Finite Group Databasebeta

← #27144824

Subgroups — #27144824

Order 12416.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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#20260632C212yesmaximal
5#20260630C212yesmaximal
6C4×#13095490C212yesmaximal
7#20260586C212yesmaximal
8#20260586C212yesmaximal
9#391×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11#15690808C418yescomplemented
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490C2212yes—
14C2×#13095490C2212yes—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2212yes—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2212yes—
20C2×C4×Dic97C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22C2×C4×Dic97C2211yescharacteristic
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—24no—
41#13095490—42no—
42#13095490C2×C418yes—
43#13095490C2×C418yes—
44#13095490Q818yescomplemented
45C4×Dic97—24no—
46C4×Dic97—24no—
47C4×Dic97—41no—
48C4×Dic97—42no—
49C4×Dic97C2×C414yes—
50C2×Q8×C97—41no—
51C2×Q8×C97—41no—
52C2×Q8×C97C2×C414yes—
53C22×C4×C97C2312yes—
54C22×C4×C97C2311yescharacteristic
55C22×C4×C97C2311yescharacteristic
56C22×C4×C97C2311yescharacteristic
57C22×C4×C97C2312yes—
58C4⋊C4×C97—41no—
59C42×C97—41no—
60C23×C97C2411yescharacteristic
61Q8×C97—44no—
62C2×C4×C97—22no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97C22×C412yes—
74C2×C4×C97C22×C412yes—
75C2×C4×C97C22×C412yes—
76C2×C4×C97C2×Q814yes—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97C22×C414yes—
85C2×Dic97C22×C414yes—
86C2×Dic97C2×Q814yes—
87C2×Dic97C2×Q814yes—
88C2×Dic97#4114yes—
89Dic194—44no—
90C22×C97C23×C411yescharacteristic
91C22×C97C22×Q812yes—
92C22×C97C2×#4112yes—
93C22×C97H(5,𝔽2)11yescharacteristic
94C22×C97H(5,𝔽2)11yescharacteristic
95C4×C97—22no—
96C4×C97—22no—
97C4×C97—22no—
98C4×C97—24no—
99C4×C97—42no—
100Dic97—24no—
101Dic97—24no—
102Dic97—42no—
103Dic97C4×Q818yes—
104C2×C97C2×C4×Q812yes—
105C2×C97#51811yescharacteristic
106C2×C97#55412yes—
107C2×C97#55511yescharacteristic
108C2×C97Q8211yescharacteristic
109#2369—971nomaximal, Sylow
110C97#236911yescharacteristic, complemented, Sylow
111C2×C4⋊Q8—971no—
112#391—971no—
113#391—972no—
114#391D9711yescharacteristic
115#389—972no—
116#389—972no—
117C4×C4⋊C4—972no—
118C4×C4⋊C4—972no—
119C4×C4⋊C4—972no—
120C22×Q8—971no—
121C22×Q8C2×D9711yescharacteristic
122C4⋊Q8—978no—
123C2×C4⋊C4—972no—
124C2×C4⋊C4—972no—
125C2×C4⋊C4—972no—
126C2×C4⋊C4—972no—
127C2×C4⋊C4—971no—
128C2×C4⋊C4—972no—
129C2×C4⋊C4C2×D9711yescharacteristic
130C2×C42—971no—
131C2×C42—971no—
132C2×C42—972no—
133C2×C42—972no—
134C2×C42—972no—
135C2×C42C2×D9711yescharacteristic
136#95—972no—
137#95—972no—
138#95—972no—
139#95—972no—
140#95C2×D9712yes—
141#95C2×D9712yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—974no—
145C2×Q8—3881no—
146C2×Q8—3881no—
147C2×Q8C4×D9714yes—
148C22×C4—972no—
149C22×C4—972no—
150C22×C4—972no—
151C22×C4—972no—
152C22×C4C22×D9712yes—
153C22×C4C22×D9711yescharacteristic
154C22×C4C22×D9711yescharacteristic
155C22×C4C22×D9711yescharacteristic
156C22×C4C22×D9712yes—
157C4⋊C4—41no—
158C4⋊C4—978no—
159C4⋊C4—978no—
160C4⋊C4—978no—
161C4⋊C4—3882no—
162C4⋊C4—3881no—
163C4⋊C4—1944no—
164C42—41no—
165C42—974no—
166C42—3882no—
167C42—3881no—
168C42—1944no—
169C42—1944no—
170C23C23×D9711yescharacteristic
171Q8—44no—
172Q8—3884no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—974no—
185C2×C4—974no—
186C2×C4—974no—
187C2×C4—974no—
188C2×C4—974no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4—1942no—
194C2×C4—1942no—
195C2×C4—1942no—
196C2×C4C2×C4×D9712yes—
197C2×C4C2×C4×D9712yes—
198C2×C4C2×C4×D9712yes—
199C2×C4Q8×D9714yes—
200C22C22×C4×D9711yescharacteristic
201C22C2×Q8×D9712yes—
202C22#41×D9712yes—
203C22#1569085511yescharacteristic
204C22#1569085511yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—978no—
211C4—3882no—
212C4—1944no—
213C4—1944no—
214C2C4×Q8×D9712yes—
215C2#2026120811yescharacteristic
216C2#2026143912yes—
217C2#2026144711yescharacteristic
218C2#2026146511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#391D97
hypocenterC97#2369
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #15690808C4semidirect
44. #13095490Q8semidirect
110. C97#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
#202606422
#202606392
#202606322
#202606302
C4×#130954902
#202605862
#391×C972
#236997