Finite Group Databasebeta

← C2×#81729 #11665000

Subgroups — C2×#81729 #11665000

Order 768.

1970 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5731C211yesmaximal, characteristic, complemented
2C2×#5697C211yesmaximal, characteristic
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C2×#5085C211yesmaximal, characteristic, complemented
5C2×#5065C211yesmaximal, characteristic, complemented
6#81729C2116yesmaximal, direct product factor
7C2×#4812C212yesmaximal, complemented
8C2×#4807C212yesmaximal, complemented
9C2×#4817C212yesmaximal, complemented
10C2×#4788C212yesmaximal, complemented
11C2×#4713C211yesmaximal, characteristic, complemented
12C2×#4708C211yesmaximal, characteristic, complemented
13C2×#3174—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×#946C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×#215C2212yes—
19C22×#215C2212yes—
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22#5731—24no—
23#5731C2218yescomplemented
24#5697—24no—
25#5697C2218yes—
26C2×#898—22no—
27C2×#895—22no—
28C2×#895C2212yes—
29C2×#894—22no—
30C2×#893C2212yes—
31C2×#889—22no—
32C2×#874C2211yescharacteristic
33C2×#872C2211yescharacteristic
34C2×#870C2211yescharacteristic
35C3×#491C2218yescomplemented
36C2×C3×#103C2212yes—
37C2×C3×#103C2212yes—
38C2×C3×#97C2211yescharacteristic
39C2×#821C2212yes—
40C2×#821C2212yes—
41C2×#815C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#5085C2218yescomplemented
44#5065C2218yescomplemented
45C2×#808C2212yes—
46C2×#807C2212yes—
47C2×C4×C3⋊C8C2211yescharacteristic
48#4817C22116yescomplemented
49#4812C22116yescomplemented
50#4807C22116yescomplemented
51#4788C22116yescomplemented
52#4713C2218yescomplemented
53#4708C2218yescomplemented
54C2×#541—31no—
55C2×#540—31no—
56#3174—316no—
57C2×#499—31no—
58C2×#492—31no—
59C2×#491D311yescharacteristic, complemented
60C2×#490—31no—
61C2×#468—32no—
62C2×#465—32no—
63C2×#440—32no—
64C2×C4×SD16—32no—
65C2×#405—31no—
66C22×C3×Q8C2312yes—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12C2312yes—
70C22×C4×D3—22no—
71C22×C4×D3C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8C2314yes—
76C3×#125—22no—
77C3×#125C2314yes—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C42C2311yescharacteristic
82#946—24no—
83#946C2318yes—
84Q8×Dic3—24no—
85Q8×Dic3C2318yes—
86C2×#215—28no—
87C2×#215—28no—
88C2×#215C2318yes—
89C2×#215C2318yes—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4D412yes—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3C2312yes—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4D412yes—
99C2×C4×Dic3—22no—
100C2×C4×Dic3C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C2×ΓL(1,𝔽25)—24no—
104C2×ΓL(1,𝔽25)C2314yes—
105C2×D3×C8—24no—
106C2×D3×C8C2314yes—
107#898—216no—
108#895—24no—
109#895—216no—
110#895C2318yes—
111#894—216no—
112#893—24no—
113#893C2318yes—
114#889—216no—
115#874—22no—
116#874C2314yes—
117#872D414yes—
118#872C2314yes—
119#870—22no—
120#870C2314yes—
121C3×#103C2318yes—
122C3×#103C2318yes—
123C3×#97C2314yes—
124#821C2318yes—
125#821C2318yes—
126#815C2314yes—
127C8×Dic3C2314yes—
128#808C2318yes—
129#807C2318yes—
130C4×C3⋊C8C2314yes—
131C22×SD16—32no—
132C22×SD16—32no—
133C22×M16—31no—
134C23×C8—31no—
135#541—38no—
136#541—64no—
137#540—38no—
138#540—64no—
139C2×C4⋊Q8C2×D311yescharacteristic
140C2×#127—31no—
141C2×#126—62no—
142C2×#125—31no—
143C2×#125—62no—
144C2×#125C2×D311yescharacteristic
145C2×#123—62no—
146C2×C22⋊Q8—32no—
147C2×#121—32no—
148C2×#121—62no—
149C2×C4×Q8—32no—
150C2×C4×D4—32no—
151C2×#117—31no—
152#499—38no—
153#492—38no—
154#491C2×D318yescomplemented
155#490—38no—
156#468—316no—
157#465—316no—
158#440—316no—
159C4×SD16—316no—
160C2×#107—32no—
161C2×#106—32no—
162C2×#103C2×D312yes—
163C2×#103C2×D312yes—
164C2×#102—32no—
165C2×#102—32no—
166#405—38no—
167C2×#97—31no—
168C2×#97C2×D311yescharacteristic
169C2×C4×C8—31no—
170C2×C4×C8—31no—
171C23×D3—22no—
172C23×D3C2×D411yescharacteristic
173C2×C3×Q8—22no—
174C2×C3×Q8—24no—
175C2×C3×Q8—24no—
176C2×C3×Q8C2414yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C22×Dic3—22no—
182C22×Dic3C2×D412yes—
183C22×Dic3C2×D411yescharacteristic
184C2×D12—24no—
185C2×D12—24no—
186C2×D12—22no—
187C2×D12—28no—
188C2×D12—28no—
189C2×D12—84no—
190C2×D12—84no—
191C2×D12C2414yes—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—28no—
195C2×C4×D3—84no—
196C2×C4×D3C2×D412yes—
197C2×C4×D3C2412yes—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8C2412yes—
201C2×C3×C8C2412yes—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4C2414yes—
206C3×C42C2×D412yes—
207C3×C42C2412yes—
208#215—216no—
209#215—216no—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4C2×D418yes—
214C4⋊Dic3—22no—
215C4⋊Dic3—28no—
216C4⋊Dic3C2414yes—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219Dic3⋊C4C2×D418yes—
220C4×Dic3—28no—
221C4×Dic3C2×D412yes—
222C4×Dic3C2412yes—
223C2×C3⋊C8#4114yes—
224C2×C3⋊C8#4114yes—
225C2×C3⋊C8C2412yes—
226C2×C3⋊C8C2412yes—
227ΓL(1,𝔽25)—28no—
228D3×C8—28no—
229C22×Q8C22×D312yes—
230C22×D4—32no—
231C22×D4—62no—
232C22×D4—62no—
233C23×C4—31no—
234C23×C4—62no—
235C2×SD16—38no—
236C2×SD16—38no—
237C2×SD16—68no—
238C2×SD16—68no—
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—22no—
248C4⋊Q8C22×D314yes—
249#127—34no—
250#127—62no—
251#126—616no—
252#125—22no—
253#125—34no—
254#125—616no—
255#125—62no—
256#125C22×D314yes—
257#123—616no—
258C22⋊Q8—38no—
259C22⋊Q8—64no—
260#121—38no—
261#121—616no—
262#121—64no—
263C4×Q8—38no—
264C4×Q8—64no—
265C4×D4—38no—
266C4×D4—64no—
267#117—34no—
268#117—34no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C42—31no—
282C2×C42—62no—
283C2×C42C22×D311yescharacteristic
284#107—38no—
285#106—38no—
286#103C22×D318yes—
287#103C22×D318yes—
288#102—38no—
289#102—38no—
290#97—34no—
291#97C22×D314yes—
292C4×C8—34no—
293C4×C8—34no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—44no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3C22×D412yes—
302C3×Q8—24no—
303C3×Q8—24no—
304C3×Q8—44no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4C22×D414yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4112yes—
313C2×C3×C4C2×#4112yes—
314C2×C3×C4C2511yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—44no—
319C2×Dic3C22×D414yes—
320C2×Dic3C22×D412yes—
321D12—24no—
322D12—24no—
323D12—44no—
324D12—88no—
325D12—88no—
326C4×D3—24no—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—88no—
330C3×C8—22no—
331C3×C8—22no—
332C3⋊C8C2×#4114yes—
333C3⋊C8C2×#4114yes—
334C24—31no—
335C24—62no—
336C2×Q8—22no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8C23×D314yes—
340C2×D4—34no—
341C2×D4—68no—
342C2×D4—62no—
343C2×D4—68no—
344C2×D4—64no—
345C2×D4—64no—
346C2×D4—244no—
347C2×D4—244no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—32no—
351C22×C4—31no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—68no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—244no—
359C22×C4D3×D412yes—
360C22×C4C23×D311yescharacteristic
361SD16—616no—
362SD16—616no—
363M16—68no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—34no—
367C2×C8—34no—
368C2×C8—32no—
369C2×C8—32no—
370C2×C8—68no—
371C2×C8C23×D312yes—
372C2×C8C23×D312yes—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—22no—
376C4⋊C4—38no—
377C4⋊C4—34no—
378C4⋊C4—68no—
379C4⋊C4—68no—
380C4⋊C4—62no—
381C4⋊C4—68no—
382C4⋊C4C23×D314yes—
383C22⋊C4—38no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—68no—
387C42—32no—
388C42—32no—
389C42—68no—
390C42D3×D412yes—
391C42C23×D312yes—
392C22×C3#53212yes—
393C22×C3#57714yes—
394C22×C3C23×D411yescharacteristic
395C2×D3—22no—
396C2×D3—24no—
397C2×D3—44no—
398C2×D3—44no—
399C2×D3—44no—
400C2×D3—44no—
401C2×D3—88no—
402C3×C4—24no—
403C3×C4—44no—
404C3×C4—44no—
405C3×C4#53212yes—
406C3×C4C22×#4111yescharacteristic
407C3×C4C22×#4111yescharacteristic
408Dic3—22no—
409Dic3—24no—
410Dic3—44no—
411C23—32no—
412C23—62no—
413C23—64no—
414C23—64no—
415C23—124no—
416C23—124no—
417C23—124no—
418C23C2×D3×D411yescharacteristic
419Q8—24no—
420Q8—24no—
421Q8—44no—
422D4—64no—
423D4—64no—
424D4—124no—
425D4—248no—
426D4—248no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—34no—
431C2×C4—32no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—62no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—64no—
440C2×C4—124no—
441C2×C4—248no—
442C2×C4C2×D3×D414yes—
443C2×C4C2×D3×D411yescharacteristic
444C2×C4D3×#4112yes—
445C2×C4D3×#4112yes—
446C2×C4C24×D311yescharacteristic
447C8—22no—
448C8—22no—
449C8—34no—
450C8—34no—
451C2×C3#317414yescomplemented
452C2×C3C2×#53211yescharacteristic
453C2×C3C2×#57712yes—
454D3—42no—
455D3—84no—
456C22—62no—
457C22—64no—
458C22—124no—
459C22—124no—
460C22—124no—
461C22—124no—
462C22—248no—
463C22#566812yes—
464C22#577814yes—
465C22C22×D3×D411yescharacteristic
466C4—24no—
467C4—44no—
468C4—44no—
469C4—62no—
470C4—64no—
471C4—124no—
472C4#566812yes—
473C4C2×D3×#4111yescharacteristic
474C4C2×D3×#4111yescharacteristic
475C3C2×#317411yescharacteristic, complemented, Sylow
476C2—122no—
477C2—244no—
478C2#8172914yesdirect product factor
479C2C2×#566811yescharacteristic
480C2C2×#577812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#491C2
socleC23×C3C22×D4
prime radicalC2×#5697C2
hypercenterC2×#491D3
hypocenterC3C2×#3174
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×#5731C2semidirect
3. C2×C3×#491C2semidirect
4. C2×#5085C2semidirect
5. C2×#5065C2semidirect
6. #81729C2direct (N × H)
7. C2×#4812C2semidirect
8. C2×#4807C2semidirect
9. C2×#4817C2semidirect
10. C2×#4788C2semidirect
11. C2×#4713C2semidirect
12. C2×#4708C2semidirect
23. #5731C22semidirect
35. C3×#491C22semidirect
43. #5085C22semidirect
44. #5065C22semidirect
48. #4817C22semidirect
49. #4812C22semidirect
50. #4807C22semidirect
51. #4788C22semidirect
52. #4713C22semidirect
53. #4708C22semidirect
59. C2×#491D3semidirect
154. #491C2×D3semidirect
451. C2×C3#3174semidirect
475. C3C2×#3174semidirect
478. C2#81729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57312
C2×#56972
C2×C3×#4912
C2×#50852
C2×#50652
#817292
C2×#48122
C2×#48072
C2×#48172
C2×#47882
C2×#47132
C2×#47082
C2×#31743