Finite Group Databasebeta

← D4×#3753 #11885724

Subgroups — D4×#3753 #11885724

Order 1152.

1860 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D42C211yesmaximal, characteristic, complemented
2C22×#3753C212yesmaximal, complemented
3D4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4#10592916C212yesmaximal, complemented
5#10592772C211yesmaximal, characteristic, complemented
6#10592771C211yesmaximal, characteristic, complemented
7#10592737C212yesmaximal, complemented
8#10592735C212yesmaximal, complemented
9#10592714C211yesmaximal, characteristic, complemented
10C4×#3753C211yesmaximal, characteristic, complemented
11D4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12D4×#213—34nomaximal
13C22×C32×D4—21no—
14C22×C32×D4C2212yescomplemented
15C2×D4×C3⋊D3—21no—
16C22×C4⋊(C3⋊D3)C2212yescomplemented
17C32×#127C2211yescharacteristic, complemented
18C32×#121—21no—
19C32×#121C2212yescomplemented
20C32×#120—22no—
21C32×C4×D4C2211yescharacteristic
22C32×C4×D4C2211yescharacteristic, complemented
23C2×#3753—28no—
24C2×#3753C2211yescharacteristic
25C2×#3753C2212yescomplemented
26C2×#3753C2212yescomplemented
27C2×#3753C2212yescomplemented
28C2×#3753C2212yescomplemented
29#64139C2212yescomplemented
30C22×C3⋊(C3⋊C8)C2212yescomplemented
31#64104—21no—
32#64091—22no—
33#64083C2211yescharacteristic, complemented
34C4×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
35#63659C2212yescomplemented
36#63659C2211yescharacteristic, complemented
37#63657C2212yescomplemented
38#63636C2211yescharacteristic, complemented
39#63636C2212yescomplemented
40#63634C2211yescharacteristic, complemented
41C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
42C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
43C3×D42D314yescomplemented
44C22×#213—38no—
45D4×D12—34no—
46#5217—38no—
47#5073—34no—
48#5072—34no—
49#5038—38no—
50#5036—38no—
51#5015—34no—
52C4×#213—34no—
53D4×C3⋊C8—34no—
54C24×C32—22no—
55C23×C3⋊D3—22no—
56C2×C32×D4—24no—
57C2×C32×D4—22no—
58C2×C32×D4—21no—
59C2×C32×D4—22no—
60C2×C32×D4—21no—
61C2×C32×D4—42no—
62C2×C32×D4—44no—
63C2×C32×D4D411yescharacteristic
64C2×C32×D4C2312yes—
65C2×C32×D4C2311yescharacteristic
66C22×C32×C4—21no—
67C22×C32×C4C2312yes—
68C2×#638—22no—
69D4×C3⋊D3—44no—
70C2×C4⋊(C3⋊D3)—24no—
71C2×C4⋊(C3⋊D3)—21no—
72C2×C4⋊(C3⋊D3)—22no—
73C2×C4⋊(C3⋊D3)—21no—
74C2×C4⋊(C3⋊D3)—42no—
75C2×C4⋊(C3⋊D3)C2312yes—
76C2×C4⋊(C3⋊D3)C2311yescharacteristic
77C2×C4×C3⋊D3—21no—
78C32×C4⋊C4D411yescharacteristic
79C32×C4⋊C4C2311yescharacteristic
80C32×C22⋊C4—21no—
81C32×C22⋊C4—22no—
82C32×C22⋊C4D412yes—
83C32×C42C2311yescharacteristic
84#3753—24no—
85#3753—24no—
86#3753—24no—
87#3753—42no—
88#3753D414yesdirect product factor
89#3752—21no—
90#3752—22no—
91C4⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊C8)—24no—
93C2×C3⋊(C3⋊C8)C2312yes—
94C2×C3⋊(C3⋊C8)C2311yescharacteristic
95C2×C3⋊(C3⋊C8)C2311yescharacteristic
96D4×D8—91noSylow
97C22×C3×D4—24no—
98C22×C3×D4C2×D318yescomplemented
99C2×D3×D4—64no—
100C22×D12—38no—
101C3×#127C2×D314yescomplemented
102C3×#121—24no—
103C3×#121C2×D318yescomplemented
104C3×#120—28no—
105C3×C4×D4C2×D314yes—
106C3×C4×D4C2×D314yescomplemented
107C2×#213—34no—
108C2×#213—38no—
109C2×#213—38no—
110C2×#213—38no—
111C2×#213—38no—
112C2×#213—632no—
113#930—38no—
114C22×C3⋊C8—38no—
115#895—64no—
116#882—68no—
117#874—34no—
118C4×D12—34no—
119#832—34no—
120#832—38no—
121#830—38no—
122#809—34no—
123#809—38no—
124#807—34no—
125C4⋊(C3⋊C8)—34no—
126C4×C3⋊C8—34no—
127C23×C32—21no—
128C23×C32—22no—
129C23×C32—44no—
130C23×C32—44no—
131C23×C32—82no—
132C23×C32C2×D412yes—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—22no—
135C22×C3⋊D3—44no—
136C22×C3⋊D3—44no—
137C22×C3⋊D3—82no—
138C32×D4—22no—
139C32×D4—21no—
140C32×D4—22no—
141C32×D4—44no—
142C32×D4—44no—
143C32×D4—42no—
144C32×D4—81no—
145C32×D4D814yescomplemented
146C32×D4C2×D412yescomplemented
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—42no—
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D412yes—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2411yescharacteristic
155#638—44no—
156C2×C3⋊Dic3—21no—
157C4⋊(C3⋊D3)—22no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—22no—
160C4⋊(C3⋊D3)—44no—
161C4⋊(C3⋊D3)—42no—
162C4⋊(C3⋊D3)—81no—
163C4⋊(C3⋊D3)C2×D412yescomplemented
164C4×C3⋊D3—42no—
165C3⋊(C3⋊C8)—22no—
166C3⋊(C3⋊C8)—21no—
167C3⋊(C3⋊C8)C2×D412yescomplemented
168C22×D8—92no—
169D42—91no—
170D42C3⋊D311yescharacteristic, complemented
171#493—91no—
172#466—92no—
173#459—91no—
174#459—91no—
175#447—92no—
176#447—92no—
177C4×D8—91no—
178C8×D4—91no—
179C24×C3—28no—
180C23×D3—68no—
181C2×C3×D4—216no—
182C2×C3×D4—28no—
183C2×C3×D4—28no—
184C2×C3×D4—24no—
185C2×C3×D4—24no—
186C2×C3×D4—48no—
187C2×C3×D4—416no—
188C2×C3×D4#6714yes—
189C2×C3×D4C22×D318yes—
190C2×C3×D4C22×D314yes—
191C22×C3×C4—24no—
192C22×C3×C4C22×D318yes—
193C2×#67—68no—
194D3×D4—1216no—
195C2×D12—34no—
196C2×D12—38no—
197C2×D12—616no—
198C2×D12—64no—
199C2×D12—68no—
200C2×D12—64no—
201C2×D12—128no—
202C2×C4×D3—64no—
203C3×C4⋊C4#6714yes—
204C3×C4⋊C4C22×D314yes—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—28no—
207C3×C22⋊C4#6718yes—
208C3×C42C22×D314yes—
209#213—316no—
210#213—616no—
211#213—616no—
212#213—616no—
213#213—128no—
214(C2×D3)⋊C4—64no—
215(C2×D3)⋊C4—68no—
216C4⋊Dic3—34no—
217C2×C3⋊C8—34no—
218C2×C3⋊C8—34no—
219C2×C3⋊C8—38no—
220C2×C3⋊C8—616no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—88no—
230C22×C32C2×D814yescomplemented
231C22×C32C22×D411yescharacteristic
232C2×C3⋊D3—22no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3—42no—
236C2×C3⋊D3—42no—
237C2×C3⋊D3—42no—
238C2×C3⋊D3—88no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—41no—
242C32×C4C2×D812yescomplemented
243C32×C4C22×D411yescharacteristic
244C32×C4H(5,𝔽2)11yescharacteristic
245C3⋊Dic3—41no—
246C22×D4—21no—
247C22×D4—92no—
248C22×D4—181no—
249C22×D4C2×C3⋊D312yescomplemented
250C2×D8—91no—
251C2×D8—92no—
252C2×D8—92no—
253C2×D8—92no—
254C2×D8—92no—
255C2×D8—188no—
256C22×C8—92no—
257#127—91no—
258#127C2×C3⋊D311yescharacteristic, complemented
259#121—21no—
260#121—92no—
261#121—181no—
262#121C2×C3⋊D312yescomplemented
263#120—22no—
264#120—182no—
265C4×D4—91no—
266C4×D4C2×C3⋊D311yescharacteristic
267C4×D4C2×C3⋊D311yescharacteristic, complemented
268#107—91no—
269C4⋊C8—91no—
270#102—91no—
271#102—92no—
272#102—91no—
273#102—92no—
274C22⋊C8—92no—
275C4×C8—91no—
276C23×C3—24no—
277C23×C3—28no—
278C23×C3—416no—
279C23×C3—416no—
280C23×C3—88no—
281C23×C3C2×#6718yes—
282C22×D3—64no—
283C22×D3—68no—
284C22×D3—1216no—
285C22×D3—1216no—
286C22×D3—248no—
287C3×D4—28no—
288C3×D4—24no—
289C3×D4—28no—
290C3×D4—416no—
291C3×D4—416no—
292C3×D4—48no—
293C3×D4—84no—
294C3×D4#213116yescomplemented
295C3×D4D3×D418yescomplemented
296C2×C3×C4—28no—
297C2×C3×C4—28no—
298C2×C3×C4—24no—
299C2×C3×C4—48no—
300C2×C3×C4C2×#6714yes—
301C2×C3×C4C2×#6718yes—
302C2×C3×C4C2×#6714yes—
303C2×C3×C4C23×D314yes—
304#67—1216no—
305C2×Dic3—64no—
306D12—38no—
307D12—68no—
308D12—64no—
309D12—68no—
310D12—1216no—
311D12—128no—
312D12—244no—
313C4×D3—128no—
314C3⋊C8—38no—
315C3⋊C8—68no—
316C3⋊C8—64no—
317C2×C32—24no—
318C2×C32—42no—
319C2×C32—82no—
320C2×C32D4211yescharacteristic
321C2×C32C22×D811yescharacteristic
322C2×C32#57611yescharacteristic
323C3⋊D3—42no—
324C3⋊D3—82no—
325C24—22no—
326C24—182no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—91no—
335C2×D4—92no—
336C2×D4—184no—
337C2×D4—182no—
338C2×D4—182no—
339C2×D4—181no—
340C2×D4—181no—
341C2×D4—362no—
342C2×D4—364no—
343C2×D4#63811yescharacteristic
344C2×D4C22×C3⋊D312yes—
345C2×D4C22×C3⋊D311yescharacteristic
346C22×C4—21no—
347C22×C4—181no—
348C22×C4C22×C3⋊D312yes—
349D8—94no—
350D8—184no—
351D8—184no—
352D8—184no—
353D8—362no—
354C2×C8—91no—
355C2×C8—91no—
356C2×C8—92no—
357C2×C8—184no—
358C4⋊C4—91no—
359C4⋊C4#63811yescharacteristic
360C4⋊C4C22×C3⋊D311yescharacteristic
361C22⋊C4—21no—
362C22⋊C4—22no—
363C22⋊C4—181no—
364C22⋊C4—182no—
365C22⋊C4#63812yes—
366C42C22×C3⋊D311yescharacteristic
367C22×C3—28no—
368C22×C3—28no—
369C22×C3—28no—
370C22×C3—44no—
371C22×C3—44no—
372C22×C3—48no—
373C22×C3—48no—
374C22×C3—48no—
375C22×C3—832no—
376C22×C3C2×#213116yescomplemented
377C22×C3C22×#6714yes—
378C2×D3—68no—
379C2×D3—124no—
380C2×D3—124no—
381C2×D3—128no—
382C2×D3—128no—
383C2×D3—128no—
384C2×D3—2432no—
385C3×C4—28no—
386C3×C4—24no—
387C3×C4—44no—
388C3×C4C2×#21318yescomplemented
389C3×C4C2×D3×D414yes—
390C3×C4#100414yes—
391Dic3—124no—
392C32D4×D811yescharacteristic, complemented, Sylow
393C23—21no—
394C23—22no—
395C23—44no—
396C23—44no—
397C23—82no—
398C23—181no—
399C23—182no—
400C23—364no—
401C23—364no—
402C23—722no—
403C23C2×#63812yes—
404D4—22no—
405D4—21no—
406D4—22no—
407D4—44no—
408D4—44no—
409D4—42no—
410D4—81no—
411D4—92no—
412D4—182no—
413D4—181no—
414D4—182no—
415D4—364no—
416D4—364no—
417D4—362no—
418D4—721no—
419D4#375314yesdirect product factor
420D4D4×C3⋊D312yescomplemented
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—181no—
426C2×C4—362no—
427C2×C4C2×#63811yescharacteristic
428C2×C4C2×#63812yes—
429C2×C4C2×#63811yescharacteristic
430C2×C4C23×C3⋊D311yescharacteristic
431C8—92no—
432C8—182no—
433C8—181no—
434C2×C3—216no—
435C2×C3—48no—
436C2×C3—88no—
437C2×C3C22×#21314yes—
438C2×C3D4×#6714yes—
439C2×C3#583514yes—
440D3—128no—
441D3—248no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—88no—
451C22—182no—
452C22—361no—
453C22—361no—
454C22—362no—
455C22—362no—
456C22—362no—
457C22—728no—
458C22C2×#375314yescomplemented
459C22C22×#63811yescharacteristic
460C4—22no—
461C4—21no—
462C4—41no—
463C4—361no—
464C4C2×#375312yescomplemented
465C4C2×D4×C3⋊D311yescharacteristic
466C4#6436111yescharacteristic
467C3D4×#21314yescomplemented
468C2—24no—
469C2—42no—
470C2—82no—
471C2—362no—
472C2—722no—
473C2C22×#375311yescharacteristic
474C2D4×#63811yescharacteristic
475C2#1059630611yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×D42C2
socleC22×C32C22×D4
hypercenterD42C3⋊D3
hypocenterC32D4×D8
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C32×D42C2semidirect
2. C22×#3753C2semidirect
3. D4×C4⋊(C3⋊D3)C2semidirect
4. #10592916C2semidirect
5. #10592772C2semidirect
6. #10592771C2semidirect
7. #10592737C2semidirect
8. #10592735C2semidirect
9. #10592714C2semidirect
10. C4×#3753C2semidirect
11. D4×C3⋊(C3⋊C8)C2semidirect
14. C22×C32×D4C22semidirect
16. C22×C4⋊(C3⋊D3)C22semidirect
17. C32×#127C22semidirect
19. C32×#121C22semidirect
22. C32×C4×D4C22semidirect
25. C2×#3753C22semidirect
26. C2×#3753C22semidirect
27. C2×#3753C22semidirect
28. C2×#3753C22semidirect
29. #64139C22semidirect
30. C22×C3⋊(C3⋊C8)C22semidirect
33. #64083C22semidirect
34. C4×C4⋊(C3⋊D3)C22semidirect
35. #63659C22semidirect
36. #63659C22semidirect
37. #63657C22semidirect
38. #63636C22semidirect
39. #63636C22semidirect
40. #63634C22semidirect
41. C4⋊(C3⋊(C3⋊C8))C22semidirect
42. C4×C3⋊(C3⋊C8)C22semidirect
43. C3×D42D3semidirect
88. #3753D4direct (N × H)
98. C22×C3×D4C2×D3semidirect
101. C3×#127C2×D3semidirect
103. C3×#121C2×D3semidirect
106. C3×C4×D4C2×D3semidirect
145. C32×D4D8semidirect
146. C32×D4C2×D4semidirect
163. C4⋊(C3⋊D3)C2×D4semidirect
167. C3⋊(C3⋊C8)C2×D4semidirect
170. D42C3⋊D3semidirect
230. C22×C32C2×D8semidirect
242. C32×C4C2×D8semidirect
249. C22×D4C2×C3⋊D3semidirect
258. #127C2×C3⋊D3semidirect
262. #121C2×C3⋊D3semidirect
267. C4×D4C2×C3⋊D3semidirect
294. C3×D4#213semidirect
295. C3×D4D3×D4semidirect
376. C22×C3C2×#213semidirect
388. C3×C4C2×#213semidirect
392. C32D4×D8semidirect
419. D4#3753direct (N × H)
420. D4D4×C3⋊D3semidirect
458. C22C2×#3753semidirect
464. C4C2×#3753semidirect
467. C3D4×#213semidirect

Maximal subgroups

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

subgroupindex
C32×D422
C22×#37532
D4×C4⋊(C3⋊D3)2
#105929162
#105927722
#105927712
#105927372
#105927352
#105927142
C4×#37532
D4×C3⋊(C3⋊C8)2
D4×#2133