Finite Group Databasebeta

← #27144915

Subgroups — #27144915

Order 12416.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690808C211yesmaximal, characteristic
2#20260642C211yesmaximal, characteristic
3#20260639C212yesmaximal
4#20260622C212yesmaximal
5#20260619C212yesmaximal
6#20260617C212yesmaximal
7#20260616C212yesmaximal
8#20260602C212yesmaximal
9#391×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11#15690808C418yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490C2212yes—
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×C4×Dic97C2212yes—
17C2×C4×Dic97C2211yescharacteristic
18C2×C4×Dic97C2211yescharacteristic
19#15690717C2212yes—
20#15690717C2212yes—
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27C22×Q8×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C42×C97C2211yescharacteristic
30#95×C97C2212yes—
31#95×C97C2212yes—
32C2×Dic194—41no—
33C2×Dic194—41no—
34C2×Dic194C2×C414yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C4⋊Dic97—41no—
40#13095490—42no—
41#13095490C2×C418yes—
42#13095490C2×C418yes—
43C4×Dic97—42no—
44C4×Dic97—41no—
45C4×Dic97C2×C414yes—
46C2×Q8×C97—41no—
47C2×Q8×C97—41no—
48C2×Q8×C97C2×C414yes—
49C22×C4×C97C2312yes—
50C22×C4×C97C2312yes—
51C22×C4×C97C2311yescharacteristic
52C22×C4×C97C2311yescharacteristic
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97—41no—
55C42×C97—41no—
56C23×C97C2411yescharacteristic
57Q8×C97—44no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—24no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97C22×C412yes—
70C2×C4×C97C22×C412yes—
71C2×C4×C97C22×C412yes—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—24no—
81C2×Dic97C22×C414yes—
82C2×Dic97C22×C414yes—
83Dic194—44no—
84C22×C97C23×C411yescharacteristic
85C22×C97H(5,𝔽2)11yescharacteristic
86C22×C97H(5,𝔽2)11yescharacteristic
87C22×C972−1+412yes—
88C22×C972−1+412yes—
89C4×C97—22no—
90C4×C97—22no—
91C4×C97—22no—
92C4×C97—42no—
93C4×C97—42no—
94Dic97—24no—
95Dic97—24no—
96Dic97—42no—
97Dic97—42no—
98C2×C97#51811yescharacteristic
99C2×C97#51912yes—
100C2×C97#56311yescharacteristic
101C2×C97#56311yescharacteristic
102C2×C97#56312yes—
103#2381—971nomaximal, Sylow
104C97#238111yescharacteristic, complemented, Sylow
105C2×C4⋊Q8—971no—
106#391—971no—
107#391—972no—
108#391D9711yescharacteristic
109#387—972no—
110#387—972no—
111#387—972no—
112#387—972no—
113#383—972no—
114C22×Q8—971no—
115C22×Q8C2×D9711yescharacteristic
116C4⋊Q8—978no—
117C2×C4⋊C4—972no—
118C2×C4⋊C4—972no—
119C2×C4⋊C4—972no—
120C2×C4⋊C4—971no—
121C2×C4⋊C4C2×D9711yescharacteristic
122C2×C42—971no—
123C2×C42—972no—
124C2×C42—971no—
125C2×C42C2×D9711yescharacteristic
126#95—972no—
127#95—972no—
128#95—972no—
129#95—972no—
130#95—972no—
131#95—972no—
132#95—972no—
133#95—972no—
134#95C2×D9712yes—
135#95C2×D9712yes—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—974no—
139C2×Q8—3881no—
140C2×Q8—3881no—
141C2×Q8C4×D9714yes—
142C22×C4—972no—
143C22×C4—972no—
144C22×C4—972no—
145C22×C4—972no—
146C22×C4C22×D9712yes—
147C22×C4C22×D9712yes—
148C22×C4C22×D9711yescharacteristic
149C22×C4C22×D9711yescharacteristic
150C22×C4C22×D9711yescharacteristic
151C4⋊C4—41no—
152C4⋊C4—978no—
153C4⋊C4—978no—
154C4⋊C4—3882no—
155C4⋊C4—3881no—
156C42—41no—
157C42—974no—
158C42—3881no—
159C42—3882no—
160C23C23×D9711yescharacteristic
161Q8—44no—
162Q8—3884no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—974no—
175C2×C4—974no—
176C2×C4—1942no—
177C2×C4—1942no—
178C2×C4—1942no—
179C2×C4—1944no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4C2×C4×D9712yes—
186C2×C4C2×C4×D9712yes—
187C2×C4C2×C4×D9712yes—
188C22C22×C4×D9711yescharacteristic
189C22#1569085511yescharacteristic
190C22#1569085511yescharacteristic
191C22#1569085712yes—
192C22#1569085812yes—
193C4—22no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—3882no—
199C4—3882no—
200C4—1944no—
201C4—1944no—
202C2#2026120811yescharacteristic
203C2#2026121112yes—
204C2#2026148411yescharacteristic
205C2#2026148511yescharacteristic
206C2#2026148612yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#391D97
hypocenterC97#2381
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #15690808C4semidirect
104. C97#2381semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
#202606422
#202606392
#202606222
#202606192
#202606172
#202606162
#202606022
#391×C972
#238197