Finite Group Databasebeta

← #16855882

Subgroups — #16855882

Order 3968.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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#14521583C211yesmaximal, characteristic
2#14521384C211yesmaximal, characteristic, complemented
3#14521367C211yesmaximal, characteristic
4#14521356C212yesmaximal, complemented
5#14521271C212yesmaximal, complemented
6#14521205C211yesmaximal, characteristic, complemented
7#14521188C211yesmaximal, characteristic
8#14521154C212yesmaximal, complemented
9#14521147C212yesmaximal, complemented
10#14521106C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C31C211yesmaximal, characteristic, complemented
12C22×Q8×C31C2211yescharacteristic
13C4×Q8×C31—21no—
14C4×Q8×C31—21no—
15C4×Q8×C31C2212yes—
16C4×Q8×C31C2212yes—
17C2×C4⋊C4×C31—21no—
18C2×C4⋊C4×C31C2211yescharacteristic
19C2×C42×C31—21no—
20C2×C42×C31C2211yescharacteristic
21C2×#92482C2212yes—
22C2×#92481C2212yes—
23#11758507C2211yescharacteristic
24#11758506—21no—
25C4×C22⋊D31—21no—
26#11758501C2211yescharacteristic
27#11758453—21no—
28#11758452C2211yescharacteristic
29C4×D124C2211yescharacteristic
30#11758449—21no—
31#11758447C2211yescharacteristic
32C4×Dic62C2211yescharacteristic
33#11758423C2212yes—
34#11758423C2212yes—
35C22⋊(C31⋊C8)C2211yescharacteristic
36C22⋊(C31⋊C8)C2211yescharacteristic
37#11758398C2212yes—
38#11758397C2212yes—
39#11758396C2212yes—
40#11758395C2212yes—
41C4⋊(C31⋊C8)C2212yes—
42#11758391C2212yes—
43C2×Q8×C31—24no—
44C2×Q8×C31—21no—
45C2×Q8×C31C2312yes—
46C22×C4×C31—21no—
47C22×C4×C31—21no—
48C22×C4×C31D411yescharacteristic
49C22×C4×C31D411yescharacteristic
50C22×C4×C31C2311yescharacteristic
51C2×C22⋊D31—21no—
52C2×D124C2311yescharacteristic
53C2×C4×D31—21no—
54C2×Dic62C2311yescharacteristic
55C4⋊C4×C31—22no—
56C4⋊C4×C31—22no—
57C4⋊C4×C31—21no—
58C4⋊C4×C31C2312yes—
59C42×C31—22no—
60C42×C31—22no—
61C42×C31D411yescharacteristic
62C42×C31D411yescharacteristic
63C42×C31C2311yescharacteristic
64C42×C31C2311yescharacteristic
65C22⋊Dic31—21no—
66#92482—24no—
67#92481—24no—
68(C2×D31)⋊C4—21no—
69(C2×D31)⋊C4—21no—
70(C2×D31)⋊C4—21no—
71(C2×D31)⋊C4—21no—
72C4⋊Dic31C2311yescharacteristic
73C4⋊Dic31C2311yescharacteristic
74Dic31⋊C4—21no—
75Dic31⋊C4—21no—
76Dic31⋊C4—21no—
77Dic31⋊C4—21no—
78C4×Dic31—21no—
79C2×C31⋊C8C2312yes—
80C2×C31⋊C8C2312yes—
81C23×C31C2×D411yescharacteristic
82C22×D31—21no—
83Q8×C31—24no—
84Q8×C31—22no—
85Q8×C31#4114yes—
86C2×C4×C31—24no—
87C2×C4×C31—22no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—41no—
92C2×C4×C31—42no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2×D411yescharacteristic
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2411yescharacteristic
99C22⋊D31—42no—
100C2×Dic31—21no—
101C2×Dic31—21no—
102C2×Dic31—21no—
103D124—22no—
104C4×D31—42no—
105Dic62—22no—
106C31⋊C8—22no—
107C31⋊C8—22no—
108#3117—311nomaximal, Sylow
109C22×C31—21no—
110C22×C31—21no—
111C22×C31#13712yes—
112C22×C31C22×D411yescharacteristic
113C2×D31—41no—
114C2×D31—41no—
115C2×D31—41no—
116C4×C31—24no—
117C4×C31—21no—
118C4×C31—21no—
119C4×C31—21no—
120C4×C31—21no—
121C4×C31—21no—
122C4×C31—42no—
123C4×C31C2×#4112yes—
124Dic31—41no—
125Dic31—41no—
126Dic31—41no—
127#529—311no—
128C2×C4×Q8D3111yescharacteristic, complemented
129#484—311no—
130#481—311no—
131#479—312no—
132#464—312no—
133#451—311no—
134#448—311no—
135#441—312no—
136#440—312no—
137#432—311no—
138C2×C31—22no—
139C2×C31#52511yescharacteristic
140C2×C31C2×#13711yescharacteristic
141C2×C31#57511yescharacteristic
142D31—81no—
143C22×Q8C2×D3111yescharacteristic
144C2×Q16—312no—
145C2×SD16—312no—
146#126—621no—
147#125—311no—
148#124—311no—
149#123—621no—
150C22⋊Q8—311no—
151#121—311no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—311no—
155C4×Q8C2×D3112yes—
156C4×Q8C2×D3112yes—
157C4×D4—311no—
158C4×D4—621no—
159#117—621no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4C2×D3111yescharacteristic
162C2×C42—21no—
163C2×C42C2×D3111yescharacteristic
164#107—312no—
165#106—312no—
166C4⋊C8—312no—
167#103—312no—
168#103—312no—
169#103—312no—
170#102—312no—
171C22⋊C8—311no—
172C22⋊C8—311no—
173#97—312no—
174C31#311711yescharacteristic, complemented, Sylow
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—311no—
178C2×Q8C22×D3112yes—
179C2×D4—311no—
180C2×D4—621no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—621no—
184C22×C4C22⋊D3111yescharacteristic
185C22×C4C22⋊D3111yescharacteristic
186C22×C4C22×D3111yescharacteristic
187Q16—624no—
188SD16—624no—
189C2×C8—312no—
190C2×C8—312no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—311no—
195C4⋊C4—311no—
196C4⋊C4—621no—
197C4⋊C4—621no—
198C4⋊C4—621no—
199C4⋊C4—621no—
200C4⋊C4C22×D3112yes—
201C22⋊C4—621no—
202C22⋊C4—621no—
203C22⋊C4—621no—
204C22⋊C4—621no—
205C22⋊C4—621no—
206C42—22no—
207C42—22no—
208C42—621no—
209C42C22⋊D3111yescharacteristic
210C42C22⋊D3111yescharacteristic
211C42C22×D3111yescharacteristic
212C42C22×D3111yescharacteristic
213C23—621no—
214C23C2×C22⋊D3111yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—622no—
218Q8#9249514yes—
219D4—622no—
220D4—1242no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—621no—
229C2×C4—621no—
230C2×C4—621no—
231C2×C4—1242no—
232C2×C4C2×C22⋊D3111yescharacteristic
233C2×C4C2×C22⋊D3111yescharacteristic
234C2×C4C2×C22⋊D3111yescharacteristic
235C2×C4C2×C22⋊D3111yescharacteristic
236C2×C4C2×C22⋊D3111yescharacteristic
237C2×C4C23×D3111yescharacteristic
238C8—622no—
239C8—622no—
240C22—21no—
241C22—21no—
242C22—1241no—
243C22—1241no—
244C22—1241no—
245C22#1175851912yes—
246C22C22×C22⋊D3111yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—1241no—
255C4—1241no—
256C4—1241no—
257C4C2×#9249512yes—
258C2—22no—
259C2—2481no—
260C2#1452156011yescharacteristic
261C2C2×#1175851911yescharacteristic
262C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C4×Q8×C31C2
socleC22×C31C22×D4
prime radical#11758507C22
hypercenterC2×C4×Q8D31
hypocenterC31#3117
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #14521384C2semidirect
4. #14521356C2semidirect
5. #14521271C2semidirect
6. #14521205C2semidirect
8. #14521154C2semidirect
9. #14521147C2semidirect
10. #14521106C2semidirect
11. C2×C4×Q8×C31C2semidirect
128. C2×C4×Q8D31semidirect
174. C31#3117semidirect

Maximal subgroups

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

subgroupindex
#145215832
#145213842
#145213672
#145213562
#145212712
#145212052
#145211882
#145211542
#145211472
#145211062
C2×C4×Q8×C312
#311731