Finite Group Databasebeta

← #30670248

Subgroups — #30670248

Order 16768.

512 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16862205C211yesmaximal, characteristic
2#22553251C212yesmaximal, complemented
3#22553242C212yesmaximal, complemented
4#22553193C214yesmaximal, complemented
5#22553185C212yesmaximal
6#22553175C212yesmaximal
7#22553175C211yesmaximal, characteristic
8C22×C4⋊C4×C131C211yesmaximal, characteristic
9#16862205—24no—
10C2×C4⋊Dic131C2211yescharacteristic
11C2×C4⋊Dic131C2211yescharacteristic
12C2×Dic131⋊C4C2214yes—
13C2×C22⋊Dic131C2212yes—
14C2×C22⋊Dic131C2212yes—
15C2×C4×Dic131C2212yes—
16C131⋊#95C2212yes—
17C131⋊#95C2214yes—
18C131⋊#95C2214yes—
19C131⋊#95C2212yes—
20C131⋊#95C2214yes—
21C23×C4×C131C2211yescharacteristic
22C23×C4×C131C2211yescharacteristic
23C23×C4×C131C2211yescharacteristic
24C2×C4⋊C4×C131—24no—
25C2×C4⋊C4×C131C2212yes—
26C2×C4⋊C4×C131C2212yes—
27C22×Dic131C2314yes—
28C22×Dic131C2312yes—
29C22×Dic131C2312yes—
30C4⋊Dic131—22no—
31C4⋊Dic131—22no—
32Dic131⋊C4—28no—
33C22⋊Dic131—24no—
34C22⋊Dic131—24no—
35C4×Dic131—24no—
36C24×C131C2311yescharacteristic
37C22×C4×C131—22no—
38C22×C4×C131—24no—
39C22×C4×C131—24no—
40C22×C4×C131—22no—
41C22×C4×C131—24no—
42C22×C4×C131D414yes—
43C22×C4×C131C2312yes—
44C22×C4×C131C2311yescharacteristic
45C22×C4×C131C2311yescharacteristic
46C22×C4×C131C2312yes—
47C4⋊C4×C131—24no—
48C4⋊C4×C131—24no—
49C2×Dic131—22no—
50C2×Dic131—22no—
51C2×Dic131—24no—
52C2×Dic131—24no—
53C2×Dic131—24no—
54C2×Dic131—24no—
55C2×Dic131—24no—
56C23×C131—22no—
57C23×C131—22no—
58C23×C131C2×D412yes—
59C23×C131#4114yes—
60C23×C131C2411yescharacteristic
61C2×C4×C131—28no—
62C2×C4×C131—28no—
63C2×C4×C131—22no—
64C2×C4×C131—22no—
65C2×C4×C131—22no—
66C2×C4×C131—22no—
67C2×C4×C131—22no—
68C2×C4×C131—24no—
69C2×C4×C131—44no—
70C2×C4×C131C2×D412yes—
71C2×C4×C131C2×D412yes—
72C2×C4×C131#4114yes—
73C22×C131—22no—
74C22×C131—22no—
75C22×C131—28no—
76C22×C131#12314yes—
77C22×C131C22×D411yescharacteristic
78C22×C131C2×#4112yes—
79C22×C131C2×#4112yes—
80C22×C1312−1+412yes—
81C4×C131—22no—
82C4×C131—22no—
83C4×C131—24no—
84C4×C131—42no—
85Dic131—42no—
86Dic131—42no—
87Dic131—44no—
88C2×C131—24no—
89C2×C131C2×#12311yescharacteristic
90C2×C131#52511yescharacteristic
91C2×C131#53611yescharacteristic
92C2×C131#55214yes—
93C131#248411yescharacteristic, complemented, Sylow
94#2484—1311nomaximal, Sylow
95C2×#117—1311no—
96C22×C4⋊C4D13111yescharacteristic
97#400—1312no—
98#398—1312no—
99#387—1314no—
100#385—1312no—
101#381—1311no—
102#381—1312no—
103C23×C4C2×D13111yescharacteristic
104C23×C4C2×D13111yescharacteristic
105C23×C4C2×D13111yescharacteristic
106#117—2624no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—1311no—
109C2×C4⋊C4—1311no—
110C2×C4⋊C4—1314no—
111C2×C4⋊C4C2×D13112yes—
112C2×C4⋊C4C2×D13112yes—
113C2×C22⋊C4—1312no—
114C2×C22⋊C4—1312no—
115C2×C42—1312no—
116#95—1312no—
117#95—1314no—
118#95—1314no—
119#95—1312no—
120#95—1314no—
121C24C22×D13111yescharacteristic
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—1314no—
128C22×C4—1312no—
129C22×C4—1312no—
130C22×C4C22⋊D13114yes—
131C22×C4C22×D13112yes—
132C22×C4C22×D13111yescharacteristic
133C22×C4C22×D13111yescharacteristic
134C22×C4C22×D13112yes—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—2622no—
138C4⋊C4—2622no—
139C4⋊C4—2628no—
140C22⋊C4—2624no—
141C22⋊C4—2624no—
142C42—2624no—
143C23—22no—
144C23—22no—
145C23C2×C22⋊D13112yes—
146C23#1452351214yes—
147C23C23×D13111yescharacteristic
148C2×C4—28no—
149C2×C4—28no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—44no—
157C2×C4—2622no—
158C2×C4—2622no—
159C2×C4—2624no—
160C2×C4—2624no—
161C2×C4—2624no—
162C2×C4—2624no—
163C2×C4—2624no—
164C2×C4C2×C22⋊D13112yes—
165C2×C4C2×C22⋊D13112yes—
166C2×C4#1452351314yes—
167C22—22no—
168C22—28no—
169C22—22no—
170C22#1686222714yes—
171C22C22×C22⋊D13111yescharacteristic
172C22C2×#1452351212yes—
173C22C2×#1452351312yes—
174C22#1686229412yes—
175C4—22no—
176C4—22no—
177C4—24no—
178C4—42no—
179C4—5242no—
180C4—5244no—
181C4—5242no—
182C2—24no—
183C2C2×#1686222711yescharacteristic
184C2#2255382311yescharacteristic
185C2#2255388611yescharacteristic
186C2#2255399714yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C4⋊C4×C131C2
prime radicalC24×C131C23
hypercenterC22×C4⋊C4D131
hypocenterC131#2484
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22553251C2semidirect
3. #22553242C2semidirect
4. #22553193C2semidirect
93. C131#2484semidirect

Maximal subgroups

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

subgroupindex
C2×#168622052
#225532512
#225532422
#225531932
#225531852
#225531752
C22×C4⋊C4×C1312
#2484131