Finite Group Databasebeta

← C2×#81640 #11664465

Subgroups — C2×#81640 #11664465

Order 768.

2246 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5688C211yesmaximal, characteristic, complemented
2C2×D3×#124C211yesmaximal, characteristic
3C2×C3×#488C211yesmaximal, characteristic, complemented
4C2×#5068C211yesmaximal, characteristic, complemented
5C2×#5056C211yesmaximal, characteristic, complemented
6#81640C2116yesmaximal, direct product factor
7C2×#4802C212yesmaximal, complemented
8C2×#4794C212yesmaximal, complemented
9C2×#4770C212yesmaximal
10C2×#4765C212yesmaximal, complemented
11C2×#4697C211yesmaximal, characteristic, complemented
12C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
13C2×#3140—31nomaximal, Sylow
14C22×D3×Q8C2211yescharacteristic
15C22×D3×D4C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic, complemented
17C2×C3×#124C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2212yes—
21#5688—24no—
22#5688C2218yescomplemented
23D3×#124—24no—
24D3×#124C2218yes—
25C2×#896—22no—
26C2×#892C2212yes—
27C2×#888C2212yescomplemented
28C2×#887—22no—
29C2×#885—22no—
30C2×D3×C22⋊C4—22no—
31C2×#875C2211yescharacteristic
32C2×C42×D3C2211yescharacteristic
33C2×#870C2211yescharacteristic, complemented
34C3×#488C2218yescomplemented
35C2×C3×#103C2212yescomplemented
36C2×C3×#102C2212yes—
37C2×C3×C4×C8C2211yescharacteristic, complemented
38C2×#835C2212yescomplemented
39C2×#832C2212yes—
40C2×#821C2212yes—
41C2×#816C2212yescomplemented
42C2×#815C2212yescomplemented
43#5068C2218yescomplemented
44#5056C2218yescomplemented
45C2×#810C2212yescomplemented
46C2×#809C2212yes—
47C2×C4×C3⋊C8C2211yescharacteristic, complemented
48#4802C22116yescomplemented
49#4794C22116yescomplemented
50#4770C22116yes—
51#4765C22116yescomplemented
52#4697C2218yescomplemented
53C4×ΓL(1,𝔽25)C2218yescomplemented
54C2×#533—31no—
55C22×#124—31no—
56#3140—316no—
57C2×#489—32no—
58C2×#489—32no—
59C2×#488—31no—
60C2×#488—31no—
61C2×#488—31no—
62C2×#488D311yescharacteristic, complemented
63C2×#419—32no—
64C2×#418—32no—
65C2×C4×M16—31no—
66C24×D3—21no—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×D4C2311yescharacteristic
69C22×#67—21no—
70C2×D3×Q8—22no—
71C2×D3×Q8—28no—
72C2×D3×Q8C2314yes—
73C2×D3×D4—22no—
74C2×D3×D4—28no—
75C2×D3×D4C2314yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3D412yes—
79C22×C4×D3C2311yescharacteristic
80C22×Dic6—21no—
81C22×Dic6—22no—
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2312yes—
84C3×#125—22no—
85C3×#125C2314yescomplemented
86C3×#124—22no—
87C3×#124C2314yes—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—22no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—22no—
93#947—22no—
94#947C2314yes—
95#936—22no—
96#936C2314yes—
97C2×(C2×D3)⋊C4—22no—
98C2×(C2×D3)⋊C4—22no—
99C2×(C2×D3)⋊C4—22no—
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3D412yes—
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2312yes—
107C2×ΓL(1,𝔽25)—28no—
108C2×ΓL(1,𝔽25)C2318yes—
109#896—216no—
110#892—24no—
111#892C2318yes—
112#888—24no—
113#888C2318yescomplemented
114#887—216no—
115#885—216no—
116D3×C22⋊C4—216no—
117#875—22no—
118#875C2314yes—
119C42×D3D414yes—
120C42×D3C2314yes—
121#870—22no—
122#870C2314yescomplemented
123C3×#103C2318yescomplemented
124C3×#102C2318yes—
125C3×C4×C8C2314yescomplemented
126#835C2318yescomplemented
127#832C2318yes—
128#821C2318yes—
129#816C2318yescomplemented
130#815C2318yescomplemented
131#810C2318yescomplemented
132#809C2318yes—
133C4×C3⋊C8C2314yescomplemented
134C23×Q8—31no—
135C23×D4—31no—
136C22×M16—32no—
137#533—38no—
138#533—64no—
139C2×C4⋊Q8—32no—
140C2×#125—31no—
141C2×#125C2×D311yescharacteristic, complemented
142C2×#124—31no—
143C2×#124—31no—
144C2×#124—38no—
145C2×#124—31no—
146C2×#124—62no—
147C2×#124—64no—
148C2×#124C2×D311yescharacteristic
149C2×C22⋊Q8—62no—
150C2×C4×Q8—62no—
151C2×#117—32no—
152C22×C22⋊C4—62no—
153C22×C42—31no—
154#489—316no—
155#489—316no—
156#488—38no—
157#488—38no—
158#488—38no—
159#488C2×D318yescomplemented
160#419—316no—
161#418—316no—
162C2×#103—32no—
163C2×#103—32no—
164C2×#103—32no—
165C2×#103C2×D312yescomplemented
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, complemented
174C24×C3—21no—
175C23×D3—21no—
176C23×D3—24no—
177C23×D3—48no—
178C23×D3—44no—
179C23×D3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8C2412yes—
183C2×C3×D4—21no—
184C2×C3×D4—24no—
185C2×C3×D4C2412yes—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—24no—
191C2×#67—44no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D411yescharacteristic
196D3×Q8—216no—
197D3×Q8—48no—
198D3×D4—216no—
199D3×D4—48no—
200C2×D12—24no—
201C2×D12—21no—
202C2×D12C2412yes—
203C2×C4×D3—24no—
204C2×C4×D3—44no—
205C2×C4×D3—44no—
206C2×C4×D3C2×D418yes—
207C2×C4×D3C2×D412yes—
208C2×C4×D3#4118yes—
209C2×C4×D3C2412yes—
210C2×Dic6—24no—
211C2×Dic6—21no—
212C2×Dic6—24no—
213C2×Dic6—28no—
214C2×Dic6—48no—
215C2×Dic6—44no—
216C2×Dic6C2412yes—
217C2×C3×C8—24no—
218C2×C3×C8C2414yes—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4C2414yes—
222C3×C22⋊C4—28no—
223C3×C42C2×D412yes—
224C3×C42C2412yes—
225#217—28no—
226(C2×D3)⋊C4—28no—
227(C2×D3)⋊C4—28no—
228(C2×D3)⋊C4—28no—
229C4⋊Dic3—22no—
230C4⋊Dic3C2414yes—
231Dic3⋊C4—28no—
232Dic3⋊C4—28no—
233C4×Dic3—28no—
234C4×Dic3C2×D412yes—
235C4×Dic3C2×D418yes—
236C4×Dic3C2412yes—
237C2×C3⋊C8—24no—
238C2×C3⋊C8C2414yes—
239ΓL(1,𝔽25)—216no—
240C25—61no—
241C22×Q8—34no—
242C22×Q8—31no—
243C22×Q8—62no—
244C22×Q8—61no—
245C22×Q8—62no—
246C22×Q8—68no—
247C22×Q8C22×D311yescharacteristic
248C22×D4—34no—
249C22×D4—31no—
250C22×D4—68no—
251C22×D4—62no—
252C22×D4—61no—
253C22×D4C22×D311yescharacteristic
254C23×C4—31no—
255C23×C4—32no—
256C23×C4—61no—
257C2×M16—38no—
258C2×M16—68no—
259C22×C8—32no—
260C22×C8C22×D312yes—
261C4⋊Q8—38no—
262C4⋊Q8—64no—
263#125—22no—
264#125—34no—
265#125—62no—
266#125C22×D314yescomplemented
267#124—22no—
268#124—34no—
269#124—34no—
270#124—34no—
271#124—616no—
272#124—62no—
273#124—62no—
274#124—62no—
275#124C22×D314yes—
276C22⋊Q8—616no—
277C4×Q8—616no—
278#117—38no—
279#117—64no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4C22×D312yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—616no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4—62no—
290C2×C22⋊C4—62no—
291C2×C42—32no—
292C2×C42—31no—
293C2×C42—34no—
294C2×C42—34no—
295C2×C42—62no—
296C2×C42C22×D311yescharacteristic
297#103—38no—
298#103—38no—
299#103—38no—
300#103C22×D318yescomplemented
301#102—38no—
302#102—38no—
303#102—38no—
304#102C22×D318yes—
305#97—38no—
306C4×C8—34no—
307C4×C8C22×D314yescomplemented
308C23×C3—22no—
309C23×C3—42no—
310C23×C3—44no—
311C23×C3C22×D411yescharacteristic
312C22×D3—24no—
313C22×D3—22no—
314C22×D3—416no—
315C22×D3—42no—
316C22×D3—44no—
317C22×D3—48no—
318C22×D3—816no—
319C22×D3#12414yes—
320C22×D3C22×D412yes—
321C3×Q8—24no—
322C3×Q8—42no—
323C3×D4—24no—
324C3×D4—42no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—42no—
329C2×C3×C4—44no—
330C2×C3×C4C22×D414yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4114yes—
333C2×C3×C4C2511yescharacteristic
334#67—48no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—44no—
340C2×Dic3#12414yes—
341C2×Dic3C22×D414yes—
342C2×Dic3C22×D412yes—
343D12—24no—
344D12—42no—
345C4×D3—48no—
346C4×D3—48no—
347C4×D3#12418yes—
348C4×D3C2×#4118yes—
349Dic6—24no—
350Dic6—416no—
351Dic6—42no—
352Dic6—48no—
353C3×C8—24no—
354C3⋊C8—24no—
355C24—21no—
356C24—31no—
357C24—64no—
358C24—61no—
359C24—124no—
360C24—128no—
361C2×Q8—21no—
362C2×Q8—24no—
363C2×Q8—32no—
364C2×Q8—68no—
365C2×Q8—64no—
366C2×Q8—616no—
367C2×Q8—61no—
368C2×Q8—64no—
369C2×Q8—124no—
370C2×Q8—128no—
371C2×Q8—128no—
372C2×Q8C23×D312yes—
373C2×D4—21no—
374C2×D4—24no—
375C2×D4—32no—
376C2×D4—616no—
377C2×D4—64no—
378C2×D4—61no—
379C2×D4—64no—
380C2×D4—128no—
381C2×D4—124no—
382C2×D4C23×D312yes—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—31no—
386C22×C4—32no—
387C22×C4—38no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—38no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—61no—
394C22×C4—124no—
395C22×C4—124no—
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—62no—
408C4⋊C4—68no—
409C4⋊C4C23×D314yes—
410C22⋊C4—28no—
411C22⋊C4—68no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C22⋊C4—68no—
415C42—38no—
416C42—32no—
417C42—32no—
418C42—68no—
419C42D3×D412yes—
420C42C23×D312yes—
421C22×C3—42no—
422C22×C3—44no—
423C22×C3—84no—
424C22×C3C2×#12412yes—
425C22×C3#57514yes—
426C22×C3C23×D411yescharacteristic
427C2×D3—28no—
428C2×D3—24no—
429C2×D3—42no—
430C2×D3—44no—
431C2×D3—816no—
432C2×D3—84no—
433C2×D3C2×#12414yes—
434C3×C4—24no—
435C3×C4—42no—
436C3×C4—44no—
437C3×C4C2×#12412yes—
438C3×C4C22×#4112yes—
439Dic3—24no—
440Dic3—42no—
441Dic3—44no—
442Dic3C2×#12414yes—
443C23—22no—
444C23—34no—
445C23—32no—
446C23—42no—
447C23—44no—
448C23—64no—
449C23—62no—
450C23—128no—
451C23—1216no—
452C23—122no—
453C23—124no—
454C23—2416no—
455C23C2×D3×D411yescharacteristic
456Q8—24no—
457Q8—42no—
458Q8—64no—
459Q8—128no—
460Q8—1216no—
461Q8—122no—
462D4—24no—
463D4—42no—
464D4—64no—
465D4—128no—
466D4—122no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—34no—
471C2×C4—32no—
472C2×C4—38no—
473C2×C4—38no—
474C2×C4—34no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—64no—
478C2×C4—64no—
479C2×C4—62no—
480C2×C4—128no—
481C2×C4—128no—
482C2×C4—122no—
483C2×C4—124no—
484C2×C4C2×D3×D414yes—
485C2×C4C2×D3×D411yescharacteristic
486C2×C4D3×#4114yes—
487C2×C4C24×D311yescharacteristic
488C8—24no—
489C8—64no—
490C2×C3—82no—
491C2×C3#314014yescomplemented
492C2×C3C22×#12411yescharacteristic
493C2×C3C2×#57512yes—
494D3—24no—
495D3—82no—
496C22—34no—
497C22—42no—
498C22—44no—
499C22—64no—
500C22—68no—
501C22—84no—
502C22—122no—
503C22—124no—
504C22—2416no—
505C22—244no—
506C22D3×#12412yes—
507C22#576814yes—
508C22C22×D3×D411yescharacteristic
509C4—24no—
510C4—34no—
511C4—42no—
512C4—44no—
513C4—64no—
514C4—122no—
515C4—124no—
516C4D3×#12412yes—
517C4C2×D3×#4112yes—
518C3C2×#314011yescharacteristic, complemented, Sylow
519C2—64no—
520C2—82no—
521C2—242no—
522C2#8164014yesdirect product factor
523C2C2×D3×#12411yescharacteristic
524C2C2×#576812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#488C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenterC2×#488D3
hypocenterC3C2×#3140
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×#5688C2semidirect
3. C2×C3×#488C2semidirect
4. C2×#5068C2semidirect
5. C2×#5056C2semidirect
6. #81640C2direct (N × H)
7. C2×#4802C2semidirect
8. C2×#4794C2semidirect
10. C2×#4765C2semidirect
11. C2×#4697C2semidirect
12. C2×C4×ΓL(1,𝔽25)C2semidirect
16. C2×C3×#125C22semidirect
22. #5688C22semidirect
27. C2×#888C22semidirect
33. C2×#870C22semidirect
34. C3×#488C22semidirect
35. C2×C3×#103C22semidirect
37. C2×C3×C4×C8C22semidirect
38. C2×#835C22semidirect
41. C2×#816C22semidirect
42. C2×#815C22semidirect
43. #5068C22semidirect
44. #5056C22semidirect
45. C2×#810C22semidirect
47. C2×C4×C3⋊C8C22semidirect
48. #4802C22semidirect
49. #4794C22semidirect
51. #4765C22semidirect
52. #4697C22semidirect
53. C4×ΓL(1,𝔽25)C22semidirect
62. C2×#488D3semidirect
85. C3×#125C23semidirect
113. #888C23semidirect
122. #870C23semidirect
123. C3×#103C23semidirect
125. C3×C4×C8C23semidirect
126. #835C23semidirect
129. #816C23semidirect
130. #815C23semidirect
131. #810C23semidirect
133. C4×C3⋊C8C23semidirect
141. C2×#125C2×D3semidirect
159. #488C2×D3semidirect
165. C2×#103C2×D3semidirect
173. C2×C4×C8C2×D3semidirect
266. #125C22×D3semidirect
300. #103C22×D3semidirect
307. C4×C8C22×D3semidirect
491. C2×C3#3140semidirect
518. C3C2×#3140semidirect
522. C2#81640direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56882
C2×D3×#1242
C2×C3×#4882
C2×#50682
C2×#50562
#816402
C2×#48022
C2×#47942
C2×#47702
C2×#47652
C2×#46972
C2×C4×ΓL(1,𝔽25)2
C2×#31403