Finite Group Databasebeta

← #16855804

Subgroups — #16855804

Order 3968.

488 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 121 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#14521600C211yesmaximal, characteristic, complemented
2C4×D4×D31C211yesmaximal, characteristic, complemented
3#14521369C212yesmaximal, complemented
4#14521336C211yesmaximal, characteristic, complemented
5#14521245C211yesmaximal, characteristic, complemented
6#14521244C211yesmaximal, characteristic, complemented
7#14521184C212yesmaximal, complemented
8#14521160C212yesmaximal, complemented
9#14521158C212yesmaximal, complemented
10#14521105C211yesmaximal, characteristic, complemented
11C31×#476C211yesmaximal, characteristic, complemented
12C2×#92499—21no—
13C2×D4×D31C2211yescharacteristic
14C22×C4×D31—21no—
15C31×C4⋊Q8C2211yescharacteristic, complemented
16C4×D4×C31C2211yescharacteristic
17#11758524—21no—
18D4×Dic31C2211yescharacteristic
19C2×#92479C2212yes—
20C4×C22⋊D31—21no—
21C2×D248C2212yes—
22#11758472—21no—
23#11758472C2212yes—
24#11758470—21no—
25#11758470C2211yescharacteristic
26#11758470C2212yescomplemented
27C4⋊C4×D31C2211yescharacteristic
28(C22⋊D31)⋊C4—21no—
29C22⋊C4×D31—21no—
30#11758451C2211yescharacteristic, complemented
31C4×D124C2211yescharacteristic
32C42×D31C2211yescharacteristic
33C31×#106C2212yescomplemented
34C31×C4⋊C8C2211yescharacteristic, complemented
35C31×#102C2212yes—
36#11758409C2212yescomplemented
37(C2×D31)⋊C8C2212yes—
38#11758403C2212yescomplemented
39#11758397C2212yes—
40#11758397C2212yescomplemented
41#11758396C2212yescomplemented
42C4⋊(C31⋊C8)C2211yescharacteristic, complemented
43C23×D31—21no—
44C2×Q8×C31—21no—
45C2×D4×C31C2311yescharacteristic
46C22×C4×C31—21no—
47C2×C22⋊D31—21no—
48C22×Dic31—21no—
49#92499—44no—
50D4×D31—22no—
51D4×D31—22no—
52C2×D124—22no—
53C2×D124—21no—
54C2×D124—21no—
55C2×D124C2312yes—
56C2×D124C2311yescharacteristic
57C2×C4×D31—21no—
58C2×C4×D31—21no—
59C2×C4×D31—21no—
60C2×C4×D31—22no—
61C2×C4×D31—44no—
62C2×C4×D31D411yescharacteristic
63C2×C4×D31D411yescharacteristic
64C2×C4×D31C2311yescharacteristic
65C2×C8×C31C2312yes—
66C4⋊C4×C31—21no—
67C4⋊C4×C31C2312yes—
68C4⋊C4×C31C2311yescharacteristic
69C22⋊C4×C31—21no—
70C42×C31C2311yescharacteristic
71C22⋊Dic31—21no—
72#92479—24no—
73(C2×D31)⋊C4—21no—
74(C2×D31)⋊C4—21no—
75(C2×D31)⋊C4—22no—
76(C2×D31)⋊C4—22no—
77C4⋊Dic31C2311yescharacteristic
78Dic31⋊C4—21no—
79C4×Dic31D411yescharacteristic
80C4×Dic31D411yescharacteristic
81C4×Dic31C2311yescharacteristic
82C2×C31⋊C8C2312yes—
83D248—24no—
84C23×C31—21no—
85C22×D31—24no—
86C22×D31—21no—
87C22×D31—22no—
88C22×D31—42no—
89C22×D31—42no—
90C22×D31C2×D411yescharacteristic
91Q8×C31—42no—
92D4×C31—21no—
93D4×C31#4112yescomplemented
94C2×C4×C31—21no—
95C2×C4×C31—22no—
96C2×C4×C31—42no—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2411yescharacteristic
100C22⋊D31—24no—
101C2×Dic31—21no—
102C2×Dic31—42no—
103C2×Dic31C2×D411yescharacteristic
104C2×Dic31C2×D411yescharacteristic
105C2×Dic31C2×D411yescharacteristic
106D124—21no—
107D124—22no—
108D124—22no—
109D124—42no—
110D124—44no—
111D124—42no—
112D124#4112yescomplemented
113C4×D31—22no—
114C4×D31—22no—
115C4×D31—21no—
116C4×D31—21no—
117C4×D31—42no—
118C4×D31—44no—
119C4×D31—42no—
120C4×D31—42no—
121C8×C31—22no—
122C31⋊C8—22no—
123#3109—311nomaximal, Sylow
124C22×C31—22no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31C22×D411yescharacteristic
128C2×D31—21no—
129C2×D31—21no—
130C2×D31—22no—
131C2×D31—48no—
132C2×D31—41no—
133C2×D31—41no—
134C2×D31—42no—
135C2×D31—42no—
136C2×D31—42no—
137C2×D31AGL(1,ℤ/8)12yes—
138C4×C31—21no—
139C4×C31—41no—
140C4×C31—42no—
141C4×C31AGL(1,ℤ/8)12yes—
142C4×C31C2×#4111yescharacteristic
143C4×C31C2×#4111yescharacteristic
144Dic31—21no—
145Dic31—21no—
146Dic31—21no—
147Dic31—41no—
148#532—311no—
149C2×C4×D4—311no—
150#481—312no—
151#476—311no—
152#476D3111yescharacteristic, complemented
153#459—311no—
154#459—311no—
155#447—312no—
156#442—312no—
157#442—312no—
158#432—311no—
159C2×C31—42no—
160C2×C31#52511yescharacteristic
161C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
162C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
163D31—22no—
164D31—42no—
165D31—82no—
166C2×#41—621no—
167C22×D4—311no—
168C23×C4—621no—
169C2×D8—312no—
170C2×D8—312no—
171C4⋊Q8C2×D3111yescharacteristic, complemented
172#127—311no—
173#124—621no—
174#121—312no—
175#121—621no—
176C4×D4—311no—
177C4×D4—311no—
178C4×D4—312no—
179C4×D4—311no—
180C4×D4—621no—
181C4×D4—621no—
182C4×D4—621no—
183C4×D4C2×D3111yescharacteristic
184C2×C4⋊C4—311no—
185C2×C22⋊C4—621no—
186C2×C42—311no—
187#106—312no—
188#106C2×D3112yescomplemented
189C4⋊C8—311no—
190C4⋊C8C2×D3111yescharacteristic, complemented
191#102—312no—
192#102—312no—
193#102—312no—
194#102C2×D3112yes—
195C22⋊C8—312no—
196#97—312no—
197C31#310911yescharacteristic, complemented, Sylow
198C24—621no—
199#41—1244no—
200C2×Q8—21no—
201C2×D4—311no—
202C2×D4—312no—
203C2×D4—622no—
204C2×D4—621no—
205C2×D4—622no—
206C2×D4—622no—
207C2×D4—621no—
208C2×D4—621no—
209C2×D4C22×D3111yescharacteristic
210C22×C4—21no—
211C22×C4—311no—
212C22×C4—311no—
213C22×C4—311no—
214C22×C4—621no—
215C22×C4—622no—
216C22×C4—621no—
217C22×C4—621no—
218C22×C4—621no—
219C22×C4—1244no—
220D8—624no—
221D8—624no—
222C2×C8—312no—
223C2×C8C22×D3112yes—
224C4⋊C4—21no—
225C4⋊C4—311no—
226C4⋊C4—621no—
227C4⋊C4C22×D3111yescharacteristic
228C4⋊C4C22×D3112yes—
229C22⋊C4—21no—
230C22⋊C4—621no—
231C22⋊C4—622no—
232C22⋊C4—622no—
233C22⋊C4—621no—
234C22⋊C4—621no—
235C42—311no—
236C42—311no—
237C42—311no—
238C42C22×D3111yescharacteristic
239C23—21no—
240C23—311no—
241C23—624no—
242C23—621no—
243C23—622no—
244C23—1242no—
245C23—1242no—
246Q8—42no—
247D4—21no—
248D4—312no—
249D4—624no—
250D4—621no—
251D4—622no—
252D4—622no—
253D4—1242no—
254D4—1244no—
255D4—1242no—
256D4#9249912yescomplemented
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—311no—
261C2×C4—311no—
262C2×C4—311no—
263C2×C4—622no—
264C2×C4—622no—
265C2×C4—621no—
266C2×C4—621no—
267C2×C4—621no—
268C2×C4—1242no—
269C2×C4—1244no—
270C2×C4—1242no—
271C2×C4—1242no—
272C2×C4—1242no—
273C2×C4D4×D3111yescharacteristic
274C2×C4D4×D3111yescharacteristic
275C2×C4C23×D3111yescharacteristic
276C8—22no—
277C8—622no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—312no—
282C22—621no—
283C22—621no—
284C22—622no—
285C22—1248no—
286C22—1241no—
287C22—1241no—
288C22—1242no—
289C22—1242no—
290C22—1242no—
291C22C2×D4×D3111yescharacteristic
292C4—21no—
293C4—42no—
294C4—41no—
295C4—621no—
296C4—621no—
297C4—621no—
298C4—1241no—
299C4#1175849112yes—
300C4C2×#9249911yescharacteristic
301C4#41×D3111yescharacteristic
302C2—42no—
303C2—622no—
304C2—1242no—
305C2—2482no—
306C2#1452156211yescharacteristic
307C2AGL(1,ℤ/8)×D3111yescharacteristic
308C2C2×#1175849111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#476C2
socleC22×C31C22×D4
hypercenter#476D31
hypocenterC31#3109
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #14521600C2semidirect
2. C4×D4×D31C2semidirect
3. #14521369C2semidirect
4. #14521336C2semidirect
5. #14521245C2semidirect
6. #14521244C2semidirect
7. #14521184C2semidirect
8. #14521160C2semidirect
9. #14521158C2semidirect
10. #14521105C2semidirect
11. C31×#476C2semidirect
15. C31×C4⋊Q8C22semidirect
26. #11758470C22semidirect
30. #11758451C22semidirect
33. C31×#106C22semidirect
34. C31×C4⋊C8C22semidirect
36. #11758409C22semidirect
38. #11758403C22semidirect
40. #11758397C22semidirect
41. #11758396C22semidirect
42. C4⋊(C31⋊C8)C22semidirect
93. D4×C31#41semidirect
112. D124#41semidirect
152. #476D31semidirect
171. C4⋊Q8C2×D31semidirect
188. #106C2×D31semidirect
190. C4⋊C8C2×D31semidirect
197. C31#3109semidirect
256. D4#92499semidirect

Maximal subgroups

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

subgroupindex
#145216002
C4×D4×D312
#145213692
#145213362
#145212452
#145212442
#145211842
#145211602
#145211582
#145211052
C31×#4762
#310931