Finite Group Databasebeta

← C2×#81887 #11664987

Subgroups — C2×#81887 #11664987

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×#5733C212yesmaximal, complemented
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×#5089C212yesmaximal, complemented
4#81887C2116yesmaximal, direct product factor
5C2×#4867C211yesmaximal, characteristic, complemented
6C2×#4865C211yesmaximal, characteristic
7C2×#4864C212yesmaximal, complemented
8C2×#4869C211yesmaximal, characteristic, complemented
9C2×#4853C211yesmaximal, characteristic, complemented
10C2×#4852C212yesmaximal, complemented
11C2×#4696C211yesmaximal, characteristic, complemented
12C2×#4689C211yesmaximal, characteristic, complemented
13C2×#3173—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2212yes—
15C2×#947—22no—
16C2×#946—22no—
17C22×Dic12C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C22×#204C2212yes—
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22#5733—28no—
23#5733C22116yescomplemented
24C2×#898—22no—
25C2×#897C2212yes—
26C2×#895C2212yes—
27C2×#894—22no—
28C2×#893C2212yes—
29C2×#887C2212yes—
30C2×#875C2212yes—
31C2×#872C2211yescharacteristic
32C3×#498C2218yescomplemented
33C2×C3×#106C2211yescharacteristic
34C2×C3×#106C2211yescharacteristic
35C2×C3×#106C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#5089C22116yescomplemented
40C2×#810C2212yes—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C2×#809C2212yes—
44C2×#803C2211yescharacteristic
45#4869C2218yescomplemented
46#4867C2218yescomplemented
47#4865C2218yes—
48#4864C22116yescomplemented
49#4853C2218yescomplemented
50#4852C22116yescomplemented
51#4696C2218yescomplemented
52#4689C2218yescomplemented
53C2×#542—32no—
54#3173—316no—
55C2×#498D311yescharacteristic, complemented
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×#405—31no—
65C22×C3×Q8—22no—
66C22×D12—22no—
67C22×D12C2312yes—
68C22×C4×D3—22no—
69C22×C4×D3C2311yescharacteristic
70C22×Dic6C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×C4⋊Q8—24no—
74C3×C4⋊Q8C2318yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C42C2311yescharacteristic
79#947—216no—
80#946—216no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4D412yes—
87C2×C4⋊Dic3—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3—22no—
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C2×Dic12—24no—
95C2×Dic12C2314yes—
96C2×D24—24no—
97C2×D24C2314yes—
98C2×#204—28no—
99C2×#204C2318yes—
100C2×ΓL(1,𝔽25)—24no—
101C2×ΓL(1,𝔽25)C2314yes—
102C2×D3×C8—24no—
103C2×D3×C8C2314yes—
104#898—216no—
105#897—24no—
106#897C2318yes—
107#895—24no—
108#895C2318yes—
109#894—216no—
110#893—24no—
111#893C2318yes—
112#887—24no—
113#887C2318yes—
114#875—24no—
115#875C2318yes—
116#872D414yes—
117#872C2314yes—
118C3×#106C2318yes—
119C3×#106C2314yes—
120C3×#106C2314yes—
121C3×C4×C8C2314yes—
122#815C2314yes—
123C8×Dic3C2314yes—
124#810C2318yes—
125#810C2318yes—
126#809C2318yes—
127#809C2318yes—
128#803C2314yes—
129C22×Q16—31no—
130C22×SD16—32no—
131C22×D8—31no—
132C22×M16—31no—
133C23×C8—31no—
134#542—316no—
135#542—68no—
136C2×C4⋊Q8C2×D312yes—
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×#117—31no—
147#498C2×D318yescomplemented
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×#106C2×D312yes—
157C2×#106C2×D311yescharacteristic
158C2×#106C2×D311yescharacteristic
159C2×#103—32no—
160C2×#103—32no—
161C2×#102—32no—
162C2×#102—32no—
163#405—38no—
164C2×#97—31no—
165C2×#97—31no—
166C2×C4×C8—31no—
167C2×C4×C8C2×D311yescharacteristic
168C23×D3—22no—
169C23×D3C2×D411yescharacteristic
170C2×C3×Q8—28no—
171C2×C3×Q8—84no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—22no—
177C22×Dic3C2×D412yes—
178C22×Dic3C2×D411yescharacteristic
179C2×D12—24no—
180C2×D12—24no—
181C2×D12—22no—
182C2×D12—28no—
183C2×D12—84no—
184C2×D12C2414yes—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—28no—
188C2×C4×D3—84no—
189C2×C4×D3C2×D412yes—
190C2×C4×D3C2412yes—
191C2×Dic6—24no—
192C2×Dic6—24no—
193C2×Dic6—22no—
194C2×Dic6C2414yes—
195C2×C3×C8#4114yes—
196C2×C3×C8#4114yes—
197C2×C3×C8C2412yes—
198C2×C3×C8C2412yes—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4C2414yes—
203C3×C4⋊C4C2414yes—
204C3×C42C2×D412yes—
205C3×C42C2412yes—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4C2×D418yes—
212C4⋊Dic3—28no—
213Dic3⋊C4—28no—
214Dic3⋊C4C2×D418yes—
215C4×Dic3—28no—
216C4×Dic3C2×D412yes—
217C4×Dic3C2412yes—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8C2412yes—
221C2×C3⋊C8C2412yes—
222Dic12—28no—
223D24—28no—
224#204—216no—
225ΓL(1,𝔽25)—28no—
226D3×C8—28no—
227C22×Q8—22no—
228C22×Q8—32no—
229C22×D4—32no—
230C22×D4—62no—
231C23×C4—31no—
232C23×C4—62no—
233C2×Q16—34no—
234C2×Q16—64no—
235C2×SD16—38no—
236C2×SD16—68no—
237C2×D8—34no—
238C2×D8—64no—
239C2×M16—34no—
240C2×M16—64no—
241C22×C8—31no—
242C22×C8—31no—
243C22×C8—34no—
244C22×C8—64no—
245C22×C8C22×D311yescharacteristic
246C22×C8C22×D311yescharacteristic
247C4⋊Q8—24no—
248C4⋊Q8C22×D318yes—
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#117—34no—
264#117—34no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—32no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—31no—
278C2×C42—62no—
279C2×C42C22×D311yescharacteristic
280#106C22×D318yes—
281#106C22×D314yes—
282#106C22×D314yes—
283#103—38no—
284#103—38no—
285#102—38no—
286#102—38no—
287#97—34no—
288#97—34no—
289C4×C8—34no—
290C4×C8C22×D314yes—
291C23×C3C22×D411yescharacteristic
292C22×D3—22no—
293C22×D3—24no—
294C22×D3—24no—
295C22×D3—44no—
296C22×D3—44no—
297C22×D3—44no—
298C22×D3C22×D412yes—
299C3×Q8—88no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4C22×D414yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4112yes—
308C2×C3×C4C2×#4112yes—
309C2×C3×C4C2511yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—44no—
314C2×Dic3C22×D414yes—
315C2×Dic3C22×D412yes—
316D12—24no—
317D12—24no—
318D12—44no—
319D12—88no—
320C4×D3—24no—
321C4×D3—22no—
322C4×D3—22no—
323C4×D3—88no—
324Dic6—24no—
325Dic6—24no—
326Dic6—44no—
327C3×C8C2×#4114yes—
328C3×C8C2×#4114yes—
329C3⋊C8—22no—
330C3⋊C8—22no—
331C24—31no—
332C24—62no—
333C2×Q8—28no—
334C2×Q8—34no—
335C2×Q8—62no—
336C2×Q8—64no—
337C2×Q8—64no—
338C2×Q8—84no—
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—32no—
348C22×C4—31no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—68no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—244no—
356C22×C4D3×D412yes—
357C22×C4C23×D311yescharacteristic
358Q16—68no—
359SD16—616no—
360D8—68no—
361M16—68no—
362C2×C8—32no—
363C2×C8—32no—
364C2×C8—62no—
365C2×C8—62no—
366C2×C8—68no—
367C2×C8#23914yes—
368C2×C8#23914yes—
369C2×C8C23×D312yes—
370C2×C8C23×D312yes—
371C4⋊C4—28no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—38no—
375C4⋊C4—68no—
376C4⋊C4—68no—
377C4⋊C4C23×D314yes—
378C4⋊C4C23×D314yes—
379C22⋊C4—38no—
380C22⋊C4—68no—
381C22⋊C4—68no—
382C22⋊C4—68no—
383C22⋊C4—68no—
384C22⋊C4—68no—
385C42—32no—
386C42—32no—
387C42—68no—
388C42D3×D412yes—
389C42C23×D312yes—
390C22×C3#53212yes—
391C22×C3#57714yes—
392C22×C3C23×D411yescharacteristic
393C2×D3—22no—
394C2×D3—24no—
395C2×D3—44no—
396C2×D3—44no—
397C2×D3—44no—
398C2×D3—44no—
399C2×D3—88no—
400C3×C4—24no—
401C3×C4—44no—
402C3×C4—44no—
403C3×C4#53212yes—
404C3×C4C22×#4111yescharacteristic
405C3×C4C22×#4111yescharacteristic
406Dic3—22no—
407Dic3—24no—
408Dic3—44no—
409C23—32no—
410C23—62no—
411C23—64no—
412C23—64no—
413C23—124no—
414C23—124no—
415C23—124no—
416C23C2×D3×D411yescharacteristic
417Q8—64no—
418Q8—64no—
419Q8—88no—
420Q8—124no—
421D4—64no—
422D4—64no—
423D4—124no—
424D4—248no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—34no—
429C2×C4—32no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—62no—
433C2×C4—64no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—124no—
439C2×C4—248no—
440C2×C4C2×D3×D414yes—
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×#23912yes—
443C2×C4C2×#23912yes—
444C2×C4C24×D311yescharacteristic
445C8—62no—
446C8—62no—
447C8C2×#23914yes—
448C8C2×#23914yes—
449C2×C3#317314yescomplemented
450C2×C3C2×#53211yescharacteristic
451C2×C3C2×#57712yes—
452D3—42no—
453D3—84no—
454C22—64no—
455C22—62no—
456C22—124no—
457C22—124no—
458C22—124no—
459C22—124no—
460C22—248no—
461C22#572512yes—
462C22#576214yes—
463C22C22×D3×D411yescharacteristic
464C4—24no—
465C4—44no—
466C4—44no—
467C4—64no—
468C4—62no—
469C4—124no—
470C4#572512yes—
471C4C22×#23911yescharacteristic
472C4C22×#23911yescharacteristic
473C3C2×#317311yescharacteristic, complemented, Sylow
474C2—122no—
475C2—244no—
476C2#8188714yesdirect product factor
477C2C2×#572511yescharacteristic
478C2C2×#576212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#498C2
socleC23×C3C22×D4
prime radicalC2×#4865C2
hypercenterC2×#498D3
hypocenterC3C2×#3173
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
1. C2×#5733C2semidirect
2. C2×C3×#498C2semidirect
3. C2×#5089C2semidirect
4. #81887C2direct (N × H)
5. C2×#4867C2semidirect
7. C2×#4864C2semidirect
8. C2×#4869C2semidirect
9. C2×#4853C2semidirect
10. C2×#4852C2semidirect
11. C2×#4696C2semidirect
12. C2×#4689C2semidirect
23. #5733C22semidirect
32. C3×#498C22semidirect
39. #5089C22semidirect
45. #4869C22semidirect
46. #4867C22semidirect
48. #4864C22semidirect
49. #4853C22semidirect
50. #4852C22semidirect
51. #4696C22semidirect
52. #4689C22semidirect
55. C2×#498D3semidirect
147. #498C2×D3semidirect
449. C2×C3#3173semidirect
473. C3C2×#3173semidirect
476. C2#81887direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57332
C2×C3×#4982
C2×#50892
#818872
C2×#48672
C2×#48652
C2×#48642
C2×#48692
C2×#48532
C2×#48522
C2×#46962
C2×#46892
C2×#31733