Finite Group Databasebeta

← #16855809

Subgroups — #16855809

Order 3968.

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

normal subgroups only

This group has 444 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#14521604C211yesmaximal, characteristic, complemented
2C4×D4×D31C211yesmaximal, characteristic
3#14521383C212yesmaximal, complemented
4#14521330C211yesmaximal, characteristic, complemented
5#14521329C211yesmaximal, characteristic, complemented
6#14521251C211yesmaximal, characteristic, complemented
7#14521198C212yesmaximal
8#14521152C212yesmaximal, complemented
9#14521150C212yesmaximal, complemented
10#14521105C211yesmaximal, characteristic, complemented
11C31×#461C211yesmaximal, characteristic, complemented
12C2×D4×D31C2211yescharacteristic
13C22×C4×D31—21no—
14C2×SD16×C31C2212yes—
15C31×C4⋊Q8C2211yescharacteristic, complemented
16C4×D4×C31C2211yescharacteristic
17#11758523C2212yes—
18Q8×Dic31C2212yescomplemented
19C2×#92481C2212yes—
20D4×Dic31C2211yescharacteristic
21C4×C22⋊D31—21no—
22#11758474—21no—
23#11758470C2211yescharacteristic
24#11758469—21no—
25C4⋊C4×D31C2211yescharacteristic
26#11758467—21no—
27#11758464—21no—
28(C22⋊D31)⋊C4—21no—
29C22⋊C4×D31—21no—
30C4×D124C2211yescharacteristic
31C42×D31C2211yescharacteristic
32#11758446C2211yescharacteristic, complemented
33C31×C4⋊C8C2211yescharacteristic, complemented
34C31×#103C2212yescomplemented
35#11758423C2212yescomplemented
36#11758420C2212yes—
37#11758409C2212yes—
38(C2×D31)⋊C8C2212yes—
39#11758406C2212yescomplemented
40#11758403C2212yescomplemented
41#11758395C2212yescomplemented
42C4⋊(C31⋊C8)C2211yescharacteristic, complemented
43C23×D31—21no—
44C2×Q8×C31C2312yes—
45C2×D4×C31C2311yescharacteristic
46C22×C4×C31—21no—
47C2×C22⋊D31—21no—
48C22×Dic31—21no—
49D4×D31—22no—
50D4×D31—22no—
51C2×D124C2311yescharacteristic
52C2×C4×D31—21no—
53C2×C4×D31—21no—
54C2×C4×D31—21no—
55C2×C4×D31—44no—
56C2×C4×D31D411yescharacteristic
57C2×C4×D31D411yescharacteristic
58C2×C4×D31C2311yescharacteristic
59C2×Dic62—21no—
60SD16×C31—24no—
61C2×C8×C31C2312yes—
62C4⋊C4×C31—21no—
63C4⋊C4×C31—21no—
64C4⋊C4×C31C2311yescharacteristic
65C22⋊C4×C31—21no—
66C42×C31C2311yescharacteristic
67C22⋊Dic31—21no—
68#92481—24no—
69(C2×D31)⋊C4—22no—
70(C2×D31)⋊C4—21no—
71(C2×D31)⋊C4—21no—
72C4⋊Dic31—21no—
73C4⋊Dic31—22no—
74C4⋊Dic31C2312yes—
75C4⋊Dic31C2311yescharacteristic
76Dic31⋊C4—21no—
77Dic31⋊C4—22no—
78C4×Dic31—22no—
79C4×Dic31D411yescharacteristic
80C4×Dic31D411yescharacteristic
81C4×Dic31C2311yescharacteristic
82C2×C31⋊C8C2312yes—
83C23×C31—21no—
84C22×D31—24no—
85C22×D31—21no—
86C22×D31—42no—
87C22×D31—42no—
88C22×D31C2×D411yescharacteristic
89Q8×C31—22no—
90Q8×C31—22no—
91D4×C31—21no—
92D4×C31#4112yes—
93C2×C4×C31—22no—
94C2×C4×C31—21no—
95C2×C4×C31—42no—
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2411yescharacteristic
99C22⋊D31—24no—
100C2×Dic31—22no—
101C2×Dic31—21no—
102C2×Dic31—42no—
103C2×Dic31C2×D411yescharacteristic
104C2×Dic31C2×D411yescharacteristic
105C2×Dic31C2×D411yescharacteristic
106D124—21no—
107D124#4112yes—
108C4×D31—22no—
109C4×D31—22no—
110C4×D31—21no—
111C4×D31—21no—
112C4×D31—42no—
113C4×D31—42no—
114C4×D31—42no—
115Dic62—42no—
116C8×C31—22no—
117C31⋊C8—22no—
118#3110—311nomaximal, Sylow
119C22×C31—22no—
120C22×C31—41no—
121C22×C31—41no—
122C22×C31C22×D411yescharacteristic
123C2×D31—21no—
124C2×D31—21no—
125C2×D31—22no—
126C2×D31—48no—
127C2×D31—41no—
128C2×D31—41no—
129C2×D31#13712yes—
130C4×C31—21no—
131C4×C31—42no—
132C4×C31—41no—
133C4×C31AGL(1,ℤ/8)12yes—
134C4×C31C2×#4111yescharacteristic
135C4×C31C2×#4111yescharacteristic
136Dic31—21no—
137Dic31—21no—
138Dic31—21no—
139Dic31—42no—
140Dic31—41no—
141#533—311no—
142C2×C4×D4—311no—
143#484—312no—
144#474—311no—
145#474—311no—
146#461—311no—
147#461D3111yescharacteristic, complemented
148#450—312no—
149#440—312no—
150#440—312no—
151#432—311no—
152C2×C31—42no—
153C2×C31#52511yescharacteristic
154C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C31C2×#13711yescharacteristic
156D31—22no—
157D31—42no—
158C22×D4—311no—
159C23×C4—621no—
160C2×SD16—312no—
161C2×SD16C2×D3112yes—
162C4⋊Q8—311no—
163C4⋊Q8C2×D3111yescharacteristic, complemented
164#125—621no—
165C22⋊Q8—312no—
166C22⋊Q8—621no—
167C4×Q8—312no—
168C4×Q8—621no—
169C4×D4—311no—
170C4×D4—311no—
171C4×D4—311no—
172C4×D4—621no—
173C4×D4—621no—
174C4×D4C2×D3111yescharacteristic
175#117—621no—
176C2×C4⋊C4—311no—
177C2×C22⋊C4—621no—
178C2×C42—311no—
179#107—312no—
180#107—312no—
181C4⋊C8—311no—
182C4⋊C8C2×D3111yescharacteristic, complemented
183#103—312no—
184#103C2×D3112yescomplemented
185#102—312no—
186#102—312no—
187C22⋊C8—312no—
188#97—312no—
189C31#311011yescharacteristic, complemented, Sylow
190C24—621no—
191C2×Q8—621no—
192C2×Q8C22×D3112yes—
193C2×D4—311no—
194C2×D4—621no—
195C2×D4—622no—
196C2×D4—622no—
197C2×D4C22×D3111yescharacteristic
198C22×C4—21no—
199C22×C4—311no—
200C22×C4—311no—
201C22×C4—311no—
202C22×C4—621no—
203C22×C4—621no—
204C22×C4—621no—
205C22×C4—621no—
206C22×C4—1244no—
207SD16—24no—
208SD16—624no—
209C2×C8—312no—
210C2×C8C22×D3112yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—312no—
214C4⋊C4—311no—
215C4⋊C4—621no—
216C4⋊C4—622no—
217C4⋊C4—622no—
218C4⋊C4—621no—
219C4⋊C4C22×D3111yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—621no—
222C22⋊C4—622no—
223C22⋊C4—621no—
224C22⋊C4—621no—
225C42—311no—
226C42—311no—
227C42—311no—
228C42—622no—
229C42C22×D3111yescharacteristic
230C23—21no—
231C23—311no—
232C23—624no—
233C23—621no—
234C23—1242no—
235C23—1242no—
236Q8—22no—
237Q8—22no—
238Q8—1242no—
239D4—21no—
240D4—312no—
241D4—624no—
242D4—621no—
243D4#9249712yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—311no—
248C2×C4—311no—
249C2×C4—311no—
250C2×C4—622no—
251C2×C4—622no—
252C2×C4—621no—
253C2×C4—621no—
254C2×C4—622no—
255C2×C4—621no—
256C2×C4—1242no—
257C2×C4—1242no—
258C2×C4—1242no—
259C2×C4—1242no—
260C2×C4D4×D3111yescharacteristic
261C2×C4D4×D3111yescharacteristic
262C2×C4C23×D3111yescharacteristic
263C8—22no—
264C8—622no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—312no—
269C22—621no—
270C22—621no—
271C22—622no—
272C22—1248no—
273C22—1241no—
274C22—1241no—
275C22C2×D4×D3111yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—621no—
280C4—621no—
281C4—621no—
282C4—1241no—
283C4—1242no—
284C4#1175849112yes—
285C4C2×#9249711yescharacteristic
286C4#41×D3111yescharacteristic
287C2—42no—
288C2—622no—
289C2—1242no—
290C2#1452155911yescharacteristic
291C2C2×#1175849111yescharacteristic
292C2#137×D3111yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#461C2
socleC22×C31C22×D4
prime radicalC2×D4×D31C22
hypercenter#461D31
hypocenterC31#3110
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #14521604C2semidirect
3. #14521383C2semidirect
4. #14521330C2semidirect
5. #14521329C2semidirect
6. #14521251C2semidirect
8. #14521152C2semidirect
9. #14521150C2semidirect
10. #14521105C2semidirect
11. C31×#461C2semidirect
15. C31×C4⋊Q8C22semidirect
18. Q8×Dic31C22semidirect
32. #11758446C22semidirect
33. C31×C4⋊C8C22semidirect
34. C31×#103C22semidirect
35. #11758423C22semidirect
39. #11758406C22semidirect
40. #11758403C22semidirect
41. #11758395C22semidirect
42. C4⋊(C31⋊C8)C22semidirect
147. #461D31semidirect
163. C4⋊Q8C2×D31semidirect
182. C4⋊C8C2×D31semidirect
184. #103C2×D31semidirect
189. C31#3110semidirect

Maximal subgroups

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

subgroupindex
#145216042
C4×D4×D312
#145213832
#145213302
#145213292
#145212512
#145211982
#145211522
#145211502
#145211052
C31×#4612
#311031