Finite Group Databasebeta

← #11664750

Subgroups — #11664750

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×#578C211yesmaximal, characteristic, complemented
2#87293C212yesmaximal, complemented
3C2×#5837C212yesmaximal, complemented
4C2×#5752C211yesmaximal, characteristic, complemented
5#83974C216yesmaximal, complemented
6#83973C216yesmaximal, complemented
7#83906C213yesmaximal, complemented
8#83903C213yesmaximal, complemented
9#83881C213yesmaximal, complemented
10C2×#5191C213yesmaximal, complemented
11Dic3×#131C211yesmaximal, characteristic, complemented
12#60649—31nomaximal, Sylow
13C2×C3×2−1+4C2212yes—
14C2×#1009C2212yes—
15C3×#578—28no—
16C2×C3×#137C2216yescomplemented
17C2×C3×#135C2213yescomplemented
18C22×C3×Q16C2213yescomplemented
19C2×C3×#131C2211yescharacteristic
20#5837—216no—
21#5832C2216yescomplemented
22#5830C2216yescomplemented
23#5829—212no—
24Dic3×#41C2211yescharacteristic
25C2×#950C2216yescomplemented
26C2×#949C2216yescomplemented
27C2×#948C2211yescharacteristic
28#5817—24no—
29#5816C2216yescomplemented
30C2×#947C2216yescomplemented
31C2×#942C2216yescomplemented
32C22×#215C2216yescomplemented
33#5752—28no—
34C2×#908C2216yes—
35C2×#904C2213yes—
36C22×D24C2213yes—
37#5191C2216yescomplemented
38#5191C2216yescomplemented
39#5191C2213yescomplemented
40#5191C2216yescomplemented
41#5190C22112yescomplemented
42#5183C2213yescomplemented
43#5176C22112yescomplemented
44#5175C2216yescomplemented
45#5168C2216yescomplemented
46#5155C2211yescharacteristic, complemented
47#5151C2213yescomplemented
48#5122C2213yescomplemented
49Dic3×M16C2213yescomplemented
50#5101C2213yescomplemented
51C2×C8×Dic3C2213yescomplemented
52C2×#578D311yescharacteristic, complemented
53C2×#577—32no—
54C2×#576—31no—
55#3500—32no—
56#3156—33no—
57#3155—33no—
58#3154—36no—
59#3153—36no—
60#3147—33no—
61C2×#495—33no—
62C4×#131—31no—
63C3×2−1+4—28no—
64C3×2−1+4—28no—
65C2×C3×#41—26no—
66C2×C3×#41C2316yes—
67C2×C3×#41C2311yescharacteristic
68C22×C3×Q8—22no—
69C22×C3×Q8C2316yes—
70#1009—28no—
71#1009—28no—
72C2×#239—22no—
73C2×D3×D4—26no—
74C2×#235C2316yes—
75C22×D12C2316yes—
76C3×#137—224no—
77C3×#135—212no—
78C2×C3×Q16—212no—
79C2×C3×Q16C2313yes—
80C2×C3×Q16C2316yes—
81C2×C3×SD16C2316yes—
82C2×C3×D8C2311yescharacteristic
83C3×#131D418yescomplemented
84C2×C3×M16C2313yes—
85C22×C3×C8C2313yes—
86#950—224no—
87#949—224no—
88#948D418yescomplemented
89#947—212no—
90#947C2316yes—
91#947C23112yes—
92#946—212no—
93Q8×Dic3D412yes—
94#944C2316yes—
95C2×#216C2316yes—
96#942—224no—
97C2×#215—224no—
98C2×#215C2316yes—
99C2×#215C23112yes—
100#940C2316yes—
101#939—212no—
102#937—212no—
103#936C2312yes—
104D4×Dic3D416yes—
105C2×#214C2312yes—
106C2×#213C2316yes—
107#930—212no—
108#929—212no—
109C4×#67—212no—
110C2×(C2×D3)⋊C4—212no—
111#926C2313yes—
112C2×C4×Dic3C2313yes—
113C2×#208C2313yes—
114C22×C3⋊C8C2313yes—
115#908—224no—
116C2×Dic12C2311yescharacteristic
117#904—212no—
118C2×D24—212no—
119C2×D24C2313yes—
120C2×D24C2316yes—
121C2×#204C2316yes—
122#815C2316yes—
123C8×Dic3C2313yes—
124C8×Dic3C2313yes—
125C8×Dic3C2311yescharacteristic
126C8×Dic3C2313yes—
127C2×2−1+4C2×D312yes—
128C2×H(5,𝔽2)—32no—
129#578—28no—
130#577—616no—
131#576—68no—
132C2×#137—36no—
133C2×#137C2×D316yescomplemented
134C2×AGL(1,ℤ/8)—36no—
135C2×AGL(1,ℤ/8)—36no—
136C2×#135—36no—
137C2×#135—33no—
138C2×#135C2×D313yescomplemented
139C22×Q16C2×D313yescomplemented
140C22×SD16—36no—
141C22×D8—33no—
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—33no—
154#495—36no—
155#495—36no—
156#495—36no—
157#495—36no—
158#495—31no—
159#494—33no—
160#493—33no—
161#492—36no—
162#405—33no—
163#405—33no—
164C4×M16—33no—
165C2×C4×C8—33no—
166C23×D3—26no—
167C3×#41—212no—
168C3×#41—212no—
169C3×#41—424no—
170C3×#41C2×D414yes—
171C3×#41C2×D414yes—
172C2×C3×Q8—26no—
173C2×C3×Q8—212no—
174C2×C3×Q8—212no—
175C2×C3×Q8—48no—
176C2×C3×Q8C2×D411yescharacteristic
177C2×C3×Q8C2416yes—
178C2×C3×D4—26no—
179C2×C3×D4C2×D413yes—
180C2×C3×D4C2412yes—
181C22×C3×C4—26no—
182C22×C3×C4—26no—
183C22×C3×C4C2413yes—
184C2×#67—26no—
185C2×#67—212no—
186C22×Dic3C2×D416yes—
187#239—48no—
188D3×D4—424no—
189#235—212no—
190#235—212no—
191C2×D12—212no—
192C2×D12—212no—
193C2×D12—26no—
194C2×D12C2416yes—
195C2×C4×D3—26no—
196C2×Dic6C2412yes—
197C3×Q16—26no—
198C3×Q16—212no—
199C3×SD16—212no—
200C3×D8—22no—
201C3×M16C2×D4112yes—
202C2×C3×C8C2×D4112yes—
203C2×C3×C8C2411yescharacteristic
204C2×C3×C8C2413yes—
205#217—24no—
206#217C2×D416yes—
207#216—212no—
208#215—212no—
209#215—224no—
210#214—24no—
211#213—212no—
212(C2×D3)⋊C4—212no—
213(C2×D3)⋊C4—224no—
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—22no—
224D24—26no—
225D24—212no—
226#204—212no—
2272−1+4—28no—
2282−1+4—28no—
229H(5,𝔽2)—68no—
230H(5,𝔽2)—68no—
231C2×#41—26no—
232C2×#41—36no—
233C2×#41—62no—
234C2×#41C22×D316yes—
235C2×#41C22×D311yescharacteristic
236C22×Q8—22no—
237C22×Q8C22×D316yes—
238C22×D4—36no—
239C22×D4—66no—
240#137—224no—
241#137—624no—
242AGL(1,ℤ/8)—624no—
243AGL(1,ℤ/8)—624no—
244#135—212no—
245#135—624no—
246#135—612no—
247C2×Q16—212no—
248C2×Q16—36no—
249C2×Q16—31no—
250C2×Q16C22×D313yes—
251C2×Q16C22×D316yes—
252C2×SD16—36no—
253C2×SD16—312no—
254C2×SD16—32no—
255C2×SD16—36no—
256C2×SD16—624no—
257C2×SD16C22×D316yes—
258C2×D8—36no—
259C2×D8—33no—
260C2×D8—36no—
261C2×D8—612no—
262C2×D8C22×D311yescharacteristic
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—36no—
272#124—312no—
273#124—32no—
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—22no—
292C23×C3C22×D413yes—
293C22×D3—26no—
294C22×D3—412no—
295C22×D3—812no—
296C3×Q8—26no—
297C3×Q8—26no—
298C3×Q8—412no—
299C3×Q8#12714yes—
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—412no—
303C3×D4#127112yes—
304C2×C3×C4—26no—
305C2×C3×C4—412no—
306C2×C3×C4—412no—
307C2×C3×C4C22×D413yes—
308C2×C3×C4C22×D416yes—
309C2×C3×C4C22×D416yes—
310C2×C3×C4C2511yescharacteristic
311#67—412no—
312#67—812no—
313C2×Dic3—22no—
314C2×Dic3—46no—
315C2×Dic3C22×D412yes—
316C2×Dic3C22×D416yes—
317D12—26no—
318D12—26no—
319D12—412no—
320C4×D3—412no—
321Dic6—22no—
322Dic6—22no—
323C3×C8C22×D416yes—
324C3×C8C22×D412yes—
325C3⋊C8C22×D412yes—
326C3⋊C8C22×D416yes—
327C24—66no—
328#41—212no—
329#41—212no—
330#41—424no—
331#41—612no—
332#41—612no—
333#41—128no—
334#41D3×D414yes—
335#41C2×#6714yes—
336C2×Q8—26no—
337C2×Q8—212no—
338C2×Q8—212no—
339C2×Q8—32no—
340C2×Q8—48no—
341C2×Q8D3×D411yescharacteristic
342C2×Q8C23×D316yes—
343C2×D4—26no—
344C2×D4—36no—
345C2×D4—612no—
346C2×D4—66no—
347C2×D4—612no—
348C2×D4—66no—
349C2×D4—612no—
350C2×D4—1224no—
351C2×D4D3×D413yes—
352C2×D4C23×D312yes—
353C22×C4—26no—
354C22×C4—26no—
355C22×C4—36no—
356C22×C4—66no—
357C22×C4C23×D313yes—
358Q16—26no—
359Q16—212no—
360Q16—612no—
361Q16—62no—
362SD16—212no—
363SD16—64no—
364SD16—612no—
365SD16—624no—
366SD16—612no—
367D8—22no—
368D8—612no—
369D8—66no—
370D8—612no—
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—612no—
383C22⋊C4—624no—
384C22⋊C4—64no—
385C42—36no—
386C42—31no—
387C42—33no—
388C22×C3—26no—
389C22×C3—44no—
390C22×C3—42no—
391C22×C3C2×#12716yes—
392C22×C3C23×D411yescharacteristic
393C2×D3—412no—
394C2×D3—46no—
395C2×D3—1612no—
396C3×C4—46no—
397C3×C4C2×#12716yes—
398C3×C4C23×D411yescharacteristic
399C3×C4C23×D411yescharacteristic
400Dic3—22no—
401Dic3—26no—
402Dic3—42no—
403C23—22no—
404C23—66no—
405C23—1212no—
406C23—2412no—
407C23C2×D3×D413yes—
408Q8—26no—
409Q8—26no—
410Q8—412no—
411Q8—62no—
412Q8—62no—
413Q8#94014yes—
414D4—22no—
415D4—22no—
416D4—412no—
417D4—66no—
418D4—66no—
419D4—1212no—
420D4—1212no—
421D4—2412no—
422D4#940112yes—
423C2×C4—26no—
424C2×C4—32no—
425C2×C4—36no—
426C2×C4—412no—
427C2×C4—412no—
428C2×C4—62no—
429C2×C4—126no—
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—82no—
441C2×C3C22×#12711yescharacteristic
442C2×C3#358412yes—
443D3—86no—
444C22—26no—
445C22—44no—
446C22—42no—
447C22—1212no—
448C22—126no—
449C22—4812no—
450C22C2×#94016yes—
451C22C22×D3×D411yescharacteristic
452C4—46no—
453C4—62no—
454C4—66no—
455C4—122no—
456C4C2×#94016yes—
457C4C22×D3×D411yescharacteristic
458C4C23×#6711yescharacteristic
459C3#6064911yescharacteristic, complemented, Sylow
460C2—26no—
461C2—82no—
462C2—246no—
463C2#8721012yes—
464C2C22×#94011yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#578C2
socleC22×C3C23×D4
hypercenterC2×#578D3
hypocenterC3#60649
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×C3×#578C2semidirect
2. #87293C2semidirect
3. C2×#5837C2semidirect
4. C2×#5752C2semidirect
5. #83974C2semidirect
6. #83973C2semidirect
7. #83906C2semidirect
8. #83903C2semidirect
9. #83881C2semidirect
10. C2×#5191C2semidirect
11. Dic3×#131C2semidirect
16. C2×C3×#137C22semidirect
17. C2×C3×#135C22semidirect
18. C22×C3×Q16C22semidirect
21. #5832C22semidirect
22. #5830C22semidirect
25. C2×#950C22semidirect
26. C2×#949C22semidirect
29. #5816C22semidirect
30. C2×#947C22semidirect
31. C2×#942C22semidirect
32. C22×#215C22semidirect
37. #5191C22semidirect
38. #5191C22semidirect
39. #5191C22semidirect
40. #5191C22semidirect
41. #5190C22semidirect
42. #5183C22semidirect
43. #5176C22semidirect
44. #5175C22semidirect
45. #5168C22semidirect
46. #5155C22semidirect
47. #5151C22semidirect
48. #5122C22semidirect
49. Dic3×M16C22semidirect
50. #5101C22semidirect
51. C2×C8×Dic3C22semidirect
52. C2×#578D3semidirect
83. C3×#131D4semidirect
88. #948D4semidirect
133. C2×#137C2×D3semidirect
138. C2×#135C2×D3semidirect
139. C22×Q16C2×D3semidirect
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×#5782
#872932
C2×#58372
C2×#57522
#839742
#839732
#839062
#839032
#838812
C2×#51912
Dic3×#1312
#606493