Finite Group Databasebeta

← #16856108

Subgroups — #16856108

Order 3968.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#14521598C211yesmaximal, characteristic
2#125×D31C211yesmaximal, characteristic, complemented
3#14521416C212yesmaximal, complemented
4#14521412C212yesmaximal, complemented
5#14521406C211yesmaximal, characteristic, complemented
6#14521404C211yesmaximal, characteristic, complemented
7#14521403C211yesmaximal, characteristic, complemented
8#14521049C212yesmaximal, complemented
9#14521047C212yesmaximal, complemented
10C4×#92469C211yesmaximal, characteristic, complemented
11C31×#488C211yesmaximal, characteristic, complemented
12C2×#92499C2211yescharacteristic
13C2×#92497C2211yescharacteristic
14C31×#125C2211yescharacteristic
15C31×#124C2211yescharacteristic
16#11758521C2211yescharacteristic
17#11758517C2211yescharacteristic
18C2×#92469C2212yes—
19C4⋊C4×D31—22no—
20C4⋊C4×D31C2212yes—
21#11758467C2212yes—
22#11758466—22no—
23#11758461—22no—
24(C22⋊D31)⋊C4—22no—
25#11758452C2211yescharacteristic
26C42×D31C2211yescharacteristic
27#11758447C2211yescharacteristic
28C31×#103C2212yes—
29C31×#102C2212yes—
30C4×C8×C31C2211yescharacteristic
31#11758423C2212yes—
32#11758420C2212yes—
33#11758409C2212yes—
34#11758404C2212yes—
35#11758403C2212yes—
36#11758398C2212yes—
37#11758397C2212yes—
38C4×C31⋊C8C2211yescharacteristic
39C2×Q8×C31C2311yescharacteristic
40C2×D4×C31C2311yescharacteristic
41C2×C22⋊D31—21no—
42C2×C22⋊D31—22no—
43C22×Dic31—21no—
44#92499—24no—
45#92497—24no—
46C2×D124—21no—
47C2×D124C2311yescharacteristic
48C2×C4×D31—22no—
49C2×C4×D31—21no—
50C2×C4×D31D412yes—
51C2×C4×D31C2311yescharacteristic
52C2×Dic62C2311yescharacteristic
53C2×C8×C31C2312yes—
54C4⋊C4×C31—22no—
55C4⋊C4×C31C2312yes—
56C22⋊C4×C31—22no—
57C42×C31C2311yescharacteristic
58(C2×D31)⋊C4—22no—
59C4⋊Dic31—22no—
60C4⋊Dic31C2312yes—
61Dic31⋊C4—22no—
62Dic31⋊C4—22no—
63Dic31⋊C4—22no—
64Dic31⋊C4—22no—
65C4×Dic31D412yes—
66C4×Dic31C2311yescharacteristic
67C2×C31⋊C8C2312yes—
68#92469—24no—
69C23×C31—21no—
70C22×D31—21no—
71C22×D31C2×D411yescharacteristic
72Q8×C31—22no—
73D4×C31—22no—
74C2×C4×C31—22no—
75C2×C4×C31—21no—
76C2×C4×C31C2×D412yes—
77C2×C4×C31C2411yescharacteristic
78C22⋊D31—44no—
79C22⋊D31—42no—
80C2×Dic31—22no—
81C2×Dic31—21no—
82C2×Dic31—42no—
83C2×Dic31C2×D412yes—
84C2×Dic31C2×D411yescharacteristic
85D124—22no—
86D124—42no—
87C4×D31—42no—
88C4×D31—44no—
89C4×D31—42no—
90C4×D31#4114yes—
91Dic62—22no—
92C8×C31—22no—
93C31⋊C8—22no—
94#3142—311nomaximal, Sylow
95C22×C31—41no—
96C22×C31—42no—
97C22×C31C22×D411yescharacteristic
98C2×D31—22no—
99C2×D31—41no—
100C2×D31—42no—
101C2×D31#12412yes—
102C4×C31—22no—
103C4×C31—41no—
104C4×C31—42no—
105C4×C31C2×#4112yes—
106Dic31—22no—
107Dic31—41no—
108Dic31—42no—
109Dic31#12412yes—
110#532—311no—
111C2×#125—311no—
112#491—312no—
113#490—312no—
114#488—311no—
115#488—311no—
116#488—311no—
117#488D3111yescharacteristic, complemented
118#417—312no—
119#417—312no—
120C4×M16—311no—
121C2×C31—81no—
122C2×C31C2×#12411yescharacteristic
123C2×C31#57512yes—
124D31—22no—
125D31—81no—
126C2×#41—311no—
127C2×#41—311no—
128C2×M16—312no—
129C4⋊Q8—311no—
130#127—311no—
131#125—312no—
132#125—311no—
133#125—622no—
134#125C2×D3111yescharacteristic
135#124—311no—
136#124C2×D3111yescharacteristic
137#121—622no—
138C4×D4—622no—
139C2×C4⋊C4—312no—
140C2×C4⋊C4—622no—
141C2×C42—311no—
142#103—312no—
143#103—312no—
144#103—312no—
145#103C2×D3112yes—
146#102—312no—
147#102—312no—
148#102—312no—
149#102C2×D3112yes—
150#97—312no—
151C4×C8—311no—
152C4×C8C2×D3111yescharacteristic
153C31#314211yescharacteristic, complemented, Sylow
154#41—624no—
155#41—624no—
156C2×Q8—311no—
157C2×Q8C22×D3111yescharacteristic
158C2×D4—311no—
159C2×D4—621no—
160C2×D4—622no—
161C2×D4—621no—
162C2×D4C22×D3111yescharacteristic
163C22×C4—311no—
164C22×C4—312no—
165C22×C4—621no—
166C22×C4—622no—
167C22×C4—621no—
168M16—624no—
169C2×C8—312no—
170C2×C8C22×D3112yes—
171C4⋊C4—22no—
172C4⋊C4—312no—
173C4⋊C4—622no—
174C4⋊C4—622no—
175C4⋊C4—622no—
176C4⋊C4—622no—
177C4⋊C4—622no—
178C4⋊C4C22×D3112yes—
179C22⋊C4—22no—
180C22⋊C4—622no—
181C42—312no—
182C42—311no—
183C42C22×D3111yescharacteristic
184C23—21no—
185C23—311no—
186C23—621no—
187Q8—22no—
188Q8—622no—
189D4—22no—
190D4—622no—
191D4—1242no—
192D4—1244no—
193D4—1242no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—311no—
197C2×C4—314no—
198C2×C4—312no—
199C2×C4—621no—
200C2×C4—622no—
201C2×C4—1242no—
202C2×C4—1242no—
203C2×C4—1244no—
204C2×C4—1242no—
205C2×C4D4×D3112yes—
206C2×C4C23×D3111yescharacteristic
207C8—22no—
208C8—622no—
209C22—41no—
210C22—42no—
211C22—312no—
212C22—622no—
213C22—1241no—
214C22—1242no—
215C22C2×D4×D3111yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—312no—
220C4—622no—
221C4—1241no—
222C4—1242no—
223C4#41×D3112yes—
224C2—81no—
225C2—622no—
226C2—2481no—
227C2#124×D3111yescharacteristic
228C2#1452184612yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#488C2
socleC22×C31C22×D4
prime radical#14521598C2
hypercenter#488D31
hypocenterC31#3142
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D31C2semidirect
3. #14521416C2semidirect
4. #14521412C2semidirect
5. #14521406C2semidirect
6. #14521404C2semidirect
7. #14521403C2semidirect
8. #14521049C2semidirect
9. #14521047C2semidirect
10. C4×#92469C2semidirect
11. C31×#488C2semidirect
117. #488D31semidirect
153. C31#3142semidirect

Maximal subgroups

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

subgroupindex
#145215982
#125×D312
#145214162
#145214122
#145214062
#145214042
#145214032
#145210492
#145210472
C4×#924692
C31×#4882
#314231