Finite Group Databasebeta

← #186671

Subgroups — #186671

Order 512.

2208 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, characteristic
8#8451C214yesmaximal, complemented
9#8448C214yesmaximal, complemented
10#3502—24no—
11#3502—24no—
12#3502—24no—
13#3502C418yescomplemented
14C2×Q82—21no—
15C2×Q82C2212yes—
16C2×#557—21no—
17C2×#557—22no—
18C2×#557—22no—
19C2×#557C2212yes—
20C2×#544—22no—
21C2×#544—21no—
22C2×#533—22no—
23C2×#533C2211yescharacteristic
24C2×#533C2211yescharacteristic
25C2×#519—21no—
26C22×C4×Q8C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103—21no—
31C2×#384C2211yescharacteristic
32#1990—21no—
33#1990C2212yes—
34#1956—21no—
35#1956C2212yes—
36#1872—22no—
37#1872—22no—
38#1803—22no—
39#1803—22no—
40#1803C2214yes—
41#1803C2214yes—
42C2×#340—21no—
43#1542—24no—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1481—24no—
47#1479—24no—
48C2×Q8⋊C8C2214yes—
49C2×Q8⋊C8C2214yes—
50C23×Q8D411yescharacteristic
51C23×C8—21no—
52Q82—28no—
53Q82—28no—
54Q82—24no—
55Q82—24no—
56Q82—24no—
57Q82C2×C418yescomplemented
58#557—28no—
59#557—216no—
60#557—216no—
61#557—28no—
62#557—24no—
63#557—216no—
64#557—216no—
65#557—24no—
66#557—24no—
67#557C2×C418yescomplemented
68#544—216no—
69#544—28no—
70#544—216no—
71#544—28no—
72#533—216no—
73#533—216no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533C2×C414yes—
79#533C2×C414yes—
80#533D414yes—
81#533D414yes—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8D411yescharacteristic
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—22no—
95C2×#125—24no—
96C2×#125—22no—
97C2×#125—21no—
98C2×#125—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8D412yes—
103C2×C22⋊Q8D412yes—
104#519—28no—
105#519—28no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2×C414yes—
120C2×C4×Q8D414yes—
121C2×C4×Q8D412yes—
122C2×C4×Q8D412yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×#117—24no—
126C2×#117—21no—
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#431—22no—
136#431—22no—
137#431—22no—
138#431—22no—
139C2×C4⋊C8—22no—
140C2×C4⋊C8—22no—
141C2×C4⋊C8—21no—
142C2×C4⋊C8—21no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—44no—
149C2×C22⋊C8—21no—
150C2×C22⋊C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8C2312yes—
153#384—21no—
154#384—21no—
155#384—21no—
156#384C2312yes—
157#340—24no—
158#340—24no—
159#340—24no—
160#333—24no—
161#332—24no—
162Q8⋊C8—28no—
163Q8⋊C8—28no—
164C22×Q8—24no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—21no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—21no—
171C22×Q8C22⋊C414yes—
172C22×Q8SD1614yes—
173C22×Q8Q1614yes—
174C22×Q8C2×D412yes—
175C22×Q8C2×D412yes—
176C22×Q8C2×D412yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—41no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8—216no—
192C4⋊Q8—232no—
193C4⋊Q8—28no—
194C4⋊Q8—216no—
195C4⋊Q8—28no—
196C4⋊Q8—216no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8C22⋊C414yes—
206C4⋊Q8C22⋊C414yes—
207C4⋊Q8C22×C412yes—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C2×D412yes—
210C4⋊Q8C2×D412yes—
211#125—232no—
212#125—216no—
213#125—232no—
214#125—28no—
215#125—216no—
216#125—216no—
217#125—28no—
218#125—24no—
219#125—24no—
220#125—24no—
221#125—24no—
222C22⋊Q8—232no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8C22⋊C418yes—
230C22⋊Q8C22⋊C418yes—
231C4×Q8—28no—
232C4×Q8—216no—
233C4×Q8—28no—
234C4×Q8—216no—
235C4×Q8—232no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—28no—
241C4×Q8—28no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8C22⋊C418yes—
257C4×Q8C22⋊C418yes—
258C4×Q8C22×C414yes—
259C4×Q8C22×C414yes—
260C4×Q8C2×D414yes—
261C4×Q8C2×D414yes—
262#117—232no—
263#117—24no—
264#117—24no—
265#117—24no—
266#117—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C22⋊C414yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C22×C412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2411yescharacteristic
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326#103—48no—
327#103—84no—
328#103—84no—
329C22⋊C8—42no—
330C22⋊C8—42no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—42no—
334C24#12011yescharacteristic
335C2×Q8—216no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—28no—
346C2×Q8—24no—
347C2×Q8—28no—
348C2×Q8—24no—
349C2×Q8—28no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—81no—
357C2×Q8—81no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8#103116yescomplemented
361C2×Q8C2×C22⋊C414yes—
362C2×Q8C2×C22⋊C414yes—
363C2×Q8C2×C22⋊C414yes—
364C2×Q8C2×SD1614yes—
365C2×Q8C2×Q1614yes—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4#12012yes—
390C22×C4#12011yescharacteristic
391C22×C4#12011yescharacteristic
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12011yescharacteristic
395C22×C4C2×SD1612yes—
396C22×C4C2×Q1612yes—
397C22×C4#13712yes—
398C22×C4#13712yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—84no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4C2×C22⋊C414yes—
446C4⋊C4C2×C22⋊C414yes—
447C4⋊C4C2×C22⋊C414yes—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C42—216no—
451C42—216no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—28no—
457C42—28no—
458C42C2×C22⋊C412yes—
459C42C2×C22⋊C412yes—
460C42C2×C22⋊C412yes—
461C42#12012yes—
462C42#12012yes—
463C42#12012yes—
464C42C23×C411yescharacteristic
465C42C22×D411yescharacteristic
466C42C22×D411yescharacteristic
467C42C22×D411yescharacteristic
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23#37912yes—
473C23#44812yes—
474C23#45112yes—
475C23C2×#12011yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—24no—
479Q8—216no—
480Q8—28no—
481Q8—44no—
482Q8—44no—
483Q8—44no—
484Q8—44no—
485Q8—82no—
486Q8—84no—
487Q8—816no—
488Q8C2×#10318yes—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—216no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—164no—
523C2×C4#37912yes—
524C2×C4#37912yes—
525C2×C4#37912yes—
526C2×C4C2×#10314yes—
527C2×C4#41914yes—
528C2×C4#44814yes—
529C2×C4#44812yes—
530C2×C4#45114yes—
531C2×C4#45112yes—
532C2×C4C22×C22⋊C411yescharacteristic
533C2×C4C22×C22⋊C411yescharacteristic
534C2×C4C22×C22⋊C411yescharacteristic
535C2×C4C2×#12011yescharacteristic
536C2×C4C2×#12011yescharacteristic
537C2×C4C2×#12011yescharacteristic
538C2×C4C22×SD1611yescharacteristic
539C2×C4C22×Q1611yescharacteristic
540C2×C4C2×#13711yescharacteristic
541C2×C4C2×#13711yescharacteristic
542C2×C4#57712yes—
543C2×C4#57812yes—
544C8—82no—
545C8—82no—
546C8—82no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22#178714yes—
554C22C2×#37911yescharacteristic
555C22C2×#44811yescharacteristic
556C22C2×#45111yescharacteristic
557C22#301011yescharacteristic
558C22#301111yescharacteristic
559C22#302111yescharacteristic
560C22#302211yescharacteristic
561C4—28no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—162no—
572C4#178714yes—
573C4C22×#10311yescharacteristic
574C4C2×#41911yescharacteristic
575C4#289812yes—
576C4C2×#44811yescharacteristic
577C4C2×#45111yescharacteristic
578C4#301312yes—
579C2—24no—
580C2#1891411yescharacteristic
581C2#1891511yescharacteristic
582C2#1892411yescharacteristic
583C2#1892511yescharacteristic
584C2C2×#178711yescharacteristic
585C2#2060311yescharacteristic
586C2#2061811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #8451C2semidirect
9. #8448C2semidirect
13. #3502C4semidirect
57. Q82C2×C4semidirect
67. #557C2×C4semidirect
360. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#35022
#218602
C2×#18032
C2×#15422
C2×#14812
C2×#14792
#84512
#84482