Finite Group Databasebeta

← #396592

Subgroups — #396592

Order 512.

1536 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×#2074C211yesmaximal, characteristic, complemented
2C2×#1892C212yesmaximal
3C22×#336C211yesmaximal, characteristic
4#14702C211yesmaximal, characteristic
5#14682C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11268C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11052C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2212yes—
13C22×C22⋊Q8C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23#2074—24no—
24#1892—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336—24no—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—22no—
40C22⋊#95—22no—
41C22⋊#95—22no—
42C23×Q8C2311yescharacteristic
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C23×C8—24no—
46C23×C8—24no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42—24no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—42no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—42no—
79C2×#103—42no—
80#391—22no—
81#391—22no—
82#391—42no—
83#382—22no—
84#382—22no—
85#382—42no—
86#380—24no—
87#380—24no—
88#380—24no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336D414yes—
108C25C2×D411yescharacteristic
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8C2×D412yes—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4#4112yes—
143C23×C4#4114yes—
144C23×C4#4114yes—
145C23×C4C2411yescharacteristic
146C22×C8—216no—
147C22×C8—22no—
148C22×C8—22no—
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×C8—48no—
160C22×C8—48no—
161C22×C8—48no—
162C22×C8—48no—
163C22×C8—48no—
164C22×C8—48no—
165C22×C8C2×D414yes—
166C22×C8#4114yes—
167C22⋊Q8—216no—
168C22⋊Q8—48no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—48no—
213C2×C42—48no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42#4112yes—
217#103—48no—
218#103—48no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—44no—
228#95—44no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24#12112yes—
236C24#12312yes—
237C24#12412yes—
238C24#13514yes—
239C24#13714yes—
240C24C22×D411yescharacteristic
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—216no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—48no—
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—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4#12114yes—
331C22×C4#12112yes—
332C22×C4#12318yes—
333C22×C4#12312yes—
334C22×C4#12412yes—
335C22×C4#12618yes—
336C22×C4#13514yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)12yes—
346C22×C42−1+411yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—432no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—42no—
364C4⋊C4—48no—
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—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C22⋊C4—48no—
378C22⋊C4—88no—
379C22⋊C4—84no—
380C42—24no—
381C42—24no—
382C42—48no—
383C42—42no—
384C42—42no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—22no—
392C23—24no—
393C23—22no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—24no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23#39714yes—
404C23#46418yes—
405C23#48518yes—
406C23#48814yes—
407C23#49114yes—
408C23C2×#12111yescharacteristic
409C23C2×#12111yescharacteristic
410C23C2×#12311yescharacteristic
411C23C2×#12411yescharacteristic
412C23#53211yescharacteristic
413C23#53411yescharacteristic
414C23#53711yescharacteristic
415C23C2×#13512yes—
416C23C2×#13512yes—
417C23C2×#13712yes—
418C23#57512yes—
419Q8—44no—
420Q8—44no—
421Q8—82no—
422Q8—82no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—432no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#39714yes—
485C2×C4C2×#12312yes—
486C2×C4#52512yes—
487C2×C4C2×#12612yes—
488C2×C4#52912yes—
489C2×C4#53812yes—
490C2×C4#53912yes—
491C2×C4D4211yescharacteristic
492C2×C4#54612yes—
493C2×C4#54612yes—
494C2×C4#54711yescharacteristic
495C2×C4#54811yescharacteristic
496C2×C4#55112yes—
497C2×C4#55112yes—
498C2×C4#55212yes—
499C2×C4#55312yes—
500C2×C4#55311yescharacteristic
501C8—44no—
502C8—44no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—28no—
508C22—28no—
509C22—28no—
510C22—28no—
511C22—24no—
512C22—24no—
513C22#207418yescomplemented
514C22C2×#39711yescharacteristic
515C22#241211yescharacteristic
516C22#241611yescharacteristic
517C22#249811yescharacteristic
518C22#254311yescharacteristic
519C22#263911yescharacteristic
520C22#285011yescharacteristic
521C22C2×#46412yes—
522C22#303512yes—
523C22#305012yes—
524C22#305612yes—
525C22C2×#48512yes—
526C22#309112yes—
527C22C2×#48811yescharacteristic
528C22#313111yescharacteristic
529C22C2×#49111yescharacteristic
530C22#313511yescharacteristic
531C22#316812yes—
532C22#318112yes—
533C22#319012yes—
534C22#319112yes—
535C22#324812yes—
536C22#325012yes—
537C4—24no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—161no—
543C4—162no—
544C4—162no—
545C4—161no—
546C4#241812yes—
547C4#241912yes—
548C4#248112yes—
549C4#251012yes—
550C2—28no—
551C2#1473511yescharacteristic
552C2C2×#207411yescharacteristic
553C2#2362711yescharacteristic
554C2#2455412yes—
555C2#2475512yes—
556C2#2786012yes—
557C2#2857612yes—
558C2#3018611yescharacteristic
559C2#3019011yescharacteristic
560C2#3304411yescharacteristic
561C2#3305311yescharacteristic
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×#2074C2semidirect
6. #11358C2semidirect
7. #11315C2semidirect
8. #11268C2semidirect
9. #11190C2semidirect
10. #11052C2semidirect
513. C22#2074semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
C2×#18922
C22×#3362
#147022
#146822
#113582
#113152
#112682
#111902
#110522
#107212