Finite Group Databasebeta

← #11662507

Subgroups — #11662507

Order 768.

2824 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5752C211yesmaximal, characteristic, complemented
2C2×D3×#131C211yesmaximal, characteristic, complemented
3#85900C212yesmaximal, complemented
4C3×#2913C211yesmaximal, characteristic, complemented
5#82563C212yesmaximal, complemented
6#79213C216yesmaximal, complemented
7#79211C216yesmaximal, complemented
8#79183C213yesmaximal, complemented
9#79180C213yesmaximal, complemented
10#79150C213yesmaximal, complemented
11C2×#4883C213yesmaximal, complemented
12#60458—31nomaximal, Sylow
13C2×#1009C2212yes—
14C2×D3×#41C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16C3×#520C2212yes—
17C2×#948C2211yescharacteristic
18#5752—28no—
19D3×#131—28no—
20C2×#908C2216yescomplemented
21C2×#907C2213yescomplemented
22C2×D3×M16C2213yescomplemented
23C2×#904C2213yescomplemented
24C22×D24C2213yescomplemented
25C2×#901C2213yescomplemented
26C22×D3×C8C2213yescomplemented
27#5577—22no—
28#5576—22no—
29#5554—26no—
30D4×D12—26no—
31#5525C2216yescomplemented
32#5524C2216yescomplemented
33#5510C2216yescomplemented
34C2×#895C2216yescomplemented
35C3×#428C2216yescomplemented
36C3×#427C2213yescomplemented
37C2×C3×#107C2213yescomplemented
38#5002C2216yescomplemented
39#5001C2216yescomplemented
40#4966C2216yescomplemented
41C2×#809C2216yescomplemented
42#4887C2213yescomplemented
43#4887C2211yescharacteristic, complemented
44#4886C22112yescomplemented
45#4883C2213yescomplemented
46#4883C2216yescomplemented
47#4883C2216yescomplemented
48#4883C2216yescomplemented
49#4883C2213yescomplemented
50#4865C22112yescomplemented
51#4864C2216yescomplemented
52#4864C2216yescomplemented
53C2×#576—31no—
54C22×#131—31no—
55#3488—32no—
56#3062—36no—
57#3061—33no—
58#3058—33no—
59#3055—36no—
60#3050—33no—
61C2×#466—33no—
62#2913D311yescharacteristic, complemented
63#2897—32no—
64C2×C3×#41C2311yescharacteristic
65#1009—28no—
66#1009—28no—
67D3×#41—24no—
68D3×#41—24no—
69C2×#239—22no—
70C2×#239D411yescharacteristic
71C2×D3×Q8D411yescharacteristic
72C2×#237D413yes—
73C2×D3×D4—26no—
74C2×D3×D4D413yes—
75C2×#235C2313yes—
76C2×#235C2316yes—
77C22×D12—26no—
78C22×D12C2316yes—
79C22×C4×D3C2313yes—
80C3×#131D418yescomplemented
81C2×C3×M16C2313yes—
82C22×C3×C8C2313yes—
83C3×C4×Q8—22no—
84C3×C4×D4—26no—
85C3×#117C2316yes—
86C2×C3×C4⋊C4C2316yes—
87#948—24no—
88#930—212no—
89C2×(C2×D3)⋊C4—26no—
90C2×#208C2313yes—
91C22×C3⋊C8C2313yes—
92#908—224no—
93#907—212no—
94D3×M16—212no—
95C2×Dic12C2311yescharacteristic
96#904—212no—
97C2×D24—212no—
98C2×D24C2316yes—
99C2×D24C2313yes—
100C2×#204C2316yes—
101#901—212no—
102C2×ΓL(1,𝔽25)C2316yes—
103C2×D3×C8—212no—
104C2×D3×C8C2313yes—
105C2×D3×C8C2313yes—
106C2×D3×C8C2311yescharacteristic
107C2×D3×C8C2313yes—
108#897—26no—
109#897C2312yes—
110#897C2316yes—
111#895—26no—
112#895—26no—
113#895—26no—
114#895C2316yes—
115#895C2316yes—
116#895C23112yes—
117#886—26no—
118#882—26no—
119#882—212no—
120#875—26no—
121#874—26no—
122C4×D12—26no—
123C4×D12—26no—
124C3×#107C2313yes—
125C3×#107C2311yescharacteristic
126C3×#107C2316yes—
127C3×#106C2316yes—
128#810C2312yes—
129#810C2316yes—
130#809C2316yes—
131#809C2316yes—
132#809C23112yes—
133C2×H(5,𝔽2)—32no—
134C22×#41—31no—
135#576—68no—
136C2×AGL(1,ℤ/8)—36no—
137C2×#135—33no—
138C22×D8—33no—
139C2×#131—31no—
140C2×#131—33no—
141C2×#131—33no—
142C2×#131—68no—
143C2×#131C2×D311yescharacteristic
144C22×M16—33no—
145C23×C8—33no—
146#550—62no—
147#548—62no—
148#546—66no—
149D42—66no—
150#537—36no—
151#535—36no—
152#525—36no—
153C2×#121—36no—
154#520C2×D312yes—
155#469—312no—
156#468—312no—
157#467—31no—
158#467—33no—
159#466—33no—
160#466—36no—
161#466—36no—
162#466—36no—
163#466—33no—
164#465—36no—
165#465—36no—
166#428C2×D316yescomplemented
167#427C2×D313yescomplemented
168C2×#107C2×D313yescomplemented
169#418—36no—
170#417—36no—
171#416—36no—
172C2×#102—36no—
173C23×D3—26no—
174C23×D3C2×D413yes—
175C3×#41C2×D414yes—
176C3×#41#4114yes—
177C2×C3×Q8C2×D411yescharacteristic
178C2×C3×D4C2×D413yes—
179C22×C3×C4—26no—
180C22×C3×C4C2413yes—
181C2×#67—26no—
182C2×#67C2×D416yes—
183C22×Dic3C2×D413yes—
184#239—24no—
185#239—48no—
186D3×Q8—24no—
187#237—212no—
188D3×D4—212no—
189D3×D4—424no—
190#235—26no—
191#235—26no—
192#235—212no—
193#235—212no—
194C2×D12—212no—
195C2×D12—212no—
196C2×D12—26no—
197C2×D12—26no—
198C2×D12—26no—
199C2×D12—212no—
200C2×D12—812no—
201C2×D12C2×D413yes—
202C2×D12C2416yes—
203C2×C4×D3—26no—
204C2×C4×D3—26no—
205C2×C4×D3—26no—
206C2×C4×D3C2×D413yes—
207C2×C4×D3C2×D413yes—
208C2×C4×D3C2411yescharacteristic
209C2×C4×D3C2413yes—
210C2×Dic6C2×D413yes—
211C2×Dic6C2412yes—
212C3×M16C2×D4112yescomplemented
213C2×C3×C8C2×D4112yescomplemented
214C2×C3×C8C2413yes—
215C2×C3×C8C2411yescharacteristic
216C3×C4⋊C4—26no—
217C3×C4⋊C4C2412yes—
218C3×C4⋊C4C2416yes—
219C3×C22⋊C4—26no—
220C3×C42—26no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—26no—
223(C2×D3)⋊C4—26no—
224(C2×D3)⋊C4—26no—
225(C2×D3)⋊C4—212no—
226C4⋊Dic3—22no—
227C4⋊Dic3—26no—
228#208—26no—
229C2×C3⋊C8—26no—
230C2×C3⋊C8C2411yescharacteristic
231C2×C3⋊C8C2413yes—
232Dic12—22no—
233D24—26no—
234D24—212no—
235#204—212no—
236ΓL(1,𝔽25)—212no—
237D3×C8—22no—
238D3×C8—26no—
239D3×C8—26no—
240D3×C8—26no—
241H(5,𝔽2)—68no—
242H(5,𝔽2)—68no—
243C2×#41—31no—
244C2×#41—33no—
245C2×#41—33no—
246C2×#41—36no—
247C2×#41—64no—
248C2×#41—64no—
249C2×#41—62no—
250C2×#41C22×D311yescharacteristic
251C22×Q8—31no—
252C22×D4—33no—
253C22×D4—36no—
254C22×D4—66no—
255C22×D4—66no—
256C23×C4—33no—
257AGL(1,ℤ/8)—624no—
258#135—612no—
259C2×Q16—31no—
260C2×SD16—36no—
261C2×D8—33no—
262C2×D8—36no—
263C2×D8—612no—
264#131—64no—
265#131—612no—
266#131—612no—
267#131D1218yescomplemented
268C2×M16—33no—
269C2×M16—36no—
270C2×M16—612no—
271C2×M16C22×D313yes—
272C22×C8—33no—
273C22×C8—31no—
274C22×C8—33no—
275C22×C8—33no—
276C22×C8—33no—
277C22×C8—612no—
278C22×C8C22×D313yes—
279#127—66no—
280#124—66no—
281#123—66no—
282C22⋊Q8—32no—
283C22⋊Q8—36no—
284C22⋊Q8—66no—
285#121—36no—
286#121—312no—
287#121—36no—
288#121—66no—
289#121—66no—
290#121—66no—
291#121—612no—
292#120—612no—
293#120—66no—
294C4×Q8—22no—
295C4×D4—26no—
296C4×D4—66no—
297C4×D4—66no—
298#117C22×D316yes—
299C2×C4⋊C4C22×D316yes—
300C2×C22⋊C4—66no—
301#107C22×D313yes—
302#107C22×D316yes—
303#107C22×D311yescharacteristic
304#106C22×D316yes—
305#103—32no—
306#103—36no—
307#102—36no—
308#102—312no—
309#102—36no—
310C23×C3C22×D413yes—
311C22×D3—26no—
312C22×D3—26no—
313C22×D3—26no—
314C22×D3—43no—
315C22×D3—412no—
316C22×D3—86no—
317C22×D3—86no—
318C22×D3C22×D413yes—
319C22×D3C22×D411yescharacteristic
320C3×Q8#12114yes—
321C3×D4#121112yes—
322C2×C3×C4—22no—
323C2×C3×C4—26no—
324C2×C3×C4—86no—
325C2×C3×C4C22×D416yes—
326C2×C3×C4C22×D413yes—
327C2×C3×C4C2×#4116yes—
328C2×C3×C4C2511yescharacteristic
329#67—212no—
330#67—412no—
331C2×Dic3—26no—
332C2×Dic3—22no—
333C2×Dic3C22×D413yes—
334C2×Dic3C22×D411yescharacteristic
335D12—26no—
336D12—26no—
337D12—26no—
338D12—412no—
339D12—412no—
340D12—86no—
341D12—812no—
342C4×D3—26no—
343C4×D3—26no—
344C4×D3—21no—
345C4×D3—23no—
346C4×D3—21no—
347C4×D3—23no—
348C4×D3—412no—
349Dic6—26no—
350Dic6—22no—
351Dic6—22no—
352C3×C8C22×D412yes—
353C3×C8C22×D416yes—
354C3⋊C8—21no—
355C3⋊C8—23no—
356C24—33no—
357C24—66no—
358#41—64no—
359#41—612no—
360#41—66no—
361#41—66no—
362#41—612no—
363#41—612no—
364#41—128no—
365#41C2×D1214yes—
366#41#23914yes—
367C2×Q8—33no—
368C2×Q8—32no—
369C2×Q8—64no—
370C2×Q8D3×D411yescharacteristic
371C2×D4—36no—
372C2×D4—33no—
373C2×D4—36no—
374C2×D4—66no—
375C2×D4—66no—
376C2×D4—612no—
377C2×D4—612no—
378C2×D4—66no—
379C2×D4—612no—
380C2×D4—66no—
381C2×D4—612no—
382C2×D4—1224no—
383C2×D4—2412no—
384C2×D4D3×D413yes—
385C22×C4—26no—
386C22×C4—33no—
387C22×C4—33no—
388C22×C4—33no—
389C22×C4—31no—
390C22×C4—33no—
391C22×C4—66no—
392C22×C4—66no—
393C22×C4—66no—
394C22×C4C23×D313yes—
395Q16—62no—
396SD16—612no—
397D8—66no—
398D8—612no—
399M16—66no—
400M16—612no—
401M16C2×D12112yescomplemented
402C2×C8—31no—
403C2×C8—33no—
404C2×C8—66no—
405C2×C8—62no—
406C2×C8—66no—
407C2×C8—66no—
408C2×C8—66no—
409C2×C8C2×D12112yescomplemented
410C2×C8C23×D313yes—
411C2×C8C23×D311yescharacteristic
412C4⋊C4—26no—
413C4⋊C4—62no—
414C4⋊C4—66no—
415C4⋊C4C23×D312yes—
416C4⋊C4C23×D316yes—
417C22⋊C4—26no—
418C22⋊C4—66no—
419C22⋊C4—612no—
420C22⋊C4—66no—
421C22⋊C4—62no—
422C22⋊C4—66no—
423C42—26no—
424C22×C3—23no—
425C22×C3—23no—
426C22×C3C2×#12116yes—
427C22×C3C23×D411yescharacteristic
428C2×D3—21no—
429C2×D3—21no—
430C2×D3—23no—
431C2×D3—23no—
432C2×D3—23no—
433C2×D3—412no—
434C2×D3—46no—
435C2×D3—46no—
436C2×D3—46no—
437C2×D3—1612no—
438C2×D3#57612yescomplemented
439C3×C4—42no—
440C3×C4—46no—
441C3×C4C2×#12116yes—
442C3×C4C23×D411yescharacteristic
443C3×C4C22×#4111yescharacteristic
444Dic3—23no—
445Dic3—21no—
446Dic3—42no—
447C23—31no—
448C23—33no—
449C23—66no—
450C23—66no—
451C23—66no—
452C23—123no—
453C23—1212no—
454C23—246no—
455C23—246no—
456C23C2×D3×D413yes—
457Q8—66no—
458Q8—62no—
459Q8—62no—
460Q8#89514yes—
461D4—612no—
462D4—66no—
463D4—66no—
464D4—66no—
465D4—1212no—
466D4—1212no—
467D4—1212no—
468D4—2412no—
469D4—246no—
470D4#895112yes—
471C2×C4—22no—
472C2×C4—26no—
473C2×C4—31no—
474C2×C4—33no—
475C2×C4—66no—
476C2×C4—61no—
477C2×C4—63no—
478C2×C4—61no—
479C2×C4—63no—
480C2×C4—66no—
481C2×C4—66no—
482C2×C4—62no—
483C2×C4—86no—
484C2×C4—1212no—
485C2×C4C22×D1216yes—
486C2×C4C2×D3×D413yes—
487C2×C4C2×#23916yes—
488C2×C4C24×D311yescharacteristic
489C8—61no—
490C8—63no—
491C8C22×D1212yes—
492C8C22×D1216yes—
493C2×C3—26no—
494C2×C3C22×#12111yescharacteristic
495C2×C3C2×#57611yescharacteristic
496C2×C3#358411yescharacteristic
497D3—22no—
498D3—43no—
499D3—86no—
500C22—23no—
501C22—23no—
502C22—32no—
503C22—61no—
504C22—61no—
505C22—63no—
506C22—63no—
507C22—63no—
508C22—1212no—
509C22—126no—
510C22—126no—
511C22—126no—
512C22—4812no—
513C22C2×#89516yes—
514C22C22×D3×D411yescharacteristic
515C4—46no—
516C4—42no—
517C4—61no—
518C4—63no—
519C4—122no—
520C4C2×#89516yes—
521C4C23×D1211yescharacteristic
522C4C22×#23911yescharacteristic
523C3#6045811yescharacteristic, complemented, Sylow
524C2—26no—
525C2—62no—
526C2—123no—
527C2—246no—
528C2C22×#89511yescharacteristic
529C2D3×#57611yescharacteristic
530C2#8721011yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2913C2
socleC22×C3C23×D4
hypercenter#2913D3
hypocenterC3#60458
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. C2×#5752C2semidirect
2. C2×D3×#131C2semidirect
3. #85900C2semidirect
4. C3×#2913C2semidirect
5. #82563C2semidirect
6. #79213C2semidirect
7. #79211C2semidirect
8. #79183C2semidirect
9. #79180C2semidirect
10. #79150C2semidirect
11. C2×#4883C2semidirect
20. C2×#908C22semidirect
21. C2×#907C22semidirect
22. C2×D3×M16C22semidirect
23. C2×#904C22semidirect
24. C22×D24C22semidirect
25. C2×#901C22semidirect
26. C22×D3×C8C22semidirect
31. #5525C22semidirect
32. #5524C22semidirect
33. #5510C22semidirect
34. C2×#895C22semidirect
35. C3×#428C22semidirect
36. C3×#427C22semidirect
37. C2×C3×#107C22semidirect
38. #5002C22semidirect
39. #5001C22semidirect
40. #4966C22semidirect
41. C2×#809C22semidirect
42. #4887C22semidirect
43. #4887C22semidirect
44. #4886C22semidirect
45. #4883C22semidirect
46. #4883C22semidirect
47. #4883C22semidirect
48. #4883C22semidirect
49. #4883C22semidirect
50. #4865C22semidirect
51. #4864C22semidirect
52. #4864C22semidirect
62. #2913D3semidirect
80. C3×#131D4semidirect
166. #428C2×D3semidirect
167. #427C2×D3semidirect
168. C2×#107C2×D3semidirect
212. C3×M16C2×D4semidirect
213. C2×C3×C8C2×D4semidirect
267. #131D12semidirect
401. M16C2×D12semidirect
409. C2×C8C2×D12semidirect
438. C2×D3#576semidirect
523. C3#60458semidirect

Maximal subgroups

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

subgroupindex
C2×#57522
C2×D3×#1312
#859002
C3×#29132
#825632
#792132
#792112
#791832
#791802
#791502
C2×#48832
#604583