Finite Group Databasebeta

← #27144537

Subgroups — #27144537

Order 12416.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15690802C211yesmaximal, characteristic
2#20260632C212yesmaximal
3#20260622C212yesmaximal
4#20260620C212yesmaximal
5#20260618C212yesmaximal
6#20260617C212yesmaximal
7C4×C4⋊Dic97C211yesmaximal, characteristic
8#20260586C212yesmaximal
9#383×C97C211yesmaximal, characteristic
10#15690802C418yescomplemented
11C2×C4⋊Dic97C2212yes—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2212yes—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27C2×C42×C97C2211yescharacteristic
28#95×C97C2212yes—
29#95×C97C2212yes—
30#95×C97C2211yescharacteristic
31#95×C97C2211yescharacteristic
32C22×Dic97C2312yes—
33C22×Dic97C2312yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C4⋊Dic97—24no—
37C4⋊Dic97C2×C414yes—
38C4⋊Dic97C2×C414yes—
39#13095490—24no—
40#13095490—42no—
41#13095490C2×C418yes—
42#13095490C2×C418yes—
43C4×Dic97—24no—
44C4×Dic97—41no—
45C4×Dic97—42no—
46C4×Dic97—41no—
47C22×C4×C97C2312yes—
48C22×C4×C97C2312yes—
49C22×C4×C97C2311yescharacteristic
50C22×C4×C97C2311yescharacteristic
51C22×C4×C97C2311yescharacteristic
52C42×C97C2×C414yes—
53C23×C97C2411yescharacteristic
54C2×C4×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97—22no—
57C2×C4×C97—21no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97C22×C412yes—
65C2×C4×C97C22×C412yes—
66C2×C4×C97C22×C412yes—
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×Dic97C22×C414yes—
77C2×Dic97C22×C414yes—
78C2×Dic97#4114yes—
79C22×C97C23×C411yescharacteristic
80C22×C97C2×#4112yes—
81C22×C97H(5,𝔽2)11yescharacteristic
82C22×C972−1+412yes—
83C22×C972−1+411yescharacteristic
84C4×C97—22no—
85C4×C97—22no—
86C4×C97—22no—
87C4×C97—42no—
88C4×C97—42no—
89Dic97—24no—
90Dic97—24no—
91Dic97—24no—
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×#125—971no—
101#389—972no—
102#387—972no—
103#387—972no—
104#387—972no—
105#387—972no—
106#383D9711yescharacteristic
107C4×C4⋊C4—971no—
108C4×C4⋊C4—972no—
109#125—978no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4—972no—
112C2×C4⋊C4—971no—
113C2×C4⋊C4—971no—
114C2×C4⋊C4—972no—
115C2×C4⋊C4—972no—
116C2×C4⋊C4—972no—
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—972no—
126#95—972no—
127#95C2×D9712yes—
128#95C2×D9711yescharacteristic
129#95C2×D9712yes—
130#95C2×D9711yescharacteristic
131C22×C4—972no—
132C22×C4—972no—
133C22×C4—972no—
134C22×C4—972no—
135C22×C4C22×D9712yes—
136C22×C4C22×D9712yes—
137C22×C4C22×D9711yescharacteristic
138C22×C4C22×D9711yescharacteristic
139C22×C4C22×D9711yescharacteristic
140C4⋊C4—978no—
141C4⋊C4—978no—
142C4⋊C4—974no—
143C4⋊C4—974no—
144C4⋊C4—3882no—
145C4⋊C4—1944no—
146C4⋊C4—1944no—
147C42—3881no—
148C42—3881no—
149C42—3882no—
150C42—1944no—
151C42C4×D9714yes—
152C23C23×D9711yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—974no—
164C2×C4—974no—
165C2×C4—974no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4—1942no—
169C2×C4—1942no—
170C2×C4—1942no—
171C2×C4—1942no—
172C2×C4—1942no—
173C2×C4—1942no—
174C2×C4—1942no—
175C2×C4C2×C4×D9712yes—
176C2×C4C2×C4×D9712yes—
177C2×C4C2×C4×D9712yes—
178C22C22×C4×D9711yescharacteristic
179C22#41×D9712yes—
180C22#1569085611yescharacteristic
181C22#1569085712yes—
182C22#1569085711yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—3882no—
189C4—1944no—
190C4—1944no—
191C4—1944no—
192C2#2026120311yescharacteristic
193C2#2026121211yescharacteristic
194C2#2026121511yescharacteristic
195C2#2026132612yes—
196C2#2026134212yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#383×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#383D97
hypocenterC97#2336
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
10. #15690802C4semidirect
99. C97#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#156908022
#202606322
#202606222
#202606202
#202606182
#202606172
C4×C4⋊Dic972
#202605862
#383×C972
#233697