Finite Group Databasebeta

← #6816673

Subgroups — #6816673

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C212yesmaximal, complemented
2#36151C211yesmaximal, characteristic
3D4×#125C211yesmaximal, characteristic
4#33317C211yesmaximal, characteristic, complemented
5#31087C214yesmaximal, complemented
6#31079C214yesmaximal, complemented
7#29883C212yesmaximal, complemented
8#28725C211yesmaximal, characteristic, complemented
9#28695C211yesmaximal, characteristic, complemented
10#26057C214yesmaximal, complemented
11#25819C212yesmaximal, complemented
12#25677C212yesmaximal, complemented
13#25024C214yesmaximal
14#21657C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic
16C2×#533C2212yes—
17C22×#125C2212yes—
18#3236—216no—
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#501C2212yes—
22C2×#491C2212yes—
23C2×#491C2211yescharacteristic
24C2×#484C2214yes—
25C2×#483C2214yes—
26C2×#479C2214yes—
27C2×#479C2214yes—
28C2×#432C2212yes—
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2212yes—
32#2851C2212yes—
33#2552C2212yes—
34#2536C2212yes—
35#2506—22no—
36#2489—22no—
37#2473—22no—
38#2473—22no—
39#2453—22no—
40#2442C2212yes—
41#2442C2212yes—
42#2421C2214yes—
43#2415—22no—
44#2362—22no—
45#2351C2212yes—
46D4×C4⋊C4—22no—
47D4×C4⋊C4C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#125C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2088C2214yes—
52#2087C2214yes—
53#2085C2214yes—
54#2077C2214yes—
55#2075C2214yes—
56#2074C2212yes—
57#2074C2212yes—
58#2059C2214yes—
59#2059C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#1993C2212yes—
63#1992C2212yes—
64#1992C2211yescharacteristic
65#1960C2211yescharacteristic
66#1955C2212yes—
67#1949C2214yes—
68#1940C2212yes—
69#1933C2214yes—
70#1923C2214yes—
71#1865C2214yes—
72#1863C2214yes—
73#1853C2214yes—
74#1852C2214yes—
75#1847C2214yes—
76#1846C2212yes—
77#1846C2212yes—
78#1836C2211yescharacteristic
79#1803C2214yes—
80#1767C2211yescharacteristic
81#1763C2214yes—
82#1749C2212yes—
83C4×#97C2211yescharacteristic
84#533—28no—
85#533—44no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#125—28no—
89C2×#125—22no—
90C2×#125—22no—
91C2×#125—22no—
92C2×#125—44no—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×#125C2312yes—
96C2×#125C2311yescharacteristic
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—22no—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2314yes—
103C2×C4×D4—22no—
104C2×C4×D4D412yes—
105C2×C4×D4C2311yescharacteristic
106C2×#117—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C42C2312yes—
112#501—28no—
113#501—28no—
114#501—44no—
115#491—28no—
116#491D418yescomplemented
117#484—216no—
118#483—216no—
119#479—216no—
120#479—216no—
121#432—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×#97C2311yescharacteristic
139C2×#97C2311yescharacteristic
140C2×C4×C8C2312yes—
141#400—24no—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#397—24no—
150#394—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#389—22no—
156#389—22no—
157#389—24no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#388—24no—
167#388—24no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—24no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2312yes—
181#382—22no—
182#382C2311yescharacteristic
183#382C2311yescharacteristic
184#382C2311yescharacteristic
185#382C2312yes—
186#382C2312yes—
187#380—22no—
188#380—22no—
189#380—22no—
190#380C2312yes—
191#380C2314yes—
192#380C2312yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2312yes—
211C22×Q8—22no—
212C22×Q8C2412yes—
213C22×D4C2×D411yescharacteristic
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2412yes—
219C22×C8C2414yes—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C4⋊Q8—24no—
223C4⋊Q8—22no—
224C4⋊Q8—24no—
225C4⋊Q8C2×D414yes—
226#125—24no—
227#125—24no—
228#125—24no—
229#125—22no—
230#125—24no—
231#125—42no—
232#125—42no—
233#125—48no—
234#125—48no—
235#125—48no—
236#125C2×D414yes—
237C22⋊Q8—28no—
238C22⋊Q8—44no—
239C22⋊Q8—48no—
240C22⋊Q8—48no—
241C4×Q8—28no—
242C4×Q8—28no—
243C4×Q8—44no—
244C4×Q8—48no—
245C4×D4—24no—
246C4×D4—44no—
247C4×D4—48no—
248#117—48no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
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—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2412yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—28no—
305#107—28no—
306#107—44no—
307#106—28no—
308#106—28no—
309#106—44no—
310C4⋊C8—28no—
311C4⋊C8—24no—
312#103—28no—
313#103—28no—
314#103—28no—
315#103C2×D4116yes—
316C22⋊C8—28no—
317C22⋊C8—28no—
318#97—24no—
319#97—22no—
320#97—24no—
321#97C2×D414yes—
322C4×C8—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95C2×D411yescharacteristic
339#95C2×D411yescharacteristic
340#95C2×D412yes—
341#95C2×D412yes—
342C24C22×D412yes—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8C22×D414yes—
350C2×D4—24no—
351C2×D4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C2511yescharacteristic
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8C22×D414yes—
391C2×C82−1+414yes—
392C4⋊C4—22no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4C22×D414yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C42—24no—
433C42—24no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C42—48no—
440C42—44no—
441C42—44no—
442C42C22×D412yes—
443C42H(5,𝔽2)12yes—
444C422−1+414yes—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23#57514yes—
452C23C23×D411yescharacteristic
453Q8—28no—
454Q8—44no—
455Q8—88no—
456D4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—84no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4#53612yes—
500C2×C4#53612yes—
501C2×C4D4214yes—
502C2×C4C23×D411yescharacteristic
503C2×C4C2×H(5,𝔽2)11yescharacteristic
504C2×C4C2×2−1+412yes—
505C2×C4C2×2−1+412yes—
506C8—24no—
507C8—24no—
508C8—44no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22C2×#53611yescharacteristic
517C22C2×D4211yescharacteristic
518C22#349911yescharacteristic
519C22C2×#57512yes—
520C22#358612yes—
521C4—24no—
522C4—22no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—44no—
528C4—42no—
529C4—44no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4#349912yes—
536C4#355212yes—
537C2—44no—
538C2#3936011yescharacteristic
539C2#6086111yescharacteristic
540C2#6088411yescharacteristic
541C2#6106812yes—
542C2#6250512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3236C2semidirect
4. #33317C2semidirect
5. #31087C2semidirect
6. #31079C2semidirect
7. #29883C2semidirect
8. #28725C2semidirect
9. #28695C2semidirect
10. #26057C2semidirect
11. #25819C2semidirect
12. #25677C2semidirect
14. #21657C2semidirect
116. #491D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
#361512
D4×#1252
#333172
#310872
#310792
#298832
#287252
#286952
#260572
#258192
#256772
#250242
#216572
#212712