Finite Group Databasebeta

← #16855845

Subgroups — #16855845

Order 3968.

450 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C4×D124C211yesmaximal, characteristic, complemented
2#14521377C211yesmaximal, characteristic, complemented
3#14521360C211yesmaximal, characteristic, complemented
4#14521346C212yesmaximal, complemented
5#14521261C212yesmaximal, complemented
6#14521199C211yesmaximal, characteristic, complemented
7#14521182C211yesmaximal, characteristic
8#14521159C212yesmaximal, complemented
9#14521147C212yesmaximal, complemented
10#14521106C211yesmaximal, characteristic, complemented
11C31×#529C211yesmaximal, characteristic, complemented
12C22×D124C2211yescharacteristic
13C22×C4×D31—21no—
14C31×#126—21no—
15C31×#125C2211yescharacteristic, complemented
16C31×#124C2211yescharacteristic, complemented
17C31×#123—21no—
18C31×C22⋊Q8C2211yescharacteristic
19C31×#121C2211yescharacteristic
20C4×Q8×C31C2211yescharacteristic, complemented
21C4×D4×C31—21no—
22C4×D4×C31C2211yescharacteristic, complemented
23C31×#117—21no—
24C2×C42×C31C2211yescharacteristic
25C2×#92481C2212yes—
26C2×#92479C2212yes—
27C2×(C2×D31)⋊C4—21no—
28C2×C4⋊Dic31C2211yescharacteristic
29C4×D124—21no—
30C4×D124—21no—
31C4×D124C2212yes—
32C4×D124C2212yes—
33#11758423C2212yescomplemented
34#11758420C2212yescomplemented
35C22⋊(C31⋊C8)C2211yescharacteristic
36C22⋊(C31⋊C8)C2211yescharacteristic
37#11758397C2212yes—
38#11758397C2212yes—
39#11758396C2212yescomplemented
40#11758395C2212yescomplemented
41C4⋊(C31⋊C8)C2212yescomplemented
42#11758391C2212yescomplemented
43C23×D31—21no—
44C2×Q8×C31C2311yescharacteristic
45C2×D4×C31—21no—
46C2×D4×C31C2311yescharacteristic
47C22×C4×C31—21no—
48C22×C4×C31D411yescharacteristic
49C22×C4×C31D411yescharacteristic
50C22×C4×C31C2311yescharacteristic
51C22×Dic31—21no—
52C2×D124—21no—
53C2×D124—24no—
54C2×D124C2312yes—
55C2×C4×D31—22no—
56C2×C4×D31—22no—
57C2×C4×D31—44no—
58C4⋊C4×C31—21no—
59C4⋊C4×C31—21no—
60C4⋊C4×C31—21no—
61C4⋊C4×C31—21no—
62C4⋊C4×C31C2311yescharacteristic
63C4⋊C4×C31C2311yescharacteristic
64C22⋊C4×C31—21no—
65C22⋊C4×C31—21no—
66C22⋊C4×C31—21no—
67C22⋊C4×C31—21no—
68C22⋊C4×C31—21no—
69C42×C31—21no—
70C42×C31D411yescharacteristic
71C42×C31D411yescharacteristic
72C42×C31C2311yescharacteristic
73C42×C31C2311yescharacteristic
74#92481—24no—
75#92479—24no—
76(C2×D31)⋊C4—22no—
77(C2×D31)⋊C4—22no—
78C4⋊Dic31—21no—
79C4⋊Dic31C2312yes—
80C2×C31⋊C8C2312yes—
81C2×C31⋊C8C2312yes—
82C23×C31—21no—
83C23×C31C2×D411yescharacteristic
84C22×D31—24no—
85C22×D31—22no—
86C22×D31—42no—
87C22×D31—42no—
88Q8×C31—22no—
89D4×C31—22no—
90D4×C31—42no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—21no—
96C2×C4×C31—41no—
97C2×C4×C31—42no—
98C2×C4×C31C2×D411yescharacteristic
99C2×C4×C31C2×D411yescharacteristic
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2×D411yescharacteristic
102C2×C4×C31C2×D411yescharacteristic
103C2×C4×C31C2411yescharacteristic
104C2×Dic31—22no—
105C2×Dic31—42no—
106D124—24no—
107D124—22no—
108D124#4114yes—
109C4×D31—44no—
110C4×D31—44no—
111C31⋊C8—22no—
112C31⋊C8—22no—
113#3114—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31AGL(1,ℤ/8)12yes—
120C22×C31C22×D411yescharacteristic
121C2×D31—24no—
122C2×D31—48no—
123C2×D31—42no—
124C2×D31—42no—
125C4×C31—21no—
126C4×C31—21no—
127C4×C31—21no—
128C4×C31—21no—
129C4×C31—21no—
130C4×C31—41no—
131C4×C31—41no—
132C4×C31—41no—
133C4×C31C2×#4112yes—
134Dic31—42no—
135#529D3111yescharacteristic, complemented
136C2×C4×D4—311no—
137#483—311no—
138#480—311no—
139#478—312no—
140#463—312no—
141#450—311no—
142#447—311no—
143#442—312no—
144#440—312no—
145#432—311no—
146C2×C31—22no—
147C2×C31—81no—
148C2×C31#52511yescharacteristic
149C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C31#57511yescharacteristic
151D31—44no—
152C22×D4—311no—
153C23×C4—621no—
154C2×SD16—312no—
155C2×D8—312no—
156#126—21no—
157#125C2×D3111yescharacteristic, complemented
158#124C2×D3111yescharacteristic, complemented
159#123—21no—
160C22⋊Q8C2×D3111yescharacteristic
161#121C2×D3111yescharacteristic
162C4×Q8C2×D3111yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—312no—
165C4×D4—312no—
166C4×D4—621no—
167C4×D4—621no—
168C4×D4C2×D3111yescharacteristic, complemented
169#117—21no—
170C2×C4⋊C4—311no—
171C2×C22⋊C4—621no—
172C2×C42C2×D3111yescharacteristic
173#107—312no—
174#106—312no—
175C4⋊C8—312no—
176#103—312no—
177#102—312no—
178#102—312no—
179#102—312no—
180C22⋊C8—311no—
181C22⋊C8—311no—
182#97—312no—
183C31#311411yescharacteristic, complemented, Sylow
184C24—621no—
185C2×Q8C22×D3111yescharacteristic
186C2×D4—21no—
187C2×D4—312no—
188C2×D4—624no—
189C2×D4—621no—
190C2×D4C22×D3111yescharacteristic
191C22×C4—21no—
192C22×C4—621no—
193C22×C4—622no—
194C22×C4—622no—
195C22×C4—1244no—
196C22×C4C22⋊D3111yescharacteristic
197C22×C4C22⋊D3111yescharacteristic
198C22×C4C22×D3111yescharacteristic
199SD16—624no—
200D8—624no—
201C2×C8—312no—
202C2×C8—312no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—312no—
208C4⋊C4—621no—
209C4⋊C4C22×D3111yescharacteristic
210C4⋊C4C22×D3111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—622no—
217C22⋊C4—622no—
218C42—21no—
219C42C22⋊D3111yescharacteristic
220C42C22⋊D3111yescharacteristic
221C42C22×D3111yescharacteristic
222C42C22×D3111yescharacteristic
223C23—21no—
224C23—624no—
225C23—622no—
226C23—1242no—
227C23—1242no—
228C23C2×C22⋊D3111yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—314no—
233D4—624no—
234D4—622no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—622no—
243C2×C4—1244no—
244C2×C4—1244no—
245C2×C4—1242no—
246C2×C4C2×C22⋊D3111yescharacteristic
247C2×C4C2×C22⋊D3111yescharacteristic
248C2×C4C2×C22⋊D3111yescharacteristic
249C2×C4C2×C22⋊D3111yescharacteristic
250C2×C4C2×C22⋊D3111yescharacteristic
251C2×C4C23×D3111yescharacteristic
252C8—622no—
253C8—622no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—624no—
260C22—1248no—
261C22—1242no—
262C22—1242no—
263C22#1175852612yes—
264C22C22×C22⋊D3111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—1242no—
274C4#41×D3112yes—
275C2—22no—
276C2—81no—
277C2—1244no—
278C2#1452155611yescharacteristic
279C2C2×#1175852611yescharacteristic
280C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#529C2
socleC22×C31C22×D4
prime radical#14521182C2
hypercenter#529D31
hypocenterC31#3114
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D124C2semidirect
2. #14521377C2semidirect
3. #14521360C2semidirect
4. #14521346C2semidirect
5. #14521261C2semidirect
6. #14521199C2semidirect
8. #14521159C2semidirect
9. #14521147C2semidirect
10. #14521106C2semidirect
11. C31×#529C2semidirect
15. C31×#125C22semidirect
16. C31×#124C22semidirect
20. C4×Q8×C31C22semidirect
22. C4×D4×C31C22semidirect
33. #11758423C22semidirect
34. #11758420C22semidirect
39. #11758396C22semidirect
40. #11758395C22semidirect
41. C4⋊(C31⋊C8)C22semidirect
42. #11758391C22semidirect
135. #529D31semidirect
157. #125C2×D31semidirect
158. #124C2×D31semidirect
162. C4×Q8C2×D31semidirect
168. C4×D4C2×D31semidirect
183. C31#3114semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D1242
#145213772
#145213602
#145213462
#145212612
#145211992
#145211822
#145211592
#145211472
#145211062
C31×#5292
#311431