Finite Group Databasebeta

← #16855841

Subgroups — #16855841

Order 3968.

380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 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#14521606C211yesmaximal, characteristic
2#14521379C212yesmaximal
3#14521334C212yesmaximal, complemented
4#14521259C212yesmaximal, complemented
5#14521205C212yesmaximal
6#14521154C214yesmaximal, complemented
7#14521106C211yesmaximal, characteristic
8C2×C4×Q8×C31C211yesmaximal, characteristic
9C22×Q8×C31C2211yescharacteristic
10C4×Q8×C31—21no—
11C4×Q8×C31—21no—
12C4×Q8×C31C2212yes—
13C4×Q8×C31C2212yes—
14C2×C4⋊C4×C31—21no—
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C42×C31—21no—
17C2×C42×C31C2211yescharacteristic
18C2×#92482C2214yes—
19#11758503—21no—
20#11758501—21no—
21#11758501C2212yes—
22#11758447—21no—
23#11758446C2211yescharacteristic
24#11758446C2211yescharacteristic
25C4×Dic62—21no—
26C4×Dic62C2212yes—
27#11758423C2214yes—
28C22⋊(C31⋊C8)C2212yes—
29#11758398C2214yes—
30#11758396C2214yes—
31C4⋊(C31⋊C8)C2212yes—
32#11758391C2212yes—
33C2×Q8×C31—24no—
34C2×Q8×C31—21no—
35C2×Q8×C31C2312yes—
36C22×C4×C31—21no—
37C22×C4×C31—21no—
38C22×C4×C31D411yescharacteristic
39C22×C4×C31D411yescharacteristic
40C22×C4×C31C2311yescharacteristic
41C2×Dic62—21no—
42C2×Dic62C2312yes—
43C4⋊C4×C31—22no—
44C4⋊C4×C31—22no—
45C4⋊C4×C31—21no—
46C4⋊C4×C31C2312yes—
47C42×C31—22no—
48C42×C31—22no—
49C42×C31D411yescharacteristic
50C42×C31D411yescharacteristic
51C42×C31C2311yescharacteristic
52C42×C31C2311yescharacteristic
53C22⋊Dic31—22no—
54#92482—28no—
55C4⋊Dic31—21no—
56C4⋊Dic31—21no—
57C4⋊Dic31—21no—
58C4⋊Dic31C2312yes—
59Dic31⋊C4—22no—
60Dic31⋊C4—22no—
61C4×Dic31—22no—
62C2×C31⋊C8C2314yes—
63C23×C31C2×D411yescharacteristic
64Q8×C31—24no—
65Q8×C31—22no—
66Q8×C31#4114yes—
67C2×C4×C31—24no—
68C2×C4×C31—22no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31—22no—
72C2×C4×C31—22no—
73C2×C4×C31—42no—
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31C2×D411yescharacteristic
76C2×C4×C31C2×D411yescharacteristic
77C2×C4×C31C2×D411yescharacteristic
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2411yescharacteristic
80C2×Dic31—22no—
81C2×Dic31—22no—
82Dic62—24no—
83Dic62—42no—
84C31⋊C8—24no—
85#3113—311nomaximal, Sylow
86C22×C31—21no—
87C22×C31—21no—
88C22×C31#13712yes—
89C22×C31C22×D411yescharacteristic
90C4×C31—24no—
91C4×C31—21no—
92C4×C31—21no—
93C4×C31—21no—
94C4×C31—21no—
95C4×C31—42no—
96C4×C31#13712yes—
97C4×C31C2×#4112yes—
98Dic31—42no—
99Dic31—42no—
100#533—311no—
101C2×C4×Q8D3111yescharacteristic
102#483—312no—
103#475—312no—
104#462—312no—
105#451—312no—
106#441—314no—
107#432—311no—
108C2×C31—22no—
109C2×C31#52511yescharacteristic
110C2×C31C2×#13711yescharacteristic
111C2×C31C2×#13711yescharacteristic
112C22×Q8C2×D3111yescharacteristic
113C2×Q16—314no—
114C4⋊Q8—311no—
115C4⋊Q8—311no—
116#125—621no—
117C22⋊Q8—312no—
118C22⋊Q8—621no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—312no—
122C4×Q8—621no—
123C4×Q8C2×D3112yes—
124C4×Q8C2×D3112yes—
125#117—621no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C2×D3111yescharacteristic
128C2×C42—21no—
129C2×C42C2×D3111yescharacteristic
130#106—314no—
131C4⋊C8—312no—
132#103—314no—
133#103—314no—
134C22⋊C8—312no—
135#97—312no—
136C31#311311yescharacteristic, complemented, Sylow
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8—312no—
140C2×Q8—621no—
141C2×Q8C22×D3112yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4C22⋊D3111yescharacteristic
145C22×C4C22⋊D3111yescharacteristic
146C22×C4C22×D3111yescharacteristic
147Q16—628no—
148C2×C8—314no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—312no—
153C4⋊C4—622no—
154C4⋊C4—622no—
155C4⋊C4—621no—
156C4⋊C4—621no—
157C4⋊C4—621no—
158C4⋊C4C22×D3112yes—
159C22⋊C4—622no—
160C42—22no—
161C42—22no—
162C42—622no—
163C42C22⋊D3111yescharacteristic
164C42C22⋊D3111yescharacteristic
165C42C22×D3111yescharacteristic
166C42C22×D3111yescharacteristic
167C23C2×C22⋊D3111yescharacteristic
168Q8—24no—
169Q8—22no—
170Q8—624no—
171Q8—1242no—
172Q8#9249514yes—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—622no—
181C2×C4—622no—
182C2×C4C2×C22⋊D3111yescharacteristic
183C2×C4C2×C22⋊D3111yescharacteristic
184C2×C4C2×C22⋊D3111yescharacteristic
185C2×C4C2×C22⋊D3111yescharacteristic
186C2×C4C2×C22⋊D3111yescharacteristic
187C2×C4C23×D3111yescharacteristic
188C8—624no—
189C22—21no—
190C22—21no—
191C22#1175852812yes—
192C22C22×C22⋊D3111yescharacteristic
193C4—24no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—1242no—
200C4—1242no—
201C4#1175851912yes—
202C4C2×#9249512yes—
203C2—22no—
204C2#1452156011yescharacteristic
205C2C2×#1175852811yescharacteristic
206C2C2×#1175851911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C4×Q8×C31C2
socleC22×C31C22×D4
prime radicalC23×C31C2×D4
hypercenterC2×C4×Q8D31
hypocenterC31#3113
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
3. #14521334C2semidirect
4. #14521259C2semidirect
6. #14521154C2semidirect
136. C31#3113semidirect

Maximal subgroups

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

subgroupindex
#145216062
#145213792
#145213342
#145212592
#145212052
#145211542
#145211062
C2×C4×Q8×C312
#311331