Finite Group Databasebeta

← C2×#20260984 #27151327

Subgroups — C2×#20260984 #27151327

Order 12416.

544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#13095494C211yesmaximal, characteristic, complemented
3C2×#15690786C211yesmaximal, characteristic, complemented
4C2×#15690785C211yesmaximal, characteristic
5#20260984C218yesmaximal, direct product factor
6C2×#15690743C211yesmaximal, characteristic, complemented
7C2×#15690732C211yesmaximal, characteristic, complemented
8C2×#103×C97C211yesmaximal, characteristic, complemented
9C22×D388—21no—
10C22×D388C2211yescharacteristic
11C22×C4×D97C2211yescharacteristic
12C2×#13095498—24no—
13C2×#13095498C2214yescomplemented
14C2×#13095494—24no—
15C2×#13095494C2214yescomplemented
16C22×#11715895C2211yescharacteristic
17#15690786—22no—
18#15690786C2214yescomplemented
19#15690785—22no—
20#15690785C2214yes—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490—21no—
23C2×#13095488—21no—
24C2×#13095488—21no—
25#15690743C2214yescomplemented
26#15690732C2214yescomplemented
27C22×Q8×C97C2211yescharacteristic
28C22×C8×C97C2211yescharacteristic
29C2×C4⋊C4×C97C2211yescharacteristic
30#103×C97C2214yescomplemented
31C23×D97—21no—
32C23×D97D411yescharacteristic
33C2×D388—22no—
34C2×D388—22no—
35C2×D388—24no—
36C2×D388—21no—
37C2×D388—44no—
38C2×D388C2312yes—
39C22×Dic97—21no—
40C22×Dic97D411yescharacteristic
41C2×C4×D97—22no—
42C2×C4×D97—22no—
43C2×C4×D97D412yes—
44C2×C4×D97C2312yes—
45#13095498—28no—
46#13095494—28no—
47C2×#11715895D414yescomplemented
48C2×#11715895C2312yes—
49C4⋊Dic97—21no—
50C4⋊Dic97C2312yes—
51#13095490—24no—
52#13095488—24no—
53#13095488—24no—
54C2×Q8×C97—21no—
55C2×Q8×C97—22no—
56C2×Q8×C97—22no—
57C2×Q8×C97C2312yes—
58C22×C4×C97—21no—
59C22×C4×C97—21no—
60C22×C4×C97C2311yescharacteristic
61C2×C8×C97—22no—
62C2×C8×C97C2312yes—
63C4⋊C4×C97—21no—
64C4⋊C4×C97C2312yes—
65C23×C97C2×D411yescharacteristic
66C22×D97—22no—
67C22×D97—22no—
68C22×D97—22no—
69C22×D97—22no—
70C22×D97—42no—
71C22×D97—42no—
72C22×D97—42no—
73C22×D97C2×D412yes—
74Q8×C97—22no—
75Q8×C97—22no—
76Q8×C97—42no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—42no—
80C2×C4×C97—42no—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D412yes—
83C2×C4×C97#4112yes—
84C2×C4×C97C2411yescharacteristic
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—42no—
88C2×Dic97C2×D412yes—
89D388—22no—
90D388—22no—
91D388—48no—
92D388—42no—
93C4×D97—24no—
94C4×D97—22no—
95C4×D97—22no—
96C8×C97—22no—
97#11715895C2×D414yescomplemented
98C22×C97#12112yes—
99C22×C97AGL(1,ℤ/8)12yes—
100C22×C97#13712yes—
101C22×C97C22×D411yescharacteristic
102C2×D97—22no—
103C2×D97—22no—
104C2×D97—22no—
105C2×D97—44no—
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—84no—
110C4×C97—42no—
111C4×C97—42no—
112C4×C97#12112yes—
113C4×C97C22×D411yescharacteristic
114C4×C97C2×#4111yescharacteristic
115Dic97—22no—
116Dic97—42no—
117C2×C97#46814yescomplemented
118C2×C97C2×#12111yescharacteristic
119C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C97C2×#13711yescharacteristic
121D97—42no—
122D97—82no—
123C2×#468—971nomaximal, Sylow
124C97C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16—971no—
126C22×M16—971no—
127C2×C22⋊Q8—971no—
128C2×#121—971no—
129#468—978no—
130C2×#107—971no—
131C2×#103D9711yescharacteristic, complemented
132C2×#102—971no—
133C22×Q8C2×D9711yescharacteristic
134C22×D4—971no—
135C22×D4—1941no—
136C23×C4—971no—
137C2×SD16—974no—
138C2×SD16—1944no—
139C2×M16—974no—
140C2×M16—1944no—
141C22×C8—971no—
142C22×C8C2×D9711yescharacteristic
143C22⋊Q8—974no—
144C22⋊Q8—1942no—
145#121—974no—
146#121—1942no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4—1941no—
149C2×C4⋊C4C2×D9711yescharacteristic
150C2×C22⋊C4—1941no—
151C2×C22⋊C4—1941no—
152#107—974no—
153#103C2×D9714yescomplemented
154#102—974no—
155C24—971no—
156C24—1941no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D9712yes—
161C2×D4—972no—
162C2×D4—3884no—
163C2×D4—1944no—
164C2×D4—1941no—
165C2×D4—1942no—
166C2×D4—1942no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—971no—
170C22×C4—972no—
171C22×C4—972no—
172C22×C4—1941no—
173C22×C4—1942no—
174C22×C4—1942no—
175C22×C4C22×D9711yescharacteristic
176SD16—1948no—
177M16—1948no—
178C2×C8—22no—
179C2×C8—974no—
180C2×C8—972no—
181C2×C8C22×D9712yes—
182C4⋊C4—21no—
183C4⋊C4—972no—
184C4⋊C4—1944no—
185C4⋊C4—1941no—
186C4⋊C4C22×D9712yes—
187C22⋊C4—1944no—
188C22⋊C4—1944no—
189C23—972no—
190C23—3882no—
191C23—3882no—
192C23—3882no—
193C23—1942no—
194C23—1942no—
195C23—1942no—
196C23—1942no—
197C23D4×D9711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—3888no—
202D4—3882no—
203D4—1942no—
204D4—1942no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—972no—
210C2×C4—3882no—
211C2×C4—1942no—
212C2×C4—1942no—
213C2×C4—1944no—
214C2×C4—1942no—
215C2×C4—1942no—
216C2×C4D4×D9711yescharacteristic
217C2×C4D4×D9712yes—
218C2×C4#1309551112yes—
219C2×C4C23×D9711yescharacteristic
220C8—22no—
221C8—974no—
222C22—7764no—
223C22—3882no—
224C22—3882no—
225C22—3882no—
226C22—3884no—
227C22—1942no—
228C22—1942no—
229C22—1942no—
230C22#1569078312yes—
231C22#1569083712yes—
232C22#1569084212yes—
233C22C2×D4×D9711yescharacteristic
234C4—42no—
235C4—42no—
236C4—3882no—
237C4—1942no—
238C4#1569078312yes—
239C4C2×D4×D9711yescharacteristic
240C4C2×#1309551111yescharacteristic
241C2—7762no—
242C2—3882no—
243C2#2026098414yesdirect product factor
244C2C2×#1569078311yescharacteristic
245C2C2×#1569083711yescharacteristic
246C2C2×#1569084211yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#103×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690785C2
hypercenterC2×#103D97
hypocenterC97C2×#468
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#13095498C2semidirect
2. C22×#13095494C2semidirect
3. C2×#15690786C2semidirect
5. #20260984C2direct (N × H)
6. C2×#15690743C2semidirect
7. C2×#15690732C2semidirect
8. C2×#103×C97C2semidirect
13. C2×#13095498C22semidirect
15. C2×#13095494C22semidirect
18. #15690786C22semidirect
25. #15690743C22semidirect
26. #15690732C22semidirect
30. #103×C97C22semidirect
47. C2×#11715895D4semidirect
97. #11715895C2×D4semidirect
117. C2×C97#468semidirect
124. C97C2×#468semidirect
131. C2×#103D97semidirect
153. #103C2×D97semidirect
243. C2#20260984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954982
C22×#130954942
C2×#156907862
C2×#156907852
#202609842
C2×#156907432
C2×#156907322
C2×#103×C972
C2×#46897