Finite Group Databasebeta

← #16856106

Subgroups — #16856106

Order 3968.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521600C211yesmaximal, characteristic
2C4⋊Q8×D31C211yesmaximal, characteristic, complemented
3#14521410C214yesmaximal, complemented
4#14521399C211yesmaximal, characteristic, complemented
5#14521397C211yesmaximal, characteristic, complemented
6#14521396C211yesmaximal, characteristic, complemented
7#14521049C214yesmaximal, complemented
8C4×#92469C211yesmaximal, characteristic, complemented
9C31×#487C211yesmaximal, characteristic, complemented
10C2×#92499C2212yes—
11C2×Q8×D31—21no—
12C31×C4⋊Q8C2211yescharacteristic
13C31×C4⋊Q8C2211yescharacteristic
14#11758524C2212yes—
15#11758521—21no—
16C2×#92469C2212yes—
17#11758472—21no—
18#11758472—21no—
19#11758470—21no—
20#11758470—21no—
21C4⋊C4×D31—21no—
22C4⋊C4×D31C2212yes—
23#11758465—21no—
24#11758465C2212yes—
25#11758451C2211yescharacteristic
26C42×D31C2211yescharacteristic
27#11758446C2211yescharacteristic
28C31×#103C2214yes—
29C4×C8×C31C2211yescharacteristic
30#11758423C2214yes—
31#11758409C2214yes—
32#11758403C2212yes—
33#11758397C2214yes—
34C4×C31⋊C8C2211yescharacteristic
35C2×Q8×C31—21no—
36C2×Q8×C31C2312yes—
37#92499—28no—
38Q8×D31—44no—
39C2×D124—22no—
40C2×D124—21no—
41C2×D124—21no—
42C2×D124C2312yes—
43C2×C4×D31—22no—
44C2×C4×D31—22no—
45C2×C4×D31D411yescharacteristic
46C2×C4×D31D411yescharacteristic
47C2×C4×D31C2311yescharacteristic
48C2×Dic62—21no—
49C2×Dic62—22no—
50C2×C8×C31C2312yes—
51C4⋊C4×C31—21no—
52C4⋊C4×C31—21no—
53C4⋊C4×C31—21no—
54C4⋊C4×C31C2312yes—
55C42×C31C2311yescharacteristic
56(C2×D31)⋊C4—22no—
57(C2×D31)⋊C4—22no—
58C4⋊Dic31—21no—
59C4⋊Dic31C2312yes—
60Dic31⋊C4—22no—
61Dic31⋊C4—22no—
62C4×Dic31D411yescharacteristic
63C4×Dic31D411yescharacteristic
64C4×Dic31C2311yescharacteristic
65C2×C31⋊C8C2312yes—
66#92469—24no—
67C22×D31—22no—
68C22×D31C2×D411yescharacteristic
69Q8×C31—24no—
70Q8×C31—42no—
71C2×C4×C31—22no—
72C2×C4×C31—22no—
73C2×C4×C31C2×D411yescharacteristic
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31C2411yescharacteristic
76C2×Dic31—22no—
77C2×Dic31C2×D411yescharacteristic
78C2×Dic31C2×D411yescharacteristic
79C2×Dic31C2×D411yescharacteristic
80D124—24no—
81D124—44no—
82D124—42no—
83D124—42no—
84C4×D31—22no—
85C4×D31—22no—
86C4×D31—44no—
87C4×D31—44no—
88C4×D31#4114yes—
89Dic62—44no—
90Dic62—42no—
91C8×C31—22no—
92C31⋊C8—22no—
93#3141—311nomaximal, Sylow
94C22×C31C22×D411yescharacteristic
95C2×D31—21no—
96C2×D31—21no—
97C2×D31—42no—
98C2×D31—42no—
99C2×D31—42no—
100C2×D31#12412yes—
101C4×C31—42no—
102C4×C31—42no—
103C4×C31AGL(1,ℤ/8)12yes—
104C4×C31#13712yes—
105C4×C31C2×#4112yes—
106Dic31—21no—
107Dic31—21no—
108Dic31—42no—
109Dic31#12412yes—
110#532—311no—
111C2×C4⋊Q8—311no—
112#489—314no—
113#487—311no—
114#487D3111yescharacteristic, complemented
115#486—311no—
116#486—311no—
117#417—314no—
118C4×M16—311no—
119C2×C31C2×#12411yescharacteristic
120C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C31C2×#13711yescharacteristic
122D31—22no—
123D31—82no—
124C2×#41—312no—
125C22×Q8—621no—
126C2×M16—312no—
127C4⋊Q8—312no—
128C4⋊Q8—311no—
129C4⋊Q8—621no—
130C4⋊Q8—621no—
131C4⋊Q8C2×D3111yescharacteristic
132C4⋊Q8C2×D3111yescharacteristic
133#127—311no—
134#124—312no—
135#121—621no—
136#121—621no—
137C4×D4—621no—
138C4×D4—621no—
139C2×C4⋊C4—312no—
140C2×C4⋊C4—621no—
141C2×C42—311no—
142#103—314no—
143#103C2×D3114yes—
144#102—314no—
145#102—314no—
146#97—312no—
147C4×C8—311no—
148C4×C8C2×D3111yescharacteristic
149C31#314111yescharacteristic, complemented, Sylow
150#41—628no—
151C2×Q8—21no—
152C2×Q8—622no—
153C2×Q8—621no—
154C2×Q8—1244no—
155C2×Q8C22×D3112yes—
156C2×D4—312no—
157C2×D4—622no—
158C2×D4—621no—
159C2×D4—621no—
160C22×C4—311no—
161C22×C4—311no—
162C22×C4—311no—
163C22×C4—622no—
164C22×C4—622no—
165M16—624no—
166C2×C8—312no—
167C2×C8C22×D3112yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—312no—
172C4⋊C4—622no—
173C4⋊C4—622no—
174C4⋊C4—621no—
175C4⋊C4C22×D3112yes—
176C22⋊C4—622no—
177C22⋊C4—622no—
178C42—311no—
179C42—311no—
180C42—311no—
181C42C22×D3111yescharacteristic
182C23—311no—
183C23—622no—
184Q8—24no—
185Q8—42no—
186Q8—1244no—
187Q8—1242no—
188D4—624no—
189D4—1244no—
190D4—1242no—
191D4—1242no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—311no—
195C2×C4—314no—
196C2×C4—311no—
197C2×C4—311no—
198C2×C4—622no—
199C2×C4—622no—
200C2×C4—622no—
201C2×C4—1244no—
202C2×C4—1244no—
203C2×C4D4×D3111yescharacteristic
204C2×C4D4×D3111yescharacteristic
205C2×C4C23×D3111yescharacteristic
206C8—22no—
207C8—622no—
208C22—312no—
209C22—621no—
210C22—621no—
211C22—1242no—
212C22—1242no—
213C22—1242no—
214C22C2×D4×D3111yescharacteristic
215C4—42no—
216C4—42no—
217C4—312no—
218C4—621no—
219C4—621no—
220C4—1242no—
221C4#1175849112yes—
222C4#1175849512yes—
223C4#41×D3112yes—
224C2—622no—
225C2—2482no—
226C2#124×D3111yescharacteristic
227C2C2×#1175849111yescharacteristic
228C2C2×#1175849511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#487C2
socleC22×C31C22×D4
prime radical#14521600C2
hypercenter#487D31
hypocenterC31#3141
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D31C2semidirect
3. #14521410C2semidirect
4. #14521399C2semidirect
5. #14521397C2semidirect
6. #14521396C2semidirect
7. #14521049C2semidirect
8. C4×#92469C2semidirect
9. C31×#487C2semidirect
114. #487D31semidirect
149. C31#3141semidirect

Maximal subgroups

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

subgroupindex
#145216002
C4⋊Q8×D312
#145214102
#145213992
#145213972
#145213962
#145210492
C4×#924692
C31×#4872
#314131