Finite Group Databasebeta

← #30671153

Subgroups — #30671153

Order 16768.

452 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 161 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×Dic262C211yesmaximal, characteristic
2#22553245C214yesmaximal
3#22553217C211yesmaximal, characteristic
4#22553217C211yesmaximal, characteristic
5#22553193C214yesmaximal
6#22553181C211yesmaximal, characteristic
7#22553177C211yesmaximal, characteristic
8C4×C4⋊Dic131C211yesmaximal, characteristic
9C2×#125×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C4×Dic262—24no—
12C2×C4⋊Dic131C2212yes—
13C2×C4⋊Dic131C2211yescharacteristic
14C2×Dic131⋊C4C2214yes—
15C2×Dic131⋊C4C2212yes—
16C2×C4×Dic131C2211yescharacteristic
17C2×C4×Dic131C2211yescharacteristic
18C2×C4×Dic131C2212yes—
19C131⋊#95C2212yes—
20C131⋊#95C2214yes—
21C131⋊#95C2214yes—
22C131⋊#95C2214yes—
23#125×C131—24no—
24C2×C4⋊C4×C131C2211yescharacteristic
25C2×C4⋊C4×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2212yes—
27C2×C4⋊C4×C131C2212yes—
28C2×C42×C131C2211yescharacteristic
29C2×Dic262—22no—
30C2×Dic262—22no—
31C2×Dic262Q814yes—
32C22×Dic131C2312yes—
33C22×Dic131C2312yes—
34C22×Dic131C2312yes—
35C22×Dic131C2312yes—
36C4⋊Dic131—24no—
37C4⋊Dic131—22no—
38Dic131⋊C4—28no—
39Dic131⋊C4—24no—
40C4×Dic131—24no—
41C4×Dic131—22no—
42C4×Dic131—22no—
43C22×C4×C131C2312yes—
44C22×C4×C131C2312yes—
45C22×C4×C131C2311yescharacteristic
46C22×C4×C131C2311yescharacteristic
47C22×C4×C131C2311yescharacteristic
48C4⋊C4×C131—24no—
49C4⋊C4×C131—24no—
50C4⋊C4×C131—22no—
51C4⋊C4×C131—22no—
52C42×C131D414yes—
53Dic262—28no—
54C2×Dic131—22no—
55C2×Dic131—22no—
56C2×Dic131—22no—
57C2×Dic131—22no—
58C2×Dic131—22no—
59C2×Dic131—22no—
60C2×Dic131—24no—
61C2×Dic131—22no—
62C2×Dic131—22no—
63C2×Dic131C2×Q814yes—
64C2×Dic131#4114yes—
65C23×C131C2411yescharacteristic
66C2×C4×C131—22no—
67C2×C4×C131—22no—
68C2×C4×C131—22no—
69C2×C4×C131—22no—
70C2×C4×C131—24no—
71C2×C4×C131—24no—
72C2×C4×C131C2×D412yes—
73C2×C4×C131C2×D412yes—
74C2×C4×C131C2×D412yes—
75C2×C4×C131C2×Q812yes—
76C2×C4×C131#4112yes—
77C22×C131C22×D411yescharacteristic
78C22×C131C22×Q811yescharacteristic
79C22×C131C2×#4112yes—
80C22×C131C2×#4111yescharacteristic
81C22×C1312−1+412yes—
82C4×C131—22no—
83C4×C131—22no—
84C4×C131—42no—
85C4×C131—42no—
86C4×C131C22⋊Q814yes—
87Dic131—24no—
88Dic131—24no—
89Dic131—42no—
90Dic131—42no—
91C2×C131C2×C22⋊Q811yescharacteristic
92C2×C131#53211yescharacteristic
93C2×C131#53611yescharacteristic
94C2×C131#55212yes—
95C2×C131#55712yes—
96C131#256511yescharacteristic, complemented, Sylow
97#2565—1311nomaximal, Sylow
98C2×#125D13111yescharacteristic
99C2×C4×Q8—1311no—
100#398—1314no—
101#391—1311no—
102#391—1311no—
103#387—1314no—
104#384—1311no—
105#382—1311no—
106C4×C4⋊C4—1311no—
107C22×Q8—1311no—
108#125—24no—
109C4×Q8—2624no—
110C2×C4⋊C4—1311no—
111C2×C4⋊C4—1312no—
112C2×C4⋊C4—1312no—
113C2×C4⋊C4—1314no—
114C2×C4⋊C4C2×D13111yescharacteristic
115C2×C4⋊C4C2×D13111yescharacteristic
116C2×C4⋊C4C2×D13112yes—
117C2×C4⋊C4C2×D13112yes—
118C2×C42—1312no—
119C2×C42—1311no—
120C2×C42—1311no—
121C2×C42C2×D13111yescharacteristic
122#95—1312no—
123#95—1314no—
124#95—1314no—
125#95—1314no—
126C2×Q8—2622no—
127C2×Q8—2622no—
128C2×Q8—1314no—
129C22×C4—1312no—
130C22×C4—1312no—
131C22×C4—1312no—
132C22×C4—1312no—
133C22×C4C22×D13112yes—
134C22×C4C22×D13112yes—
135C22×C4C22×D13111yescharacteristic
136C22×C4C22×D13111yescharacteristic
137C22×C4C22×D13111yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—2628no—
143C4⋊C4—2624no—
144C4⋊C4—2624no—
145C4⋊C4—2622no—
146C42—2624no—
147C42—2622no—
148C42—2622no—
149C42C22⋊D13114yes—
150C23C23×D13111yescharacteristic
151Q8—2628no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—2622no—
159C2×C4—2622no—
160C2×C4—2622no—
161C2×C4—2622no—
162C2×C4—2622no—
163C2×C4—2624no—
164C2×C4—2622no—
165C2×C4—2622no—
166C2×C4—2622no—
167C2×C4—1314no—
168C2×C4—1314no—
169C2×C4C2×C22⋊D13112yes—
170C2×C4C2×C22⋊D13112yes—
171C2×C4C2×C22⋊D13112yes—
172C2×C4Q8×D13112yes—
173C2×C4#1452351412yes—
174C22C22×C22⋊D13111yescharacteristic
175C22C2×Q8×D13111yescharacteristic
176C22#41×D13112yes—
177C22C2×#1452351411yescharacteristic
178C22#1686229312yes—
179C4—22no—
180C4—22no—
181C4—42no—
182C4—42no—
183C4—2624no—
184C4—2624no—
185C4—5242no—
186C4—5242no—
187C4#1686222214yes—
188C2C2×#1686222211yescharacteristic
189C2#2255385811yescharacteristic
190C2#2255388311yescharacteristic
191C2#2255400012yes—
192C2#2255403912yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, prime radical, cosocleC23×C131C24
FittingC2×#125×C131C2
hypercenterC2×#125D131
hypocenterC131#2565
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
96. C131#2565semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2622
#225532452
#225532172
#225531932
#225531812
#225531772
C4×C4⋊Dic1312
C2×#125×C1312
#2565131