Finite Group Databasebeta

← #392970

Subgroups — #392970

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic
3C2×#1960C211yesmaximal, characteristic, complemented
4C2×#1956C211yesmaximal, characteristic, complemented
5C2×#1894C212yesmaximal
6C2×#1892C212yesmaximal
7#14340C211yesmaximal, characteristic
8#10928C212yesmaximal, complemented
9#10926C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×#117C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C2×#405—24no—
22C2×#405—24no—
23C2×#405—24no—
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2621—24no—
29#2621C418yescomplemented
30C2×#398—22no—
31C2×#397—22no—
32C2×#391C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37#1960—24no—
38#1956—24no—
39#1894—28no—
40#1892—28no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1441—22no—
46#1439—22no—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16D414yes—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8D414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×C414yes—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8C2×C418yes—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117C2×C414yes—
85C2×#117D414yes—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4D412yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110#405—232no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97D414yes—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121C2×C4×C8—22no—
122C2×C4×C8D414yes—
123#398—216no—
124#397—216no—
125#391—24no—
126#391C2×C418yes—
127#384—22no—
128#384—22no—
129#382—22no—
130#382—22no—
131#382—22no—
132#382C2×C414yes—
133#380—24no—
134#380—24no—
135C2×#95—22no—
136C2×#95—24no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144C25C2×D411yescharacteristic
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8C22×C414yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4C22×C412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C2×M16—28no—
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C22×C8—28no—
189C22×C8—28no—
190C22×C8—28no—
191C22×C8—28no—
192C22×C8—28no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—28no—
210C22×C8—28no—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8#4112yes—
220C22×C8#4112yes—
221C22×C8#4112yes—
222C22×C8#4112yes—
223C4⋊Q8—28no—
224C4⋊Q8—28no—
225C4⋊Q8—44no—
226C4⋊Q8—44no—
227C22⋊Q8—216no—
228C22⋊Q8—216no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231#117—28no—
232#117—28no—
233#117—28no—
234#117—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—84no—
262C2×C4⋊C4C22⋊C418yes—
263C2×C4⋊C4C22×C414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—84no—
271C2×C22⋊C4C22⋊C418yes—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C22×C412yes—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42#4112yes—
283C2×C42#4112yes—
284#103—48no—
285#103—48no—
286#103—48no—
287#103—48no—
288#97—28no—
289#97—28no—
290C4×C8—28no—
291C4×C8—28no—
292#95—28no—
293#95—216no—
294#95—44no—
295#95—44no—
296#95—84no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—24no—
302C24—24no—
303C24C2×C22⋊C412yes—
304C24#12412yes—
305C24#12712yes—
306C24C22×D411yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C4×D414yes—
395C22×C4C4×D414yes—
396C22×C4#12114yes—
397C22×C4#12114yes—
398C22×C4#12114yes—
399C22×C4#12114yes—
400C22×C4#12412yes—
401C22×C4#12712yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416M16—232no—
417C2×C8—232no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8#12114yes—
443C2×C8#12114yes—
444C2×C8#12114yes—
445C2×C8#12114yes—
446C4⋊C4—216no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—416no—
464C22⋊C4—88no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42C4×D414yes—
470C42C4×D414yes—
471C42#12014yes—
472C42#12314yes—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23#39014yes—
490C23C22×C22⋊C411yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12411yescharacteristic
494C23C2×#12711yescharacteristic
495C23#53211yescharacteristic
496C23#53211yescharacteristic
497C23#57714yes—
498C23#57814yes—
499Q8—44no—
500Q8—44no—
501Q8—44no—
502Q8—44no—
503Q8—82no—
504Q8—82no—
505Q8—82no—
506Q8—82no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4#38618yes—
553C2×C4#39014yes—
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C4×#4111yescharacteristic
559C2×C4C4×#4111yescharacteristic
560C2×C4C2×#12011yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C2×C4#52511yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4#52511yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#52911yescharacteristic
575C2×C4#52911yescharacteristic
576C2×C4#52911yescharacteristic
577C2×C4#52911yescharacteristic
578C2×C4#53211yescharacteristic
579C2×C4#53211yescharacteristic
580C2×C4#53211yescharacteristic
581C2×C4#53211yescharacteristic
582C8—24no—
583C8—24no—
584C8—24no—
585C8—24no—
586C22—24no—
587C22—28no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22C2×#38611yescharacteristic
596C22C2×#39011yescharacteristic
597C22#229711yescharacteristic
598C22#285011yescharacteristic
599C22#285011yescharacteristic
600C22#285611yescharacteristic
601C22#285611yescharacteristic
602C22#289814yes—
603C22#305712yes—
604C22#305912yes—
605C22#306314yes—
606C22#313714yes—
607C22#315412yes—
608C22#315612yes—
609C22#317412yes—
610C22#317712yes—
611C22#317814yes—
612C4—24no—
613C4—28no—
614C4—44no—
615C4—84no—
616C4—84no—
617C4—162no—
618C4—162no—
619C4C4×#12111yescharacteristic
620C4C4×#12111yescharacteristic
621C4C4×#12111yescharacteristic
622C4C4×#12111yescharacteristic
623C4#284711yescharacteristic
624C4#284711yescharacteristic
625C4#285411yescharacteristic
626C4#285411yescharacteristic
627C2—44no—
628C2#1469811yescharacteristic
629C2#2097412yes—
630C2#2120512yes—
631C2#2141312yes—
632C2#3304612yes—
633C2#3305612yes—
634C2#3317811yescharacteristic
635C2#3318011yescharacteristic
636C2#3318812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1960C2semidirect
4. C2×#1956C2semidirect
8. #10928C2semidirect
9. #10926C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect
29. #2621C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#26212
C2×#19602
C2×#19562
C2×#18942
C2×#18922
#143402
#109282
#109262
#108432
#108422