Finite Group Databasebeta

← #363885

Subgroups — #363885

Order 512.

2732 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#1901C212yesmaximal, complemented
3#20898C212yesmaximal
4C2×#1821C211yesmaximal, characteristic, complemented
5C22×#343C211yesmaximal, characteristic, complemented
6#11383C212yesmaximal, complemented
7#10968C212yesmaximal, complemented
8#10763C212yesmaximal, complemented
9#10728C212yesmaximal, complemented
10C23×M16C2212yes—
11C2×#552—28no—
12C2×#552—24no—
13C2×#552—24no—
14C2×#552—24no—
15C2×#552—24no—
16C22×#126C412yes—
17C22×#125C412yes—
18C22×#125C2211yescharacteristic
19C22×#123C412yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×#117C412yes—
25C22×#117C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C22×#101C2214yes—
28C2×#407C2212yes—
29C2×#407C2214yes—
30#1901—28no—
31#1870—24no—
32#1869—24no—
33#1821—24no—
34C2×#343—24no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2214yes—
40#1728C2214yes—
41C23×Q8C2311yescharacteristic
42C23×D4C2×C411yescharacteristic
43C24×C4C2×C412yes—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—28no—
49C22×M16—24no—
50C22×M16—44no—
51C22×M16C2314yes—
52C22×M16C2314yes—
53C23×C8—22no—
54#552—232no—
55#552—232no—
56#552—232no—
57#552—232no—
58C2×#126—28no—
59C2×#126—28no—
60C2×#126—28no—
61C2×#125—28no—
62C2×#125—28no—
63C2×#125—28no—
64C2×#125—24no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125—24no—
68C2×#123—24no—
69C2×#123—24no—
70C2×#123—28no—
71C2×#123—28no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8D418yes—
81C2×C4×Q8—24no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—24no—
85C2×C4×D4—24no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4D414yes—
89C2×C4×D4D414yes—
90C2×#117—28no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117—28no—
96C2×#117—24no—
97C2×#117—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2×C412yes—
102C22×C4⋊C4C2×C412yes—
103C22×C4⋊C4C2×C412yes—
104C22×C4⋊C4C2×C412yes—
105C22×C4⋊C4C2×C412yes—
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C412yes—
108C22×C4⋊C4D414yes—
109C22×C4⋊C4D414yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2×C412yes—
114C22×C22⋊C4C2×C412yes—
115C22×C22⋊C4C2×C411yescharacteristic
116C22×C22⋊C4C2×C412yes—
117C22×C22⋊C4C2×C411yescharacteristic
118C22×C42C2×C412yes—
119C22×C42C2×C412yes—
120C22×C42C2311yescharacteristic
121C2×#101—28no—
122C2×#101—28no—
123C2×#101—44no—
124#407—24no—
125#407—24no—
126#407—28no—
127#407—44no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—28no—
130C2×C22⋊C8—22no—
131#343—24no—
132#343—24no—
133#343—28no—
134#343—24no—
135#343—24no—
136#336—28no—
137C25C22×C411yescharacteristic
138C25C22×C411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C2×D414yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—44no—
149C22×D4C22⋊C412yes—
150C22×D4C22⋊C412yes—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—48no—
164C23×C4C22⋊C418yes—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D414yes—
174C23×C4C2×D414yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2411yescharacteristic
180C2×M16—24no—
181C2×M16—216no—
182C2×M16—24no—
183C2×M16—24no—
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—48no—
193C2×M16—416no—
194C2×M16—48no—
195C2×M16—44no—
196C2×M16#4118yes—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—48no—
200C22×C8—44no—
201#126—264no—
202#125—264no—
203#125—216no—
204#125—216no—
205#123—232no—
206#123—232no—
207C22⋊Q8—216no—
208C22⋊Q8—216no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C4×Q8—216no—
214C4×Q8—216no—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219#117—216no—
220#117—216no—
221#117—232no—
222#117—232no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4C22⋊C418yes—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22⋊C418yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4C22⋊C412yes—
274C2×C22⋊C4C22⋊C412yes—
275C2×C22⋊C4C22⋊C418yes—
276C2×C22⋊C4C22⋊C414yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42C2×D414yes—
287#101—416no—
288C22⋊C8—416no—
289C22⋊C8—44no—
290C22⋊C8—44no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24C2×C22⋊C412yes—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C412yes—
304C24C2×C22⋊C412yes—
305C24C2×C22⋊C411yescharacteristic
306C24C23×C411yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—28no—
321C2×D4—44no—
322C2×D4—48no—
323C2×D4—48no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—416no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—416no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4C2×C22⋊C414yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C414yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C414yes—
387C22×C4#12014yes—
388C22×C4#12014yes—
389C22×C4#12014yes—
390C22×C4#12014yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C2×#4114yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
398M16—28no—
399M16—28no—
400M16—416no—
401M16—44no—
402M16—44no—
403M16—44no—
404M16—44no—
405M16—44no—
406M16—44no—
407C2×C8—28no—
408C2×C8—416no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—216no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—216no—
425C4⋊C4—28no—
426C4⋊C4—216no—
427C22⋊C4—216no—
428C22⋊C4—216no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C42—216no—
435C42—216no—
436C42—24no—
437C42—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—48no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—84no—
466C23—84no—
467C23#37912yes—
468C23#37912yes—
469C23#41314yes—
470C23C22×C22⋊C411yescharacteristic
471C23C22×C22⋊C411yescharacteristic
472C23C22×C22⋊C411yescharacteristic
473C23C4×#4112yes—
474C23#52012yes—
475Q8—44no—
476Q8—44no—
477Q8—48no—
478D4—48no—
479D4—48no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—416no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—416no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4#37914yes—
526C2×C4C2×#12012yes—
527C2×C4C2×#12012yes—
528C2×C4#52512yes—
529C2×C4#53212yes—
530C2×C4#53511yescharacteristic
531C2×C4#53611yescharacteristic
532C2×C4#53712yes—
533C2×C4#54614yes—
534C2×C4#54614yes—
535C2×C4#54714yes—
536C2×C4#54714yes—
537C8—48no—
538C8—84no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—28no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—44no—
552C22—48no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—48no—
559C22—82no—
560C22—84no—
561C22—82no—
562C22#178514yes—
563C22C2×#37911yescharacteristic
564C22#229711yescharacteristic
565C22#231111yescharacteristic
566C22#233912yes—
567C22#233912yes—
568C22C2×#41312yes—
569C22#288912yes—
570C4—24no—
571C4—42no—
572C4—44no—
573C4—44no—
574C4—42no—
575C4—44no—
576C4—42no—
577C4—44no—
578C4#240412yes—
579C4#240612yes—
580C4#243014yes—
581C2—21no—
582C2—22no—
583C2—21no—
584C2—24no—
585C2—42no—
586C2—82no—
587C2#1395611yescharacteristic
588C2C2×#178511yescharacteristic
589C2#2059311yescharacteristic
590C2#2248014yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1901C2semidirect
4. C2×#1821C2semidirect
5. C22×#343C2semidirect
6. #11383C2semidirect
7. #10968C2semidirect
8. #10763C2semidirect
9. #10728C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#19012
#208982
C2×#18212
C22×#3432
#113832
#109682
#107632
#107282