Finite Group Databasebeta

← #6519683

Subgroups — #6519683

Order 512.

2156 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic
3C2×#2547C211yesmaximal, characteristic
4#34213C211yesmaximal, characteristic, complemented
5#28124C214yesmaximal, complemented
6#25365C214yesmaximal, complemented
7#25273C214yesmaximal, complemented
8#24729C214yesmaximal, complemented
9#24629C214yesmaximal, complemented
10#24217C214yesmaximal
11#21951C211yesmaximal, characteristic, complemented
12#21804C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2212yes—
17C22×C4×D4C2211yescharacteristic
18#3209—28no—
19C2×#494C2212yescomplemented
20C2×#487C2212yescomplemented
21#3101—28no—
22C2×#483C2212yes—
23C2×#476C2212yes—
24C2×#464C2218yes—
25C2×#462C2214yescomplemented
26C2×#461C2212yes—
27C2×#451C2214yes—
28C2×#450C2212yes—
29C2×C4×SD16C2214yes—
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2547—22no—
33#2547—22no—
34#2547C2214yes—
35#2547C2214yes—
36C2×#397—22no—
37#2370—22no—
38#2369C2212yescomplemented
39#2365C2212yescomplemented
40#2355—22no—
41#2355C2212yescomplemented
42Q8×C4⋊C4C2212yescomplemented
43#2329—22no—
44#2317—21no—
45C4×C22⋊Q8C2214yes—
46C2×#391C2212yes—
47#2285C2211yescharacteristic
48C2×#380—21no—
49C2×#380—21no—
50#2274—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2066C2218yescomplemented
53#2056C2218yescomplemented
54#2042C2218yes—
55#2015C2218yes—
56#1983C2218yes—
57#1981C2214yescomplemented
58#1956C2212yescomplemented
59#1956C2212yescomplemented
60#1939C2214yescomplemented
61#1939C2214yescomplemented
62#1927C2218yescomplemented
63#1878C2218yescomplemented
64#1823C2212yes—
65#1802C2212yes—
66#1802C2212yes—
67#1797C2214yes—
68#1787C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×Q16C2318yes—
75C22×SD16C2314yes—
76#533—24no—
77#533D418yes—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#125—21no—
87C2×#124—24no—
88C2×#124—48no—
89C2×#124C2318yes—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8—21no—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2314yes—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×#117—21no—
102C2×#117—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#494—28no—
114#487—28no—
115#483—28no—
116#476—28no—
117#464—416no—
118#462—216no—
119#461—28no—
120#451—216no—
121#450—28no—
122C4×SD16—216no—
123#431—24no—
124#431—24no—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2318yes—
129C2×#103C2314yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×#103C2318yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#397—28no—
141#397—28no—
142#391—24no—
143#391—24no—
144#391C2314yes—
145#391C2318yes—
146#389—22no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#387—24no—
151#387—24no—
152#387—24no—
153#385—24no—
154#384—22no—
155#384C2311yescharacteristic
156#384C2311yescharacteristic
157#382—22no—
158#382—21no—
159#380—21no—
160#380—24no—
161#380—24no—
162#380—24no—
163#380—24no—
164#380C2312yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—24no—
178C2×#95—22no—
179C2×#95—22no—
180C2×#95D412yes—
181#340C2314yes—
182#340C2314yes—
183#336C2318yes—
184C25—21no—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8C2414yes—
192C22×Q8C2412yes—
193C22×D4—24no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—42no—
198C22×D4C2412yes—
199C23×C4—22no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C2×Q16—216no—
212C2×Q16—48no—
213C2×Q16—48no—
214C2×SD16—28no—
215C2×SD16—28no—
216C2×SD16—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—44no—
226C4⋊Q8—44no—
227C4⋊Q8C2×D414yes—
228C4⋊Q8C2×D414yes—
229#125—28no—
230#124—216no—
231#124—48no—
232#124—416no—
233C22⋊Q8—28no—
234C22⋊Q8—24no—
235C22⋊Q8C2×D418yes—
236C4×Q8—28no—
237C4×Q8—24no—
238C4×Q8—48no—
239C4×Q8C2×D418yes—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—44no—
245C4×D4—44no—
246#117—28no—
247#117—44no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—44no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42C2×D414yes—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297C2×C42C2414yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#106—28no—
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—48no—
304#103—28no—
305#103—216no—
306#103—48no—
307#103—48no—
308#102—28no—
309#102—28no—
310#102—44no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C4×C8—24no—
314C4×C8—24no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—24no—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24C22×D411yescharacteristic
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—28no—
340C2×Q8—24no—
341C2×Q8—28no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—81no—
346C2×Q8—81no—
347C2×Q8—88no—
348C2×Q8—88no—
349C2×Q8C22×D414yes—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4H(5,𝔽2)14yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—816no—
390C22×C4#12114yes—
391C22×C4#12114yes—
392C22×C4#13514yes—
393C22×C4#13714yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2511yescharacteristic
404Q16—816no—
405SD16—416no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4C22×D414yes—
438C22⋊C4—28no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C42—28no—
445C42—28no—
446C42—44no—
447C42—44no—
448C42—84no—
449C42—84no—
450C42#12114yes—
451C42#12114yes—
452C42C22×D412yes—
453C42C22×D412yes—
454C42C2×#4112yes—
455C42C2×#4112yes—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—48no—
462C23—48no—
463C23—48no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23C2×#12112yes—
475C23C2×#13512yes—
476C23C2×#13712yes—
477C23C23×D411yescharacteristic
478Q8—28no—
479Q8—44no—
480Q8—48no—
481Q8—48no—
482Q8—84no—
483Q8—88no—
484D4—24no—
485D4—24no—
486D4—48no—
487D4—42no—
488D4—42no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—88no—
525C2×C4—84no—
526C2×C4—88no—
527C2×C4—88no—
528C2×C4#464116yescomplemented
529C2×C4C2×#12112yes—
530C2×C4C2×#12112yes—
531C2×C4C2×#12112yes—
532C2×C4C2×#12112yes—
533C2×C4C2×#12112yes—
534C2×C4C2×#13512yes—
535C2×C4C2×#13512yes—
536C2×C4C2×#13712yes—
537C2×C4C2×#13712yes—
538C2×C4C23×D411yescharacteristic
539C2×C4C22×#4111yescharacteristic
540C2×C4C2×H(5,𝔽2)12yes—
541C2×C4#58512yes—
542C8—44no—
543C8—44no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C2×#46414yes—
559C22C22×#12111yescharacteristic
560C22#348411yescharacteristic
561C22#348711yescharacteristic
562C22C22×#13511yescharacteristic
563C22C22×#13711yescharacteristic
564C22#358611yescharacteristic
565C22#358611yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—44no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4—82no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4C2×#46414yes—
578C4C2×#46414yes—
579C4#348812yes—
580C4#349412yes—
581C2—24no—
582C2—84no—
583C2#3556311yescharacteristic
584C2C22×#46411yescharacteristic
585C2#6042711yescharacteristic
586C2#6156511yescharacteristic
587C2#6158611yescharacteristic
588C2#6188211yescharacteristic
589C2#6188911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3209C2semidirect
4. #34213C2semidirect
5. #28124C2semidirect
6. #25365C2semidirect
7. #25273C2semidirect
8. #24729C2semidirect
9. #24629C2semidirect
11. #21951C2semidirect
12. #21804C2semidirect
19. C2×#494C22semidirect
20. C2×#487C22semidirect
25. C2×#462C22semidirect
38. #2369C22semidirect
39. #2365C22semidirect
41. #2355C22semidirect
42. Q8×C4⋊C4C22semidirect
52. #2066C22semidirect
53. #2056C22semidirect
57. #1981C22semidirect
58. #1956C22semidirect
59. #1956C22semidirect
60. #1939C22semidirect
61. #1939C22semidirect
62. #1927C22semidirect
63. #1878C22semidirect
528. C2×C4#464semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#31012
C2×#25472
#342132
#281242
#253652
#252732
#247292
#246292
#242172
#219512
#218042
#207322