Finite Group Databasebeta

← #11860338

Subgroups — #11860338

Order 1152.

1251 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3832C211yesmaximal, characteristic, complemented
2C2×#63941C211yesmaximal, characteristic
3C4×#3802C211yesmaximal, characteristic, complemented
4C3×#5026C212yesmaximal, complemented
5#10591571C212yesmaximal, complemented
6#10591554C211yesmaximal, characteristic, complemented
7#10590594C211yesmaximal, characteristic, complemented
8#10590582C211yesmaximal, characteristic, complemented
9#10590459C212yesmaximal, complemented
10#10590346C212yesmaximal, complemented
11#10590247C211yesmaximal, characteristic, complemented
12#79773—32nomaximal
13C2×#3832C2211yescharacteristic
14C2×#3802C2211yescharacteristic
15C2×C4×D32—21no—
16C32×C4×Q8C2211yescharacteristic, complemented
17Q8×C3⋊Dic3C2211yescharacteristic, complemented
18#64102—21no—
19#64102C2211yescharacteristic, complemented
20#64101—21no—
21#64101C2211yescharacteristic
22C4×C4⋊(C3⋊D3)—21no—
23C4×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
24C42×C3⋊D3—21no—
25C42×C3⋊D3C2211yescharacteristic
26C2×C3×#215C2212yes—
27C3×C4×D12C2212yes—
28#63941C418yescomplemented
29#63921—21no—
30#63902—21no—
31C4×#625—21no—
32#63900C2211yescharacteristic
33#63898C2212yes—
34C2×#3714C2212yes—
35C2×#3712C2211yescharacteristic
36C2×#3712C2211yescharacteristic
37C3×#835C2212yescomplemented
38C3×#809C2212yes—
39C3×#807C2212yescomplemented
40C3×#803C2212yescomplemented
41#63573C2212yescomplemented
42#63564C2212yescomplemented
43#63563C2212yes—
44#63553C2212yescomplemented
45C2×#914—32no—
46C4×#239—32no—
47C4×#239—61no—
48C4×D3×D4—32no—
49C3×#440—32no—
50Dic3×SD16—32no—
51#5026D312yescomplemented
52#5014—32no—
53#4881—32no—
54#4872—32no—
55#4803—32no—
56#4788—32no—
57#4770—32no—
58#4756—32no—
59#4711—32no—
60#4707—32no—
61C22×D32—21no—
62C2×C32×Q8C2311yescharacteristic
63#3832—22no—
64#3832C2×C414yescomplemented
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)C2311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3—21no—
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3D411yescharacteristic
71C2×C4×C3⋊D3D411yescharacteristic
72C2×C4×C3⋊D3C2311yescharacteristic
73C2×C3×D12C2312yes—
74C2×C3×C4×D3—22no—
75C2×#625—21no—
76C2×#624—21no—
77C2×D3×Dic3—22no—
78#3802—22no—
79#3802C2×C414yescomplemented
80C4×D32—44no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4C2311yescharacteristic
83C32×C42—21no—
84C32×C42C2311yescharacteristic
85#3752—21no—
86#3752—21no—
87#3752—21no—
88C4⋊(C3⋊Dic3)—21no—
89C4⋊(C3⋊Dic3)C2311yescharacteristic
90C4×C3⋊Dic3—21no—
91C4×C3⋊Dic3—21no—
92C4×C3⋊Dic3D411yescharacteristic
93C4×C3⋊Dic3D411yescharacteristic
94C4×C3⋊Dic3C2311yescharacteristic
95C3×#215C2×C418yescomplemented
96C3×(C2×D3)⋊C4—22no—
97C3×C4⋊Dic3C2312yes—
98C2×C3×C3⋊C8C2312yes—
99C2×C3×C3⋊C8C2312yes—
100#3724—21no—
101(C2×D3)⋊Dic3—22no—
102#3714C2×C418yescomplemented
103#3712—22no—
104#3712C2×C414yescomplemented
105C4×AGL(1,ℤ/8)—91noSylow
106C2×#239—32no—
107C2×#239—61no—
108C2×D3×D4—32no—
109C22×C4×D3—62no—
110C2×C3×SD16—32no—
111C3×C4×Q8—21no—
112C3×C4×Q8C2×D312yescomplemented
113C3×C4×D4—32no—
114Q8×Dic3—32no—
115Q8×Dic3—61no—
116C2×#215C2×D312yes—
117D4×Dic3—32no—
118C2×#213—32no—
119C4×#67—62no—
120#914—316no—
121C2×D24—32no—
122C2×ΓL(1,𝔽25)—32no—
123C2×ΓL(1,𝔽25)—32no—
124#893—32no—
125#893—32no—
126#893—61no—
127#893—62no—
128#893—62no—
129#892—32no—
130#892—61no—
131#892—62no—
132#892—62no—
133D3×C4⋊C4—32no—
134#67⋊C4—62no—
135D3×C22⋊C4—62no—
136C4×D12—32no—
137C4×D12—61no—
138C4×D12—62no—
139C4×D12—62no—
140C4×D12C2×D312yes—
141C42×D3—32no—
142C42×D3—61no—
143C42×D3—62no—
144C42×D3—62no—
145C3×#107—32no—
146C3×#103—32no—
147C3×#102—32no—
148C3×#97—32no—
149#835C2×D312yescomplemented
150#832—32no—
151#821—32no—
152#821—32no—
153#817—32no—
154C8×Dic3—32no—
155C8×Dic3—32no—
156#809—32no—
157#809C2×D312yes—
158#808—32no—
159#807C2×D312yescomplemented
160#803C2×D312yescomplemented
161C22×C3⋊D3—21no—
162C22×C3⋊D3C2×D411yescharacteristic
163C22×C3×D3—22no—
164C2×D32—24no—
165C2×D32—42no—
166C2×D32—42no—
167C32×Q8—21no—
168C32×Q8C22×C412yescomplemented
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2411yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3C2×D411yescharacteristic
176C2×C3⋊Dic3C2×D411yescharacteristic
177C2×C3⋊Dic3C2×D411yescharacteristic
178C4⋊(C3⋊D3)—24no—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)C22×C412yescomplemented
181C4×C3⋊D3—24no—
182C4×C3⋊D3—22no—
183C4×C3⋊D3—41no—
184C4×C3⋊D3—42no—
185C4×C3⋊D3—42no—
186C4×C3⋊D3C22×C412yes—
187C4×C3⋊D3#4112yes—
188C2×C3×Dic3—22no—
189C3×D12—22no—
190C3×D12C22×C414yes—
191C3×C4×D3—44no—
192#625—24no—
193#624—42no—
194D3×Dic3—44no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8C22×C414yescomplemented
197C2×AGL(1,ℤ/8)—91no—
198C4×#41—91no—
199C2×C4×D4—91no—
200#442—92no—
201#440—32no—
202C4×SD16—92no—
203C4×D8—92no—
204#428—91no—
205#418—91no—
206#417—91no—
207C4×M16—91no—
208C23×D3—62no—
209C2×C3×Q8—21no—
210C2×C3×Q8C22×D312yes—
211C2×C3×D4—32no—
212C22×C3×C4—62no—
213C2×#67—62no—
214C22×Dic3—62no—
215#239—38no—
216#239—64no—
217#239—64no—
218#239—64no—
219D3×D4—38no—
220D3×D4—64no—
221C2×D12—32no—
222C2×D12—61no—
223C2×D12—62no—
224C2×D12—62no—
225C2×D12C22×D312yes—
226C2×C4×D3—22no—
227C2×C4×D3—32no—
228C2×C4×D3—32no—
229C2×C4×D3—32no—
230C2×C4×D3—61no—
231C2×C4×D3—61no—
232C2×C4×D3—61no—
233C2×C4×D3—62no—
234C2×C4×D3—62no—
235C2×C4×D3—62no—
236C2×C4×D3—62no—
237C2×C4×D3—62no—
238C2×C4×D3—62no—
239C2×C4×D3—62no—
240C2×C4×D3—62no—
241C2×C4×D3—128no—
242C3×SD16—38no—
243C2×C3×C8—32no—
244C2×C3×C8—32no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4C22×D312yes—
250C3×C22⋊C4—62no—
251C3×C42—22no—
252C3×C42—22no—
253C3×C42—21no—
254C3×C42C22×D312yes—
255#217—62no—
256#215C4×D318yescomplemented
257#213—38no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—62no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—62no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—62no—
264(C2×D3)⋊C4—62no—
265(C2×D3)⋊C4—62no—
266C4⋊Dic3—32no—
267C4⋊Dic3—61no—
268C4⋊Dic3—62no—
269C4⋊Dic3—62no—
270C4⋊Dic3C22×D312yes—
271Dic3⋊C4—62no—
272C4×Dic3—32no—
273C4×Dic3—32no—
274C4×Dic3—32no—
275C4×Dic3—61no—
276C4×Dic3—61no—
277C4×Dic3—61no—
278C4×Dic3—62no—
279C4×Dic3—62no—
280C4×Dic3—62no—
281C4×Dic3—62no—
282C2×C3⋊C8C22×D312yes—
283C2×C3⋊C8C22×D312yes—
284D24—38no—
285ΓL(1,𝔽25)—38no—
286ΓL(1,𝔽25)—64no—
287C22×C32C22×D411yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—41no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3C4×D412yes—
294C2×C3×D3—24no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297D32—48no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4C23×C411yescharacteristic
303C32×C4C2×#4111yescharacteristic
304C3⋊Dic3—21no—
305C3⋊Dic3—41no—
306C3⋊Dic3C4×D412yes—
307C3⋊Dic3AGL(1,ℤ/8)12yescomplemented
308C3×Dic3—42no—
309C2×#41—91no—
310C22×D4—91no—
311C23×C4—181no—
312AGL(1,ℤ/8)—98no—
313C2×SD16—32no—
314C2×D8—92no—
315C2×M16—91no—
316C2×M16—91no—
317C4×Q8—91no—
318C4×Q8D3211yescharacteristic, complemented
319C4×D4—32no—
320C4×D4—92no—
321C4×D4—91no—
322C4×D4—91no—
323C4×D4—181no—
324C4×D4—181no—
325C4×D4—181no—
326C4×D4—181no—
327#117—91no—
328#117—181no—
329C2×C4⋊C4—91no—
330C2×C22⋊C4—181no—
331C2×C42—91no—
332C2×C42—181no—
333#107—32no—
334#106—92no—
335#103—32no—
336#102—32no—
337#102—92no—
338#102—92no—
339#97—32no—
340C4×C8—92no—
341C23×C3—62no—
342C22×D3—22no—
343C22×D3—32no—
344C22×D3—61no—
345C22×D3—68no—
346C22×D3—62no—
347C22×D3—62no—
348C22×D3—124no—
349C22×D3—124no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8C2×C4×D314yescomplemented
354C3×D4—34no—
355C3×D4—62no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—62no—
364C2×C3×C4—124no—
365C2×C3×C4D3×D412yes—
366C2×C3×C4D3×D412yes—
367C2×C3×C4C23×D312yes—
368#67—68no—
369C2×Dic3—22no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—62no—
377C2×Dic3—62no—
378C2×Dic3—124no—
379D12—22no—
380D12—34no—
381D12—62no—
382D12—68no—
383D12—68no—
384D12—62no—
385D12—62no—
386D12C2×C4×D314yes—
387C4×D3—34no—
388C4×D3—34no—
389C4×D3—44no—
390C4×D3—62no—
391C4×D3—62no—
392C4×D3—64no—
393C4×D3—68no—
394C4×D3—68no—
395C4×D3—122no—
396C4×D3—124no—
397C4×D3—124no—
398C4×D3—124no—
399C4×D3—122no—
400C4×D3—124no—
401C4×D3—122no—
402C4×D3—124no—
403C4×D3—124no—
404C3×C8—34no—
405C3×C8—62no—
406C3⋊C8—22no—
407C3⋊C8C2×C4×D314yescomplemented
408C2×C32C2×C4×D411yescharacteristic
409C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C32#57511yescharacteristic
411C3⋊D3—22no—
412C3⋊D3—42no—
413C3×D3—44no—
414C24—181no—
415#41—94no—
416#41—182no—
417C2×Q8C2×D3211yescharacteristic
418C2×D4—32no—
419C2×D4—94no—
420C2×D4—91no—
421C2×D4—181no—
422C2×D4—182no—
423C2×D4—181no—
424C22×C4—62no—
425C22×C4—91no—
426C22×C4—91no—
427C22×C4—91no—
428C22×C4—181no—
429C22×C4—181no—
430C22×C4—181no—
431C22×C4—182no—
432C22×C4—181no—
433C22×C4—364no—
434SD16—38no—
435D8—98no—
436M16—94no—
437M16—182no—
438C2×C8—32no—
439C2×C8—32no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—91no—
443C4⋊C4—181no—
444C4⋊C4—181no—
445C4⋊C4C2×D3211yescharacteristic
446C22⋊C4—62no—
447C22⋊C4—181no—
448C22⋊C4—181no—
449C22⋊C4—182no—
450C22⋊C4—181no—
451C42—21no—
452C42—91no—
453C42—91no—
454C42—91no—
455C42—181no—
456C42—181no—
457C42C2×D3211yescharacteristic
458C22×C3—21no—
459C22×C3—64no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3C2×D3×D412yes—
463C2×D3—24no—
464C2×D3—34no—
465C2×D3—42no—
466C2×D3—42no—
467C2×D3—62no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—1216no—
475C2×D3—122no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—122no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—24no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—122no—
490C3×C4C22×C4×D312yes—
491C3×C4D3×#4112yes—
492Dic3—34no—
493Dic3—34no—
494Dic3—42no—
495Dic3—62no—
496Dic3—62no—
497Dic3—62no—
498Dic3—121no—
499Dic3—122no—
500Dic3—122no—
501C32C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
502C23—62no—
503C23—91no—
504C23—184no—
505C23—181no—
506C23—362no—
507C23—362no—
508Q8—21no—
509Q8C4×D3212yescomplemented
510D4—34no—
511D4—62no—
512D4—92no—
513D4—184no—
514D4—184no—
515D4—181no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—62no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—91no—
524C2×C4—124no—
525C2×C4—182no—
526C2×C4—184no—
527C2×C4—181no—
528C2×C4—361no—
529C2×C4—362no—
530C2×C4—362no—
531C2×C4—364no—
532C2×C4—362no—
533C2×C4#381111yescharacteristic
534C2×C4#381111yescharacteristic
535C2×C4C22×D3211yescharacteristic
536C8—34no—
537C8—62no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—124no—
542C2×C3C4×D3×D412yes—
543C2×C3D3×AGL(1,ℤ/8)12yes—
544C2×C3#577712yes—
545D3—44no—
546D3—64no—
547D3—122no—
548D3—124no—
549D3—124no—
550C22—64no—
551C22—92no—
552C22—122no—
553C22—122no—
554C22—181no—
555C22—181no—
556C22—182no—
557C22—368no—
558C22—361no—
559C22—361no—
560C22C2×#381111yescharacteristic
561C4—21no—
562C4—22no—
563C4—21no—
564C4—41no—
565C4—92no—
566C4—92no—
567C4—122no—
568C4—181no—
569C4—361no—
570C4C2×C4×D3211yescharacteristic
571C4#6430611yescharacteristic
572C3—21no—
573C3#7977312yescomplemented
574C2—124no—
575C2—182no—
576C2—362no—
577C2C4×#381111yescharacteristic
578C2#1059532311yescharacteristic
579C2#1059545811yescharacteristic
subgroupG/H
centerC22C2×#3811
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC2×#63941C2
cosocleC2×C32×C4C24
hypercenterC4×Q8D32
hypocenterC32C4×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×#3832C2semidirect
3. C4×#3802C2semidirect
4. C3×#5026C2semidirect
5. #10591571C2semidirect
6. #10591554C2semidirect
7. #10590594C2semidirect
8. #10590582C2semidirect
9. #10590459C2semidirect
10. #10590346C2semidirect
11. #10590247C2semidirect
16. C32×C4×Q8C22semidirect
17. Q8×C3⋊Dic3C22semidirect
19. #64102C22semidirect
23. C4×C4⋊(C3⋊D3)C22semidirect
28. #63941C4semidirect
37. C3×#835C22semidirect
39. C3×#807C22semidirect
40. C3×#803C22semidirect
41. #63573C22semidirect
42. #63564C22semidirect
44. #63553C22semidirect
51. #5026D3semidirect
64. #3832C2×C4semidirect
79. #3802C2×C4semidirect
95. C3×#215C2×C4semidirect
102. #3714C2×C4semidirect
104. #3712C2×C4semidirect
112. C3×C4×Q8C2×D3semidirect
149. #835C2×D3semidirect
159. #807C2×D3semidirect
160. #803C2×D3semidirect
168. C32×Q8C22×C4semidirect
180. C4⋊(C3⋊D3)C22×C4semidirect
196. C3×C3⋊C8C22×C4semidirect
256. #215C4×D3semidirect
307. C3⋊Dic3AGL(1,ℤ/8)semidirect
318. C4×Q8D32semidirect
353. C3×Q8C2×C4×D3semidirect
407. C3⋊C8C2×C4×D3semidirect
501. C32C4×AGL(1,ℤ/8)semidirect
509. Q8C4×D32semidirect
573. C3#79773semidirect

Maximal subgroups

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

subgroupindex
C4×#38322
C2×#639412
C4×#38022
C3×#50262
#105915712
#105915542
#105905942
#105905822
#105904592
#105903462
#105902472
#797733