Finite Group Databasebeta

← D4×#3832 #11902768

Subgroups — D4×#3832 #11902768

Order 1152.

4944 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3832C212yesmaximal, complemented
2C32×D4×Q8C211yesmaximal, characteristic, complemented
3#10596288C212yesmaximal, complemented
4#10596197C213yesmaximal, complemented
5#10596196C213yesmaximal, complemented
6#10596101C216yesmaximal, complemented
7#10596098C216yesmaximal, complemented
8#10596048C211yesmaximal, characteristic, complemented
9C4×#3832C211yesmaximal, characteristic, complemented
10D4×C4⋊(C3⋊D3)C213yesmaximal, complemented
11C4×D4×C3⋊D3C213yesmaximal, complemented
12D4×#239—34nomaximal
13C22×C32×Q8C2212yescomplemented
14C2×#3832—216no—
15C2×#3832C2211yescharacteristic
16C2×#3832C2216yescomplemented
17C2×#3832C2216yescomplemented
18C2×#3832C2212yescomplemented
19C2×D4×C3⋊D3C2213yes—
20C22×C4⋊(C3⋊D3)C2216yescomplemented
21C22×C4×C3⋊D3C2216yescomplemented
22C32×C4⋊Q8C2213yescomplemented
23C32×C22⋊Q8C2216yescomplemented
24C32×C4×Q8C2211yescharacteristic, complemented
25C32×C4×D4C2213yes—
26#64156C2216yescomplemented
27#64155C2216yescomplemented
28Q8×C3⋊Dic3C2211yescharacteristic, complemented
29D4×C3⋊Dic3C2211yescharacteristic
30#64139C2216yescomplemented
31C4×#638C2216yescomplemented
32#64104C2216yescomplemented
33#64104C2216yescomplemented
34#64104C2216yescomplemented
35#64103C22112yescomplemented
36#64102C2216yescomplemented
37#64102C2213yescomplemented
38#64102C2213yescomplemented
39#64101C2213yescomplemented
40C4⋊C4×C3⋊D3C2213yescomplemented
41#64091C22112yescomplemented
42#64090C2216yescomplemented
43C22⋊C4×C3⋊D3C2216yescomplemented
44#64083C2213yescomplemented
45C4×C4⋊(C3⋊D3)C2213yescomplemented
46C4×C4⋊(C3⋊D3)C2213yescomplemented
47C42×C3⋊D3C2213yescomplemented
48C22×#239—38no—
49C3×D4×Q8D314yescomplemented
50#5817—38no—
51#5726—312no—
52#5725—312no—
53#5630—324no—
54#5627—324no—
55#5577—34no—
56C4×#239—34no—
57D4×D12—312no—
58C4×D3×D4—312no—
59C23×C3⋊D3C2316yes—
60C2×C32×Q8—28no—
61C2×C32×Q8C2316yes—
62C2×C32×Q8C2311yescharacteristic
63C2×C32×D4C2311yescharacteristic
64C22×C32×C4C2316yes—
65C2×#638C2316yes—
66C22×C3⋊Dic3C2312yes—
67#3832—28no—
68#3832—224no—
69#3832—224no—
70#3832D418yesdirect product factor
71D4×C3⋊D3—212no—
72C2×C4⋊(C3⋊D3)—224no—
73C2×C4⋊(C3⋊D3)C2313yes—
74C2×C4⋊(C3⋊D3)C2316yes—
75C2×C4⋊(C3⋊D3)C23112yes—
76C2×C4⋊(C3⋊D3)C2316yes—
77C2×C4⋊(C3⋊D3)C2313yes—
78C2×C4×C3⋊D3—224no—
79C2×C4×C3⋊D3C2313yes—
80C2×C4×C3⋊D3C2316yes—
81C2×C4×C3⋊D3C2316yes—
82C2×C4×C3⋊D3C2316yes—
83C2×C4×C3⋊D3C2313yes—
84C2×C4×C3⋊D3C2313yes—
85C32×C4⋊C4C2313yes—
86C32×C4⋊C4C2313yes—
87C32×C4⋊C4C2316yes—
88C32×C22⋊C4C2316yes—
89C32×C42C2313yes—
90#3757C2312yes—
91#3752C2316yes—
92#3752C2316yes—
93#3752C2316yes—
94#3752C23112yes—
95C4⋊(C3⋊Dic3)C2311yescharacteristic
96C4⋊(C3⋊Dic3)C2313yes—
97#3750C2316yes—
98C4×C3⋊Dic3C2311yescharacteristic
99C4×C3⋊Dic3C2313yes—
100C4×C3⋊Dic3C2313yes—
101D4×#41—91noSylow
102C22×C3×Q8C2×D318yescomplemented
103C2×#239—34no—
104C2×#239—324no—
105C2×#239—324no—
106C2×#239—38no—
107C2×#239—664no—
108C2×D3×D4—312no—
109C22×D12—324no—
110C22×C4×D3—324no—
111C3×C4⋊Q8C2×D3112yescomplemented
112C3×C22⋊Q8C2×D3124yescomplemented
113C3×C4×Q8C2×D314yescomplemented
114C3×C4×D4C2×D3112yes—
115#947—324no—
116#946—324no—
117Q8×Dic3—34no—
118D4×Dic3—34no—
119#930—324no—
120C4×#67—324no—
121#895—324no—
122#895—324no—
123#895—324no—
124#894—348no—
125#893—312no—
126#893—324no—
127#893—312no—
128#892—312no—
129D3×C4⋊C4—312no—
130#882—348no—
131#67⋊C4—324no—
132D3×C22⋊C4—324no—
133#874—312no—
134C4×D12—312no—
135C4×D12—312no—
136C42×D3—312no—
137C23×C32C2412yes—
138C22×C3⋊D3—212no—
139C22×C3⋊D3—212no—
140C22×C3⋊D3—46no—
141C22×C3⋊D3C2416yes—
142C22×C3⋊D3C2413yes—
143C32×Q8—212no—
144C32×Q8C2×D414yescomplemented
145C32×D4#4114yescomplemented
146C2×C32×C4—212no—
147C2×C32×C4C2413yes—
148C2×C32×C4C2416yes—
149C2×C32×C4C2413yes—
150C2×C32×C4C2411yescharacteristic
151#638—212no—
152C2×C3⋊Dic3—24no—
153C2×C3⋊Dic3C2411yescharacteristic
154C2×C3⋊Dic3C2413yes—
155C2×C3⋊Dic3C2412yes—
156C2×C3⋊Dic3C2411yescharacteristic
157C4⋊(C3⋊D3)—224no—
158C4⋊(C3⋊D3)—212no—
159C4⋊(C3⋊D3)—26no—
160C4⋊(C3⋊D3)—46no—
161C4⋊(C3⋊D3)C2×D4112yescomplemented
162C4×C3⋊D3—212no—
163C4×C3⋊D3—212no—
164C4×C3⋊D3—26no—
165C4×C3⋊D3—212no—
166C4×C3⋊D3—43no—
167C4×C3⋊D3C2×D4112yescomplemented
168C22×#41—92no—
169#550—91no—
170D4×Q8C3⋊D311yescharacteristic, complemented
171#548—92no—
172#547—93no—
173#546—96no—
174D42—93no—
175#532—93no—
176#525—96no—
177C4×#41—91no—
178C2×C4×D4—93no—
179C23×D3—324no—
180C2×C3×Q8—232no—
181C2×C3×Q8C22×D3124yes—
182C2×C3×Q8C22×D314yes—
183C2×C3×D4C22×D314yes—
184C22×C3×C4C22×D3124yes—
185C2×#67—324no—
186C22×Dic3—38no—
187#239—332no—
188#239—696no—
189#239—696no—
190#239—632no—
191D3×D4—648no—
192C2×D12—312no—
193C2×D12—312no—
194C2×D12—324no—
195C2×D12—348no—
196C2×D12—324no—
197C2×D12—696no—
198C2×C4×D3—312no—
199C2×C4×D3—312no—
200C2×C4×D3—324no—
201C2×C4×D3—324no—
202C2×C4×D3—324no—
203C2×C4×D3—312no—
204C2×C4×D3—696no—
205C3×C4⋊C4C22×D3112yes—
206C3×C4⋊C4C22×D3112yes—
207C3×C4⋊C4C22×D3124yes—
208C3×C22⋊C4C22×D3124yes—
209C3×C42C22×D3112yes—
210#217—38no—
211(C2×D3)⋊C4—324no—
212(C2×D3)⋊C4—324no—
213(C2×D3)⋊C4—324no—
214(C2×D3)⋊C4—348no—
215C4⋊Dic3—34no—
216C4⋊Dic3—312no—
217Dic3⋊C4—324no—
218C4×Dic3—34no—
219C4×Dic3—312no—
220C4×Dic3—312no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32C2×#4114yescomplemented
224C22×C32C2511yescharacteristic
225C2×C3⋊D3—23no—
226C2×C3⋊D3—23no—
227C2×C3⋊D3—26no—
228C2×C3⋊D3—26no—
229C2×C3⋊D3—26no—
230C2×C3⋊D3—424no—
231C2×C3⋊D3C22×D416yes—
232C32×C4—26no—
233C32×C4—23no—
234C32×C4C22×D416yes—
235C32×C4C2×#4112yescomplemented
236C3⋊Dic3—22no—
237C3⋊Dic3—23no—
238C3⋊Dic3—21no—
239C3⋊Dic3C22×D412yes—
240C2×#41—91no—
241C2×#41—96no—
242C2×#41—96no—
243C2×#41—92no—
244C2×#41—1816no—
245C22×Q8C2×C3⋊D312yescomplemented
246C22×D4—93no—
247C22×D4—96no—
248C23×C4—96no—
249C4⋊Q8C2×C3⋊D313yescomplemented
250#127—93no—
251#124—96no—
252#123—912no—
253C22⋊Q8—96no—
254C22⋊Q8C2×C3⋊D316yescomplemented
255#121—96no—
256#121—96no—
257#121—96no—
258#121—96no—
259#120—912no—
260C4×Q8—91no—
261C4×Q8C2×C3⋊D311yescharacteristic, complemented
262C4×D4—91no—
263C4×D4—96no—
264C4×D4—93no—
265C4×D4—93no—
266C4×D4—96no—
267C4×D4—96no—
268C4×D4—93no—
269C4×D4—93no—
270C4×D4C2×C3⋊D313yes—
271#117—93no—
272C2×C4⋊C4—93no—
273C2×C22⋊C4—96no—
274C2×C42—93no—
275C23×C3C23×D318yes—
276C22×D3—312no—
277C22×D3—324no—
278C22×D3—648no—
279C22×D3—648no—
280C22×D3—1224no—
281C3×Q8—248no—
282C3×Q8D3×D4116yescomplemented
283C3×D4#239116yescomplemented
284C2×C3×C4—248no—
285C2×C3×C4C23×D3112yes—
286C2×C3×C4C23×D3124yes—
287C2×C3×C4C23×D3112yes—
288C2×C3×C4C23×D314yes—
289#67—648no—
290C2×Dic3—34no—
291C2×Dic3—34no—
292C2×Dic3—312no—
293C2×Dic3—38no—
294C2×Dic3—616no—
295D12—348no—
296D12—696no—
297D12—648no—
298D12—624no—
299D12—1224no—
300C4×D3—348no—
301C4×D3—648no—
302C4×D3—648no—
303C4×D3—648no—
304C4×D3—624no—
305C4×D3—1212no—
306C2×C32—24no—
307C2×C32C23×D411yescharacteristic
308C2×C32C22×#4111yescharacteristic
309C2×C32#58511yescharacteristic
310C3⋊D3—26no—
311C3⋊D3—46no—
312C24—96no—
313#41—98no—
314#41—1824no—
315#41—1824no—
316#41—188no—
317C2×Q8—28no—
318C2×Q8C22×C3⋊D316yes—
319C2×Q8C22×C3⋊D311yescharacteristic
320C2×D4—96no—
321C2×D4—93no—
322C2×D4—93no—
323C2×D4—912no—
324C2×D4—96no—
325C2×D4—96no—
326C2×D4—1824no—
327C2×D4—1812no—
328C2×D4C22×C3⋊D311yescharacteristic
329C22×C4—93no—
330C22×C4—92no—
331C22×C4—93no—
332C22×C4—96no—
333C22×C4—96no—
334C22×C4—96no—
335C22×C4—93no—
336C22×C4—1824no—
337C22×C4C22×C3⋊D316yes—
338C4⋊C4—91no—
339C4⋊C4—96no—
340C4⋊C4—93no—
341C4⋊C4C22×C3⋊D313yes—
342C4⋊C4C22×C3⋊D313yes—
343C4⋊C4C22×C3⋊D316yes—
344C22⋊C4—92no—
345C22⋊C4—96no—
346C22⋊C4—96no—
347C22⋊C4—96no—
348C22⋊C4—912no—
349C22⋊C4C22×C3⋊D316yes—
350C42—91no—
351C42—93no—
352C42—93no—
353C42C22×C3⋊D313yes—
354C22×C3—28no—
355C22×C3—28no—
356C22×C3C2×#239116yescomplemented
357C22×C3C24×D314yes—
358C2×D3—324no—
359C2×D3—612no—
360C2×D3—612no—
361C2×D3—624no—
362C2×D3—624no—
363C2×D3—624no—
364C2×D3—1296no—
365C3×C4—224no—
366C3×C4—212no—
367C3×C4C2×D3×D4124yes—
368C3×C4C2×#23918yescomplemented
369Dic3—38no—
370Dic3—68no—
371Dic3—612no—
372Dic3—64no—
373C32D4×#4111yescharacteristic, complemented, Sylow
374C23—93no—
375C23—96no—
376C23—1812no—
377C23—1812no—
378C23—366no—
379C23C23×C3⋊D312yes—
380Q8—212no—
381Q8D4×C3⋊D314yescomplemented
382D4—912no—
383D4—1824no—
384D4—1812no—
385D4—1812no—
386D4—186no—
387D4—366no—
388D4#383214yesdirect product factor
389C2×C4—212no—
390C2×C4—91no—
391C2×C4—912no—
392C2×C4—91no—
393C2×C4—93no—
394C2×C4—92no—
395C2×C4—1812no—
396C2×C4—1812no—
397C2×C4—184no—
398C2×C4—1812no—
399C2×C4—186no—
400C2×C4—363no—
401C2×C4C23×C3⋊D313yes—
402C2×C4C23×C3⋊D316yes—
403C2×C4C23×C3⋊D313yes—
404C2×C4C23×C3⋊D311yescharacteristic
405C2×C3—216no—
406C2×C3C22×D3×D414yes—
407C2×C3C22×#23914yes—
408C2×C3#596814yes—
409D3—624no—
410D3—1224no—
411C22—22no—
412C22—22no—
413C22—96no—
414C22—183no—
415C22—183no—
416C22—186no—
417C22—186no—
418C22—186no—
419C22—3624no—
420C22C2×#383214yescomplemented
421C22C24×C3⋊D311yescharacteristic
422C4—26no—
423C4—23no—
424C4—92no—
425C4—182no—
426C4—183no—
427C4—181no—
428C4C2×D4×C3⋊D316yes—
429C4C2×#383212yescomplemented
430C3D4×#23914yescomplemented
431C2—24no—
432C2—186no—
433C2—366no—
434C2C22×D4×C3⋊D311yescharacteristic
435C2C22×#383211yescharacteristic
436C2#1059676011yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC32×D4×Q8C2
hypercenterD4×Q8C3⋊D3
hypocenterC32D4×#41
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#3832C2semidirect
2. C32×D4×Q8C2semidirect
3. #10596288C2semidirect
4. #10596197C2semidirect
5. #10596196C2semidirect
6. #10596101C2semidirect
7. #10596098C2semidirect
8. #10596048C2semidirect
9. C4×#3832C2semidirect
10. D4×C4⋊(C3⋊D3)C2semidirect
11. C4×D4×C3⋊D3C2semidirect
13. C22×C32×Q8C22semidirect
16. C2×#3832C22semidirect
17. C2×#3832C22semidirect
18. C2×#3832C22semidirect
20. C22×C4⋊(C3⋊D3)C22semidirect
21. C22×C4×C3⋊D3C22semidirect
22. C32×C4⋊Q8C22semidirect
23. C32×C22⋊Q8C22semidirect
24. C32×C4×Q8C22semidirect
26. #64156C22semidirect
27. #64155C22semidirect
28. Q8×C3⋊Dic3C22semidirect
30. #64139C22semidirect
31. C4×#638C22semidirect
32. #64104C22semidirect
33. #64104C22semidirect
34. #64104C22semidirect
35. #64103C22semidirect
36. #64102C22semidirect
37. #64102C22semidirect
38. #64102C22semidirect
39. #64101C22semidirect
40. C4⋊C4×C3⋊D3C22semidirect
41. #64091C22semidirect
42. #64090C22semidirect
43. C22⋊C4×C3⋊D3C22semidirect
44. #64083C22semidirect
45. C4×C4⋊(C3⋊D3)C22semidirect
46. C4×C4⋊(C3⋊D3)C22semidirect
47. C42×C3⋊D3C22semidirect
49. C3×D4×Q8D3semidirect
70. #3832D4direct (N × H)
102. C22×C3×Q8C2×D3semidirect
111. C3×C4⋊Q8C2×D3semidirect
112. C3×C22⋊Q8C2×D3semidirect
113. C3×C4×Q8C2×D3semidirect
144. C32×Q8C2×D4semidirect
145. C32×D4#41semidirect
161. C4⋊(C3⋊D3)C2×D4semidirect
167. C4×C3⋊D3C2×D4semidirect
170. D4×Q8C3⋊D3semidirect
223. C22×C32C2×#41semidirect
235. C32×C4C2×#41semidirect
245. C22×Q8C2×C3⋊D3semidirect
249. C4⋊Q8C2×C3⋊D3semidirect
254. C22⋊Q8C2×C3⋊D3semidirect
261. C4×Q8C2×C3⋊D3semidirect
282. C3×Q8D3×D4semidirect
283. C3×D4#239semidirect
356. C22×C3C2×#239semidirect
368. C3×C4C2×#239semidirect
373. C32D4×#41semidirect
381. Q8D4×C3⋊D3semidirect
388. D4#3832direct (N × H)
420. C22C2×#3832semidirect
429. C4C2×#3832semidirect
430. C3D4×#239semidirect

Maximal subgroups

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

subgroupindex
C22×#38322
C32×D4×Q82
#105962882
#105961972
#105961962
#105961012
#105960982
#105960482
C4×#38322
D4×C4⋊(C3⋊D3)2
C4×D4×C3⋊D32
D4×#2393