Finite Group Databasebeta

← #16854939

Subgroups — #16854939

Order 3968.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D31C211yesmaximal, characteristic, complemented
2C2×#11758491C212yesmaximal, complemented
3#14521562C211yesmaximal, characteristic, complemented
4#14521199C211yesmaximal, characteristic, complemented
5#14521198C212yesmaximal, complemented
6#14521185C211yesmaximal, characteristic, complemented
7#14521184C212yesmaximal, complemented
8#14521183C211yesmaximal, characteristic, complemented
9#14521052C212yesmaximal, complemented
10#14521011C211yesmaximal, characteristic, complemented
11C31×#450C211yesmaximal, characteristic, complemented
12C24×D31—21no—
13C22×D4×C31C2211yescharacteristic
14C22×C22⋊D31—21no—
15C2×#92499C2211yescharacteristic, complemented
16C2×D4×D31—28no—
17C2×D4×D31—21no—
18C2×D4×D31—21no—
19C2×D4×D31C2212yes—
20C2×D4×D31C2212yes—
21C22×D124C2211yescharacteristic
22C22×C4×D31C2211yescharacteristic
23C2×SD16×C31C2212yescomplemented
24C31×C22⋊Q8C2211yescharacteristic, complemented
25#11758523C2211yescharacteristic, complemented
26C2×#92481C2212yescomplemented
27C2×#92479C2212yescomplemented
28#11758507C2211yescharacteristic, complemented
29#11758491—28no—
30C2×D248C2212yescomplemented
31C2×#92469C2212yescomplemented
32#11758472C2211yescharacteristic, complemented
33#11758471—21no—
34#11758470—21no—
35#11758469C2211yescharacteristic, complemented
36#11758459—21no—
37(C22⋊D31)⋊C4—21no—
38C31×#102C2212yescomplemented
39C31×C22⋊C8C2211yescharacteristic, complemented
40#11758420C2212yescomplemented
41C22⋊(C31⋊C8)C2211yescharacteristic, complemented
42#11758409C2212yescomplemented
43(C2×D31)⋊C8C2212yescomplemented
44#11758397C2212yescomplemented
45C24×C31—21no—
46C23×D31—28no—
47C23×D31—21no—
48C23×D31—22no—
49C23×D31—22no—
50C23×D31—44no—
51C23×D31—44no—
52C23×D31D411yescharacteristic
53C2×Q8×C31C2311yescharacteristic, complemented
54C2×D4×C31—24no—
55C2×D4×C31—21no—
56C2×D4×C31C2312yes—
57C22×C4×C31C2311yescharacteristic
58C2×C22⋊D31—28no—
59C2×C22⋊D31—21no—
60C2×C22⋊D31—22no—
61C2×C22⋊D31—22no—
62C22×Dic31D411yescharacteristic
63#92499—24no—
64D4×D31—28no—
65D4×D31—28no—
66D4×D31—24no—
67D4×D31—24no—
68D4×D31—24no—
69D4×D31D418yescomplemented
70C2×D124—21no—
71C2×D124—24no—
72C2×D124—21no—
73C2×D124—21no—
74C2×D124C2312yes—
75C2×D124C2311yescharacteristic, complemented
76C2×C4×D31—24no—
77C2×C4×D31—21no—
78C2×C4×D31D411yescharacteristic
79C2×C4×D31D411yescharacteristic
80C2×C4×D31C2311yescharacteristic
81C2×C4×D31C2311yescharacteristic
82SD16×C31—24no—
83C2×C8×C31C2312yescomplemented
84C4⋊C4×C31—21no—
85C4⋊C4×C31C2311yescharacteristic, complemented
86C22⋊C4×C31—21no—
87#92481—24no—
88#92479—24no—
89(C2×D31)⋊C4—21no—
90(C2×D31)⋊C4—21no—
91(C2×D31)⋊C4—21no—
92(C2×D31)⋊C4—21no—
93C4⋊Dic31C2311yescharacteristic, complemented
94Dic31⋊C4—21no—
95C4×Dic31—21no—
96C2×C31⋊C8C2312yescomplemented
97D248—24no—
98#92469—24no—
99C23×C31—24no—
100C23×C31—22no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31C2×D411yescharacteristic
104C22×D31—22no—
105C22×D31—22no—
106C22×D31—22no—
107C22×D31—24no—
108C22×D31—28no—
109C22×D31—22no—
110C22×D31—28no—
111C22×D31—21no—
112C22×D31—432no—
113C22×D31—44no—
114C22×D31—44no—
115C22×D31—44no—
116C22×D31—42no—
117C22×D31—42no—
118C22×D31—44no—
119C22×D31C2×D411yescharacteristic
120C22×D31C2×D411yescharacteristic
121Q8×C31—22no—
122D4×C31—24no—
123D4×C31—22no—
124D4×C31C2×D414yes—
125C2×C4×C31—22no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31C2×D411yescharacteristic
129C2×C4×C31C2411yescharacteristic
130C22⋊D31—28no—
131C22⋊D31—28no—
132C22⋊D31—42no—
133C2×Dic31—22no—
134C2×Dic31—21no—
135C2×Dic31C2×D411yescharacteristic
136C2×Dic31C2×D411yescharacteristic
137D124—24no—
138D124—22no—
139D124—22no—
140D124—42no—
141D124—42no—
142D124C2×D414yes—
143C4×D31—22no—
144C4×D31—22no—
145C4×D31—22no—
146C4×D31—42no—
147C4×D31C2×D414yes—
148C8×C31—22no—
149C31⋊C8—22no—
150#3003—311nomaximal, Sylow
151C22×C31—21no—
152C22×C31—21no—
153C22×C31—24no—
154C22×C31—48no—
155C22×C31—42no—
156C22×C31—42no—
157C22×C31AGL(1,ℤ/8)12yescomplemented
158C22×C31C22×D411yescharacteristic
159C2×D31—28no—
160C2×D31—21no—
161C2×D31—21no—
162C2×D31—21no—
163C2×D31—21no—
164C2×D31—24no—
165C2×D31—416no—
166C2×D31—416no—
167C2×D31—48no—
168C2×D31—42no—
169C2×D31—42no—
170C2×D31—41no—
171C2×D31—41no—
172C2×D31—41no—
173C2×D31#12012yes—
174C2×D31AGL(1,ℤ/8)12yescomplemented
175C4×C31—21no—
176C4×C31—41no—
177C4×C31—41no—
178C4×C31C22×D412yes—
179Dic31—21no—
180Dic31—41no—
181Dic31#12012yes—
182C23×D4—311no—
183C2×AGL(1,ℤ/8)—312no—
184#525—311no—
185#450—312no—
186#450—311no—
187#450D3111yescharacteristic, complemented
188#447—311no—
189#447—312no—
190#447—311no—
191#418—312no—
192#407—311no—
193C2×C31—22no—
194C2×C31—44no—
195C2×C31C2×#12011yescharacteristic
196C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
197C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
198D31—22no—
199D31—22no—
200D31—44no—
201D31—81no—
202C25—621no—
203C2×#41—311no—
204C22×D4—312no—
205C22×D4—312no—
206C22×D4—311no—
207C22×D4—621no—
208C22×D4—621no—
209C22×D4—621no—
210C22×D4—628no—
211C22×D4C2×D3111yescharacteristic
212C23×C4—311no—
213AGL(1,ℤ/8)—628no—
214C2×SD16—312no—
215C2×SD16C2×D3112yescomplemented
216C2×D8—312no—
217C2×D8—312no—
218C2×M16—312no—
219#123—621no—
220C22⋊Q8—311no—
221C22⋊Q8C2×D3111yescharacteristic, complemented
222#121—311no—
223#121—311no—
224#120—621no—
225C4×D4—621no—
226C4×D4—621no—
227#117—311no—
228#102—312no—
229#102—312no—
230#102—312no—
231#102C2×D3112yescomplemented
232C22⋊C8—311no—
233C22⋊C8—312no—
234C22⋊C8C2×D3111yescharacteristic, complemented
235C31#300311yescharacteristic, complemented, Sylow
236C24—21no—
237C24—311no—
238C24—622no—
239C24—622no—
240C24—628no—
241C24—621no—
242C24—1244no—
243C24—1244no—
244#41—624no—
245C2×Q8C22×D3111yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—318no—
249C2×D4—312no—
250C2×D4—311no—
251C2×D4—624no—
252C2×D4—621no—
253C2×D4—621no—
254C2×D4—624no—
255C2×D4—622no—
256C2×D4—622no—
257C2×D4—624no—
258C2×D4—628no—
259C2×D4—628no—
260C2×D4—628no—
261C2×D4—624no—
262C2×D4—621no—
263C2×D4—621no—
264C2×D4C22×D3112yes—
265C22×C4—311no—
266C22×C4—311no—
267C22×C4—311no—
268C22×C4—311no—
269C22×C4—311no—
270C22×C4—624no—
271C22×C4—621no—
272C22×C4C22×D3111yescharacteristic
273SD16—24no—
274SD16—624no—
275D8—624no—
276D8—624no—
277M16—624no—
278C2×C8—312no—
279C2×C8C22×D3112yescomplemented
280C4⋊C4—21no—
281C4⋊C4—311no—
282C4⋊C4—621no—
283C4⋊C4C22×D3111yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—621no—
286C22⋊C4—621no—
287C22⋊C4—621no—
288C22⋊C4—621no—
289C42—621no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—311no—
295C23—311no—
296C23—622no—
297C23—622no—
298C23—628no—
299C23—624no—
300C23—628no—
301C23—622no—
302C23—622no—
303C23—621no—
304C23—1242no—
305C23—1244no—
306C23—12432no—
307C23—1244no—
308C23—1244no—
309C23—1244no—
310C23—1242no—
311C23D4×D3111yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—314no—
316D4—624no—
317D4—628no—
318D4—628no—
319D4—622no—
320D4—622no—
321D4—1242no—
322D4—1242no—
323D4—1242no—
324D4D4×D3114yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—311no—
329C2×C4—314no—
330C2×C4—311no—
331C2×C4—622no—
332C2×C4—622no—
333C2×C4—622no—
334C2×C4—622no—
335C2×C4—621no—
336C2×C4—1242no—
337C2×C4D4×D3111yescharacteristic
338C2×C4C23×D3111yescharacteristic
339C8—22no—
340C8—622no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—312no—
348C22—312no—
349C22—628no—
350C22—621no—
351C22—621no—
352C22—621no—
353C22—621no—
354C22—624no—
355C22—1248no—
356C22—12416no—
357C22—12416no—
358C22—1242no—
359C22—1242no—
360C22—1241no—
361C22—1241no—
362C22—1241no—
363C22#1175849112yescomplemented
364C22C2×D4×D3111yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—312no—
369C4—621no—
370C4—1241no—
371C4C2×D4×D3112yes—
372C2—22no—
373C2—44no—
374C2—622no—
375C2—622no—
376C2—1244no—
377C2—2481no—
378C2#120×D3111yescharacteristic
379C2AGL(1,ℤ/8)×D3111yescharacteristic
380C2C2×#1175849111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#450C2
socleC22×C31C22×D4
hypercenter#450D31
hypocenterC31#3003
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D31C2semidirect
2. C2×#11758491C2semidirect
3. #14521562C2semidirect
4. #14521199C2semidirect
5. #14521198C2semidirect
6. #14521185C2semidirect
7. #14521184C2semidirect
8. #14521183C2semidirect
9. #14521052C2semidirect
10. #14521011C2semidirect
11. C31×#450C2semidirect
15. C2×#92499C22semidirect
23. C2×SD16×C31C22semidirect
24. C31×C22⋊Q8C22semidirect
25. #11758523C22semidirect
26. C2×#92481C22semidirect
27. C2×#92479C22semidirect
28. #11758507C22semidirect
30. C2×D248C22semidirect
31. C2×#92469C22semidirect
32. #11758472C22semidirect
35. #11758469C22semidirect
38. C31×#102C22semidirect
39. C31×C22⋊C8C22semidirect
40. #11758420C22semidirect
41. C22⋊(C31⋊C8)C22semidirect
42. #11758409C22semidirect
43. (C2×D31)⋊C8C22semidirect
44. #11758397C22semidirect
53. C2×Q8×C31C23semidirect
69. D4×D31D4semidirect
75. C2×D124C23semidirect
83. C2×C8×C31C23semidirect
85. C4⋊C4×C31C23semidirect
93. C4⋊Dic31C23semidirect
96. C2×C31⋊C8C23semidirect
157. C22×C31AGL(1,ℤ/8)semidirect
174. C2×D31AGL(1,ℤ/8)semidirect
187. #450D31semidirect
215. C2×SD16C2×D31semidirect
221. C22⋊Q8C2×D31semidirect
231. #102C2×D31semidirect
234. C22⋊C8C2×D31semidirect
235. C31#3003semidirect
245. C2×Q8C22×D31semidirect
279. C2×C8C22×D31semidirect
283. C4⋊C4C22×D31semidirect
363. C22#11758491semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D312
C2×#117584912
#145215622
#145211992
#145211982
#145211852
#145211842
#145211832
#145210522
#145210112
C31×#4502
#300331