Finite Group Databasebeta

← #401674

Subgroups — #401674

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1862C212yesmaximal
4C2×#1802C211yesmaximal, characteristic
5#15196C211yesmaximal, characteristic, complemented
6#11228C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11208C212yesmaximal, complemented
9#11202C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2349—24no—
20#2349—24no—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#385—21no—
25C2×#385—21no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C22×#95—21no—
31#2083—24no—
32#1862—28no—
33#1802—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1444—21no—
44#1441—21no—
45#1441—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2211yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C2×C4×D4D414yes—
67C2×C4×D4D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102Q818yes—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—24no—
101#389—24no—
102#389D414yes—
103#385—28no—
104#385—28no—
105#384—22no—
106#384—22no—
107#382—22no—
108#382—22no—
109#380—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C2×#95—28no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—24no—
127#340—42no—
128#340—42no—
129#336—22no—
130#336—24no—
131#336—42no—
132C25—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C2×D412yes—
139C22×D4C2×D412yes—
140C22×D4C2×Q814yes—
141C22×D4#4112yes—
142C22×D4#4114yes—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×Q814yes—
184C4×D4—28no—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—28no—
189C4×D4—44no—
190C4×D4—48no—
191C4×D4—48no—
192C4×D4—44no—
193C4×D4—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×Q814yes—
236C2×C4⋊C4#4112yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42#4112yes—
258#107—48no—
259C4⋊C8—44no—
260#102—216no—
261#102—48no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—24no—
267#95—22no—
268#95—42no—
269#95—44no—
270#95—42no—
271#95—82no—
272#95—82no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24C22×D411yescharacteristic
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4#12114yes—
299C2×D4C22⋊Q818yes—
300C2×D4#12314yes—
301C2×D4#12518yes—
302C2×D4C4⋊Q814yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—216no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—88no—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#13512yes—
377C22×C4#13712yes—
378C22×C4#13712yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q812yes—
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C4⋊C4#12114yes—
428C22⋊C4—216no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—41no—
439C42—41no—
440C42—42no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—24no—
445C23—28no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23C2×#12011yescharacteristic
465C23C2×#12111yescharacteristic
466C23#52511yescharacteristic
467C23#54612yes—
468C23D4×Q812yes—
469C23C2×#13511yescharacteristic
470C23C2×#13711yescharacteristic
471C23C2×#13711yescharacteristic
472C23#57511yescharacteristic
473C23#57612yes—
474C23#57612yes—
475D4—24no—
476D4—24no—
477D4—28no—
478D4—28no—
479D4—24no—
480D4#39818yes—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—816no—
527C2×C4—816no—
528C2×C4—816no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#45214yes—
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×C22⋊Q812yes—
548C2×C4C2×#12311yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4C2×#12512yes—
552C2×C4C2×C4⋊Q811yescharacteristic
553C2×C4#53411yescharacteristic
554C2×C4#53611yescharacteristic
555C2×C4#53712yes—
556C2×C4#53912yes—
557C2×C4#54612yes—
558C2×C4D4×Q812yes—
559C2×C4#55212yes—
560C2×C4#55212yes—
561C2×C4#55412yes—
562C2×C4#55412yes—
563C8—44no—
564C8—82no—
565C22—28no—
566C22—48no—
567C22—44no—
568C22—44no—
569C22—48no—
570C22—48no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22#240511yescharacteristic
577C22#241712yes—
578C22#243411yescharacteristic
579C22#247312yes—
580C22#284711yescharacteristic
581C22C2×#45211yescharacteristic
582C22#300711yescharacteristic
583C22#300811yescharacteristic
584C22#300811yescharacteristic
585C22#303511yescharacteristic
586C22#303911yescharacteristic
587C22#304112yes—
588C22#310711yescharacteristic
589C22#311611yescharacteristic
590C22#312212yes—
591C22#329712yes—
592C22#331412yes—
593C22#331412yes—
594C22#332212yes—
595C22#338012yes—
596C22#338512yes—
597C22#338512yes—
598C22#338912yes—
599C4—24no—
600C4—42no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—42no—
605C4—81no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4—162no—
610C4C2×#39811yescharacteristic
611C4#241712yes—
612C4#247312yes—
613C4#248411yescharacteristic
614C4#258311yescharacteristic
615C4#261811yescharacteristic
616C2—48no—
617C2#1482111yescharacteristic
618C2#2403911yescharacteristic
619C2#2404911yescharacteristic
620C2#2482012yes—
621C2#2483312yes—
622C2#2583911yescharacteristic
623C2#2585011yescharacteristic
624C2#2730912yes—
625C2#2732012yes—
626C2#3290711yescharacteristic
627C2#3291111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
5. #15196C2semidirect
6. #11228C2semidirect
7. #11219C2semidirect
8. #11208C2semidirect
9. #11202C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#20832
C2×#18622
C2×#18022
#151962
#112282
#112192
#112082
#112022
#107882