Finite Group Databasebeta

← #27151259

Subgroups — #27151259

Order 12416.

490 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13095498C211yesmaximal, characteristic, complemented
2#20261299C211yesmaximal, characteristic
3#20261213C211yesmaximal, characteristic, complemented
4#20260959C214yesmaximal, complemented
5#20260933C214yesmaximal, complemented
6#20260763C211yesmaximal, characteristic, complemented
7#20260741C211yesmaximal, characteristic, complemented
8#20260731C211yesmaximal, characteristic, complemented
9C2×#137×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C2×#13095498—24no—
12C2×#13095498C2214yes—
13C22×#11715895C2211yescharacteristic
14#15690806C2212yes—
15#15690799C2212yes—
16#15690784—22no—
17#15690778—22no—
18Q8×Dic97—22no—
19Q8×Dic97C2214yescomplemented
20#15690769—22no—
21#15690769C2211yescharacteristic
22#15690769C2211yescharacteristic
23#15690739C2214yescomplemented
24#15690734C2212yescomplemented
25#15690732C2214yes—
26#15690731C2212yescomplemented
27C2×#41×C97C2211yescharacteristic
28C22×Q8×C97C2211yescharacteristic
29#137×C97—24no—
30C2×Q16×C97C2212yescomplemented
31C2×SD16×C97C2212yescomplemented
32C2×M16×C97C2211yescharacteristic
33C23×D97—21no—
34C2×D388—22no—
35C2×D388—22no—
36C2×D388—21no—
37C2×D388C2312yes—
38C2×C22⋊D97—24no—
39#13095498—28no—
40C2×#11715895—22no—
41C2×#11715895C2312yes—
42C4⋊Dic97—24no—
43C4⋊Dic97D412yes—
44C4⋊Dic97C2312yes—
45C22⋊Dic97—21no—
46C22⋊Dic97D412yes—
47#13095488—24no—
48C4×Dic97—21no—
49C4×Dic97—24no—
50C4×Dic97C2312yes—
51#41×C97—22no—
52#41×C97—22no—
53C2×Q8×C97—21no—
54C2×Q8×C97—22no—
55C2×Q8×C97D414yescomplemented
56C2×Q8×C97C2311yescharacteristic
57C2×Q8×C97C2312yes—
58C2×D4×C97—21no—
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97—21no—
61C22×C4×C97—21no—
62C22×C4×C97C2311yescharacteristic
63Q16×C97—24no—
64SD16×C97—24no—
65M16×C97—22no—
66C2×C8×C97C2312yes—
67C23×C97—21no—
68C23×C97C2×D411yescharacteristic
69C22×D97—22no—
70C22×D97—42no—
71C22×D97—82no—
72Q8×C97—24no—
73Q8×C97—22no—
74Q8×C97—21no—
75Q8×C97—21no—
76Q8×C97C2×D414yes—
77D4×C97—21no—
78D4×C97—21no—
79D4×C97—42no—
80C2×C4×C97—24no—
81C2×C4×C97—22no—
82C2×C4×C97—21no—
83C2×C4×C97—42no—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D412yes—
86C2×C4×C97#4112yes—
87C2×C4×C97C2411yescharacteristic
88C22⋊D97—84no—
89C2×Dic97—22no—
90C2×Dic97C2×D414yes—
91D388—22no—
92D388—22no—
93D388—42no—
94C8×C97—22no—
95#11715895—22no—
96C22×C97—22no—
97C22×C97—41no—
98C22×C97—42no—
99C22×C97#12112yes—
100C22×C97C22×D411yescharacteristic
101C2×D97—42no—
102C2×D97—44no—
103C2×D97—84no—
104C4×C97—24no—
105C4×C97—41no—
106C4×C97#12112yes—
107C4×C97C22×D411yescharacteristic
108C4×C97C2×#4111yescharacteristic
109Dic97—24no—
110Dic97—42no—
111C2×C97—22no—
112C2×C97—81no—
113C2×C97C2×#12111yescharacteristic
114C2×C97#57712yes—
115D97—82no—
116#3043—971nomaximal, Sylow
117C97#304311yescharacteristic, complemented, Sylow
118C2×#137D9711yescharacteristic, complemented
119C22×SD16—971no—
120#535—971no—
121#519—971no—
122#464—974no—
123#460—974no—
124#424—971no—
125#418—971no—
126#416—971no—
127C2×#41C2×D9711yescharacteristic
128C22×Q8C2×D9711yescharacteristic
129C22×D4—971no—
130#137—24no—
131C2×Q16C2×D9712yescomplemented
132C2×SD16—974no—
133C2×SD16—1944no—
134C2×SD16C2×D9712yescomplemented
135C2×M16C2×D9711yescharacteristic
136C22×C8—971no—
137#127—972no—
138#124—972no—
139#121—1942no—
140#120—1942no—
141C4×Q8—974no—
142C4×Q8—1942no—
143#117—971no—
144#117—971no—
145#117—1942no—
146C4⋊C8—974no—
147#103—972no—
148#102—972no—
149#102—974no—
150C24—1941no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8C22⋊D9714yescomplemented
156C2×Q8C22×D9711yescharacteristic
157C2×Q8C22×D9712yes—
158C2×D4—21no—
159C2×D4—972no—
160C2×D4—1944no—
161C2×D4—1941no—
162C2×D4—1942no—
163C2×D4—1942no—
164C2×D4C22×D9711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22×D9711yescharacteristic
168Q16—24no—
169SD16—24no—
170SD16—1948no—
171M16—22no—
172C2×C8—972no—
173C2×C8—1942no—
174C2×C8C22×D9712yes—
175C4⋊C4—972no—
176C4⋊C4—972no—
177C4⋊C4—1944no—
178C22⋊C4—972no—
179C22⋊C4—1944no—
180C22⋊C4—1941no—
181C42—972no—
182C42—1944no—
183C42—1941no—
184C23—21no—
185C23—7762no—
186C23—3882no—
187C23—1942no—
188C23D4×D9711yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—21no—
192Q8—21no—
193Q8C2×C22⋊D9714yes—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—7764no—
198D4—3882no—
199D4—1942no—
200D4—1942no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—974no—
206C2×C4—1942no—
207C2×C4D4×D9711yescharacteristic
208C2×C4C2×C22⋊D9712yes—
209C2×C4#1309551012yes—
210C2×C4C23×D9711yescharacteristic
211C8—22no—
212C8—1942no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—7764no—
217C22—3882no—
218C22—3884no—
219C22#1569078212yes—
220C22C2×D4×D9711yescharacteristic
221C4—24no—
222C4—41no—
223C4—3882no—
224C4—1944no—
225C4#1569078212yes—
226C4C22×C22⋊D9711yescharacteristic
227C4C2×#1309551011yescharacteristic
228C2—22no—
229C2—81no—
230C2—7762no—
231C2C2×#1569078211yescharacteristic
232C2#2026154412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radical#20261299C2
hypercenterC2×#137D97
hypocenterC97#3043
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13095498C2semidirect
3. #20261213C2semidirect
4. #20260959C2semidirect
5. #20260933C2semidirect
6. #20260763C2semidirect
7. #20260741C2semidirect
8. #20260731C2semidirect
9. C2×#137×C97C2semidirect
19. Q8×Dic97C22semidirect
23. #15690739C22semidirect
24. #15690734C22semidirect
26. #15690731C22semidirect
30. C2×Q16×C97C22semidirect
31. C2×SD16×C97C22semidirect
55. C2×Q8×C97D4semidirect
117. C97#3043semidirect
118. C2×#137D97semidirect
131. C2×Q16C2×D97semidirect
134. C2×SD16C2×D97semidirect
155. C2×Q8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954982
#202612992
#202612132
#202609592
#202609332
#202607632
#202607412
#202607312
C2×#137×C972
#304397