Finite Group Databasebeta

← D31×#481 #16855607

Subgroups — D31×#481 #16855607

Order 3968.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#121×D31C211yesmaximal, characteristic
2C2×C4⋊C4×D31C211yesmaximal, characteristic, complemented
3#14521369C212yesmaximal, complemented
4#14521368C211yesmaximal, characteristic, complemented
5#14521367C211yesmaximal, characteristic, complemented
6#14521366C212yesmaximal, complemented
7#14521365C211yesmaximal, characteristic, complemented
8#106×D31C212yesmaximal, complemented
9#102×D31C212yesmaximal, complemented
10C22⋊C8×D31C211yesmaximal, characteristic, complemented
11C31×#481C211yesmaximal, characteristic, complemented
12C2×D4×D31—21no—
13C2×D4×D31C2211yescharacteristic
14C22×C4×D31—21no—
15C22×C4×D31C2211yescharacteristic
16C31×#121C2211yescharacteristic
17C2×C4⋊C4×C31C2211yescharacteristic, complemented
18#11758516—21no—
19#11758515C2211yescharacteristic
20#11758507C2211yescharacteristic
21C2×C4⋊Dic31C2211yescharacteristic, complemented
22C2×Dic31⋊C4—21no—
23C2×C8×D31C2212yes—
24#11758472C2211yescharacteristic
25C4⋊C4×D31—21no—
26C4⋊C4×D31—21no—
27C4⋊C4×D31C2212yes—
28C4⋊C4×D31C2211yescharacteristic
29C4⋊C4×D31C2212yescomplemented
30#11758461—21no—
31C22⋊C4×D31—21no—
32C31×#106C2212yescomplemented
33C31×#102C2212yescomplemented
34C31×C22⋊C8C2211yescharacteristic, complemented
35#11758420C2212yescomplemented
36C22⋊(C31⋊C8)C2211yescharacteristic, complemented
37#11758409C2212yescomplemented
38(C2×D31)⋊C8C2212yescomplemented
39#11758405C2212yescomplemented
40#11758397C2212yescomplemented
41#11758396C2212yescomplemented
42#11758396C2212yescomplemented
43C23×D31—21no—
44C23×D31D411yescharacteristic
45C2×D4×C31—21no—
46C2×D4×C31C2311yescharacteristic
47C22×C4×C31—21no—
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—21no—
50C2×C22⋊D31—21no—
51C2×C22⋊D31—21no—
52C22×Dic31—21no—
53C22×Dic31D411yescharacteristic
54D4×D31—24no—
55D4×D31—44no—
56C2×D124—21no—
57C2×D124C2311yescharacteristic
58C2×C4×D31—24no—
59C2×C4×D31—21no—
60C2×C4×D31—22no—
61C2×C4×D31—22no—
62C2×C4×D31—44no—
63C2×C4×D31D411yescharacteristic
64C2×C4×D31D411yescharacteristic
65C2×C4×D31C2311yescharacteristic
66C2×C4×D31C2311yescharacteristic
67C2×C8×C31C2312yes—
68C4⋊C4×C31—21no—
69C4⋊C4×C31C2311yescharacteristic
70C4⋊C4×C31C2312yes—
71C22⋊C4×C31—21no—
72C22⋊Dic31—21no—
73(C2×D31)⋊C4—21no—
74(C2×D31)⋊C4—21no—
75C4⋊Dic31—21no—
76C4⋊Dic31C2311yescharacteristic
77C4⋊Dic31C2312yes—
78Dic31⋊C4—21no—
79Dic31⋊C4—22no—
80Dic31⋊C4—22no—
81C2×C31⋊C8C2312yes—
82C8×D31—24no—
83C23×C31—21no—
84C23×C31C2×D411yescharacteristic
85C22×D31—22no—
86C22×D31—22no—
87C22×D31—21no—
88C22×D31—42no—
89C22×D31—42no—
90C22×D31—42no—
91C22×D31SD1614yes—
92C22×D31C2×D411yescharacteristic
93C22×D31C2×D411yescharacteristic
94D4×C31—22no—
95D4×C31—42no—
96C2×C4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—21no—
99C2×C4×C31—42no—
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2411yescharacteristic
102C22⋊D31—42no—
103C22⋊D31—42no—
104C22⋊D31—42no—
105C2×Dic31—22no—
106C2×Dic31—22no—
107C2×Dic31—21no—
108C2×Dic31—42no—
109C2×Dic31C2×D411yescharacteristic
110C2×Dic31C2×D411yescharacteristic
111D124—22no—
112D124—42no—
113C4×D31—22no—
114C4×D31—22no—
115C4×D31—22no—
116C4×D31—44no—
117C4×D31—42no—
118C4×D31—44no—
119C4×D31#4114yes—
120C8×C31—22no—
121C31⋊C8—22no—
122C2×#481—311nomaximal, Sylow
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31C2×SD1612yes—
129C22×C31C22×D411yescharacteristic
130C2×D31—21no—
131C2×D31—21no—
132C2×D31—28no—
133C2×D31—41no—
134C2×D31—41no—
135C2×D31—41no—
136C2×D31—84no—
137C2×D31#12312yes—
138C2×D31C2×SD1612yes—
139C2×D31C2×SD1612yes—
140C2×D31AGL(1,ℤ/8)12yes—
141C4×C31—21no—
142C4×C31—41no—
143C4×C31—42no—
144C4×C31C2×#4112yes—
145Dic31—21no—
146Dic31—41no—
147Dic31—42no—
148Dic31#12312yes—
149C2×#121—311no—
150C22×C4⋊C4—311no—
151#481—311no—
152#481—311no—
153#481—312no—
154#481—312no—
155#481—311no—
156#481D3111yescharacteristic, direct product factor
157C2×#106—312no—
158C2×#102—312no—
159C2×C22⋊C8—311no—
160C2×C31—22no—
161C2×C31—81no—
162C2×C31C2×#12311yescharacteristic
163C2×C31C22×SD1611yescharacteristic
164C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
165D31—22no—
166D31—81no—
167D31#48114yesdirect product factor
168C22×D4—311no—
169C22×D4—621no—
170C23×C4—311no—
171C23×C4—621no—
172C22×C8—312no—
173#121—311no—
174#121—311no—
175#121—311no—
176#121—621no—
177#121—621no—
178#121C2×D3111yescharacteristic
179C2×C4⋊C4—311no—
180C2×C4⋊C4—312no—
181C2×C4⋊C4—312no—
182C2×C4⋊C4—311no—
183C2×C4⋊C4—621no—
184C2×C4⋊C4—621no—
185C2×C4⋊C4—621no—
186C2×C4⋊C4C2×D3111yescharacteristic, complemented
187C2×C22⋊C4—621no—
188#106—312no—
189#106—312no—
190#106—312no—
191#106C2×D3112yescomplemented
192#102—312no—
193#102—312no—
194#102—312no—
195#102C2×D3112yescomplemented
196C22⋊C8—311no—
197C22⋊C8—312no—
198C22⋊C8C2×D3111yescharacteristic, complemented
199C31C2×#48111yescharacteristic, complemented, Sylow
200C24—311no—
201C24—621no—
202C2×D4—21no—
203C2×D4—311no—
204C2×D4—621no—
205C2×D4—624no—
206C2×D4—621no—
207C2×D4—621no—
208C2×D4—621no—
209C2×D4—1244no—
210C2×D4C22×D3111yescharacteristic
211C22×C4—21no—
212C22×C4—311no—
213C22×C4—311no—
214C22×C4—311no—
215C22×C4—311no—
216C22×C4—311no—
217C22×C4—624no—
218C22×C4—621no—
219C22×C4—622no—
220C22×C4—622no—
221C22×C4—621no—
222C22×C4—1244no—
223C22×C4C22×D3111yescharacteristic
224C2×C8—312no—
225C2×C8—624no—
226C2×C8C22×D3112yes—
227C4⋊C4—21no—
228C4⋊C4—312no—
229C4⋊C4—311no—
230C4⋊C4—622no—
231C4⋊C4—622no—
232C4⋊C4—621no—
233C4⋊C4—621no—
234C4⋊C4C22×D3112yes—
235C4⋊C4C22×D3111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—621no—
238C22⋊C4—621no—
239C22⋊C4—621no—
240C23—21no—
241C23—311no—
242C23—314no—
243C23—311no—
244C23—622no—
245C23—622no—
246C23—621no—
247C23—1242no—
248C23—1242no—
249C23—1242no—
250C23D4×D3111yescharacteristic
251D4—22no—
252D4—42no—
253D4—622no—
254D4—1242no—
255D4—1242no—
256D4—1242no—
257D4—1242no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—311no—
263C2×C4—314no—
264C2×C4—311no—
265C2×C4—622no—
266C2×C4—622no—
267C2×C4—622no—
268C2×C4—622no—
269C2×C4—621no—
270C2×C4—622no—
271C2×C4—1244no—
272C2×C4—1242no—
273C2×C4—1244no—
274C2×C4—1242no—
275C2×C4D4×D3111yescharacteristic
276C2×C4C23×D3111yescharacteristic
277C8—22no—
278C8—622no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—312no—
285C22—312no—
286C22—312no—
287C22—312no—
288C22—628no—
289C22—621no—
290C22—621no—
291C22—1241no—
292C22—1241no—
293C22—1241no—
294C22—2484no—
295C22SD16×D3112yes—
296C22C2×D4×D3111yescharacteristic
297C4—21no—
298C4—41no—
299C4—42no—
300C4—312no—
301C4—621no—
302C4—1241no—
303C4—1242no—
304C4#41×D3112yes—
305C2—22no—
306C2—81no—
307C2—314no—
308C2—622no—
309C2—2481no—
310C2#123×D3111yescharacteristic
311C2C2×SD16×D3111yescharacteristic
312C2AGL(1,ℤ/8)×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#481C2
socleC22×C31C22×D4
prime radical#121×D31C2
hypercenter#481D31
hypocenterC31C2×#481
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D31C2semidirect
3. #14521369C2semidirect
4. #14521368C2semidirect
5. #14521367C2semidirect
6. #14521366C2semidirect
7. #14521365C2semidirect
8. #106×D31C2semidirect
9. #102×D31C2semidirect
10. C22⋊C8×D31C2semidirect
11. C31×#481C2semidirect
17. C2×C4⋊C4×C31C22semidirect
21. C2×C4⋊Dic31C22semidirect
29. C4⋊C4×D31C22semidirect
32. C31×#106C22semidirect
33. C31×#102C22semidirect
34. C31×C22⋊C8C22semidirect
35. #11758420C22semidirect
36. C22⋊(C31⋊C8)C22semidirect
37. #11758409C22semidirect
38. (C2×D31)⋊C8C22semidirect
39. #11758405C22semidirect
40. #11758397C22semidirect
41. #11758396C22semidirect
42. #11758396C22semidirect
156. #481D31direct (N × H)
167. D31#481direct (N × H)
186. C2×C4⋊C4C2×D31semidirect
191. #106C2×D31semidirect
195. #102C2×D31semidirect
198. C22⋊C8C2×D31semidirect
199. C31C2×#481semidirect

Maximal subgroups

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

subgroupindex
#121×D312
C2×C4⋊C4×D312
#145213692
#145213682
#145213672
#145213662
#145213652
#106×D312
#102×D312
C22⋊C8×D312
C31×#4812
C2×#48131