Finite Group Databasebeta

← #10948001

Subgroups — #10948001

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86341C211yesmaximal, characteristic
2#78694C212yesmaximal, complemented
3C3×#2129C211yesmaximal, characteristic, complemented
4#71871C211yesmaximal, characteristic, complemented
5#70060C212yesmaximal, complemented
6#69157C214yesmaximal, complemented
7D3×#354C214yesmaximal, complemented
8#33361—31nomaximal, Sylow
9C2×#1009C2211yescharacteristic
10C2×#1007C411yescharacteristic, complemented
11C3×#536C2211yescharacteristic
12#5832C2211yescharacteristic
13C2×#947C411yescharacteristic, complemented
14#5725C412yescomplemented
15#5724C412yescomplemented
16#5674C2212yes—
17D3×#124C2212yes—
18#5654—24no—
19#5633—22no—
20#5630—22no—
21#5525C411yescharacteristic, complemented
22#5522C411yescharacteristic, complemented
23#5521—22no—
24C3×#410C2212yes—
25#5240C2212yes—
26#5006—22no—
27D3×#99—24no—
28D3×#99C2218yes—
29C3×#354C2214yescomplemented
30#4541C2214yescomplemented
31#4474C2214yescomplemented
32#4473C2214yescomplemented
33#3500—31no—
34#2883—32no—
35#2129—31no—
36#2129D311yescharacteristic, complemented
37#2127—34no—
38#2126—32no—
39C2×#354—34no—
40C2×C3×#41C2311yescharacteristic
41C22×C3×Q8C2×C411yescharacteristic, complemented
42#1009—44no—
43#1007—44no—
44C2×#239—21no—
45C2×#239C2×C412yes—
46C2×#239C2×C412yescomplemented
47C2×D3×Q8—21no—
48C2×D3×Q8C2×C412yescomplemented
49C2×D3×D4D414yes—
50C2×D3×D4C2312yes—
51C2×#235—22no—
52C2×#235C2×C411yescharacteristic
53C2×#235C2×C411yescharacteristic, complemented
54C2×#235D412yes—
55C22×D12C2×C411yescharacteristic
56C22×D12D412yes—
57C3×C4⋊Q8C2×C412yescomplemented
58C3×#124C2312yes—
59C3×#123—22no—
60C3×C22⋊Q8—22no—
61C3×#117—22no—
62C3×#117C2×C411yescharacteristic, complemented
63#947—22no—
64#947C2×C412yescomplemented
65#947C2×C412yescomplemented
66#947C2×C411yescharacteristic, complemented
67#944C2×C411yescharacteristic, complemented
68#940C2311yescharacteristic
69#939—22no—
70#936C2311yescharacteristic
71C4×#67—22no—
72C2×(C2×D3)⋊C4—22no—
73#926—22no—
74C2×C4×Dic3C2×C411yescharacteristic, complemented
75#896—24no—
76#895—22no—
77#895—24no—
78#894—24no—
79#893—21no—
80#893—24no—
81#893—44no—
82#887—21no—
83#885—24no—
84#884—24no—
85#882—22no—
86#882—24no—
87#67⋊C4—22no—
88D3×C22⋊C4—24no—
89D3×C22⋊C4—28no—
90#875C2×C412yescomplemented
91#874C2×C412yescomplemented
92C4×D12—44no—
93#872C2×C412yescomplemented
94C42×D3C2×C412yescomplemented
95C3×#99—22no—
96C3×#99C2314yes—
97#834—22no—
98#834C2314yes—
99#806—28no—
100C2×2−1+4—31no—
101C2×H(5,𝔽2)—31no—
102#548—62no—
103#546—62no—
104#546—64no—
105#536—32no—
106#536C2×D311yescharacteristic
107#535—32no—
108#535—31no—
109#532—31no—
110#532—32no—
111C2×#124—31no—
112C2×#124—32no—
113#518—62no—
114C4×#41—31no—
115#410—32no—
116#410—62no—
117#410C2×D312yes—
118C2×#99—38no—
119C2×#99—64no—
120#354—34no—
121#354—34no—
122#354C2×D314yescomplemented
123#353—34no—
124C23×D3C22⋊C412yes—
125C23×D3C2×D414yes—
126C3×#41—42no—
127C2×C3×Q8—21no—
128C2×C3×Q8—42no—
129C2×C3×Q8C22×C412yes—
130C2×C3×Q8C22×C411yescharacteristic
131C2×C3×D4C2×D412yes—
132C2×C3×D4C2411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D412yes—
136C2×#67—21no—
137C2×#67—22no—
138C2×#67C22⋊C412yes—
139C2×#67C2×D414yes—
140C22×Dic3—21no—
141#239—44no—
142#239—44no—
143#239—44no—
144D3×Q8—44no—
145D3×Q8—44no—
146D3×D4—44no—
147D3×D4—48no—
148#235—24no—
149#235—42no—
150#235—42no—
151#235—48no—
152#235—42no—
153C2×D12—24no—
154C2×D12—24no—
155C2×D12—22no—
156C2×D12—24no—
157C2×D12—42no—
158C2×D12C22⋊C414yes—
159C2×D12C22⋊C412yes—
160C2×D12C22⋊C411yescharacteristic
161C2×D12C22×C411yescharacteristic
162C2×D12C22×C412yes—
163C2×C4×D3—22no—
164C2×C4×D3—21no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—48no—
168C2×C4×D3C22⋊C414yes—
169C2×C4×D3C22×C412yes—
170C2×Dic6—22no—
171C2×Dic6C22⋊C411yescharacteristic
172C2×Dic6C22×C411yescharacteristic
173C3×C4⋊C4—21no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—42no—
176C3×C22⋊C4—21no—
177C3×C22⋊C4—24no—
178C3×C22⋊C4—24no—
179C3×C42—42no—
180C3×C42C22×C412yes—
181#217—22no—
182#217—24no—
183(C2×D3)⋊C4—22no—
184(C2×D3)⋊C4—24no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—24no—
187(C2×D3)⋊C4—22no—
188(C2×D3)⋊C4—48no—
189C4⋊Dic3—42no—
190Dic3⋊C4—22no—
191Dic3⋊C4—22no—
192C4×Dic3—21no—
193C4×Dic3—22no—
194C4×Dic3—42no—
195C4×Dic3C22×C411yescharacteristic
196C4×Dic3C22×C411yescharacteristic
1972−1+4—124no—
198H(5,𝔽2)—124no—
199C2×#41—32no—
200C2×#41—31no—
201C2×#41—32no—
202C2×#41—32no—
203C2×#41—31no—
204C2×#41—62no—
205C2×#41—61no—
206C2×#41C22×D311yescharacteristic
207C22×Q8—32no—
208C22×Q8—61no—
209C22×Q8C4×D311yescharacteristic, complemented
210C22×D4—34no—
211C22×D4—32no—
212C22×D4—31no—
213C22×D4—32no—
214C4⋊Q8—31no—
215C4⋊Q8C4×D312yescomplemented
216#127—31no—
217#127—32no—
218#124—32no—
219#124—32no—
220#124—31no—
221#124—31no—
222#124—32no—
223#124—64no—
224#124—62no—
225#124C22×D312yes—
226#123—22no—
227#123—64no—
228C22⋊Q8—22no—
229C22⋊Q8—64no—
230#121—64no—
231#121—64no—
232#121—62no—
233#121—62no—
234#120—62no—
235#120—64no—
236C4×Q8—61no—
237C4×D4—62no—
238C4×D4—64no—
239C4×D4—61no—
240C4×D4—62no—
241C4×D4—124no—
242C4×D4—124no—
243#117—22no—
244#117—32no—
245#117—62no—
246#117C4×D311yescharacteristic, complemented
247C2×C22⋊C4—64no—
248C2×C22⋊C4—68no—
249C2×C22⋊C4—62no—
250C2×C42—31no—
251C2×C42—32no—
252#99—22no—
253#99—34no—
254#99—68no—
255#99—62no—
256#99C22×D314yes—
257C23×C3C2×C22⋊C411yescharacteristic
258C23×C3C22×D412yes—
259C22×D3—28no—
260C22×D3—21no—
261C22×D3—42no—
262C22×D3—84no—
263C22×D3—82no—
264C22×D3C2×C22⋊C414yes—
265C22×D3C2×C22⋊C412yes—
266C3×Q8—41no—
267C3×Q8—42no—
268C3×Q8—42no—
269C3×D4—42no—
270C3×D4—41no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—42no—
277C2×C3×C4—44no—
278C2×C3×C4C2×C22⋊C412yes—
279C2×C3×C4C2×C22⋊C411yescharacteristic
280C2×C3×C4C23×C411yescharacteristic
281#67—24no—
282#67—44no—
283#67—42no—
284#67—82no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—44no—
289C2×Dic3—81no—
290C2×Dic3C2×C22⋊C412yes—
291D12—28no—
292D12—22no—
293D12—22no—
294D12—42no—
295D12—41no—
296D12—44no—
297D12—84no—
298D12#9914yes—
299C4×D3—24no—
300C4×D3—44no—
301C4×D3—44no—
302C4×D3—44no—
303C4×D3—42no—
304C4×D3—82no—
305C4×D3—168no—
306Dic6—22no—
307Dic6—44no—
308Dic6—41no—
309C24—34no—
310C24—32no—
311#41—42no—
312#41—64no—
313#41—122no—
314#41—122no—
315#41—122no—
316#41—128no—
317#41—124no—
318#41—124no—
319#41—124no—
320C2×Q8—21no—
321C2×Q8—31no—
322C2×Q8—31no—
323C2×Q8—42no—
324C2×Q8—62no—
325C2×Q8—124no—
326C2×Q8—124no—
327C2×Q8C2×C4×D312yes—
328C2×Q8C2×C4×D311yescharacteristic
329C2×D4—32no—
330C2×D4—34no—
331C2×D4—34no—
332C2×D4—31no—
333C2×D4—32no—
334C2×D4—32no—
335C2×D4—31no—
336C2×D4—64no—
337C2×D4—64no—
338C2×D4—62no—
339C2×D4—64no—
340C2×D4—61no—
341C2×D4—62no—
342C2×D4—122no—
343C2×D4—124no—
344C2×D4—128no—
345C2×D4D3×D412yes—
346C2×D4C23×D311yescharacteristic
347C22×C4—21no—
348C22×C4—32no—
349C22×C4—34no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—128no—
356C22×C4C2×C4×D311yescharacteristic
357C22×C4D3×D412yes—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—42no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—122no—
364C22⋊C4—21no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—64no—
368C22⋊C4—64no—
369C22⋊C4—64no—
370C22⋊C4—62no—
371C22⋊C4—62no—
372C22⋊C4—62no—
373C22⋊C4—64no—
374C22⋊C4—128no—
375C42—31no—
376C42—31no—
377C42—42no—
378C42—62no—
379C42—61no—
380C42—122no—
381C42C2×C4×D312yes—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—42no—
386C22×C3C22×C22⋊C411yescharacteristic
387C2×D3—24no—
388C2×D3—42no—
389C2×D3—44no—
390C2×D3—42no—
391C2×D3—41no—
392C2×D3—88no—
393C2×D3—162no—
394C2×D3C2×#9914yes—
395C3×C4—21no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—84no—
400C3×C4C2×#9912yes—
401Dic3—22no—
402Dic3—41no—
403Dic3—41no—
404Dic3—41no—
405Dic3—84no—
406C23—32no—
407C23—34no—
408C23—68no—
409C23—61no—
410C23—122no—
411C23—244no—
412C23—242no—
413C23D3×C22⋊C411yescharacteristic
414C23C2×D3×D412yes—
415Q8—41no—
416Q8—42no—
417Q8—42no—
418Q8—62no—
419Q8—121no—
420Q8—124no—
421D4—34no—
422D4—42no—
423D4—41no—
424D4—68no—
425D4—64no—
426D4—62no—
427D4—62no—
428D4—122no—
429D4—121no—
430D4—124no—
431D4—124no—
432D4—122no—
433D4—244no—
434D4—242no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—32no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—64no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—124no—
448C2×C4—124no—
449C2×C4—124no—
450C2×C4—122no—
451C2×C4—124no—
452C2×C4—242no—
453C2×C4—241no—
454C2×C4—488no—
455C2×C4D3×C22⋊C412yes—
456C2×C4D3×C22⋊C411yescharacteristic
457C2×C4C22×C4×D311yescharacteristic
458C2×C3—21no—
459C2×C3—42no—
460C2×C3—41no—
461C2×C3C22×#9911yescharacteristic
462D3—24no—
463D3—44no—
464D3—81no—
465C22—22no—
466C22—21no—
467C22—34no—
468C22—41no—
469C22—42no—
470C22—64no—
471C22—122no—
472C22—124no—
473C22—122no—
474C22—121no—
475C22—248no—
476C22—482no—
477C22C2×D3×C22⋊C411yescharacteristic
478C4—21no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—62no—
483C4—84no—
484C4—121no—
485C4—121no—
486C4—121no—
487C4—244no—
488C4D3×#9912yes—
489C3#3336111yescharacteristic, complemented, Sylow
490C2—21no—
491C2—42no—
492C2—41no—
493C2—64no—
494C2—124no—
495C2—241no—
496C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2129C2
socleC2×C3C22×#99
prime radical#86341C2
cosocleC2×C3×D4C24
hypercenter#2129D3
hypocenterC3#33361
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2129

Complements

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

NH ≅ G/Ntype
2. #78694C2semidirect
3. C3×#2129C2semidirect
4. #71871C2semidirect
5. #70060C2semidirect
6. #69157C2semidirect
7. D3×#354C2semidirect
10. C2×#1007C4semidirect
13. C2×#947C4semidirect
14. #5725C4semidirect
15. #5724C4semidirect
21. #5525C4semidirect
22. #5522C4semidirect
29. C3×#354C22semidirect
30. #4541C22semidirect
31. #4474C22semidirect
32. #4473C22semidirect
36. #2129D3semidirect
41. C22×C3×Q8C2×C4semidirect
46. C2×#239C2×C4semidirect
48. C2×D3×Q8C2×C4semidirect
53. C2×#235C2×C4semidirect
57. C3×C4⋊Q8C2×C4semidirect
62. C3×#117C2×C4semidirect
64. #947C2×C4semidirect
65. #947C2×C4semidirect
66. #947C2×C4semidirect
67. #944C2×C4semidirect
74. C2×C4×Dic3C2×C4semidirect
90. #875C2×C4semidirect
91. #874C2×C4semidirect
93. #872C2×C4semidirect
94. C42×D3C2×C4semidirect
122. #354C2×D3semidirect
209. C22×Q8C4×D3semidirect
215. C4⋊Q8C4×D3semidirect
246. #117C4×D3semidirect
489. C3#33361semidirect

Maximal subgroups

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

subgroupindex
#863412
#786942
C3×#21292
#718712
#700602
#691572
D3×#3542
#333613