Finite Group Databasebeta

← #16855621

Subgroups — #16855621

Order 3968.

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

normal subgroups only

This group has 420 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#14521559C211yesmaximal, characteristic
2C2×#11758469C211yesmaximal, characteristic, complemented
3#14521393C212yesmaximal, complemented
4#14521384C211yesmaximal, characteristic, complemented
5#14521380C211yesmaximal, characteristic, complemented
6#14521372C212yesmaximal, complemented
7#14521360C211yesmaximal, characteristic, complemented
8#14521084C212yesmaximal, complemented
9#14521042C212yesmaximal, complemented
10C22⋊C8×D31C211yesmaximal, characteristic, complemented
11C31×#481C211yesmaximal, characteristic, complemented
12C2×#92497C2211yescharacteristic
13C22×C4×D31C2211yescharacteristic
14C31×#121C2211yescharacteristic
15C2×C4⋊C4×C31C2211yescharacteristic
16#11758515C2211yescharacteristic
17#11758514—21no—
18D4×Dic31—21no—
19C2×(C2×D31)⋊C4—21no—
20C2×C4⋊Dic31C2211yescharacteristic
21#11758501C2211yescharacteristic
22C2×C4×Dic31—21no—
23C2×C8×D31C2212yes—
24#11758474C2211yescharacteristic
25#11758469—21no—
26#11758469—21no—
27#11758469C2212yes—
28#11758469C2212yes—
29#11758469C2211yescharacteristic
30#11758460—21no—
31(C22⋊D31)⋊C4—21no—
32C31×#106C2212yescomplemented
33C31×#102C2212yescomplemented
34C31×C22⋊C8C2211yescharacteristic
35#11758420C2212yescomplemented
36C22⋊(C31⋊C8)C2211yescharacteristic
37(C2×D31)⋊C8C2212yes—
38#11758405C2212yescomplemented
39#11758404C2212yescomplemented
40#11758398C2212yescomplemented
41#11758395C2212yescomplemented
42#11758395C2212yescomplemented
43C23×D31D411yescharacteristic
44C2×D4×C31—21no—
45C2×D4×C31C2311yescharacteristic
46C22×C4×C31—21no—
47C22×C4×C31C2311yescharacteristic
48C2×C22⋊D31—21no—
49C2×C22⋊D31—21no—
50C22×Dic31—21no—
51C22×Dic31—21no—
52C22×Dic31D411yescharacteristic
53#92497—24no—
54C2×C4×D31—24no—
55C2×C4×D31D411yescharacteristic
56C2×C4×D31D411yescharacteristic
57C2×C4×D31C2311yescharacteristic
58C2×C4×D31C2311yescharacteristic
59C2×Dic62C2311yescharacteristic
60C2×C8×C31C2312yes—
61C4⋊C4×C31—21no—
62C4⋊C4×C31C2311yescharacteristic
63C4⋊C4×C31C2312yes—
64C22⋊C4×C31—21no—
65C22⋊Dic31—21no—
66C22⋊Dic31—21no—
67(C2×D31)⋊C4—21no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—22no—
70(C2×D31)⋊C4—22no—
71C4⋊Dic31—21no—
72C4⋊Dic31—21no—
73C4⋊Dic31C2311yescharacteristic
74C4⋊Dic31C2312yes—
75Dic31⋊C4—21no—
76C4×Dic31—21no—
77C4×Dic31—22no—
78C4×Dic31—22no—
79C2×C31⋊C8C2312yes—
80C8×D31—24no—
81C23×C31—21no—
82C23×C31C2×D411yescharacteristic
83C22×D31—22no—
84C22×D31—22no—
85C22×D31—41no—
86C22×D31C2×D411yescharacteristic
87C22×D31C2×D411yescharacteristic
88D4×C31—22no—
89D4×C31—42no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—21no—
93C2×C4×C31—42no—
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2411yescharacteristic
96C22⋊D31—42no—
97C22⋊D31—42no—
98C2×Dic31—22no—
99C2×Dic31—22no—
100C2×Dic31—21no—
101C2×Dic31—21no—
102C2×Dic31—42no—
103C2×Dic31—42no—
104C2×Dic31C2×D411yescharacteristic
105C2×Dic31C2×D411yescharacteristic
106C4×D31—22no—
107C4×D31—22no—
108C4×D31—22no—
109C4×D31#4114yes—
110Dic62—22no—
111C8×C31—22no—
112C31⋊C8—22no—
113#3091—311nomaximal, Sylow
114C22×C31—21no—
115C22×C31—21no—
116C22×C31—41no—
117C22×C31—41no—
118C22×C31—41no—
119C22×C31#13512yes—
120C22×C31C22×D411yescharacteristic
121C2×D31—21no—
122C2×D31—21no—
123C2×D31—21no—
124C2×D31—21no—
125C2×D31—21no—
126C2×D31—44no—
127C2×D31#12312yes—
128C4×C31—21no—
129C4×C31—41no—
130C4×C31—42no—
131C4×C31C2×#4112yes—
132Dic31—21no—
133Dic31—41no—
134Dic31—42no—
135Dic31—41no—
136Dic31#12312yes—
137#525—311no—
138C2×#117—311no—
139#485—312no—
140#484—311no—
141#483—311no—
142#482—312no—
143#481D3111yescharacteristic, complemented
144#480—311no—
145#427—312no—
146#416—312no—
147C2×C22⋊C8—311no—
148C2×C31—22no—
149C2×C31—81no—
150C2×C31C2×#12311yescharacteristic
151C2×C31C2×#13511yescharacteristic
152C2×C31#57511yescharacteristic
153D31—22no—
154D31—41no—
155C2×#41—311no—
156C23×C4—311no—
157C22×C8—312no—
158#123—621no—
159C22⋊Q8—311no—
160C22⋊Q8—311no—
161#121—311no—
162#121C2×D3111yescharacteristic
163#120—621no—
164C4×D4—621no—
165C4×D4—621no—
166#117—312no—
167#117—312no—
168#117—311no—
169#117—621no—
170#117—621no—
171C2×C4⋊C4—311no—
172C2×C4⋊C4C2×D3111yescharacteristic
173C2×C22⋊C4—621no—
174C2×C42—621no—
175#107—312no—
176#107—312no—
177#106—312no—
178#106C2×D3112yescomplemented
179#103—312no—
180#103—312no—
181#102—312no—
182#102C2×D3112yescomplemented
183C22⋊C8—311no—
184C22⋊C8—312no—
185C22⋊C8C2×D3111yescharacteristic
186C31#309111yescharacteristic, complemented, Sylow
187C24—311no—
188#41—624no—
189C2×Q8—311no—
190C2×D4—21no—
191C2×D4—621no—
192C2×D4—621no—
193C2×D4C22×D3111yescharacteristic
194C22×C4—21no—
195C22×C4—311no—
196C22×C4—311no—
197C22×C4—311no—
198C22×C4—311no—
199C22×C4—311no—
200C22×C4—624no—
201C22×C4—621no—
202C22×C4—621no—
203C22×C4C22×D3111yescharacteristic
204C2×C8—312no—
205C2×C8—624no—
206C2×C8C22×D3112yes—
207C4⋊C4—21no—
208C4⋊C4—312no—
209C4⋊C4—311no—
210C4⋊C4—621no—
211C4⋊C4—621no—
212C4⋊C4—621no—
213C4⋊C4C22×D3112yes—
214C4⋊C4C22×D3111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—622no—
217C22⋊C4—622no—
218C22⋊C4—621no—
219C22⋊C4—621no—
220C22⋊C4—621no—
221C22⋊C4—621no—
222C42—622no—
223C42—622no—
224C42—621no—
225C23—21no—
226C23—311no—
227C23—311no—
228C23—622no—
229C23—622no—
230C23—1241no—
231C23D4×D3111yescharacteristic
232Q8—622no—
233D4—22no—
234D4—42no—
235D4—1242no—
236D4—1242no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—311no—
242C2×C4—314no—
243C2×C4—311no—
244C2×C4—622no—
245C2×C4—622no—
246C2×C4—622no—
247C2×C4—622no—
248C2×C4—622no—
249C2×C4—621no—
250C2×C4—621no—
251C2×C4—1242no—
252C2×C4—1242no—
253C2×C4D4×D3111yescharacteristic
254C2×C4C23×D3111yescharacteristic
255C8—22no—
256C8—622no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—312no—
263C22—621no—
264C22—621no—
265C22—621no—
266C22—621no—
267C22—621no—
268C22—1244no—
269C22#1175849312yes—
270C22C2×D4×D3111yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—312no—
275C4—621no—
276C4—1241no—
277C4—1242no—
278C4—1241no—
279C4#41×D3112yes—
280C2—22no—
281C2—81no—
282C2—622no—
283C2—1241no—
284C2#123×D3111yescharacteristic
285C2C2×#1175849311yescharacteristic
286C2#1452184511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#481C2
socleC22×C31C22×D4
prime radical#14521559C2
hypercenter#481D31
hypocenterC31#3091
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×#11758469C2semidirect
3. #14521393C2semidirect
4. #14521384C2semidirect
5. #14521380C2semidirect
6. #14521372C2semidirect
7. #14521360C2semidirect
8. #14521084C2semidirect
9. #14521042C2semidirect
10. C22⋊C8×D31C2semidirect
11. C31×#481C2semidirect
32. C31×#106C22semidirect
33. C31×#102C22semidirect
35. #11758420C22semidirect
38. #11758405C22semidirect
39. #11758404C22semidirect
40. #11758398C22semidirect
41. #11758395C22semidirect
42. #11758395C22semidirect
143. #481D31semidirect
178. #106C2×D31semidirect
182. #102C2×D31semidirect
186. C31#3091semidirect

Maximal subgroups

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

subgroupindex
#145215592
C2×#117584692
#145213932
#145213842
#145213802
#145213722
#145213602
#145210842
#145210422
C22⋊C8×D312
C31×#4812
#309131