Finite Group Databasebeta

← #16855837

Subgroups — #16855837

Order 3968.

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

normal subgroups only

This group has 380 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
1C2×C4×Dic62C211yesmaximal, characteristic
2#14521377C212yesmaximal
3#14521331C212yesmaximal, complemented
4#14521256C212yesmaximal, complemented
5#14521204C212yesmaximal
6#14521154C214yesmaximal, complemented
7#14521106C211yesmaximal, characteristic
8C31×#533C211yesmaximal, characteristic
9C22×Dic62C2211yescharacteristic
10C31×C4⋊Q8C2211yescharacteristic
11C31×C4⋊Q8C2211yescharacteristic
12C31×#125—21no—
13C31×C22⋊Q8—21no—
14C31×C22⋊Q8C2212yes—
15C4×Q8×C31—21no—
16C4×Q8×C31C2212yes—
17C31×#117—21no—
18C2×C42×C31C2211yescharacteristic
19C2×#92482C2214yes—
20C2×C4⋊Dic31C2211yescharacteristic
21C2×Dic31⋊C4—21no—
22C2×C4×Dic31—21no—
23C4×Dic62—21no—
24C4×Dic62—21no—
25C4×Dic62C2212yes—
26C4×Dic62C2212yes—
27#11758423C2214yes—
28C22⋊(C31⋊C8)C2212yes—
29#11758398C2214yes—
30#11758396C2214yes—
31C4⋊(C31⋊C8)C2212yes—
32#11758391C2212yes—
33C2×Q8×C31—21no—
34C2×Q8×C31C2312yes—
35C22×C4×C31D411yescharacteristic
36C22×C4×C31D411yescharacteristic
37C22×C4×C31C2311yescharacteristic
38C22×Dic31—21no—
39C22×Dic31—21no—
40C2×Dic62—21no—
41C2×Dic62—24no—
42C2×Dic62C2312yes—
43C4⋊C4×C31—21no—
44C4⋊C4×C31—21no—
45C4⋊C4×C31—21no—
46C4⋊C4×C31—22no—
47C4⋊C4×C31—22no—
48C4⋊C4×C31C2312yes—
49C22⋊C4×C31—22no—
50C42×C31—22no—
51C42×C31D411yescharacteristic
52C42×C31D411yescharacteristic
53C42×C31C2311yescharacteristic
54C42×C31C2311yescharacteristic
55#92482—28no—
56C4⋊Dic31—21no—
57C4⋊Dic31C2312yes—
58Dic31⋊C4—22no—
59Dic31⋊C4—22no—
60C4×Dic31—22no—
61C4×Dic31—22no—
62C2×C31⋊C8C2314yes—
63C23×C31C2×D411yescharacteristic
64Q8×C31—24no—
65Q8×C31—42no—
66C2×C4×C31—22no—
67C2×C4×C31—22no—
68C2×C4×C31—22no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31C2×D411yescharacteristic
72C2×C4×C31C2×D411yescharacteristic
73C2×C4×C31C2×D411yescharacteristic
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31C2×D411yescharacteristic
76C2×C4×C31C2411yescharacteristic
77C2×Dic31—24no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—42no—
81Dic62—24no—
82Dic62—22no—
83Dic62#4114yes—
84C31⋊C8—24no—
85#3113—311nomaximal, Sylow
86C22×C31—21no—
87C22×C31—21no—
88C22×C31#13712yes—
89C22×C31C22×D411yescharacteristic
90C4×C31—21no—
91C4×C31—21no—
92C4×C31—21no—
93C4×C31—21no—
94C4×C31—42no—
95C4×C31—42no—
96C4×C31#13712yes—
97C4×C31C2×#4112yes—
98Dic31—24no—
99Dic31—42no—
100#533D3111yescharacteristic
101C2×C4×Q8—311no—
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×Q8—311no—
113C2×Q16—314no—
114C4⋊Q8C2×D3111yescharacteristic
115C4⋊Q8C2×D3111yescharacteristic
116#125—21no—
117C22⋊Q8—21no—
118C22⋊Q8C2×D3112yes—
119C4×Q8—21no—
120C4×Q8—312no—
121C4×Q8—312no—
122C4×Q8—621no—
123C4×Q8—621no—
124C4×Q8C2×D3112yes—
125#117—21no—
126C2×C4⋊C4—311no—
127C2×C4⋊C4—621no—
128C2×C42—621no—
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—21no—
138C2×Q8—312no—
139C2×Q8—624no—
140C2×Q8—621no—
141C2×Q8C22×D3112yes—
142C22×C4—621no—
143C22×C4—621no—
144C22×C4C22⋊D3111yescharacteristic
145C22×C4C22⋊D3111yescharacteristic
146C22×C4C22×D3111yescharacteristic
147Q16—628no—
148C2×C8—314no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—312no—
155C4⋊C4—622no—
156C4⋊C4—622no—
157C4⋊C4—621no—
158C4⋊C4C22×D3112yes—
159C22⋊C4—22no—
160C42—22no—
161C42—622no—
162C42—622no—
163C42C22⋊D3111yescharacteristic
164C42C22⋊D3111yescharacteristic
165C42C22×D3111yescharacteristic
166C42C22×D3111yescharacteristic
167C23C2×C22⋊D3111yescharacteristic
168Q8—24no—
169Q8—42no—
170Q8—314no—
171Q8—624no—
172Q8—622no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—624no—
179C2×C4—622no—
180C2×C4—622no—
181C2×C4—1242no—
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#1175851912yes—
192C22C22×C22⋊D3111yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—624no—
200C4—1242no—
201C4#1175852812yes—
202C4#41×D3112yes—
203C2—22no—
204C2#1452155611yescharacteristic
205C2C2×#1175852811yescharacteristic
206C2C2×#1175851911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#533C2
socleC22×C31C22×D4
prime radicalC23×C31C2×D4
hypercenter#533D31
hypocenterC31#3113
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #14521331C2semidirect
4. #14521256C2semidirect
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
C2×C4×Dic622
#145213772
#145213312
#145212562
#145212042
#145211542
#145211062
C31×#5332
#311331