Finite Group Databasebeta

← #27151371

Subgroups — #27151371

Order 12416.

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

normal subgroups only

This group has 492 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
1C2×#15690816C211yesmaximal, characteristic, complemented
2#20261309C211yesmaximal, characteristic, complemented
3#20261302C211yesmaximal, characteristic
4#20260978C214yesmaximal, complemented
5#20260961C214yesmaximal, complemented
6#20260773C211yesmaximal, characteristic, complemented
7#20260744C211yesmaximal, characteristic, complemented
8#20260740C211yesmaximal, characteristic, complemented
9C22×SD16×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11#15690816—24no—
12C2×#13095495C2211yescharacteristic
13C2×#13095494C2212yescomplemented
14C2×C8×D97C2212yescomplemented
15#15690808—21no—
16#15690806—21no—
17#15690799—21no—
18#15690797—21no—
19#15690793—22no—
20#15690786C2214yescomplemented
21#15690781C2214yes—
22#15690779—22no—
23#15690769C2211yescharacteristic
24#15690769C2211yescharacteristic
25#15690744C2212yescomplemented
26#15690742C2212yescomplemented
27#15690734C2214yescomplemented
28#15690731C2214yes—
29C22×Q8×C97C2211yescharacteristic
30C22×D4×C97C2211yescharacteristic
31C2×SD16×C97—24no—
32C2×SD16×C97C2212yes—
33C2×SD16×C97C2212yes—
34C22×C8×C97C2211yescharacteristic
35#13095511—22no—
36#13095511—22no—
37C2×Dic194D411yescharacteristic
38C2×D388D411yescharacteristic
39C2×C22⋊D97—24no—
40C2×C22⋊D97D412yes—
41C2×C4×D97C2312yes—
42#13095495—22no—
43#13095494—24no—
44C2×#11715895C2312yes—
45C8×D97—24no—
46C4⋊Dic97C2312yes—
47C4⋊Dic97C2312yes—
48#13095490—24no—
49C22⋊Dic97—21no—
50C22⋊Dic97—21no—
51C22⋊Dic97—24no—
52#13095488—24no—
53C4×Dic97—21no—
54C4×Dic97—21no—
55C24×C97—21no—
56C2×Q8×C97—21no—
57C2×Q8×C97—22no—
58C2×Q8×C97—22no—
59C2×Q8×C97C2312yes—
60C2×D4×C97—22no—
61C2×D4×C97—21no—
62C2×D4×C97—22no—
63C2×D4×C97C2312yes—
64C22×C4×C97—21no—
65C22×C4×C97C2311yescharacteristic
66SD16×C97—24no—
67SD16×C97—24no—
68C2×C8×C97D414yescomplemented
69C2×C8×C97C2311yescharacteristic
70C2×C8×C97C2311yescharacteristic
71C23×C97—22no—
72C23×C97—42no—
73C23×C97—82no—
74C23×C97C2×D411yescharacteristic
75C22×D97C2×D412yes—
76Q8×C97—22no—
77Q8×C97—22no—
78Q8×C97—42no—
79D4×C97—22no—
80D4×C97—22no—
81D4×C97—42no—
82C2×C4×C97—22no—
83C2×C4×C97—42no—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D412yes—
86C2×C4×C97#4112yes—
87C2×C4×C97C2411yescharacteristic
88C22⋊D97—24no—
89C22⋊D97—84no—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97C2×D412yes—
93D388—22no—
94C4×D97—22no—
95C4×D97—22no—
96Dic194—22no—
97C8×C97C2×D412yes—
98C8×C97C2×D412yes—
99#11715895—22no—
100C22×C97—22no—
101C22×C97—42no—
102C22×C97—44no—
103C22×C97—84no—
104C22×C97#12112yes—
105C22×C97C22×D411yescharacteristic
106C2×D97—22no—
107C2×D97—24no—
108C4×C97—42no—
109C4×C97#12112yes—
110C4×C97C22×D411yescharacteristic
111C4×C97C2×#4111yescharacteristic
112Dic97—22no—
113Dic97—42no—
114Dic97—42no—
115C2×C97—22no—
116C2×C97—82no—
117C2×C97C2×#12111yescharacteristic
118C2×C97#57712yes—
119D97—42no—
120#3057—971nomaximal, Sylow
121C97#305711yescharacteristic, complemented, Sylow
122C22×SD16D9711yescharacteristic, complemented
123C2×#131—971no—
124#536—971no—
125#535—971no—
126#468—974no—
127#465—974no—
128#427—971no—
129#419—971no—
130#418—971no—
131C2×#41—971no—
132C22×Q8C2×D9711yescharacteristic
133C22×D4C2×D9711yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D9712yes—
136C2×SD16C2×D9712yes—
137#131—1944no—
138C2×M16—971no—
139C2×M16—972no—
140C22×C8—972no—
141C22×C8C2×D9711yescharacteristic
142C4⋊Q8—1941no—
143#127—1941no—
144#124—1941no—
145#124—1941no—
146#123—1942no—
147C22⋊Q8—974no—
148#121—974no—
149#120—1942no—
150#117—971no—
151#117—971no—
152#107—972no—
153#106—972no—
154#103—974no—
155#102—974no—
156C24—21no—
157#41—1942no—
158#41—1942no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—971no—
163C2×Q8C22×D9712yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—972no—
168C2×D4—971no—
169C2×D4—1944no—
170C2×D4C22×D9712yes—
171C22×C4—21no—
172C22×C4—972no—
173C22×C4C22×D9711yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—1942no—
177M16—1944no—
178C2×C8—972no—
179C2×C8—1944no—
180C2×C8C22⋊D9714yescomplemented
181C2×C8C22×D9711yescharacteristic
182C2×C8C22×D9711yescharacteristic
183C4⋊C4—972no—
184C4⋊C4—972no—
185C4⋊C4—1944no—
186C22⋊C4—1944no—
187C22⋊C4—1944no—
188C22⋊C4—1941no—
189C22⋊C4—1941no—
190C42—1941no—
191C42—1941no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—972no—
196C23D4×D9711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—1942no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—7764no—
205D4—1944no—
206D4—1942no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—972no—
210C2×C4—1942no—
211C2×C4—1942no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4D4×D9711yescharacteristic
215C2×C4C2×C22⋊D9712yes—
216C2×C4#1309551012yes—
217C2×C4C23×D9711yescharacteristic
218C8—1942no—
219C8C2×C22⋊D9712yes—
220C8C2×C22⋊D9712yes—
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—1942no—
226C22—1944no—
227C22#1569078112yes—
228C22C2×D4×D9711yescharacteristic
229C4—42no—
230C4—3882no—
231C4—3882no—
232C4—1942no—
233C4#1569078112yes—
234C4C22×C22⋊D9711yescharacteristic
235C4C2×#1309551011yescharacteristic
236C2—22no—
237C2—82no—
238C2—3882no—
239C2C2×#1569078111yescharacteristic
240C2#2026154112yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC22×C97C22×D4
prime radical#20261302C2
hypercenterC22×SD16D97
hypocenterC97#3057
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690816C2semidirect
2. #20261309C2semidirect
4. #20260978C2semidirect
5. #20260961C2semidirect
6. #20260773C2semidirect
7. #20260744C2semidirect
8. #20260740C2semidirect
9. C22×SD16×C97C2semidirect
13. C2×#13095494C22semidirect
14. C2×C8×D97C22semidirect
20. #15690786C22semidirect
25. #15690744C22semidirect
26. #15690742C22semidirect
27. #15690734C22semidirect
68. C2×C8×C97D4semidirect
121. C97#3057semidirect
122. C22×SD16D97semidirect
180. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908162
#202613092
#202613022
#202609782
#202609612
#202607732
#202607442
#202607402
C22×SD16×C972
#305797