Finite Group Databasebeta

← C2×#78566 #11664985

Subgroups — C2×#78566 #11664985

Order 768.

1970 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5534C212yesmaximal, complemented
2C2×C3×#405C211yesmaximal, characteristic, complemented
3C2×#5137C211yesmaximal, characteristic, complemented
4C2×#5135C211yesmaximal, characteristic
5C2×#5114C212yesmaximal, complemented
6#78566C2116yesmaximal, direct product factor
7C2×#4716C211yesmaximal, characteristic, complemented
8C2×#4714C212yesmaximal, complemented
9C2×#4711C211yesmaximal, characteristic, complemented
10C2×C4×#204C212yesmaximal, complemented
11C2×#4696C211yesmaximal, characteristic, complemented
12C2×#4680C211yesmaximal, characteristic, complemented
13C2×#3173—31nomaximal, Sylow
14C22×C3×M16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#930C2212yes—
18C2×#924C2212yes—
19C22×Dic12C2211yescharacteristic
20C22×D24C2211yescharacteristic
21C22×#204C2212yes—
22#5534—28no—
23#5534C22116yescomplemented
24C2×#898—22no—
25C2×#896—22no—
26C2×#885—22no—
27C2×#883—22no—
28C2×#875C2212yes—
29C2×C4×D12C2212yes—
30C2×#869C2212yes—
31C2×C4×Dic6C2212yes—
32C3×#405C2218yescomplemented
33C2×C3×#97C2211yescharacteristic
34C2×C3×#97C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C2×C3×C4×C8C2211yescharacteristic
37#5137C2218yescomplemented
38#5135C2218yes—
39#5114C22116yescomplemented
40C2×#821C2212yes—
41C2×#821C2212yes—
42C2×#817C2212yes—
43C2×#817C2211yescharacteristic
44C2×#817C2211yescharacteristic
45C2×#816C2212yes—
46C2×#816C2212yes—
47#4716C2218yescomplemented
48#4714C22116yescomplemented
49#4711C2218yescomplemented
50#4696C2218yescomplemented
51C4×#204C22116yescomplemented
52#4680C2218yescomplemented
53C2×#542—32no—
54#3173—316no—
55C2×#498—31no—
56C2×#495—31no—
57C2×#489—32no—
58C2×#470—31no—
59C2×#469—31no—
60C2×#465—32no—
61C2×#442—31no—
62C2×#441—31no—
63C2×C4×SD16—32no—
64C2×#405D311yescharacteristic, complemented
65C23×C3×C4C2311yescharacteristic
66C22×#67—22no—
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×Dic6—22no—
70C22×Dic6C2312yes—
71C2×C3×M16—24no—
72C2×C3×M16C2314yes—
73C22×C3×C8—24no—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2314yes—
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×#117D414yes—
80C3×#117C2314yes—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C22⋊C4D412yes—
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—22no—
86#930—24no—
87#930C2318yes—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95#924—24no—
96#924C2318yes—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—22no—
100C2×Dic12—24no—
101C2×Dic12C2314yes—
102C2×D24—24no—
103C2×D24C2314yes—
104C2×#204—28no—
105C2×#204C2318yes—
106#898—216no—
107#896—216no—
108#885—216no—
109#883—216no—
110#875—24no—
111#875C2318yes—
112C4×D12—24no—
113C4×D12C2318yes—
114#869—24no—
115#869C2318yes—
116C4×Dic6—24no—
117C4×Dic6C2318yes—
118C3×#97C2314yes—
119C3×#97C2314yes—
120C3×C4×C8C2314yes—
121C3×C4×C8C2314yes—
122#821C2318yes—
123#821C2318yes—
124#817C2314yes—
125#817C2314yes—
126#817C2318yes—
127#816C2318yes—
128#816C2318yes—
129C22×Q16—31no—
130C22×SD16—32no—
131C22×D8—31no—
132C22×M16C2×D311yescharacteristic
133C23×C8C2×D311yescharacteristic
134#542—316no—
135#542—68no—
136C2×C4⋊Q8—32no—
137C2×#126—62no—
138C2×#124—32no—
139C2×#124—62no—
140C2×#123—62no—
141C2×C22⋊Q8—32no—
142C2×C22⋊Q8—62no—
143C2×#121—32no—
144C2×C4×Q8—32no—
145C2×C4×D4—32no—
146C2×#117C2×D311yescharacteristic
147#498—38no—
148#495—38no—
149#489—316no—
150#470—38no—
151#469—38no—
152#465—316no—
153#442—38no—
154#441—38no—
155C4×SD16—316no—
156C2×#106—32no—
157C2×#106—31no—
158C2×#106—31no—
159C2×#103—32no—
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163#405C2×D318yescomplemented
164C2×#97C2×D311yescharacteristic
165C2×#97C2×D311yescharacteristic
166C2×C4×C8C2×D311yescharacteristic
167C2×C4×C8C2×D311yescharacteristic
168C24×C3C2×D411yescharacteristic
169C23×D3—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2412yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—28no—
179C2×#67—84no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C2×D12—24no—
184C2×D12—24no—
185C2×D12—22no—
186C2×D12C2414yes—
187C2×C4×D3—28no—
188C2×C4×D3—84no—
189C2×Dic6—24no—
190C2×Dic6—24no—
191C2×Dic6—22no—
192C2×Dic6—28no—
193C2×Dic6—84no—
194C2×Dic6C2414yes—
195C3×M16—28no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—28no—
199C2×C3×C8#4114yes—
200C2×C3×C8#4114yes—
201C2×C3×C8C2412yes—
202C2×C3×C8C2412yes—
203C2×C3×C8C2412yes—
204C2×C3×C8C2412yes—
205C3×C4⋊C4C2×D418yes—
206C3×C22⋊C4C2×D418yes—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209C3×C42C2412yes—
210C3×C42C2412yes—
211#217—28no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—28no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—28no—
219C4⋊Dic3C2414yes—
220C4⋊Dic3C2414yes—
221Dic3⋊C4—28no—
222Dic3⋊C4—28no—
223C4×Dic3—28no—
224Dic12—28no—
225D24—28no—
226#204—216no—
227C22×Q8—32no—
228C22×Q8—62no—
229C22×D4—32no—
230C22×D4—62no—
231C23×C4—62no—
232C23×C4C22×D311yescharacteristic
233C2×Q16—34no—
234C2×Q16—64no—
235C2×SD16—38no—
236C2×SD16—68no—
237C2×D8—34no—
238C2×D8—64no—
239C2×M16—24no—
240C2×M16C22×D314yes—
241C22×C8—24no—
242C22×C8C22×D311yescharacteristic
243C22×C8C22×D314yes—
244C22×C8C22×D311yescharacteristic
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247C4⋊Q8—38no—
248C4⋊Q8—64no—
249#126—616no—
250#124—38no—
251#124—616no—
252#124—64no—
253#123—616no—
254C22⋊Q8—38no—
255C22⋊Q8—64no—
256C22⋊Q8—616no—
257#121—38no—
258#121—64no—
259C4×Q8—38no—
260C4×Q8—64no—
261C4×D4—38no—
262C4×D4—64no—
263#117D1214yes—
264#117C22×D314yes—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4D1212yes—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4D1212yes—
277C2×C42—62no—
278C2×C42C22×D311yescharacteristic
279C2×C42C22×D311yescharacteristic
280#106—38no—
281#106—34no—
282#106—34no—
283#103—38no—
284#103—38no—
285#102—38no—
286#102—38no—
287#97C22×D314yes—
288#97C22×D314yes—
289C4×C8C22×D314yes—
290C4×C8C22×D314yes—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3C22×D412yes—
294C23×C3C22×D411yescharacteristic
295C22×D3—24no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3—44no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4C22×D414yes—
306C2×C3×C4C22×D412yes—
307C2×C3×C4C22×D414yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4112yes—
310C2×C3×C4C2×#4112yes—
311C2×C3×C4C2511yescharacteristic
312#67—88no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—44no—
317C2×Dic3—44no—
318C2×Dic3—44no—
319D12—24no—
320D12—24no—
321D12—44no—
322C4×D3—88no—
323Dic6—24no—
324Dic6—24no—
325Dic6—44no—
326Dic6—88no—
327C3×C8—22no—
328C3×C8—22no—
329C3×C8C2×#4114yes—
330C3×C8C2×#4114yes—
331C24—62no—
332C24C2×D1211yescharacteristic
333C2×Q8—34no—
334C2×Q8—62no—
335C2×Q8—68no—
336C2×Q8—64no—
337C2×Q8—64no—
338C2×Q8—244no—
339C2×D4—34no—
340C2×D4—68no—
341C2×D4—62no—
342C2×D4—64no—
343C2×D4—64no—
344C2×D4—244no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—68no—
351C22×C4—244no—
352C22×C4C2×D1212yes—
353C22×C4C2×D1212yes—
354C22×C4C2×D1212yes—
355C22×C4C2×D1211yescharacteristic
356C22×C4C23×D312yes—
357C22×C4C23×D311yescharacteristic
358Q16—68no—
359SD16—616no—
360D8—68no—
361M16—28no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—28no—
365C2×C8#23514yes—
366C2×C8#23514yes—
367C2×C8C23×D312yes—
368C2×C8C23×D312yes—
369C2×C8C23×D312yes—
370C2×C8C23×D312yes—
371C4⋊C4—34no—
372C4⋊C4—34no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—68no—
376C4⋊C4—68no—
377C4⋊C4—68no—
378C4⋊C4C2×D1218yes—
379C22⋊C4—68no—
380C22⋊C4—68no—
381C22⋊C4—68no—
382C22⋊C4—68no—
383C22⋊C4—68no—
384C22⋊C4C2×D1218yes—
385C42—68no—
386C42C2×D1212yes—
387C42C2×D1212yes—
388C42C23×D312yes—
389C42C23×D312yes—
390C22×C3—22no—
391C22×C3—24no—
392C22×C3—44no—
393C22×C3#53212yes—
394C22×C3#57714yes—
395C22×C3C23×D411yescharacteristic
396C2×D3—44no—
397C2×D3—44no—
398C2×D3—44no—
399C2×D3—88no—
400C3×C4—22no—
401C3×C4—24no—
402C3×C4—24no—
403C3×C4#53212yes—
404C3×C4C22×#4111yescharacteristic
405C3×C4C22×#4111yescharacteristic
406Dic3—44no—
407Dic3—44no—
408Dic3—44no—
409C23—22no—
410C23—24no—
411C23—64no—
412C23—124no—
413C23—124no—
414C23—124no—
415C23C22×D1212yes—
416C23C22×D1211yescharacteristic
417Q8—64no—
418Q8—64no—
419Q8—124no—
420Q8—248no—
421D4—64no—
422D4—64no—
423D4—124no—
424D4—248no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—124no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—248no—
438C2×C4C22×D1214yes—
439C2×C4C22×D1214yes—
440C2×C4C22×D1212yes—
441C2×C4C22×D1211yescharacteristic
442C2×C4C2×#23512yes—
443C2×C4C2×#23512yes—
444C2×C4C24×D311yescharacteristic
445C8—22no—
446C8—22no—
447C8C2×#23514yes—
448C8C2×#23514yes—
449C2×C3—42no—
450C2×C3#317314yescomplemented
451C2×C3C2×#53211yescharacteristic
452C2×C3C2×#57712yes—
453D3—84no—
454C22—24no—
455C22—22no—
456C22—44no—
457C22—124no—
458C22—124no—
459C22—124no—
460C22—248no—
461C22#547712yes—
462C22#575114yes—
463C22C23×D1211yescharacteristic
464C4—24no—
465C4—24no—
466C4—22no—
467C4—124no—
468C4—124no—
469C4—124no—
470C4#547712yes—
471C4C22×#23511yescharacteristic
472C4C22×#23511yescharacteristic
473C3C2×#317311yescharacteristic, complemented, Sylow
474C2—42no—
475C2—244no—
476C2#7856614yesdirect product factor
477C2C2×#547711yescharacteristic
478C2C2×#575112yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#405C2
socleC23×C3C22×D4
prime radicalC2×#5135C2
hypercenterC2×#405D3
hypocenterC3C2×#3173
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×#5534C2semidirect
2. C2×C3×#405C2semidirect
3. C2×#5137C2semidirect
5. C2×#5114C2semidirect
6. #78566C2direct (N × H)
7. C2×#4716C2semidirect
8. C2×#4714C2semidirect
9. C2×#4711C2semidirect
10. C2×C4×#204C2semidirect
11. C2×#4696C2semidirect
12. C2×#4680C2semidirect
23. #5534C22semidirect
32. C3×#405C22semidirect
37. #5137C22semidirect
39. #5114C22semidirect
47. #4716C22semidirect
48. #4714C22semidirect
49. #4711C22semidirect
50. #4696C22semidirect
51. C4×#204C22semidirect
52. #4680C22semidirect
64. C2×#405D3semidirect
163. #405C2×D3semidirect
450. C2×C3#3173semidirect
473. C3C2×#3173semidirect
476. C2#78566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55342
C2×C3×#4052
C2×#51372
C2×#51352
C2×#51142
#785662
C2×#47162
C2×#47142
C2×#47112
C2×C4×#2042
C2×#46962
C2×#46802
C2×#31733