Finite Group Databasebeta

← C2×#78567 #11664997

Subgroups — C2×#78567 #11664997

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×#5535C211yesmaximal, characteristic
2C2×#5518C211yesmaximal, characteristic, complemented
3C2×C3×#405C211yesmaximal, characteristic, complemented
4C2×#5134C212yesmaximal, complemented
5C2×#5114C212yesmaximal, complemented
6#78567C2116yesmaximal, direct product factor
7C2×#4703C211yesmaximal, characteristic, complemented
8C2×#4713C211yesmaximal, characteristic, complemented
9C2×#4710C212yesmaximal, complemented
10C2×C4×#204C212yesmaximal, complemented
11C2×#4693C211yesmaximal, characteristic, complemented
12C2×#4683C211yesmaximal, characteristic, complemented
13C2×#3174—31nomaximal, Sylow
14C22×C3×M16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#930C2212yes—
18C2×#924C2212yes—
19C22×#204C2212yes—
20C22×#204C2212yes—
21#5535—24no—
22#5535C2218yes—
23#5518—24no—
24#5518C2218yescomplemented
25C2×#894—22no—
26C2×#889—22no—
27C2×#884—22no—
28C2×#879—22no—
29C2×#874C2211yescharacteristic
30C2×C4×D12C2212yes—
31C2×#870C2211yescharacteristic
32C2×#870C2211yescharacteristic
33C2×#869C2211yescharacteristic
34C2×C4×Dic6C2212yes—
35C3×#405C2218yescomplemented
36C2×C3×#97C2211yescharacteristic
37C2×C3×#97C2211yescharacteristic
38C2×C3×C4×C8C2211yescharacteristic
39C2×C3×C4×C8C2211yescharacteristic
40#5134C22116yescomplemented
41#5114C22116yescomplemented
42C2×#821C2212yes—
43C2×#821C2212yes—
44C2×#818C2212yes—
45C2×#817C2212yes—
46C2×#816C2212yes—
47C2×#816C2212yes—
48#4713C2218yescomplemented
49#4710C22116yescomplemented
50#4703C2218yescomplemented
51#4693C2218yescomplemented
52C4×#204C22116yescomplemented
53#4683C2218yescomplemented
54C2×#541—31no—
55C2×#540—31no—
56#3174—316no—
57C2×#499—31no—
58C2×#492—31no—
59C2×#491—31no—
60C2×#490—31no—
61C2×#468—32no—
62C2×#465—32no—
63C2×#440—32no—
64C2×C4×SD16—32no—
65C2×#405D311yescharacteristic, complemented
66C23×C3×C4C2311yescharacteristic
67C22×#67—22no—
68C22×D12—22no—
69C22×D12C2312yes—
70C22×C4×D3—22no—
71C22×Dic6C2312yes—
72C2×C3×M16—24no—
73C2×C3×M16C2314yes—
74C22×C3×C8—24no—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2314yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#117D414yes—
81C3×#117C2314yes—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—22no—
87#930—24no—
88#930C2318yes—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2312yes—
94#924—24no—
95#924C2318yes—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×C4×Dic3—22no—
101C2×#204—28no—
102C2×#204—28no—
103C2×#204C2318yes—
104C2×#204C2318yes—
105#894—216no—
106#889—216no—
107#884—216no—
108#879—216no—
109#874—22no—
110#874C2314yes—
111C4×D12—24no—
112C4×D12C2318yes—
113#870—22no—
114#870—22no—
115#870C2314yes—
116#870C2314yes—
117#869—22no—
118#869C2314yes—
119C4×Dic6—24no—
120C4×Dic6C2318yes—
121C3×#97C2314yes—
122C3×#97C2314yes—
123C3×C4×C8C2314yes—
124C3×C4×C8C2314yes—
125#821C2318yes—
126#821C2318yes—
127#818C2318yes—
128#817C2318yes—
129#816C2318yes—
130#816C2318yes—
131C22×SD16—32no—
132C22×SD16—32no—
133C22×M16C2×D311yescharacteristic
134C23×C8C2×D311yescharacteristic
135#541—38no—
136#541—64no—
137#540—38no—
138#540—64no—
139C2×C4⋊Q8—31no—
140C2×#127—31no—
141C2×#126—62no—
142C2×#125—31no—
143C2×#125—31no—
144C2×#125—62no—
145C2×#123—62no—
146C2×C22⋊Q8—32no—
147C2×#121—32no—
148C2×#121—62no—
149C2×C4×Q8—32no—
150C2×C4×D4—32no—
151C2×#117C2×D311yescharacteristic
152#499—38no—
153#492—38no—
154#491—38no—
155#490—38no—
156#468—316no—
157#465—316no—
158#440—316no—
159C4×SD16—316no—
160C2×#107—32no—
161C2×#106—32no—
162C2×#103—32no—
163C2×#103—32no—
164C2×#102—32no—
165C2×#102—32no—
166#405C2×D318yescomplemented
167C2×#97C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169C2×C4×C8C2×D311yescharacteristic
170C2×C4×C8C2×D311yescharacteristic
171C24×C3C2×D411yescharacteristic
172C23×D3—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2412yes—
180C22×C3×C4C2411yescharacteristic
181C2×#67—28no—
182C2×#67—84no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C2×D12—24no—
187C2×D12—24no—
188C2×D12—22no—
189C2×D12—28no—
190C2×D12—84no—
191C2×D12C2414yes—
192C2×C4×D3—28no—
193C2×C4×D3—84no—
194C2×Dic6—24no—
195C2×Dic6—24no—
196C2×Dic6—22no—
197C2×Dic6C2414yes—
198C3×M16—28no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—28no—
202C2×C3×C8#4114yes—
203C2×C3×C8#4114yes—
204C2×C3×C8C2412yes—
205C2×C3×C8C2412yes—
206C2×C3×C8C2412yes—
207C2×C3×C8C2412yes—
208C3×C4⋊C4C2×D418yes—
209C3×C22⋊C4C2×D418yes—
210C3×C42C2×D412yes—
211C3×C42C2×D412yes—
212C3×C42C2412yes—
213C3×C42C2412yes—
214#217—28no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—28no—
217C4⋊Dic3—22no—
218C4⋊Dic3—28no—
219C4⋊Dic3—22no—
220C4⋊Dic3C2414yes—
221C4⋊Dic3C2414yes—
222Dic3⋊C4—28no—
223Dic3⋊C4—28no—
224Dic3⋊C4—28no—
225Dic3⋊C4—28no—
226C4×Dic3—28no—
227#204—216no—
228#204—216no—
229C22×Q8—32no—
230C22×D4—32no—
231C22×D4—62no—
232C22×D4—62no—
233C23×C4—62no—
234C23×C4C22×D311yescharacteristic
235C2×SD16—38no—
236C2×SD16—38no—
237C2×SD16—68no—
238C2×SD16—68no—
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—34no—
248C4⋊Q8—62no—
249#127—34no—
250#127—62no—
251#126—616no—
252#125—34no—
253#125—34no—
254#125—616no—
255#125—62no—
256#125—62no—
257#123—616no—
258C22⋊Q8—38no—
259C22⋊Q8—64no—
260#121—38no—
261#121—64no—
262#121—616no—
263C4×Q8—38no—
264C4×Q8—64no—
265C4×D4—38no—
266C4×D4—64no—
267#117D1214yes—
268#117C22×D314yes—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4D1212yes—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4D1212yes—
281C2×C42—62no—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284#107—38no—
285#106—38no—
286#103—38no—
287#103—38no—
288#102—38no—
289#102—38no—
290#97C22×D314yes—
291#97C22×D314yes—
292C4×C8C22×D314yes—
293C4×C8C22×D314yes—
294C23×C3—22no—
295C23×C3—24no—
296C23×C3C22×D412yes—
297C23×C3C22×D411yescharacteristic
298C22×D3—24no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3—44no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4C22×D414yes—
309C2×C3×C4C22×D414yes—
310C2×C3×C4C22×D412yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4112yes—
313C2×C3×C4C2×#4112yes—
314C2×C3×C4C2511yescharacteristic
315#67—88no—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—44no—
320C2×Dic3—44no—
321C2×Dic3—44no—
322D12—24no—
323D12—24no—
324D12—44no—
325D12—88no—
326C4×D3—88no—
327Dic6—24no—
328Dic6—24no—
329Dic6—44no—
330C3×C8—22no—
331C3×C8—22no—
332C3×C8C2×#4114yes—
333C3×C8C2×#4114yes—
334C24—62no—
335C24C2×D1211yescharacteristic
336C2×Q8—34no—
337C2×Q8—62no—
338C2×Q8—64no—
339C2×Q8—64no—
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—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—68no—
354C22×C4—244no—
355C22×C4C2×D1212yes—
356C22×C4C2×D1212yes—
357C22×C4C2×D1212yes—
358C22×C4C2×D1211yescharacteristic
359C22×C4C23×D312yes—
360C22×C4C23×D311yescharacteristic
361SD16—616no—
362SD16—616no—
363M16—28no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—28no—
367C2×C8#23514yes—
368C2×C8#23514yes—
369C2×C8C23×D312yes—
370C2×C8C23×D312yes—
371C2×C8C23×D312yes—
372C2×C8C23×D312yes—
373C4⋊C4—34no—
374C4⋊C4—34no—
375C4⋊C4—62no—
376C4⋊C4—62no—
377C4⋊C4—68no—
378C4⋊C4—68no—
379C4⋊C4—68no—
380C4⋊C4—68no—
381C4⋊C4—68no—
382C4⋊C4C2×D1218yes—
383C22⋊C4—68no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4C2×D1218yes—
387C42—68no—
388C42C2×D1212yes—
389C42C2×D1212yes—
390C42C23×D312yes—
391C42C23×D312yes—
392C22×C3—22no—
393C22×C3—24no—
394C22×C3—44no—
395C22×C3#53212yes—
396C22×C3#57714yes—
397C22×C3C23×D411yescharacteristic
398C2×D3—44no—
399C2×D3—44no—
400C2×D3—44no—
401C2×D3—88no—
402C3×C4—22no—
403C3×C4—24no—
404C3×C4—24no—
405C3×C4#53212yes—
406C3×C4C22×#4111yescharacteristic
407C3×C4C22×#4111yescharacteristic
408Dic3—44no—
409Dic3—44no—
410Dic3—44no—
411C23—22no—
412C23—24no—
413C23—64no—
414C23—124no—
415C23—124no—
416C23—124no—
417C23C22×D1212yes—
418C23C22×D1211yescharacteristic
419Q8—64no—
420Q8—64no—
421Q8—124no—
422D4—64no—
423D4—64no—
424D4—124no—
425D4—248no—
426D4—248no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—248no—
440C2×C4C22×D1214yes—
441C2×C4C22×D1214yes—
442C2×C4C22×D1212yes—
443C2×C4C22×D1211yescharacteristic
444C2×C4C2×#23512yes—
445C2×C4C2×#23512yes—
446C2×C4C24×D311yescharacteristic
447C8—22no—
448C8—22no—
449C8C2×#23514yes—
450C8C2×#23514yes—
451C2×C3—42no—
452C2×C3#317414yescomplemented
453C2×C3C2×#53211yescharacteristic
454C2×C3C2×#57712yes—
455D3—84no—
456C22—24no—
457C22—22no—
458C22—44no—
459C22—124no—
460C22—124no—
461C22—124no—
462C22—248no—
463C22#547712yes—
464C22#575114yes—
465C22C23×D1211yescharacteristic
466C4—24no—
467C4—24no—
468C4—22no—
469C4—124no—
470C4—124no—
471C4—124no—
472C4#547712yes—
473C4C22×#23511yescharacteristic
474C4C22×#23511yescharacteristic
475C3C2×#317411yescharacteristic, complemented, Sylow
476C2—42no—
477C2—244no—
478C2#7856714yesdirect product factor
479C2C2×#547711yescharacteristic
480C2C2×#575112yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#405C2
socleC23×C3C22×D4
prime radicalC2×#5535C2
hypercenterC2×#405D3
hypocenterC3C2×#3174
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
2. C2×#5518C2semidirect
3. C2×C3×#405C2semidirect
4. C2×#5134C2semidirect
5. C2×#5114C2semidirect
6. #78567C2direct (N × H)
7. C2×#4703C2semidirect
8. C2×#4713C2semidirect
9. C2×#4710C2semidirect
10. C2×C4×#204C2semidirect
11. C2×#4693C2semidirect
12. C2×#4683C2semidirect
24. #5518C22semidirect
35. C3×#405C22semidirect
40. #5134C22semidirect
41. #5114C22semidirect
48. #4713C22semidirect
49. #4710C22semidirect
50. #4703C22semidirect
51. #4693C22semidirect
52. C4×#204C22semidirect
53. #4683C22semidirect
65. C2×#405D3semidirect
166. #405C2×D3semidirect
452. C2×C3#3174semidirect
475. C3C2×#3174semidirect
478. C2#78567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55352
C2×#55182
C2×C3×#4052
C2×#51342
C2×#51142
#785672
C2×#47032
C2×#47132
C2×#47102
C2×C4×#2042
C2×#46932
C2×#46832
C2×#31743