Finite Group Databasebeta

← #16856316

Subgroups — #16856316

Order 3968.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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
2#14521584C211yesmaximal, characteristic
3#14521459C211yesmaximal, characteristic, complemented
4#14521441C211yesmaximal, characteristic, complemented
5#14521409C211yesmaximal, characteristic, complemented
6#14521306C212yesmaximal, complemented
7#14521298C212yesmaximal
8#14521141C212yesmaximal, complemented
9#14521134C212yesmaximal, complemented
10#14521001C211yesmaximal, characteristic, complemented
11C31×#491C211yesmaximal, characteristic, complemented
12C31×C4⋊Q8C2211yescharacteristic
13C31×#125C2211yescharacteristic
14#11758523C2212yes—
15Q8×Dic31C2212yes—
16C2×#92482C2212yes—
17C2×#92481C2212yes—
18C2×#92469C2212yes—
19#11758475—21no—
20#11758474—21no—
21#11758474C2211yescharacteristic
22#11758472C2211yescharacteristic
23#11758471—21no—
24#11758470—21no—
25#11758470C2211yescharacteristic
26#11758469—21no—
27#11758469—21no—
28#11758467—21no—
29#11758464—21no—
30#11758464C2211yescharacteristic
31#11758452C2211yescharacteristic
32C42×D31C2211yescharacteristic
33#11758446C2211yescharacteristic
34C31×#103C2212yes—
35C31×#103C2212yes—
36C31×#97C2211yescharacteristic
37#11758409C2212yes—
38#11758404C2212yes—
39C8×Dic31C2212yes—
40#11758396C2212yes—
41#11758395C2212yes—
42#11758391C2211yescharacteristic
43C2×Q8×C31C2312yes—
44C2×D124—21no—
45C2×D124C2311yescharacteristic
46C2×C4×D31—21no—
47C2×C4×D31D411yescharacteristic
48C2×C4×D31D411yescharacteristic
49C2×C4×D31C2311yescharacteristic
50C2×Dic62—21no—
51C2×Dic62C2311yescharacteristic
52C2×C8×C31C2312yes—
53C4⋊C4×C31—21no—
54C4⋊C4×C31—21no—
55C4⋊C4×C31—21no—
56C4⋊C4×C31—21no—
57C4⋊C4×C31C2311yescharacteristic
58C4⋊C4×C31C2311yescharacteristic
59C42×C31C2311yescharacteristic
60#92482—24no—
61#92481—24no—
62(C2×D31)⋊C4—22no—
63(C2×D31)⋊C4—21no—
64(C2×D31)⋊C4—21no—
65(C2×D31)⋊C4—21no—
66(C2×D31)⋊C4—21no—
67(C2×D31)⋊C4—21no—
68C4⋊Dic31—21no—
69C4⋊Dic31—22no—
70C4⋊Dic31—21no—
71C4⋊Dic31C2312yes—
72Dic31⋊C4—22no—
73Dic31⋊C4—21no—
74C4×Dic31—22no—
75C4×Dic31—21no—
76C4×Dic31D411yescharacteristic
77C4×Dic31D411yescharacteristic
78C4×Dic31C2311yescharacteristic
79C2×C31⋊C8C2312yes—
80#92469—24no—
81C22×D31—21no—
82C22×D31C2×D411yescharacteristic
83Q8×C31—22no—
84Q8×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2411yescharacteristic
91C2×Dic31—22no—
92C2×Dic31—21no—
93C2×Dic31C2×D411yescharacteristic
94C2×Dic31C2×D411yescharacteristic
95C2×Dic31C2×D411yescharacteristic
96D124—22no—
97D124—42no—
98C4×D31—22no—
99C4×D31—22no—
100C4×D31—41no—
101C4×D31—42no—
102Dic62—22no—
103Dic62—42no—
104C8×C31—22no—
105C31⋊C8#4114yes—
106#3172—311nomaximal, Sylow
107C22×C31C22×D411yescharacteristic
108C2×D31—21no—
109C2×D31—21no—
110C2×D31—21no—
111C2×D31—41no—
112C2×D31—41no—
113C2×D31—41no—
114C4×C31—21no—
115C4×C31—21no—
116C4×C31—42no—
117C4×C31—41no—
118C4×C31—41no—
119C4×C31C2×#4112yes—
120Dic31—21no—
121Dic31—21no—
122Dic31—41no—
123Dic31—42no—
124Dic31#13712yes—
125#533—311no—
126#529—311no—
127#501—311no—
128#497—311no—
129#491D3111yescharacteristic, complemented
130#489—311no—
131#470—312no—
132#468—312no—
133C4×Q16—312no—
134C4×SD16—312no—
135C4×M16—311no—
136C2×C31#53211yescharacteristic
137C2×C31C2×#13711yescharacteristic
138C2×C31#57511yescharacteristic
139D31—41no—
140D31—81no—
141C2×Q16—312no—
142C2×SD16—312no—
143C2×M16—312no—
144C4⋊Q8—311no—
145C4⋊Q8C2×D3111yescharacteristic
146#126—621no—
147#125—621no—
148#125C2×D3111yescharacteristic
149#124—311no—
150#123—621no—
151C22⋊Q8—312no—
152C22⋊Q8—311no—
153C22⋊Q8—621no—
154#121—311no—
155C4×Q8—312no—
156C4×Q8—311no—
157C4×Q8—621no—
158C4×D4—311no—
159C4×D4—621no—
160#117—621no—
161#117—621no—
162C2×C42—311no—
163#107—312no—
164#106—312no—
165#103—312no—
166#103C2×D3112yes—
167#103C2×D3112yes—
168#102—312no—
169#97—311no—
170#97C2×D3111yescharacteristic
171C4×C8—312no—
172C31#317211yescharacteristic, complemented, Sylow
173C2×Q8—311no—
174C2×Q8—621no—
175C2×Q8C22×D3112yes—
176C2×D4—311no—
177C2×D4—621no—
178C22×C4—311no—
179C22×C4—311no—
180C22×C4—311no—
181C22×C4—621no—
182Q16—624no—
183SD16—624no—
184M16—624no—
185C2×C8—312no—
186C2×C8C22×D3112yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—312no—
192C4⋊C4—621no—
193C4⋊C4—622no—
194C4⋊C4—621no—
195C4⋊C4—622no—
196C4⋊C4—621no—
197C4⋊C4C22×D3111yescharacteristic
198C4⋊C4C22×D3111yescharacteristic
199C22⋊C4—621no—
200C22⋊C4—621no—
201C22⋊C4—622no—
202C22⋊C4—621no—
203C22⋊C4—621no—
204C22⋊C4—621no—
205C42—311no—
206C42—311no—
207C42—311no—
208C42—621no—
209C42—622no—
210C42C22×D3111yescharacteristic
211C23—311no—
212C23—621no—
213Q8—22no—
214Q8—22no—
215Q8—622no—
216Q8—1242no—
217D4—622no—
218D4—1242no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—311no—
223C2×C4—311no—
224C2×C4—311no—
225C2×C4—622no—
226C2×C4—622no—
227C2×C4—621no—
228C2×C4—622no—
229C2×C4—1241no—
230C2×C4—1242no—
231C2×C4D4×D3111yescharacteristic
232C2×C4D4×D3111yescharacteristic
233C2×C4C23×D3111yescharacteristic
234C8—22no—
235C8—314no—
236C22—621no—
237C22—621no—
238C22—621no—
239C22—1241no—
240C22—1241no—
241C22—1241no—
242C22C2×D4×D3111yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—312no—
249C4—621no—
250C4—621no—
251C4—1242no—
252C4—1241no—
253C4#41×D3112yes—
254C2—1241no—
255C2—2481no—
256C2#1452159811yescharacteristic
257C2#137×D3111yescharacteristic
258C2#1452184811yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#491C2
socleC22×C31C22×D4
prime radical#11758472C22
hypercenter#491D31
hypocenterC31#3172
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #14521604C2semidirect
3. #14521459C2semidirect
4. #14521441C2semidirect
5. #14521409C2semidirect
6. #14521306C2semidirect
8. #14521141C2semidirect
9. #14521134C2semidirect
10. #14521001C2semidirect
11. C31×#491C2semidirect
129. #491D31semidirect
172. C31#3172semidirect

Maximal subgroups

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

subgroupindex
#145216042
#145215842
#145214592
#145214412
#145214092
#145213062
#145212982
#145211412
#145211342
#145210012
C31×#4912
#317231