Finite Group Databasebeta

← #27144489

Subgroups — #27144489

Order 12416.

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

normal subgroups only

This group has 484 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×#15690789C211yesmaximal, characteristic
2#20260639C212yesmaximal, complemented
3#20260629C212yesmaximal
4#20260623C212yesmaximal
5#20260618C212yesmaximal, complemented
6#20260617C212yesmaximal, complemented
7#20260586C212yesmaximal, complemented
8C4×C22⋊Dic97C211yesmaximal, characteristic
9#381×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11#15690789C418yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490C2212yes—
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×C22⋊Dic97C2212yes—
17C2×C22⋊Dic97C2212yes—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27C23×C4×C97C2211yescharacteristic
28C2×C22⋊C4×C97C2211yescharacteristic
29C2×C22⋊C4×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⋊Dic97C2×C414yes—
40#13095490—24no—
41#13095490—42no—
42#13095490C2×C418yes—
43C22⋊Dic97—24no—
44C22⋊Dic97C2×C418yes—
45C4×Dic97—24no—
46C4×Dic97—41no—
47C4×Dic97—42no—
48C4×Dic97—41no—
49C24×C97C2311yescharacteristic
50C22×C4×C97—41no—
51C22×C4×C97—41no—
52C22×C4×C97C2×C414yes—
53C22×C4×C97C2312yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C22×C4×C97C2311yescharacteristic
57C22⋊C4×C97—41no—
58C22⋊C4×C97—41no—
59C23×C97—22no—
60C23×C97—21no—
61C23×C97—21no—
62C23×C97—22no—
63C23×C97C22×C412yes—
64C23×C97C2411yescharacteristic
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—44no—
76C2×C4×C97C22×C412yes—
77C2×C4×C97C22×C412yes—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97C22×C414yes—
88C2×Dic97C22×C414yes—
89C2×Dic97#4114yes—
90Dic194—44no—
91C22×C97—22no—
92C22×C97—22no—
93C22×C97—22no—
94C22×C97—42no—
95C22×C97—42no—
96C22×C97C23×C411yescharacteristic
97C22×C97C2×#4112yes—
98C22×C97H(5,𝔽2)11yescharacteristic
99C22×C97H(5,𝔽2)11yescharacteristic
100C22×C972−1+412yes—
101C4×C97—22no—
102C4×C97—22no—
103C4×C97—42no—
104C4×C97—42no—
105Dic97—24no—
106Dic97—24no—
107Dic97—24no—
108Dic97—42no—
109C2×C97—42no—
110C2×C97C4×#4111yescharacteristic
111C2×C97#51811yescharacteristic
112C2×C97#51911yescharacteristic
113C2×C97#53912yes—
114C2×C97#54212yes—
115#2332—971nomaximal, Sylow
116C97#233211yescharacteristic, complemented, Sylow
117C2×C22⋊Q8—971no—
118#391—972no—
119#388—972no—
120#388—972no—
121#387—972no—
122#387—972no—
123#381D9711yescharacteristic
124C4×C4⋊C4—972no—
125C4×C22⋊C4—971no—
126C22×Q8—971no—
127C23×C4C2×D9711yescharacteristic
128C22⋊Q8—978no—
129C2×C4⋊C4—972no—
130C2×C4⋊C4—971no—
131C2×C4⋊C4—972no—
132C2×C4⋊C4—972no—
133C2×C22⋊C4—972no—
134C2×C22⋊C4—972no—
135C2×C22⋊C4C2×D9711yescharacteristic
136C2×C22⋊C4C2×D9711yescharacteristic
137C2×C42—971no—
138C2×C42—971no—
139C2×C42—972no—
140C2×C42—972no—
141#95—972no—
142#95—972no—
143#95—972no—
144#95—972no—
145#95—972no—
146#95C2×D9712yes—
147#95C2×D9712yes—
148C24C22×D9711yescharacteristic
149C2×Q8—974no—
150C2×Q8—3881no—
151C2×Q8—3881no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—972no—
155C22×C4—972no—
156C22×C4—972no—
157C22×C4—972no—
158C22×C4C4×D9714yes—
159C22×C4C22×D9712yes—
160C22×C4C22×D9712yes—
161C22×C4C22×D9711yescharacteristic
162C22×C4C22×D9711yescharacteristic
163C4⋊C4—978no—
164C4⋊C4—974no—
165C4⋊C4—3882no—
166C4⋊C4—1944no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—978no—
170C22⋊C4—1944no—
171C42—3881no—
172C42—3881no—
173C42—3882no—
174C42—1944no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C2×C4×D9712yes—
180C23C23×D9711yescharacteristic
181Q8—3884no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4—974no—
194C2×C4—974no—
195C2×C4—974no—
196C2×C4—1942no—
197C2×C4—1942no—
198C2×C4—1942no—
199C2×C4—1942no—
200C2×C4—1942no—
201C2×C4—1942no—
202C2×C4—1942no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4C2×C4×D9712yes—
206C2×C4C2×C4×D9712yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C22×C4×D9711yescharacteristic
213C22#41×D9712yes—
214C22#1569085511yescharacteristic
215C22#1569085511yescharacteristic
216C22#1569085712yes—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—42no—
221C4—3882no—
222C4—1944no—
223C4—1944no—
224C4—1944no—
225C2—42no—
226C2#2026120311yescharacteristic
227C2#2026120811yescharacteristic
228C2#2026121211yescharacteristic
229C2#2026132812yes—
230C2#2026135312yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#381×C97C2
socle, cosocleC23×C97C24
prime radicalC24×C97C23
hypercenter#381D97
hypocenterC97#2332
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #20260639C2semidirect
5. #20260618C2semidirect
6. #20260617C2semidirect
7. #20260586C2semidirect
11. #15690789C4semidirect
116. C97#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#156907892
#202606392
#202606292
#202606232
#202606182
#202606172
#202605862
C4×C22⋊Dic972
#381×C972
#233297