Finite Group Databasebeta

← #27151677

Subgroups — #27151677

Order 12416.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15690784C211yesmaximal, characteristic
2C2×#15690769C211yesmaximal, characteristic, complemented
3#20261060C214yesmaximal, complemented
4#20261054C212yesmaximal, complemented
5#20261049C212yesmaximal, complemented
6#20260777C212yesmaximal, complemented
7#20260741C212yesmaximal, complemented
8#407×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×C22⋊D97—21no—
11#15690784—21no—
12#15690784—21no—
13#15690784C2212yes—
14#15690784C2212yes—
15#15690769—21no—
16#15690769—21no—
17#15690769C2212yes—
18#15690769C2212yescomplemented
19C2×C4⋊Dic97C2211yescharacteristic
20C2×C4⋊Dic97C2211yescharacteristic, complemented
21C2×C4⋊Dic97C2211yescharacteristic, complemented
22C2×C22⋊Dic97—21no—
23C2×#13095488—21no—
24C2×C4×Dic97—21no—
25#15690744C2214yescomplemented
26#15690742C2214yescomplemented
27#15690732C2214yescomplemented
28#15690732C2214yescomplemented
29C23×C4×C97C2211yescharacteristic
30C2×M16×C97C2212yes—
31C22⋊C8×C97C2212yescomplemented
32C22⋊C8×C97C2212yescomplemented
33C23×D97—21no—
34C2×D388—24no—
35C2×D388—21no—
36C2×D388C2312yes—
37C2×C22⋊D97—22no—
38C2×C22⋊D97—22no—
39C2×C22⋊D97—44no—
40C22×Dic97—21no—
41C22×Dic97—21no—
42C22×Dic97—21no—
43C4⋊Dic97—21no—
44C4⋊Dic97—21no—
45C4⋊Dic97—21no—
46C4⋊Dic97C2312yes—
47C4⋊Dic97C2312yes—
48C4⋊Dic97C2312yes—
49C22⋊Dic97—22no—
50C22⋊Dic97—22no—
51#13095488—22no—
52#13095488—22no—
53C4×Dic97—22no—
54C4×Dic97—22no—
55C24×C97D411yescharacteristic
56C22×C4×C97—24no—
57C22×C4×C97D411yescharacteristic
58C22×C4×C97D411yescharacteristic
59C22×C4×C97D411yescharacteristic
60C22×C4×C97C2311yescharacteristic
61C22×C4×C97C2311yescharacteristic
62C22×C4×C97C2311yescharacteristic
63M16×C97—24no—
64C2×C8×C97C2314yes—
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—41no—
68C23×C97C2×D411yescharacteristic
69C23×C97C2×D411yescharacteristic
70C23×C97C2×D411yescharacteristic
71C22×D97—22no—
72C22×D97—42no—
73C22×D97—42no—
74C22×D97—42no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4114yes—
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—44no—
86C22⋊D97—44no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—22no—
90C2×Dic97—42no—
91C2×Dic97—42no—
92C2×Dic97—42no—
93D388—24no—
94D388—42no—
95C8×C97—24no—
96C22×C97—21no—
97C22×C97—21no—
98C22×C97—21no—
99C22×C97—21no—
100C22×C97—21no—
101C22×C97—21no—
102C22×C97—21no—
103C22×C97—44no—
104C22×C97#12312yes—
105C22×C97AGL(1,ℤ/8)12yes—
106C22×C97C22×D411yescharacteristic
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—42no—
110C2×D97—84no—
111C4×C97—21no—
112C4×C97—21no—
113C4×C97#12312yes—
114C4×C97C2×#4112yes—
115Dic97—42no—
116Dic97—42no—
117Dic97—42no—
118C2×C97—22no—
119C2×C97—22no—
120C2×C97—41no—
121C2×C97C2×#12311yescharacteristic
122C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C97#57511yescharacteristic
124D97—82no—
125#3094—971nomaximal, Sylow
126C97#309411yescharacteristic, complemented, Sylow
127C2×#121—971no—
128C2×#117—971no—
129#482—974no—
130#481—972no—
131#480—972no—
132#428—972no—
133#418—972no—
134#407D9711yescharacteristic, complemented
135C22×D4—971no—
136C22×D4—1941no—
137C23×C4C2×D9711yescharacteristic
138C2×M16C2×D9712yes—
139#121—972no—
140#121—972no—
141#121—1941no—
142#121—1941no—
143#117—972no—
144#117—972no—
145#117—1941no—
146#117—1941no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4—971no—
150C2×C22⋊C4—1941no—
151C2×C22⋊C4—1941no—
152C2×C42—1941no—
153#107—974no—
154#106—974no—
155#102—974no—
156#102—974no—
157C22⋊C8C2×D9712yescomplemented
158C22⋊C8C2×D9712yescomplemented
159C24—1941no—
160C24D38811yescharacteristic
161C2×D4—972no—
162C2×D4—3884no—
163C2×D4—1942no—
164C2×D4—1942no—
165C2×D4—1941no—
166C2×D4—1944no—
167C22×C4—24no—
168C22×C4—1941no—
169C22×C4—1941no—
170C22×C4—1941no—
171C22×C4D38811yescharacteristic
172C22×C4D38811yescharacteristic
173C22×C4D38811yescharacteristic
174C22×C4C22×D9711yescharacteristic
175C22×C4C22×D9711yescharacteristic
176C22×C4C22×D9711yescharacteristic
177M16—24no—
178C2×C8C22×D9714yes—
179C4⋊C4—972no—
180C4⋊C4—972no—
181C4⋊C4—972no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4—1941no—
185C22⋊C4—1942no—
186C22⋊C4—1942no—
187C22⋊C4—1942no—
188C22⋊C4—1942no—
189C42—1942no—
190C42—1942no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—3882no—
195C23—3882no—
196C23—3882no—
197C23—1942no—
198C23C2×D38811yescharacteristic
199C23C2×D38811yescharacteristic
200C23C2×D38811yescharacteristic
201D4—3884no—
202D4—3884no—
203D4—3882no—
204D4—1944no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—3882no—
211C2×C4—3882no—
212C2×C4—3882no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4C2×D38811yescharacteristic
217C2×C4C2×D38811yescharacteristic
218C2×C4C2×D38811yescharacteristic
219C2×C4#1309551014yes—
220C2×C4C23×D9711yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—7764no—
231C22—3882no—
232C22—3882no—
233C22—3882no—
234C22#1569079412yes—
235C22#1569083812yes—
236C22C22×D38811yescharacteristic
237C4—21no—
238C4—21no—
239C4—3882no—
240C4—3882no—
241C4—3882no—
242C4#1569079412yes—
243C4C2×#1309551012yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—7762no—
248C2C2×#1569079411yescharacteristic
249C2C2×#1569083811yescharacteristic
250C2#2026153511yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#407×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690784C2
hypercenter#407D97
hypocenterC97#3094
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
3. #20261060C2semidirect
4. #20261054C2semidirect
5. #20261049C2semidirect
6. #20260777C2semidirect
7. #20260741C2semidirect
8. #407×C97C2semidirect
18. #15690769C22semidirect
20. C2×C4⋊Dic97C22semidirect
21. C2×C4⋊Dic97C22semidirect
25. #15690744C22semidirect
26. #15690742C22semidirect
27. #15690732C22semidirect
28. #15690732C22semidirect
31. C22⋊C8×C97C22semidirect
32. C22⋊C8×C97C22semidirect
126. C97#3094semidirect
134. #407D97semidirect
157. C22⋊C8C2×D97semidirect
158. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907842
C2×#156907692
#202610602
#202610542
#202610492
#202607772
#202607412
#407×C972
#309497