Finite Group Databasebeta

← #406738

Subgroups — #406738

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#11385C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11268C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2442—24no—
18#2442—24no—
19C2×#400—21no—
20C2×#398—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#385—21no—
24C2×#382C2211yescharacteristic
25C2×#380—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1824—24no—
30C2×#340—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×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—42no—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—24no—
60C2×#125—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D414yes—
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⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#400—28no—
93#398—28no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#385—28no—
99#382—22no—
100#382—22no—
101#380—28no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380D414yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—21no—
109C2×#95—21no—
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—24no—
142C23×C4—22no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×Q812yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C22×C8—28no—
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—22no—
177C22×C8—48no—
178C22×C8—48no—
179C22×C8—44no—
180C22×C8—44no—
181C22×C8#4114yes—
182#125—48no—
183#125—44no—
184#125—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q814yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C22⋊C8—44no—
254#95—24no—
255#95—24no—
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—22no—
269C24—22no—
270C24#12012yes—
271C24C22⋊Q812yes—
272C24#12412yes—
273C24#13512yes—
274C24#13512yes—
275C24#13712yes—
276C24#13712yes—
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—22no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—48no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—84no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4#12012yes—
359C22×C4C22⋊Q814yes—
360C22×C4C22⋊Q814yes—
361C22×C4C22⋊Q812yes—
362C22×C4#12414yes—
363C22×C4#12414yes—
364C22×C4#12412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—416no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4#12314yes—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—84no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—42no—
425C42—42no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23#39314yes—
443C23#45214yes—
444C23#45214yes—
445C23#47914yes—
446C23#47914yes—
447C23#48814yes—
448C23#49114yes—
449C23C2×#12011yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23C2×#12411yescharacteristic
452C23#54612yes—
453C23#54712yes—
454C23C2×#13511yescharacteristic
455C23C2×#13511yescharacteristic
456C23C2×#13711yescharacteristic
457C23C2×#13711yescharacteristic
458C23#57612yes—
459C23#57712yes—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—816no—
508C2×C4—816no—
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—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#39314yes—
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4C2×#12411yescharacteristic
533C2×C4#52912yes—
534C2×C4#53312yes—
535C2×C4#53411yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#53912yes—
538C2×C4#54412yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#55112yes—
542C2×C4#55112yes—
543C2×C4#55212yes—
544C2×C4#55212yes—
545C8—44no—
546C8—82no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—28no—
551C22—24no—
552C22—28no—
553C22—28no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22#202618yes—
558C22C2×#39311yescharacteristic
559C22#240511yescharacteristic
560C22#240611yescharacteristic
561C22#243812yes—
562C22#248812yes—
563C22C2×#45211yescharacteristic
564C22C2×#45211yescharacteristic
565C22#300811yescharacteristic
566C22#301011yescharacteristic
567C22C2×#47911yescharacteristic
568C22C2×#47911yescharacteristic
569C22#308112yes—
570C22C2×#48811yescharacteristic
571C22C2×#49111yescharacteristic
572C22#313612yes—
573C22#329712yes—
574C22#330112yes—
575C22#331412yes—
576C22#331612yes—
577C22#333712yes—
578C22#333912yes—
579C22#334412yes—
580C22#334612yes—
581C4—24no—
582C4—44no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—84no—
587C4—81no—
588C4—82no—
589C4—81no—
590C4—82no—
591C4—162no—
592C4#243812yes—
593C4#248812yes—
594C4#250311yescharacteristic
595C4#256111yescharacteristic
596C4#262711yescharacteristic
597C4#285511yescharacteristic
598C2—28no—
599C2#1493411yescharacteristic
600C2C2×#202611yescharacteristic
601C2#2345611yescharacteristic
602C2#2403611yescharacteristic
603C2#2403911yescharacteristic
604C2#2409211yescharacteristic
605C2#2409511yescharacteristic
606C2#2600012yes—
607C2#2600212yes—
608C2#2770112yes—
609C2#2771512yes—
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. #11292C2semidirect
7. #11292C2semidirect
8. #11268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#18242
C22×#3402
#139242
#113852
#112922
#112682
#107302
#107212
#107142