Finite Group Databasebeta

← #406743

Subgroups — #406743

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#11385C212yesmaximal, complemented
6#11293C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11267C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2441—24no—
18#2441—24no—
19C2×#399—21no—
20C2×#396—21no—
21C2×#389—21no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#384C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1823—24no—
30#1736—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1445—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—42no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—42no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117D414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4Q814yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94#399—28no—
95#396—28no—
96#389—28no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#387—28no—
102#384—22no—
103#384—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380D414yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—42no—
130C25C2×D411yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×Q812yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—28no—
159C2×M16—48no—
160C2×M16—48no—
161C2×M16—44no—
162C2×M16—44no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8#4114yes—
177#125—48no—
178#125—44no—
179#125—44no—
180#117—28no—
181#117—44no—
182#117—48no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×Q814yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—82no—
245C2×C42—24no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—44no—
255C22⋊C8—44no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—42no—
262#95—82no—
263#95—82no—
264#95—82no—
265#95—82no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24#12012yes—
275C24C22⋊Q812yes—
276C24#12412yes—
277C24C22×D411yescharacteristic
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
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—42no—
346C22×C4—42no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4#12012yes—
352C22×C4C22⋊Q814yes—
353C22×C4C22⋊Q814yes—
354C22×C4C22⋊Q812yes—
355C22×C4#12414yes—
356C22×C4#12414yes—
357C22×C4#12412yes—
358C22×C4#13712yes—
359C22×C4#13712yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370M16—416no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4#12314yes—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—48no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—42no—
423C42—42no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23#39314yes—
447C23C2×#12011yescharacteristic
448C23C2×C22⋊Q811yescharacteristic
449C23C2×#12411yescharacteristic
450C23#54612yes—
451C23#54712yes—
452C23C2×#13711yescharacteristic
453C23C2×#13711yescharacteristic
454C23#57511yescharacteristic
455C23#57511yescharacteristic
456C23#57612yes—
457C23#57712yes—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—816no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#39314yes—
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×#12311yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4C2×#12411yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4#52912yes—
531C2×C4#53312yes—
532C2×C4#53411yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#53912yes—
535C2×C4#54412yes—
536C2×C4#54612yes—
537C2×C4#54712yes—
538C2×C4#55112yes—
539C2×C4#55112yes—
540C2×C4#55212yes—
541C2×C4#55212yes—
542C8—44no—
543C8—82no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22C2×#39311yescharacteristic
556C22#240511yescharacteristic
557C22#240611yescharacteristic
558C22#243812yes—
559C22#248812yes—
560C22#300711yescharacteristic
561C22#300711yescharacteristic
562C22#300811yescharacteristic
563C22#301011yescharacteristic
564C22#307811yescharacteristic
565C22#307811yescharacteristic
566C22#308112yes—
567C22#313612yes—
568C22#313911yescharacteristic
569C22#314011yescharacteristic
570C22#330712yes—
571C22#330912yes—
572C22#333112yes—
573C22#333312yes—
574C22#334412yes—
575C22#334612yes—
576C22#335112yes—
577C22#335512yes—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—82no—
583C4—84no—
584C4—81no—
585C4—82no—
586C4—82no—
587C4—81no—
588C4—162no—
589C4#243812yes—
590C4#248812yes—
591C4#250311yescharacteristic
592C4#256111yescharacteristic
593C4#262711yescharacteristic
594C4#285511yescharacteristic
595C2—44no—
596C2#1493411yescharacteristic
597C2#2345511yescharacteristic
598C2#2345611yescharacteristic
599C2#2404611yescharacteristic
600C2#2405511yescharacteristic
601C2#2410211yescharacteristic
602C2#2411111yescharacteristic
603C2#2601612yes—
604C2#2601812yes—
605C2#2770712yes—
606C2#2772512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
5. #11385C2semidirect
6. #11293C2semidirect
7. #11292C2semidirect
8. #11267C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#18232
C2×#17362
#139242
#113852
#112932
#112922
#112672
#107312
#107222
#107152