Finite Group Databasebeta

← C2×#20260983 #27151326

Subgroups — C2×#20260983 #27151326

Order 12416.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13095495C211yesmaximal, characteristic, complemented
3C2×#15690784C211yesmaximal, characteristic
4#20260983C218yesmaximal, direct product factor
5C2×#15690743C211yesmaximal, characteristic, complemented
6C2×#15690734C211yesmaximal, characteristic, complemented
7C2×#15690730C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×C22⋊D97—21no—
11C2×#13095498—24no—
12C2×#13095498C2214yescomplemented
13C2×#13095495—24no—
14C2×#13095495C2214yescomplemented
15C22×#11715895C2211yescharacteristic
16C22×#11715895C2211yescharacteristic
17#15690784—22no—
18#15690784C2214yes—
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095488—21no—
21#15690743C2214yescomplemented
22#15690734C2214yescomplemented
23#15690730C2214yescomplemented
24C22×Q8×C97C2211yescharacteristic
25C23×C4×C97C2211yescharacteristic
26C22⋊Q8×C97—22no—
27C22⋊Q8×C97C2214yescomplemented
28C2×C4⋊C4×C97—21no—
29C2×C4⋊C4×C97C2211yescharacteristic
30C2×C22⋊C4×C97—21no—
31C23×D97—21no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—21no—
35C2×D388C2312yes—
36C2×C22⋊D97—24no—
37C2×C22⋊D97—44no—
38C22×Dic97—21no—
39#13095498—28no—
40#13095495—28no—
41C2×#11715895—22no—
42C2×#11715895D414yescomplemented
43C2×#11715895C2312yes—
44C2×#11715895C2312yes—
45C4⋊Dic97—21no—
46C4⋊Dic97C2312yes—
47#13095488—24no—
48C24×C97D411yescharacteristic
49C2×Q8×C97—21no—
50C2×Q8×C97—22no—
51C2×Q8×C97—22no—
52C2×Q8×C97C2312yes—
53C22×C4×C97—22no—
54C22×C4×C97—22no—
55C22×C4×C97—21no—
56C22×C4×C97—21no—
57C22×C4×C97D411yescharacteristic
58C22×C4×C97D412yes—
59C22×C4×C97C2312yes—
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—24no—
62C4⋊C4×C97—21no—
63C4⋊C4×C97C2312yes—
64C22⋊C4×C97—24no—
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97C2×D412yes—
69C23×C97C2×D411yescharacteristic
70C22×D97—22no—
71C22×D97—42no—
72C22×D97—42no—
73C22×D97—42no—
74Q8×C97—22no—
75Q8×C97—22no—
76Q8×C97—42no—
77C2×C4×C97—22no—
78C2×C4×C97—24no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—42no—
84C2×C4×C97—42no—
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D412yes—
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97#4112yes—
89C2×C4×C97C2411yescharacteristic
90C22⋊D97—48no—
91C2×Dic97—22no—
92C2×Dic97—42no—
93D388—22no—
94D388—22no—
95D388—42no—
96#11715895—22no—
97#11715895C2×D414yescomplemented
98C22×C97—22no—
99C22×C97—22no—
100C22×C97—22no—
101C22×C97—44no—
102C22×C97#12112yes—
103C22×C97AGL(1,ℤ/8)12yes—
104C22×C97#13712yes—
105C22×C97C22×D411yescharacteristic
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—42no—
109C2×D97—84no—
110C4×C97—22no—
111C4×C97—42no—
112C4×C97—42no—
113C4×C97#12112yes—
114C4×C97C22×D411yescharacteristic
115C4×C97C2×#4111yescharacteristic
116Dic97—42no—
117C2×C97—42no—
118C2×C97#46814yescomplemented
119C2×C97C2×#12111yescharacteristic
120C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C97C2×#13711yescharacteristic
122D97—82no—
123C2×#468—971nomaximal, Sylow
124C97C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16—971no—
126C22×M16—971no—
127C2×C22⋊Q8D9711yescharacteristic, complemented
128C2×#121—971no—
129#468—978no—
130C2×#107—971no—
131C2×#103—971no—
132C2×#102—971no—
133C22×Q8C2×D9711yescharacteristic
134C22×D4—971no—
135C22×D4—1941no—
136C23×C4C2×D9711yescharacteristic
137C2×SD16—974no—
138C2×SD16—1944no—
139C2×M16—974no—
140C2×M16—1944no—
141C22×C8—971no—
142C22×C8—971no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D9714yescomplemented
145#121—974no—
146#121—1942no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4C2×D9711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—1941no—
152#107—974no—
153#103—974no—
154#102—974no—
155C24—1941no—
156C24C22⋊D9711yescharacteristic
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D9712yes—
161C2×D4—972no—
162C2×D4—3884no—
163C2×D4—1944no—
164C2×D4—1942no—
165C2×D4—1941no—
166C2×D4—1942no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1941no—
172C22×C4C22⋊D9712yes—
173C22×C4C22⋊D9711yescharacteristic
174C22×C4C22×D9712yes—
175C22×C4C22×D9711yescharacteristic
176SD16—1948no—
177M16—1948no—
178C2×C8—974no—
179C2×C8—972no—
180C2×C8—972no—
181C2×C8—1942no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—972no—
185C4⋊C4—1941no—
186C4⋊C4C22×D9712yes—
187C22⋊C4—24no—
188C22⋊C4—1944no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—3882no—
193C23—3882no—
194C23—3882no—
195C23—1942no—
196C23C2×C22⋊D9712yes—
197C23C2×C22⋊D9711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—3888no—
202D4—3882no—
203D4—1942no—
204D4—1942no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—3882no—
214C2×C4—1942no—
215C2×C4D4×D9712yes—
216C2×C4C2×C22⋊D9712yes—
217C2×C4C2×C22⋊D9711yescharacteristic
218C2×C4#1309551012yes—
219C2×C4C23×D9711yescharacteristic
220C8—974no—
221C8—1942no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—44no—
226C22—7764no—
227C22—3882no—
228C22—3882no—
229C22—3882no—
230C22#1569078212yes—
231C22#1569083512yes—
232C22#1569084112yes—
233C22C22×C22⋊D9711yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—3882no—
238C4#1569078212yes—
239C4C2×D4×D9711yescharacteristic
240C4C2×#1309551011yescharacteristic
241C2—42no—
242C2—7762no—
243C2#2026098314yesdirect product factor
244C2C2×#1569078211yescharacteristic
245C2C2×#1569083511yescharacteristic
246C2C2×#1569084111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C2×D4
prime radicalC2×#15690784C2
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#468
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13095498C2semidirect
2. C22×#13095495C2semidirect
4. #20260983C2direct (N × H)
5. C2×#15690743C2semidirect
6. C2×#15690734C2semidirect
7. C2×#15690730C2semidirect
8. C2×C22⋊Q8×C97C2semidirect
12. C2×#13095498C22semidirect
14. C2×#13095495C22semidirect
21. #15690743C22semidirect
22. #15690734C22semidirect
23. #15690730C22semidirect
27. C22⋊Q8×C97C22semidirect
42. C2×#11715895D4semidirect
97. #11715895C2×D4semidirect
118. C2×C97#468semidirect
124. C97C2×#468semidirect
127. C2×C22⋊Q8D97semidirect
144. C22⋊Q8C2×D97semidirect
243. C2#20260983direct (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×#130954952
C2×#156907842
#202609832
C2×#156907432
C2×#156907342
C2×#156907302
C2×C22⋊Q8×C972
C2×#46897