Finite Group Databasebeta

← C2×#81641 #11664481

Subgroups — C2×#81641 #11664481

Order 768.

2182 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#125C211yesmaximal, characteristic, complemented
2C2×#5668C211yesmaximal, characteristic
3C2×C3×#488C211yesmaximal, characteristic, complemented
4C2×#5068C211yesmaximal, characteristic, complemented
5C2×#5056C211yesmaximal, characteristic, complemented
6#81641C2116yesmaximal, direct product factor
7C2×#4803C212yesmaximal, complemented
8C2×#4797C212yesmaximal, complemented
9C2×#4771C212yesmaximal, complemented
10C2×#4764C212yesmaximal, complemented
11C2×#4697C211yesmaximal, characteristic, complemented
12C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
13C2×#3142—31nomaximal, Sylow
14C22×#239C2211yescharacteristic
15C22×#237C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×#944C2211yescharacteristic
19C2×#940C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2212yes—
21D3×#125—24no—
22D3×#125C2218yescomplemented
23#5668—24no—
24#5668C2218yes—
25C2×D3×C4⋊C4—22no—
26C2×D3×C4⋊C4C2212yes—
27C2×#890C2212yes—
28C2×#889—22no—
29C2×#884—22no—
30C2×#67⋊C4—22no—
31C2×#875C2211yescharacteristic
32C2×C42×D3C2211yescharacteristic
33C2×#870C2211yescharacteristic
34C3×#488C2218yescomplemented
35C2×C3×#103C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C2×#835C2212yes—
39C2×#832C2212yes—
40C2×#821C2212yes—
41C2×#816C2212yes—
42C2×#815C2212yes—
43#5068C2218yescomplemented
44#5056C2218yescomplemented
45C2×#810C2212yes—
46C2×#809C2212yes—
47C2×C4×C3⋊C8C2211yescharacteristic
48#4803C22116yescomplemented
49#4797C22116yescomplemented
50#4771C22116yescomplemented
51#4764C22116yescomplemented
52#4697C2218yescomplemented
53C4×ΓL(1,𝔽25)C2218yescomplemented
54C2×#532—31no—
55C22×#125—31no—
56#3142—316no—
57C2×#491—32no—
58C2×#490—32no—
59C2×#488—31no—
60C2×#488—31no—
61C2×#488—31no—
62C2×#488D311yescharacteristic, complemented
63C2×#417—32no—
64C2×#417—32no—
65C2×C4×M16—31no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C22×#67—21no—
69C22×#67—22no—
70C23×Dic3—21no—
71C2×#239—22no—
72C2×#239—28no—
73C2×#239C2314yes—
74C2×#237—22no—
75C2×#237—28no—
76C2×#237C2314yes—
77C22×D12—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—22no—
80C22×C4×D3—21no—
81C22×C4×D3D412yes—
82C22×C4×D3C2311yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2312yes—
85C3×#125—22no—
86C3×#125C2314yes—
87C3×#124—22no—
88C3×#124C2314yes—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—22no—
92C2×C3×C42C2311yescharacteristic
93#944—22no—
94#944C2314yes—
95#940—22no—
96#940C2314yes—
97C2×(C2×D3)⋊C4—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3C2312yes—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×C4×Dic3D412yes—
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107C2×ΓL(1,𝔽25)—28no—
108C2×ΓL(1,𝔽25)C2318yes—
109D3×C4⋊C4—24no—
110D3×C4⋊C4—216no—
111D3×C4⋊C4C2318yes—
112#890—24no—
113#890C2318yes—
114#889—216no—
115#884—216no—
116#67⋊C4—216no—
117#875—22no—
118#875C2314yes—
119C42×D3D414yes—
120C42×D3C2314yes—
121#870—22no—
122#870C2314yes—
123C3×#103C2318yes—
124C3×#102C2318yes—
125C3×C4×C8C2314yes—
126#835C2318yes—
127#832C2318yes—
128#821C2318yes—
129#816C2318yes—
130#815C2318yes—
131#810C2318yes—
132#809C2318yes—
133C4×C3⋊C8C2314yes—
134C22×#41—31no—
135C22×#41—31no—
136C22×M16—32no—
137#532—38no—
138#532—64no—
139C2×C4⋊Q8—31no—
140C2×#127—31no—
141C2×#125—38no—
142C2×#125—32no—
143C2×#125—31no—
144C2×#125—62no—
145C2×#125—64no—
146C2×#125C2×D311yescharacteristic
147C2×#124—31no—
148C2×#124C2×D311yescharacteristic
149C2×#121—62no—
150C2×C4×D4—62no—
151C22×C4⋊C4—32no—
152C22×C4⋊C4—62no—
153C22×C42—31no—
154#491—316no—
155#490—316no—
156#488—38no—
157#488—38no—
158#488—38no—
159#488C2×D318yescomplemented
160#417—316no—
161#417—316no—
162C2×#103—32no—
163C2×#103—32no—
164C2×#103—32no—
165C2×#103C2×D312yes—
166C2×#102—32no—
167C2×#102—32no—
168C2×#102—32no—
169C2×#102C2×D312yes—
170C4×M16—38no—
171C2×#97—32no—
172C2×C4×C8—31no—
173C2×C4×C8C2×D311yescharacteristic
174C24×C3—21no—
175C23×D3—21no—
176C23×D3C2×D411yescharacteristic
177C2×C3×Q8—21no—
178C2×C3×Q8—24no—
179C2×C3×Q8C2412yes—
180C2×C3×D4—21no—
181C2×C3×D4—24no—
182C2×C3×D4C2412yes—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C2×#67—24no—
188C2×#67—28no—
189C2×#67—48no—
190C2×#67—44no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—24no—
194C22×Dic3—44no—
195C22×Dic3C2×D412yes—
196C22×Dic3C2×D411yescharacteristic
197#239—216no—
198#239—48no—
199#237—216no—
200#237—48no—
201C2×D12—24no—
202C2×D12—24no—
203C2×D12—21no—
204C2×D12—44no—
205C2×D12C2412yes—
206C2×C4×D3—28no—
207C2×C4×D3—24no—
208C2×C4×D3—44no—
209C2×C4×D3—48no—
210C2×C4×D3—44no—
211C2×C4×D3C2×D418yes—
212C2×C4×D3C2×D412yes—
213C2×C4×D3#4118yes—
214C2×C4×D3C2412yes—
215C2×Dic6—24no—
216C2×Dic6—21no—
217C2×Dic6C2412yes—
218C2×C3×C8—24no—
219C2×C3×C8C2414yes—
220C3×C4⋊C4—28no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4C2414yes—
223C3×C22⋊C4—28no—
224C3×C42C2×D412yes—
225C3×C42C2412yes—
226(C2×D3)⋊C4—28no—
227C4⋊Dic3—22no—
228C4⋊Dic3—28no—
229C4⋊Dic3C2414yes—
230Dic3⋊C4—28no—
231Dic3⋊C4—28no—
232Dic3⋊C4—28no—
233Dic3⋊C4—28no—
234C4×Dic3C2×D412yes—
235C4×Dic3C2×D418yes—
236C4×Dic3C2412yes—
237C2×C3⋊C8—24no—
238C2×C3⋊C8C2414yes—
239ΓL(1,𝔽25)—216no—
240C2×#41—34no—
241C2×#41—34no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—68no—
245C2×#41—68no—
246C22×Q8—31no—
247C22×Q8C22×D311yescharacteristic
248C22×D4—31no—
249C22×D4—61no—
250C22×D4—62no—
251C22×D4—61no—
252C22×D4C22×D311yescharacteristic
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—61no—
256C23×C4—62no—
257C23×C4—61no—
258C2×M16—38no—
259C2×M16—68no—
260C22×C8—32no—
261C22×C8C22×D312yes—
262C4⋊Q8—34no—
263C4⋊Q8—62no—
264#127—34no—
265#127—62no—
266#125—22no—
267#125—38no—
268#125—34no—
269#125—616no—
270#125—64no—
271#125—62no—
272#125C22×D314yes—
273#124—22no—
274#124—34no—
275#124—62no—
276#124C22×D314yes—
277#121—616no—
278C4×D4—616no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4—38no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—616no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4C22×D312yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—62no—
292C2×C42—32no—
293C2×C42—31no—
294C2×C42—34no—
295C2×C42—34no—
296C2×C42C22×D311yescharacteristic
297#103—38no—
298#103—38no—
299#103—38no—
300#103C22×D318yes—
301#102—38no—
302#102—38no—
303#102—38no—
304#102C22×D318yes—
305#97—38no—
306C4×C8—34no—
307C4×C8C22×D314yes—
308C23×C3—22no—
309C23×C3—42no—
310C23×C3—44no—
311C23×C3C22×D411yescharacteristic
312C22×D3—24no—
313C22×D3—22no—
314C22×D3—42no—
315C22×D3—44no—
316C22×D3#12414yes—
317C22×D3C22×D412yes—
318C3×Q8—24no—
319C3×Q8—42no—
320C3×D4—24no—
321C3×D4—42no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4C22×D414yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4114yes—
330C2×C3×C4C2511yescharacteristic
331#67—416no—
332#67—48no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3—44no—
338C2×Dic3—48no—
339C2×Dic3#12414yes—
340C2×Dic3C22×D414yes—
341C2×Dic3C22×D412yes—
342D12—24no—
343D12—48no—
344D12—42no—
345C4×D3—48no—
346C4×D3—416no—
347C4×D3—48no—
348C4×D3#12418yes—
349C4×D3C2×#4118yes—
350Dic6—24no—
351Dic6—42no—
352C3×C8—24no—
353C3⋊C8—24no—
354C24—21no—
355C24—31no—
356C24—61no—
357#41—616no—
358#41—616no—
359#41—128no—
360#41—128no—
361C2×Q8—21no—
362C2×Q8—24no—
363C2×Q8—32no—
364C2×Q8—61no—
365C2×Q8—64no—
366C2×Q8C23×D312yes—
367C2×D4—21no—
368C2×D4—24no—
369C2×D4—32no—
370C2×D4—64no—
371C2×D4—68no—
372C2×D4—64no—
373C2×D4—61no—
374C2×D4—64no—
375C2×D4—124no—
376C2×D4—128no—
377C2×D4—124no—
378C2×D4C23×D312yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—31no—
382C22×C4—32no—
383C22×C4—38no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—38no—
387C22×C4—64no—
388C22×C4—68no—
389C22×C4—64no—
390C22×C4—61no—
391C22×C4—62no—
392C22×C4—124no—
393C22×C4—124no—
394C22×C4—124no—
395C22×C4—128no—
396C22×C4D3×D412yes—
397C22×C4C23×D311yescharacteristic
398M16—616no—
399C2×C8—24no—
400C2×C8—34no—
401C2×C8—64no—
402C2×C8C23×D314yes—
403C4⋊C4—28no—
404C4⋊C4—22no—
405C4⋊C4—34no—
406C4⋊C4—68no—
407C4⋊C4—68no—
408C4⋊C4—68no—
409C4⋊C4—68no—
410C4⋊C4—62no—
411C4⋊C4—68no—
412C4⋊C4C23×D314yes—
413C22⋊C4—28no—
414C22⋊C4—68no—
415C42—38no—
416C42—32no—
417C42—32no—
418C42D3×D412yes—
419C42C23×D312yes—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—84no—
423C22×C3C2×#12412yes—
424C22×C3#57514yes—
425C22×C3C23×D411yescharacteristic
426C2×D3—28no—
427C2×D3—24no—
428C2×D3—42no—
429C2×D3—44no—
430C2×D3—84no—
431C2×D3C2×#12414yes—
432C3×C4—24no—
433C3×C4—42no—
434C3×C4—44no—
435C3×C4C2×#12412yes—
436C3×C4C22×#4112yes—
437Dic3—24no—
438Dic3—42no—
439Dic3—44no—
440Dic3C2×#12414yes—
441C23—22no—
442C23—34no—
443C23—32no—
444C23—42no—
445C23—44no—
446C23—64no—
447C23—62no—
448C23—122no—
449C23—124no—
450C23C2×D3×D411yescharacteristic
451Q8—24no—
452Q8—42no—
453Q8—64no—
454Q8—122no—
455D4—24no—
456D4—42no—
457D4—64no—
458D4—128no—
459D4—1216no—
460D4—128no—
461D4—122no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—34no—
466C2×C4—32no—
467C2×C4—38no—
468C2×C4—38no—
469C2×C4—34no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—64no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—128no—
476C2×C4—128no—
477C2×C4—128no—
478C2×C4—1216no—
479C2×C4—122no—
480C2×C4—124no—
481C2×C4C2×D3×D414yes—
482C2×C4C2×D3×D411yescharacteristic
483C2×C4D3×#4114yes—
484C2×C4C24×D311yescharacteristic
485C8—24no—
486C8—64no—
487C2×C3—82no—
488C2×C3#314214yescomplemented
489C2×C3C22×#12411yescharacteristic
490C2×C3C2×#57512yes—
491D3—24no—
492D3—82no—
493C22—34no—
494C22—42no—
495C22—44no—
496C22—64no—
497C22—68no—
498C22—84no—
499C22—122no—
500C22—124no—
501C22—244no—
502C22D3×#12412yes—
503C22#576814yes—
504C22C22×D3×D411yescharacteristic
505C4—24no—
506C4—34no—
507C4—42no—
508C4—44no—
509C4—64no—
510C4—122no—
511C4—124no—
512C4D3×#12412yes—
513C4C2×D3×#4112yes—
514C3C2×#314211yescharacteristic, complemented, Sylow
515C2—64no—
516C2—82no—
517C2—242no—
518C2#8164114yesdirect product factor
519C2C2×D3×#12411yescharacteristic
520C2C2×#576812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#488C2
socleC23×C3C22×D4
prime radicalC2×#5668C2
hypercenterC2×#488D3
hypocenterC3C2×#3142
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. C2×D3×#125C2semidirect
3. C2×C3×#488C2semidirect
4. C2×#5068C2semidirect
5. C2×#5056C2semidirect
6. #81641C2direct (N × H)
7. C2×#4803C2semidirect
8. C2×#4797C2semidirect
9. C2×#4771C2semidirect
10. C2×#4764C2semidirect
11. C2×#4697C2semidirect
12. C2×C4×ΓL(1,𝔽25)C2semidirect
22. D3×#125C22semidirect
34. C3×#488C22semidirect
43. #5068C22semidirect
44. #5056C22semidirect
48. #4803C22semidirect
49. #4797C22semidirect
50. #4771C22semidirect
51. #4764C22semidirect
52. #4697C22semidirect
53. C4×ΓL(1,𝔽25)C22semidirect
62. C2×#488D3semidirect
159. #488C2×D3semidirect
488. C2×C3#3142semidirect
514. C3C2×#3142semidirect
518. C2#81641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×#1252
C2×#56682
C2×C3×#4882
C2×#50682
C2×#50562
#816412
C2×#48032
C2×#47972
C2×#47712
C2×#47642
C2×#46972
C2×C4×ΓL(1,𝔽25)2
C2×#31423