Finite Group Databasebeta

← #16855834

Subgroups — #16855834

Order 3968.

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

normal subgroups only

This group has 424 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#14521599C211yesmaximal, characteristic
2C2×C4×Dic62C211yesmaximal, characteristic, complemented
3#14521363C212yesmaximal, complemented
4#14521345C212yesmaximal, complemented
5#14521250C212yesmaximal, complemented
6#14521190C212yesmaximal, complemented
7#14521151C214yesmaximal, complemented
8C31×#432C211yesmaximal, characteristic, complemented
9C2×#92495—21no—
10C22×Dic62C2211yescharacteristic
11C2×C42×C31C2211yescharacteristic
12#11758507—21no—
13#11758507C2212yes—
14C2×C4⋊Dic31C2211yescharacteristic
15C2×Dic31⋊C4—21no—
16C2×C4×Dic31—21no—
17C2×#92470C2214yes—
18#11758452—21no—
19#11758451C2211yescharacteristic
20C4×D124—21no—
21C4×D124C2212yes—
22#11758446C2211yescharacteristic
23C4×Dic62—21no—
24C4×Dic62—21no—
25C4×Dic62C2212yes—
26C4×Dic62C2212yes—
27C31×C4⋊C8C2212yes—
28C31×C22⋊C8C2212yes—
29C31×#97C2212yes—
30#11758409C2214yes—
31#11758406C2214yes—
32#11758404C2214yes—
33C22×C4×C31D411yescharacteristic
34C22×C4×C31D411yescharacteristic
35C22×C4×C31C2311yescharacteristic
36C2×C22⋊D31—22no—
37C22×Dic31—21no—
38C22×Dic31—21no—
39#92495—44no—
40C2×D124—21no—
41C2×D124—21no—
42C2×D124C2312yes—
43C2×C4×D31—22no—
44C2×Dic62—21no—
45C2×Dic62—24no—
46C2×Dic62—21no—
47C2×Dic62C2312yes—
48C2×C8×C31C2314yes—
49C42×C31D411yescharacteristic
50C42×C31D411yescharacteristic
51C42×C31C2311yescharacteristic
52C42×C31C2311yescharacteristic
53(C2×D31)⋊C4—22no—
54(C2×D31)⋊C4—22no—
55C4⋊Dic31—21no—
56C4⋊Dic31—21no—
57C4⋊Dic31C2312yes—
58C4⋊Dic31C2312yes—
59Dic31⋊C4—22no—
60Dic31⋊C4—22no—
61C4×Dic31—22no—
62C4×Dic31—22no—
63#92470—28no—
64C23×C31C2×D411yescharacteristic
65C22×D31—22no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—22no—
69C2×C4×C31C2×D411yescharacteristic
70C2×C4×C31C2×D411yescharacteristic
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31C2×D411yescharacteristic
73C2×C4×C31C2×D411yescharacteristic
74C2×C4×C31C2411yescharacteristic
75C22⋊D31—44no—
76C2×Dic31—24no—
77C2×Dic31—22no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—42no—
81D124—24no—
82D124—42no—
83D124—42no—
84C4×D31—44no—
85Dic62—24no—
86Dic62—22no—
87Dic62—42no—
88Dic62#4114yescomplemented
89C8×C31—24no—
90#3112—311nomaximal, Sylow
91C22×C31—21no—
92C22×C31—21no—
93C22×C31AGL(1,ℤ/8)12yes—
94C22×C31C22×D411yescharacteristic
95C2×D31—42no—
96C2×D31—42no—
97C2×D31—42no—
98C4×C31—21no—
99C4×C31—21no—
100C4×C31—21no—
101C4×C31—21no—
102C4×C31#13712yes—
103C4×C31C2×#4112yes—
104Dic31—24no—
105Dic31—42no—
106Dic31—42no—
107#532—311no—
108C2×C4×Q8—311no—
109#480—312no—
110#477—312no—
111#460—312no—
112#448—312no—
113#440—314no—
114#432D3111yescharacteristic, complemented
115C2×C31—22no—
116C2×C31#52511yescharacteristic
117C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
118C2×C31C2×#13711yescharacteristic
119D31—82no—
120C2×#41—621no—
121C22×Q8—311no—
122C2×SD16—314no—
123C4⋊Q8—311no—
124#127—311no—
125#124—621no—
126#121—312no—
127#121—621no—
128C4×Q8—312no—
129C4×Q8—312no—
130C4×Q8—621no—
131C4×Q8—621no—
132C4×D4—312no—
133C4×D4—621no—
134C2×C4⋊C4—311no—
135C2×C4⋊C4—621no—
136C2×C42—621no—
137C2×C42C2×D3111yescharacteristic
138#107—314no—
139C4⋊C8C2×D3112yes—
140#103—314no—
141#102—314no—
142C22⋊C8C2×D3112yes—
143#97C2×D3112yes—
144C31#311211yescharacteristic, complemented, Sylow
145#41—1244no—
146C2×Q8—312no—
147C2×Q8—624no—
148C2×Q8—621no—
149C2×Q8—621no—
150C2×D4—312no—
151C2×D4—622no—
152C2×D4—621no—
153C2×D4—621no—
154C22×C4—621no—
155C22×C4—621no—
156C22×C4—622no—
157C22×C4D12411yescharacteristic
158C22×C4D12411yescharacteristic
159C22×C4C22×D3111yescharacteristic
160SD16—628no—
161C2×C8C22×D3114yes—
162C4⋊C4—312no—
163C4⋊C4—312no—
164C4⋊C4—622no—
165C4⋊C4—622no—
166C4⋊C4—621no—
167C4⋊C4—621no—
168C22⋊C4—622no—
169C22⋊C4—622no—
170C42—622no—
171C42—622no—
172C42D12411yescharacteristic
173C42D12411yescharacteristic
174C42C22×D3111yescharacteristic
175C42C22×D3111yescharacteristic
176C23—622no—
177C23C2×D12411yescharacteristic
178Q8—314no—
179Q8—624no—
180Q8—622no—
181Q8—1242no—
182D4—624no—
183D4—1242no—
184D4—1244no—
185D4—1242no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—624no—
190C2×C4—622no—
191C2×C4—622no—
192C2×C4—622no—
193C2×C4—1244no—
194C2×C4—1242no—
195C2×C4C2×D12411yescharacteristic
196C2×C4C2×D12411yescharacteristic
197C2×C4C2×D12411yescharacteristic
198C2×C4C2×D12411yescharacteristic
199C2×C4C2×D12411yescharacteristic
200C2×C4C23×D3111yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—1242no—
205C22—1242no—
206C22—1242no—
207C22#1175848512yes—
208C22C22×D12411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—624no—
214C4—1242no—
215C4—1242no—
216C4#1175848612yes—
217C4#41×D3112yes—
218C2—22no—
219C2—2482no—
220C2#1452156111yescharacteristic
221C2C2×#1175848511yescharacteristic
222C2C2×#1175848611yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#432C2
socleC22×C31C22×D4
prime radical#14521599C2
hypercenter#432D31
hypocenterC31#3112
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic62C2semidirect
3. #14521363C2semidirect
4. #14521345C2semidirect
5. #14521250C2semidirect
6. #14521190C2semidirect
7. #14521151C2semidirect
8. C31×#432C2semidirect
88. Dic62#41semidirect
114. #432D31semidirect
144. C31#3112semidirect

Maximal subgroups

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

subgroupindex
#145215992
C2×C4×Dic622
#145213632
#145213452
#145212502
#145211902
#145211512
C31×#4322
#311231