Finite Group Databasebeta

← #16846526

Subgroups — #16846526

Order 3968.

344 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#11758513C211yesmaximal, characteristic
2C2×#11758404C211yesmaximal, characteristic, complemented
3C2×#11758398C211yesmaximal, characteristic, complemented
4C2×#11758420C211yesmaximal, characteristic
5C2×#11758403C211yesmaximal, characteristic, complemented
6#14520918C211yesmaximal, characteristic, complemented
7C2×C31×#102C211yesmaximal, characteristic
8C22×D4×C31C2211yescharacteristic
9C22×Dic62C2211yescharacteristic
10C22×C8×C31C2211yescharacteristic
11C2×C4⋊C4×C31C2211yescharacteristic
12C2×C22⋊Dic31—22no—
13#11758513—22no—
14#11758513C414yescomplemented
15C2×C4⋊Dic31C2211yescharacteristic
16C2×Dic31⋊C4—22no—
17C2×C4×Dic31C2211yescharacteristic
18C22×C31⋊C8C2211yescharacteristic
19C31×#102—22no—
20#11758420—22no—
21#11758404—22no—
22#11758403—22no—
23#11758398—22no—
24C24×C31—21no—
25C2×D4×C31—21no—
26C2×D4×C31—21no—
27C2×D4×C31—21no—
28C2×D4×C31—21no—
29C2×D4×C31—21no—
30C2×D4×C31C2×C412yes—
31C22×C4×C31—21no—
32C22×C4×C31C2311yescharacteristic
33C22×Dic31—21no—
34C22×Dic31—21no—
35C22×Dic31D412yes—
36C2×Dic62—21no—
37C2×Dic62—21no—
38C2×Dic62—21no—
39C2×Dic62—21no—
40C2×Dic62—21no—
41C2×Dic62C2×C412yes—
42C2×C8×C31—21no—
43C2×C8×C31—21no—
44C2×C8×C31D412yes—
45C4⋊C4×C31—21no—
46C4⋊C4×C31—21no—
47C22⋊Dic31—28no—
48C4⋊Dic31—21no—
49C4⋊Dic31—21no—
50Dic31⋊C4—44no—
51C4×Dic31C2×C412yes—
52C4×Dic31D412yes—
53C2×C31⋊C8—21no—
54C2×C31⋊C8—21no—
55C2×C31⋊C8D412yes—
56C23×C31—22no—
57C23×C31—41no—
58C23×C31—41no—
59C23×C31—42no—
60C23×C31—42no—
61C23×C31C2×D411yescharacteristic
62D4×C31—22no—
63D4×C31—22no—
64D4×C31—41no—
65D4×C31—41no—
66C2×C4×C31—41no—
67C2×C4×C31—42no—
68C2×C4×C31C22×C411yescharacteristic
69C2×C4×C31C2×D411yescharacteristic
70C2×C4×C31C2×D411yescharacteristic
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31#4111yescharacteristic
73C2×C4×C31#4111yescharacteristic
74C2×Dic31—22no—
75C2×Dic31—22no—
76C2×Dic31—22no—
77C2×Dic31—41no—
78C2×Dic31—41no—
79C2×Dic31—41no—
80C2×Dic31—42no—
81C2×Dic31C22⋊C414yes—
82Dic62—22no—
83Dic62—22no—
84Dic62—41no—
85Dic62—41no—
86C8×C31—22no—
87C31⋊C8—22no—
88#1959—311nomaximal, Sylow
89C22×C31—42no—
90C22×C31—44no—
91C22×C31—82no—
92C22×C31—81no—
93C22×C31—81no—
94C22×C31C2×C22⋊C411yescharacteristic
95C22×C31#12411yescharacteristic
96C22×C31#12711yescharacteristic
97C22×C31AGL(1,ℤ/8)12yes—
98C22×C31#13712yes—
99C4×C31—81no—
100C4×C31C4×D411yescharacteristic
101C4×C31C4×D411yescharacteristic
102C4×C31#12111yescharacteristic
103C4×C31#12111yescharacteristic
104Dic31—24no—
105Dic31—42no—
106Dic31—81no—
107C2×#124—311no—
108C2×#103—311no—
109C2×#103—311no—
110C2×#102—311no—
111C2×#102D3111yescharacteristic
112C2×#97—311no—
113#384—311no—
114C2×C31—82no—
115C2×C31#39011yescharacteristic
116C2×C31#41712yes—
117C2×C31#48912yes—
118C2×C31#49611yescharacteristic
119C2×C31#49711yescharacteristic
120C22×Q8—311no—
121C22×D4C2×D3111yescharacteristic
122C22×C8—311no—
123C22×C8C2×D3111yescharacteristic
124#124—314no—
125#124—622no—
126C2×C4⋊C4—311no—
127C2×C4⋊C4—622no—
128C2×C4⋊C4C2×D3111yescharacteristic
129C2×C22⋊C4—622no—
130C2×C42—311no—
131#103—622no—
132#103—622no—
133#102—22no—
134#102—622no—
135#97—622no—
136C31#195911yescharacteristic, complemented, Sylow
137C24—21no—
138C2×Q8—312no—
139C2×Q8—621no—
140C2×Q8—621no—
141C2×Q8—621no—
142C2×Q8—621no—
143C2×Q8—621no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C4×D3112yes—
150C22×C4—21no—
151C22×C4—312no—
152C22×C4—621no—
153C22×C4—621no—
154C22×C4C22×D3111yescharacteristic
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—312no—
158C2×C8—621no—
159C2×C8—621no—
160C2×C8C22⋊D3112yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—621no—
164C4⋊C4—621no—
165C4⋊C4—1244no—
166C22⋊C4—628no—
167C42—312no—
168C42—312no—
169C23—22no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23D4×D3111yescharacteristic
175Q8—622no—
176Q8—622no—
177Q8—1241no—
178Q8—1241no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—314no—
186C2×C4—622no—
187C2×C4—622no—
188C2×C4—622no—
189C2×C4—1241no—
190C2×C4—1241no—
191C2×C4—1242no—
192C2×C4—1241no—
193C2×C4C2×C4×D3111yescharacteristic
194C2×C4#9249511yescharacteristic
195C2×C4D4×D3111yescharacteristic
196C2×C4D4×D3111yescharacteristic
197C2×C4#9249711yescharacteristic
198C2×C4C2×C22⋊D3111yescharacteristic
199C8—22no—
200C8—622no—
201C22—42no—
202C22—44no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22C22⋊C4×D3111yescharacteristic
207C22#1175846211yescharacteristic
208C22#1175848812yes—
209C22#1175849212yes—
210C22#1175851711yescharacteristic
211C4—81no—
212C4—624no—
213C4—1242no—
214C4—2481no—
215C4(C22⋊D31)⋊C411yescharacteristic
216C4#1175846111yescharacteristic
217C4C4×C22⋊D3111yescharacteristic
218C4#1175851611yescharacteristic
219C2—82no—
220C2#1452094911yescharacteristic
221C2#1452104712yes—
222C2#1452140712yes—
223C2#1452143711yescharacteristic
224C2#1452144011yescharacteristic
subgroupG/H
centerC23D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC2×C31×#102C2
socleC23×C31C2×D4
prime radicalC22×D4×C31C22
cosocleC22×C4×C31C23
hypercenterC2×#102D31
hypocenterC31#1959
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#11758404C2semidirect
3. C2×#11758398C2semidirect
5. C2×#11758403C2semidirect
6. #14520918C2semidirect
14. #11758513C4semidirect
136. C31#1959semidirect

Maximal subgroups

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

subgroupindex
C2×#117585132
C2×#117584042
C2×#117583982
C2×#117584202
C2×#117584032
#145209182
C2×C31×#1022
#195931