Finite Group Databasebeta

← #16855844

Subgroups — #16855844

Order 3968.

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

normal subgroups only

This group has 450 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#14521580C211yesmaximal, characteristic, complemented
2C2×D4×Dic31C211yesmaximal, characteristic, complemented
3#14521380C211yesmaximal, characteristic, complemented
4#14521360C211yesmaximal, characteristic, complemented
5#14521351C212yesmaximal, complemented
6#14521262C212yesmaximal, complemented
7#14521200C211yesmaximal, characteristic, complemented
8#14521161C212yesmaximal, complemented
9#14521146C212yesmaximal, complemented
10#14521103C211yesmaximal, characteristic, complemented
11C31×#447C211yesmaximal, characteristic
12C22×D4×C31C2211yescharacteristic
13C23×Dic31—21no—
14C2×D8×C31C2212yes—
15C31×#121C2211yescharacteristic
16C2×C22⋊Dic31—21no—
17#11758513C2211yescharacteristic, complemented
18#11758512—21no—
19D4×Dic31—21no—
20D4×Dic31—21no—
21D4×Dic31—21no—
22D4×Dic31C2212yes—
23D4×Dic31C2212yes—
24D4×Dic31C2211yescharacteristic, complemented
25C2×#92480C2212yes—
26C2×C4⋊Dic31C2211yescharacteristic
27#11758501C2211yescharacteristic
28C2×C4×Dic31C2211yescharacteristic
29#11758467C2211yescharacteristic, complemented
30#11758464C2211yescharacteristic, complemented
31#11758456—21no—
32#11758454—21no—
33C31×#102C2212yes—
34C31×C22⋊C8C2211yescharacteristic
35#11758420C2212yes—
36#11758420C2212yescomplemented
37C22⋊(C31⋊C8)C2211yescharacteristic
38#11758405C2212yescomplemented
39#11758404C2212yescomplemented
40#11758403C2212yescomplemented
41Dic31⋊C8C2212yescomplemented
42#11758395C2212yescomplemented
43C24×C31—21no—
44C2×D4×C31—24no—
45C2×D4×C31—21no—
46C2×D4×C31—21no—
47C2×D4×C31C2311yescharacteristic
48C2×D4×C31C2312yes—
49C22×C4×C31C2311yescharacteristic
50C22×Dic31—21no—
51C22×Dic31—21no—
52C22×Dic31—22no—
53C22×Dic31—22no—
54C22×Dic31—44no—
55C22×Dic31D411yescharacteristic
56C22×Dic31D411yescharacteristic
57C2×Dic62C2311yescharacteristic
58D8×C31—24no—
59C2×C8×C31C2312yes—
60C4⋊C4×C31C2311yescharacteristic
61C22⋊C4×C31—21no—
62C22⋊Dic31—21no—
63C22⋊Dic31—21no—
64C22⋊Dic31—21no—
65C22⋊Dic31—21no—
66C22⋊Dic31—22no—
67C22⋊Dic31—22no—
68#92480—24no—
69C4⋊Dic31—21no—
70C4⋊Dic31—21no—
71C4⋊Dic31C2312yes—
72C4⋊Dic31C2311yescharacteristic
73Dic31⋊C4—21no—
74Dic31⋊C4—21no—
75Dic31⋊C4—21no—
76C4×Dic31—21no—
77C4×Dic31D411yescharacteristic
78C4×Dic31D411yescharacteristic
79C4×Dic31C2311yescharacteristic
80C4×Dic31C2311yescharacteristic
81C2×C31⋊C8C2312yes—
82C23×C31—24no—
83C23×C31—22no—
84C23×C31—21no—
85C23×C31—42no—
86C23×C31—42no—
87C23×C31C2×D411yescharacteristic
88D4×C31—24no—
89D4×C31—22no—
90D4×C31—22no—
91D4×C31—42no—
92D4×C31#4114yes—
93C2×C4×C31—22no—
94C2×C4×C31—21no—
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2411yescharacteristic
97C2×Dic31—22no—
98C2×Dic31—21no—
99C2×Dic31—21no—
100C2×Dic31—22no—
101C2×Dic31—41no—
102C2×Dic31—42no—
103C2×Dic31—44no—
104C2×Dic31—44no—
105C2×Dic31—42no—
106C2×Dic31C2×D411yescharacteristic
107C2×Dic31C2×D411yescharacteristic
108C2×Dic31C2×D411yescharacteristic
109C2×Dic31C2×D411yescharacteristic
110Dic62—22no—
111C8×C31—22no—
112C31⋊C8—22no—
113#3114—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—24no—
117C22×C31—48no—
118C22×C31—42no—
119C22×C31—42no—
120C22×C31—41no—
121C22×C31—41no—
122C22×C31—41no—
123C22×C31AGL(1,ℤ/8)12yes—
124C22×C31C22×D411yescharacteristic
125C4×C31—21no—
126C4×C31—41no—
127C4×C31C2×#4112yes—
128Dic31—21no—
129Dic31—21no—
130Dic31—21no—
131Dic31—21no—
132Dic31—41no—
133Dic31—42no—
134Dic31—41no—
135#529—311no—
136C2×C4×D4—311no—
137#483—311no—
138#480—311no—
139#478—312no—
140#463—312no—
141#450—311no—
142#447D3111yescharacteristic
143#442—312no—
144#440—312no—
145#432—311no—
146C2×C31—22no—
147C2×C31—44no—
148C2×C31—81no—
149C2×C31#52511yescharacteristic
150C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C31#57511yescharacteristic
152C22×D4C2×D3111yescharacteristic
153C23×C4—621no—
154C2×SD16—312no—
155C2×D8C2×D3112yes—
156#126—621no—
157#125—311no—
158#124—311no—
159#123—621no—
160C22⋊Q8—311no—
161#121C2×D3111yescharacteristic
162C4×Q8—311no—
163C4×D4—312no—
164C4×D4—312no—
165C4×D4—311no—
166C4×D4—621no—
167C4×D4—621no—
168C4×D4—621no—
169#117—621no—
170C2×C4⋊C4—311no—
171C2×C22⋊C4—621no—
172C2×C42—311no—
173#107—312no—
174#106—312no—
175C4⋊C8—312no—
176#103—312no—
177#102—312no—
178#102—312no—
179#102C2×D3112yes—
180C22⋊C8—311no—
181C22⋊C8C2×D3111yescharacteristic
182#97—312no—
183C31#311411yescharacteristic, complemented, Sylow
184C24—21no—
185C2×Q8—311no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4C22×D3111yescharacteristic
190C2×D4C22×D3112yes—
191C22×C4—311no—
192C22×C4—311no—
193C22×C4—621no—
194C22×C4—622no—
195C22×C4—622no—
196C22×C4—621no—
197C22×C4—1244no—
198C22×C4C22×D3111yescharacteristic
199SD16—624no—
200D8—24no—
201C2×C8—312no—
202C2×C8C22×D3112yes—
203C4⋊C4—311no—
204C4⋊C4—312no—
205C4⋊C4—621no—
206C4⋊C4—621no—
207C4⋊C4—621no—
208C4⋊C4—621no—
209C4⋊C4—621no—
210C4⋊C4C22×D3111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—622no—
213C22⋊C4—622no—
214C22⋊C4—621no—
215C22⋊C4—621no—
216C22⋊C4—621no—
217C22⋊C4—621no—
218C42—311no—
219C42—311no—
220C42—311no—
221C42—311no—
222C42—621no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—42no—
227C23—42no—
228C23D4×D3111yescharacteristic
229Q8—622no—
230D4—24no—
231D4—22no—
232D4—22no—
233D4—42no—
234D4#9249714yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—311no—
238C2×C4—311no—
239C2×C4—311no—
240C2×C4—311no—
241C2×C4—622no—
242C2×C4—621no—
243C2×C4—621no—
244C2×C4—622no—
245C2×C4—1241no—
246C2×C4—1242no—
247C2×C4—1244no—
248C2×C4—1244no—
249C2×C4—1242no—
250C2×C4D4×D3111yescharacteristic
251C2×C4C23×D3111yescharacteristic
252C8—22no—
253C8—622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#1175848812yes—
264C22C2×D4×D3111yescharacteristic
265C4—21no—
266C4—41no—
267C4—621no—
268C4—621no—
269C4—621no—
270C4—621no—
271C4—1241no—
272C4—1242no—
273C4—1241no—
274C4C2×#9249712yes—
275C2—22no—
276C2—44no—
277C2—81no—
278C2#1452155511yescharacteristic
279C2C2×#1175848811yescharacteristic
280C2#1452184511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC31×#447C2
socleC22×C31C22×D4
hypercenter#447D31
hypocenterC31#3114
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #14521580C2semidirect
2. C2×D4×Dic31C2semidirect
3. #14521380C2semidirect
4. #14521360C2semidirect
5. #14521351C2semidirect
6. #14521262C2semidirect
7. #14521200C2semidirect
8. #14521161C2semidirect
9. #14521146C2semidirect
10. #14521103C2semidirect
17. #11758513C22semidirect
24. D4×Dic31C22semidirect
29. #11758467C22semidirect
30. #11758464C22semidirect
36. #11758420C22semidirect
38. #11758405C22semidirect
39. #11758404C22semidirect
40. #11758403C22semidirect
41. Dic31⋊C8C22semidirect
42. #11758395C22semidirect
183. C31#3114semidirect

Maximal subgroups

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

subgroupindex
#145215802
C2×D4×Dic312
#145213802
#145213602
#145213512
#145212622
#145212002
#145211612
#145211462
#145211032
C31×#4472
#311431