Finite Group Databasebeta

← #11664746

Subgroups — #11664746

Order 768.

3178 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#576C211yesmaximal, characteristic, complemented
2#87283C212yesmaximal, complemented
3C2×#5836C212yesmaximal, complemented
4C2×#5753C211yesmaximal, characteristic, complemented
5#83939C216yesmaximal, complemented
6#83938C216yesmaximal, complemented
7#83907C213yesmaximal, complemented
8#83903C213yesmaximal, complemented
9#83830C213yesmaximal, complemented
10C2×#5155C213yesmaximal, complemented
11Dic3×#131C211yesmaximal, characteristic, complemented
12#60649—31nomaximal, Sylow
13C2×C3×H(5,𝔽2)C2212yes—
14C2×#1010C2212yes—
15C3×#576—28no—
16C2×C3×AGL(1,ℤ/8)C2216yes—
17C2×C3×#135C2213yes—
18C22×C3×D8C2213yes—
19C2×C3×#131C2211yescharacteristic
20#5836—216no—
21#5832C2216yescomplemented
22#5831—24no—
23Dic3×#41C2211yescharacteristic
24#5825C2216yescomplemented
25C2×#951C2216yescomplemented
26C2×#950C2216yescomplemented
27C2×#948C2211yescharacteristic
28#5806—212no—
29#5805C2216yescomplemented
30C2×#936C2216yescomplemented
31C22×#214C2216yescomplemented
32C2×#932C2216yescomplemented
33#5753—28no—
34C2×#909C2216yescomplemented
35C22×Dic12C2213yescomplemented
36C2×#904C2213yescomplemented
37#5191C2211yescharacteristic, complemented
38#5183C2213yescomplemented
39#5175C2216yescomplemented
40#5168C2216yescomplemented
41#5167C22112yescomplemented
42#5155C2213yescomplemented
43#5155C2216yescomplemented
44#5155C2216yescomplemented
45#5155C2216yescomplemented
46#5154C22112yescomplemented
47#5151C2213yescomplemented
48#5122C2213yescomplemented
49Dic3×M16C2213yescomplemented
50#5101C2213yescomplemented
51C2×C8×Dic3C2213yescomplemented
52C2×#578—31no—
53C2×#577—32no—
54C2×#576D311yescharacteristic, complemented
55#3500—32no—
56#3156—33no—
57#3155—33no—
58#3154—36no—
59#3153—36no—
60#3147—33no—
61C2×#495—33no—
62C4×#131—31no—
63C3×H(5,𝔽2)—28no—
64C3×H(5,𝔽2)—28no—
65C2×C3×#41—22no—
66C2×C3×#41C2316yes—
67C2×C3×#41C2311yescharacteristic
68C22×C3×D4—26no—
69C22×C3×D4C2316yes—
70#1010—28no—
71#1010—28no—
72C2×D3×Q8—22no—
73C2×#237—26no—
74C2×#235C2316yes—
75C22×Dic6C2316yes—
76C3×AGL(1,ℤ/8)—224no—
77C3×#135—212no—
78C2×C3×Q16C2311yescharacteristic
79C2×C3×SD16C2316yes—
80C2×C3×D8—212no—
81C2×C3×D8C2313yes—
82C2×C3×D8C2316yes—
83C3×#131D418yescomplemented
84C2×C3×M16C2313yes—
85C22×C3×C8C2313yes—
86#953—212no—
87C2×#217—212no—
88#951—224no—
89#950—224no—
90#948D418yescomplemented
91#947C2312yes—
92Q8×Dic3D412yes—
93#944C2316yes—
94C2×#216C2316yes—
95C2×#215C2312yes—
96#940C2316yes—
97#939—212no—
98#938—212no—
99#936—212no—
100#936C23112yes—
101#936C2316yes—
102#935—212no—
103D4×Dic3D416yes—
104C2×#214—224no—
105C2×#214C23112yes—
106C2×#214C2316yes—
107#932—224no—
108C2×#213C2316yes—
109C4×#67—212no—
110#926C2313yes—
111#924—212no—
112C2×C4×Dic3C2313yes—
113C2×#208C2313yes—
114C22×C3⋊C8C2313yes—
115#909—224no—
116C2×Dic12—212no—
117C2×Dic12C2313yes—
118C2×Dic12C2316yes—
119#904—212no—
120C2×D24C2311yescharacteristic
121C2×#204C2316yes—
122#815C2316yes—
123C8×Dic3C2313yes—
124C8×Dic3C2313yes—
125C8×Dic3C2311yescharacteristic
126C8×Dic3C2313yes—
127C2×2−1+4—32no—
128C2×H(5,𝔽2)C2×D312yes—
129#578—68no—
130#577—616no—
131#576—28no—
132C2×#137—36no—
133C2×#137—36no—
134C2×AGL(1,ℤ/8)—36no—
135C2×AGL(1,ℤ/8)C2×D316yes—
136C2×#135—36no—
137C2×#135—33no—
138C2×#135C2×D313yes—
139C22×Q16—33no—
140C22×SD16—36no—
141C22×D8C2×D313yes—
142C2×#131—31no—
143C2×#131C2×D311yescharacteristic
144#548—64no—
145#546—612no—
146#536—36no—
147#535—36no—
148#532—36no—
149C2×#124—36no—
150C4×#41—31no—
151#497—312no—
152#496—312no—
153#495—31no—
154#495—36no—
155#495—33no—
156#495—36no—
157#495—36no—
158#495—36no—
159#494—33no—
160#493—33no—
161#492—36no—
162#405—33no—
163#405—33no—
164C4×M16—33no—
165C2×C4×C8—33no—
166C24×C3—26no—
167C3×#41—212no—
168C3×#41—212no—
169C3×#41—48no—
170C3×#41C2×D414yes—
171C3×#41C2×D414yes—
172C2×C3×Q8C2×D411yescharacteristic
173C2×C3×Q8C2412yes—
174C2×C3×D4—26no—
175C2×C3×D4—212no—
176C2×C3×D4—212no—
177C2×C3×D4—26no—
178C2×C3×D4—424no—
179C2×C3×D4C2×D413yes—
180C2×C3×D4C2416yes—
181C22×C3×C4—26no—
182C22×C3×C4C2413yes—
183C2×#67—26no—
184C2×#67—212no—
185C22×Dic3—26no—
186C22×Dic3C2×D416yes—
187D3×Q8—48no—
188#237—424no—
189#235—212no—
190#235—212no—
191C2×D12C2412yes—
192C2×C4×D3—26no—
193C2×Dic6—212no—
194C2×Dic6—212no—
195C2×Dic6—26no—
196C2×Dic6C2416yes—
197C3×Q16—22no—
198C3×SD16—212no—
199C3×D8—26no—
200C3×D8—212no—
201C3×M16C2×D4112yes—
202C2×C3×C8C2×D4112yes—
203C2×C3×C8C2411yescharacteristic
204C2×C3×C8C2413yes—
205#217—224no—
206#217—212no—
207#217C2×D416yes—
208#216—212no—
209#215—24no—
210#214—224no—
211#214—212no—
212#213—212no—
213(C2×D3)⋊C4—24no—
214C4⋊Dic3C2×D416yes—
215Dic3⋊C4—212no—
216C4×Dic3C2×D416yes—
217C4×Dic3C2411yescharacteristic
218C4×Dic3C2413yes—
219#208C2×D4112yes—
220C2×C3⋊C8C2×D4112yes—
221C2×C3⋊C8C2411yescharacteristic
222C2×C3⋊C8C2413yes—
223Dic12—26no—
224Dic12—212no—
225D24—22no—
226#204—212no—
2272−1+4—68no—
2282−1+4—68no—
229H(5,𝔽2)—28no—
230H(5,𝔽2)—28no—
231C2×#41—22no—
232C2×#41—36no—
233C2×#41—66no—
234C2×#41C22×D316yes—
235C2×#41C22×D311yescharacteristic
236C22×Q8—36no—
237C22×Q8—62no—
238C22×D4—26no—
239C22×D4C22×D316yes—
240#137—624no—
241#137—624no—
242AGL(1,ℤ/8)—224no—
243AGL(1,ℤ/8)—624no—
244#135—212no—
245#135—624no—
246#135—612no—
247C2×Q16—36no—
248C2×Q16—33no—
249C2×Q16—36no—
250C2×Q16—612no—
251C2×Q16C22×D311yescharacteristic
252C2×SD16—32no—
253C2×SD16—36no—
254C2×SD16—312no—
255C2×SD16—36no—
256C2×SD16—624no—
257C2×SD16C22×D316yes—
258C2×D8—212no—
259C2×D8—36no—
260C2×D8—31no—
261C2×D8C22×D313yes—
262C2×D8C22×D316yes—
263#131—38no—
264#131#6718yescomplemented
265C2×M16—33no—
266C2×M16C22×D313yes—
267C22×C8—33no—
268C22×C8C22×D313yes—
269C4⋊Q8—36no—
270#127—36no—
271#124—32no—
272#124—36no—
273#124—312no—
274#124—612no—
275#123—612no—
276C22⋊Q8—612no—
277#121—612no—
278#121—612no—
279#120—612no—
280C4×Q8—32no—
281C4×D4—36no—
282C4×D4—612no—
283#117—33no—
284C2×C22⋊C4—612no—
285C2×C42—33no—
286#97—36no—
287C4×C8—31no—
288C4×C8—33no—
289C4×C8—33no—
290C4×C8—33no—
291C23×C3—26no—
292C23×C3—412no—
293C23×C3—812no—
294C23×C3C22×D413yes—
295C22×D3—22no—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8#12714yes—
299C3×D4—26no—
300C3×D4—26no—
301C3×D4—412no—
302C3×D4—412no—
303C3×D4#127112yes—
304C2×C3×C4—22no—
305C2×C3×C4—412no—
306C2×C3×C4C22×D413yes—
307C2×C3×C4C22×D416yes—
308C2×C3×C4C22×D416yes—
309C2×C3×C4C2511yescharacteristic
310#67—412no—
311#67—812no—
312C2×Dic3—26no—
313C2×Dic3—46no—
314C2×Dic3—412no—
315C2×Dic3C22×D412yes—
316C2×Dic3C22×D416yes—
317D12—22no—
318D12—22no—
319C4×D3—412no—
320Dic6—26no—
321Dic6—26no—
322Dic6—412no—
323C3×C8C22×D416yes—
324C3×C8C22×D412yes—
325C3⋊C8C22×D412yes—
326C3⋊C8C22×D416yes—
327C24—26no—
328#41—212no—
329#41—212no—
330#41—48no—
331#41—612no—
332#41—612no—
333#41—1224no—
334#41D3×D414yes—
335#41C2×#6714yes—
336C2×Q8—36no—
337C2×Q8—66no—
338C2×Q8—612no—
339C2×Q8—612no—
340C2×Q8—128no—
341C2×Q8D3×D411yescharacteristic
342C2×Q8C23×D312yes—
343C2×D4—26no—
344C2×D4—212no—
345C2×D4—26no—
346C2×D4—212no—
347C2×D4—32no—
348C2×D4—424no—
349C2×D4—612no—
350C2×D4—66no—
351C2×D4D3×D413yes—
352C2×D4C23×D316yes—
353C22×C4—26no—
354C22×C4—36no—
355C22×C4—66no—
356C22×C4—66no—
357C22×C4C23×D313yes—
358Q16—22no—
359Q16—612no—
360Q16—66no—
361Q16—612no—
362SD16—212no—
363SD16—612no—
364SD16—624no—
365SD16—64no—
366SD16—612no—
367D8—26no—
368D8—212no—
369D8—612no—
370D8—62no—
371M16—312no—
372M16C2×#67112yes—
373C2×C8—312no—
374C2×C8—31no—
375C2×C8—33no—
376C2×C8C2×#67112yes—
377C2×C8C23×D311yescharacteristic
378C2×C8C23×D313yes—
379C4⋊C4—36no—
380C4⋊C4—612no—
381C22⋊C4—36no—
382C22⋊C4—64no—
383C22⋊C4—612no—
384C22⋊C4—624no—
385C42—36no—
386C42—31no—
387C42—33no—
388C22×C3—26no—
389C22×C3—412no—
390C22×C3—46no—
391C22×C3—1612no—
392C22×C3C2×#12716yes—
393C22×C3C23×D411yescharacteristic
394C2×D3—44no—
395C2×D3—42no—
396C3×C4—42no—
397C3×C4C2×#12716yes—
398C3×C4C23×D411yescharacteristic
399C3×C4C23×D411yescharacteristic
400Dic3—22no—
401Dic3—26no—
402Dic3—46no—
403C23—26no—
404C23—412no—
405C23—62no—
406C23—812no—
407C23C2×D3×D413yes—
408Q8—22no—
409Q8—22no—
410Q8—66no—
411Q8—66no—
412Q8—1212no—
413Q8#94014yes—
414D4—26no—
415D4—26no—
416D4—412no—
417D4—412no—
418D4—62no—
419D4—62no—
420D4—1212no—
421D4—2412no—
422D4#940112yes—
423C2×C4—22no—
424C2×C4—32no—
425C2×C4—36no—
426C2×C4—412no—
427C2×C4—66no—
428C2×C4—126no—
429C2×C4—1212no—
430C2×C4—1212no—
431C2×C4C2×D3×D416yes—
432C2×C4C2×D3×D413yes—
433C2×C4C22×#6716yes—
434C2×C4C24×D311yescharacteristic
435C8—32no—
436C8—36no—
437C8C22×#6712yes—
438C8C22×#6716yes—
439C2×C3—26no—
440C2×C3—86no—
441C2×C3C22×#12711yescharacteristic
442C2×C3#358412yes—
443D3—82no—
444C22—26no—
445C22—412no—
446C22—46no—
447C22—124no—
448C22—122no—
449C22—1612no—
450C22C2×#94016yes—
451C22C22×D3×D411yescharacteristic
452C4—42no—
453C4—62no—
454C4—66no—
455C4—126no—
456C4C2×#94016yes—
457C4C22×D3×D411yescharacteristic
458C4C23×#6711yescharacteristic
459C3#6064911yescharacteristic, complemented, Sylow
460C2—26no—
461C2—86no—
462C2—242no—
463C2#8720312yes—
464C2C22×#94011yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#576C2
socleC22×C3C23×D4
hypercenterC2×#576D3
hypocenterC3#60649
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#576

Complements

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

NH ≅ G/Ntype
1. C2×C3×#576C2semidirect
2. #87283C2semidirect
3. C2×#5836C2semidirect
4. C2×#5753C2semidirect
5. #83939C2semidirect
6. #83938C2semidirect
7. #83907C2semidirect
8. #83903C2semidirect
9. #83830C2semidirect
10. C2×#5155C2semidirect
11. Dic3×#131C2semidirect
21. #5832C22semidirect
24. #5825C22semidirect
25. C2×#951C22semidirect
26. C2×#950C22semidirect
29. #5805C22semidirect
30. C2×#936C22semidirect
31. C22×#214C22semidirect
32. C2×#932C22semidirect
34. C2×#909C22semidirect
35. C22×Dic12C22semidirect
36. C2×#904C22semidirect
37. #5191C22semidirect
38. #5183C22semidirect
39. #5175C22semidirect
40. #5168C22semidirect
41. #5167C22semidirect
42. #5155C22semidirect
43. #5155C22semidirect
44. #5155C22semidirect
45. #5155C22semidirect
46. #5154C22semidirect
47. #5151C22semidirect
48. #5122C22semidirect
49. Dic3×M16C22semidirect
50. #5101C22semidirect
51. C2×C8×Dic3C22semidirect
54. C2×#576D3semidirect
83. C3×#131D4semidirect
90. #948D4semidirect
264. #131#67semidirect
459. C3#60649semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5762
#872832
C2×#58362
C2×#57532
#839392
#839382
#839072
#839032
#838302
C2×#51552
Dic3×#1312
#606493