, complemented
8#11323C211yesmaximal, characteristic, complemented
9#11050C211yesmaximal, characteristic, complemented
10#10926C212yesmaximal, complemented
11#10861C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#106C2212yes—
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#380—22no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27#2096—24no—
28#2077—28no—
29#2035—28no—
30C4×C22⋊C8—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1436—22no—
38C4⋊#95—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—24no—
66C22×C42—42no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—42no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8D418yes—
85C2×C4×C8—24no—
86C2×C4×C8—24no—
87C2×C4×C8—42no—
88#391—22no—
89#391—22no—
90#391—24no—
91#384—22no—
92#384—22no—
93#384—42no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—48no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—24no—
104C2×#95—21no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—22no—
110#336D414yes—
111C25C2×D411yescharacteristic
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8#4112yes—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4SD1618yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D414yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
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×C8C2×D414yes—
160C22×C8C2×D414yes—
161C22×C8C2×D414yes—
162C22×C8#4114yes—
163C22⋊Q8—48no—
164C22⋊Q8—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
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—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4#4112yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—48no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—48no—
213C2×C42—48no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42#4112yes—
217C2×C42#4114yes—
218#106—48no—
219#106—48no—
220#103—48no—
221#103—48no—
222C22⋊C8—216no—
223C4×C8—48no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—48no—
229#95—44no—
230#95—44no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24#12712yes—
241C24C2×SD1614yes—
242C24C22×D411yescharacteristic
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—216no—
280C22×C4—216no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4#12114yes—
329C22×C4#12314yes—
330C22×C4#12712yes—
331C22×C4C2×SD1614yes—
332C22×C4C2×SD1614yes—
333C22×C4#13514yes—
334C22×C4#13714yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D412yes—
337C22×C4C2×#4112yes—
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8#12118yes—
359C4⋊C4—24no—
360C4⋊C4—48no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C22⋊C4—48no—
377C22⋊C4—84no—
378C22⋊C4—88no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—48no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23#49214yes—
407C23C2×#12111yescharacteristic
408C23C2×#12311yescharacteristic
409C23#52511yescharacteristic
410C23C2×#12711yescharacteristic
411C23#53211yescharacteristic
412C23#53411yescharacteristic
413C23#53611yescharacteristic
414C23C22×SD1612yes—
415C23C2×#13512yes—
416C23C2×#13712yes—
417C23#57512yes—
418Q8—44no—
419Q8—44no—
420Q8—82no—
421Q8—82no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—432no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#46518yes—
479C2×C4#48318yes—
480C2×C4C2×#12112yes—
481C2×C4#52912yes—
482C2×C4#52912yes—
483C2×C4#53212yes—
484C2×C4#53912yes—
485C2×C4#54212yes—
486C2×C4#54612yes—
487C2×C4#54612yes—
488C2×C4#54612yes—
489C2×C4#54711yescharacteristic
490C2×C4#54712yes—
491C2×C4#54712yes—
492C2×C4#54811yescharacteristic
493C2×C4#55112yes—
494C2×C4#55211yescharacteristic
495C2×C4#55611yescharacteristic
496C8—28no—
497C8—44no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22#239511yescharacteristic
510C22#241211yescharacteristic
511C22#241511yescharacteristic
512C22#250111yescharacteristic
513C22#253911yescharacteristic
514C22#266311yescharacteristic
515C22#285611yescharacteristic
516C22C2×#46512yes—
517C22#305612yes—
518C22C2×#48312yes—
519C22#309112yes—
520C22#310312yes—
521C22#310712yes—
522C22C2×#49211yescharacteristic
523C22#314711yescharacteristic
524C22#315211yescharacteristic
525C22#315311yescharacteristic
526C22#316712yes—
527C22#317212yes—
528C22#318712yes—
529C22#319012yes—
530C22#322912yes—
531C22#323312yes—
532C4—28no—
533C4—28no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4—161no—
538C4—162no—
539C4—162no—
540C4—161no—
541C4#242512yes—
542C4#242712yes—
543C4#252112yes—
544C4#255312yes—
545C2—44no—
546C2#1483911yescharacteristic
547C2#2375612yes—
548C2#2455112yes—
549C2#2470312yes—
550C2#2790511yescharacteristic
551C2#2791111yescharacteristic
552C2#2830511yescharacteristic
553C2#2830811yescharacteristic
554C2#3035912yes—
555C2#3317011yescharacteristic
556C2#3318611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2096C2semidirect
7. #11373C2semidirect
8. #11323C2semidirect
9. #11050C2semidirect
10. #10926C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#20772
C2×#20352
C2×C4×C22⋊C82
#147022
#146862
#113732
#113232
#110502
#109262
#108612