Finite Group Databasebeta

← #27144542

Subgroups — #27144542

Order 12416.

428 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 217 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#20260634C212yesmaximal
2#20260621C212yesmaximal
3#20260621C212yesmaximal
4#20260621C212yesmaximal
5#20260621C212yesmaximal
6#20260603C211yesmaximal, characteristic
7C4⋊C4×Dic97C212yesmaximal
8C4×C4⋊Dic97C211yesmaximal, characteristic
9C2×#125×C97C211yesmaximal, characteristic
10C2×C4⋊Dic97C2212yes—
11C2×C4⋊Dic97C2212yes—
12C2×C4⋊Dic97C2212yes—
13C2×C4×Dic97C2212yes—
14C2×C4×Dic97C2211yescharacteristic
15C2×C4×Dic97C2212yes—
16C2×C4×Dic97C2211yescharacteristic
17#15690717C2212yes—
18#15690717C2212yes—
19#15690717C2212yes—
20#15690717C2212yes—
21#15690717C2212yes—
22#15690717C2211yescharacteristic
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2211yescharacteristic
26#125×C97C418yescomplemented
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2212yes—
30C2×C4⋊C4×C97C2212yes—
31C2×C42×C97C2211yescharacteristic
32C22×Dic97C2312yes—
33C22×Dic97C2312yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C4⋊Dic97—24no—
37C4⋊Dic97—24no—
38C4⋊Dic97—42no—
39C4×Dic97—24no—
40C4×Dic97—41no—
41C4×Dic97—42no—
42C4×Dic97—41no—
43C22×C4×C97C2311yescharacteristic
44C22×C4×C97C2312yes—
45C22×C4×C97C2312yes—
46C22×C4×C97C2311yescharacteristic
47C22×C4×C97C2311yescharacteristic
48C4⋊C4×C97C2×C414yes—
49C4⋊C4×C97C2×C414yes—
50C4⋊C4×C97C2×C418yes—
51C4⋊C4×C97C2×C418yes—
52C42×C97C2×C414yes—
53C23×C97C2411yescharacteristic
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—21no—
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97C22×C412yes—
63C2×C4×C97C22×C412yes—
64C2×C4×C97C22×C412yes—
65C2×C4×C97C22×C414yes—
66C2×C4×C97C22×C414yes—
67C2×Dic97—22no—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97#4114yes—
79C22×C97C23×C411yescharacteristic
80C22×C97C2×#4112yes—
81C22×C97H(5,𝔽2)11yescharacteristic
82C22×C972−1+411yescharacteristic
83C22×C972−1+412yes—
84C4×C97—24no—
85C4×C97—22no—
86C4×C97—22no—
87C4×C97—22no—
88C4×C97—24no—
89Dic97—24no—
90Dic97—42no—
91Dic97—42no—
92Dic97—42no—
93C2×C97C4×#4111yescharacteristic
94C2×C97#51911yescharacteristic
95C2×C97#52011yescharacteristic
96C2×C97#53912yes—
97C2×C97#54112yes—
98#2336—971nomaximal, Sylow
99C97#233611yescharacteristic, complemented, Sylow
100C2×#125D9711yescharacteristic
101#389—972no—
102#387—972no—
103#387—972no—
104#387—972no—
105#387—972no—
106#383—971no—
107C4×C4⋊C4—972no—
108C4×C4⋊C4—971no—
109#125Dic9718yescomplemented
110C2×C4⋊C4—972no—
111C2×C4⋊C4—972no—
112C2×C4⋊C4—972no—
113C2×C4⋊C4C2×D9712yes—
114C2×C4⋊C4C2×D9711yescharacteristic
115C2×C4⋊C4C2×D9711yescharacteristic
116C2×C4⋊C4C2×D9712yes—
117C2×C42—971no—
118C2×C42—971no—
119C2×C42—972no—
120C2×C42—972no—
121C2×C42C2×D9711yescharacteristic
122#95—972no—
123#95—972no—
124#95—972no—
125#95—971no—
126#95—972no—
127#95—971no—
128#95—972no—
129#95—972no—
130#95—972no—
131C22×C4—972no—
132C22×C4—972no—
133C22×C4—972no—
134C22×C4—972no—
135C22×C4C22×D9711yescharacteristic
136C22×C4C22×D9711yescharacteristic
137C22×C4C22×D9712yes—
138C22×C4C22×D9712yes—
139C22×C4C22×D9711yescharacteristic
140C4⋊C4—3882no—
141C4⋊C4—1944no—
142C4⋊C4—1944no—
143C4⋊C4C2×Dic9718yes—
144C4⋊C4C2×Dic9714yes—
145C4⋊C4C2×Dic9714yes—
146C4⋊C4C2×Dic9718yes—
147C42—3881no—
148C42—3881no—
149C42—3882no—
150C42—1944no—
151C42C2×Dic9714yes—
152C23C23×D9711yescharacteristic
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—974no—
162C2×C4—1942no—
163C2×C4—1942no—
164C2×C4—1942no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4—1942no—
169C2×C4—1942no—
170C2×C4—1942no—
171C2×C4—1942no—
172C2×C4—1942no—
173C2×C4C22×Dic9712yes—
174C2×C4C22×Dic9712yes—
175C2×C4C22×Dic9712yes—
176C2×C4C22×Dic9714yes—
177C2×C4C22×Dic9714yes—
178C22C23×Dic9711yescharacteristic
179C22#41×D9712yes—
180C22#1569085611yescharacteristic
181C22#1569085711yescharacteristic
182C22#1569085812yes—
183C4—24no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—24no—
188C4—3882no—
189C4—3882no—
190C4—3882no—
191C4—1944no—
192C2#41×Dic9711yescharacteristic
193C2#2026121311yescharacteristic
194C2#2026121811yescharacteristic
195C2#2026133312yes—
196C2#2026134412yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC2×#125×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenterC2×#125D97
hypocenterC97#2336
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
26. #125×C97C4semidirect
99. C97#2336semidirect
109. #125Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606342
#202606212
#202606032
C4⋊C4×Dic972
C4×C4⋊Dic972
C2×#125×C972
#233697