Finite Group Databasebeta

← C2×#20260908 #27151072

Subgroups — C2×#20260908 #27151072

Order 12416.

728 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15690835C212yesmaximal, complemented
2C2×#15690799C211yesmaximal, characteristic, complemented
3#20260908C218yesmaximal, direct product factor
4C2×#15690737C212yesmaximal, complemented
5C2×#15690729C211yesmaximal, characteristic, complemented
6C2×2−1+4×C97C211yesmaximal, characteristic, complemented
7C22×D388C2211yescharacteristic
8#15690835—24no—
9#15690835C2218yescomplemented
10C2×#13095498—22no—
11C2×#13095496—22no—
12C2×#13095495C2212yes—
13#15690799—22no—
14#15690799C2214yescomplemented
15C2×#13095488—22no—
16C2×C4×Dic97C2211yescharacteristic
17#15690737C2218yescomplemented
18#15690729C2214yescomplemented
192−1+4×C97—24no—
202−1+4×C97—22no—
212−1+4×C97C2214yescomplemented
22C2×#41×C97—22no—
23C2×#41×C97—22no—
24C2×#41×C97C2212yes—
25C22×Q8×C97—22no—
26C22×Q8×C97C2211yescharacteristic
27C23×D97—21no—
28C2×D388—22no—
29C2×D388—22no—
30C2×D388—21no—
31C2×D388C2312yes—
32C22×Dic97—21no—
33#13095498—28no—
34#13095496—28no—
35#13095495—22no—
36#13095495C2314yes—
37C2×#11715895—22no—
38#13095488—28no—
39C4×Dic97—21no—
40C4×Dic97C2312yes—
41#41×C97—28no—
42#41×C97—28no—
43#41×C97—28no—
44#41×C97—28no—
45#41×C97—22no—
46#41×C97—22no—
47#41×C97D414yes—
48#41×C97C2314yes—
49C2×Q8×C97—22no—
50C2×Q8×C97—28no—
51C2×Q8×C97—28no—
52C2×Q8×C97—22no—
53C2×Q8×C97—22no—
54C2×Q8×C97D412yes—
55C2×Q8×C97D412yes—
56C2×Q8×C97C2312yes—
57C2×D4×C97—22no—
58C2×D4×C97—22no—
59C2×D4×C97D412yes—
60C22×C4×C97—24no—
61C22×C4×C97—22no—
62C22×C4×C97D412yes—
63C22×C4×C97C2311yescharacteristic
64C23×C97—22no—
65C23×C97C2×D411yescharacteristic
66C22×D97—22no—
67C22×D97—44no—
68C22×D97—41no—
69C22×D97—41no—
70Q8×C97—28no—
71Q8×C97—22no—
72Q8×C97—24no—
73Q8×C97—24no—
74Q8×C97C2×D414yes—
75D4×C97—28no—
76D4×C97—24no—
77D4×C97—24no—
78D4×C97—22no—
79D4×C97C2×D414yes—
80C2×C4×C97—216no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—24no—
85C2×C4×C97—24no—
86C2×C4×C97C2×D414yes—
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D412yes—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2411yescharacteristic
91C2×Dic97—22no—
92C2×Dic97—42no—
93D388—24no—
94D388—42no—
95#11715895—24no—
96C22×C97—22no—
97C22×C97—24no—
98C22×C97—44no—
99C22×C97#12011yescharacteristic
100C22×C97#12011yescharacteristic
101C22×C97C22×D411yescharacteristic
102C2×D97—44no—
103C2×D97—42no—
104C2×D97—82no—
105C2×D97—82no—
106C4×C97—24no—
107C4×C97—24no—
108C4×C97#12012yes—
109C4×C97C22×D412yes—
110Dic97—42no—
111C2×C97—21no—
112C2×C97—21no—
113C2×C97—44no—
114C2×C97#45412yescomplemented
115C2×C97C2×#12011yescharacteristic
116D97—82no—
117C2×#454—971nomaximal, Sylow
118C97C2×#45411yescharacteristic, complemented, Sylow
119C2×2−1+4D9711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—972no—
121C2×#124—971no—
122#454—978no—
123C2×#104—972no—
124C2×#101—971no—
1252−1+4—24no—
1262−1+4—22no—
1272−1+4C2×D9714yescomplemented
128C2×#41—22no—
129C2×#41—22no—
130C2×#41C2×D9712yes—
131C22×Q8—22no—
132C22×Q8C2×D9711yescharacteristic
133C22×D4—971no—
134AGL(1,ℤ/8)—978no—
135AGL(1,ℤ/8)—1944no—
136C2×SD16—1942no—
137C2×D8—1942no—
138C2×M16—972no—
139#124—974no—
140#124—1942no—
141C2×C22⋊C4—1942no—
142C2×C42—971no—
143#104—978no—
144#101—974no—
145C24—1941no—
146#41—28no—
147#41—28no—
148#41—28no—
149#41—28no—
150#41—22no—
151#41—22no—
152#41C22⋊D9714yes—
153#41C22×D9714yes—
154C2×Q8—22no—
155C2×Q8—28no—
156C2×Q8—28no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8C22⋊D9712yes—
160C2×Q8C22⋊D9712yes—
161C2×Q8C22×D9712yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—972no—
165C2×D4—1941no—
166C2×D4—1942no—
167C2×D4—1942no—
168C2×D4C22⋊D9712yes—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—1941no—
172C22×C4C22⋊D9712yes—
173C22×C4C22×D9711yescharacteristic
174SD16—1948no—
175D8—1948no—
176M16—974no—
177M16—1942no—
178C2×C8—1942no—
179C22⋊C4—1948no—
180C42—972no—
181C42—1941no—
182C23—22no—
183C23—3884no—
184C23—3881no—
185C23—3881no—
186C23—1942no—
187C23C2×C22⋊D9711yescharacteristic
188Q8—28no—
189Q8—22no—
190Q8—24no—
191Q8—24no—
192Q8C2×C22⋊D9714yes—
193D4—28no—
194D4—24no—
195D4—24no—
196D4—22no—
197D4—3882no—
198D4—1944no—
199D4C2×C22⋊D9714yes—
200C2×C4—216no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—3882no—
207C2×C4—1942no—
208C2×C4C2×C22⋊D9714yes—
209C2×C4C2×C22⋊D9711yescharacteristic
210C2×C4C2×C22⋊D9712yes—
211C2×C4C2×C22⋊D9712yes—
212C2×C4C23×D9711yescharacteristic
213C8—1944no—
214C22—22no—
215C22—24no—
216C22—44no—
217C22—7762no—
218C22—7762no—
219C22—3884no—
220C22—3882no—
221C22#1569077911yescharacteristic
222C22#1569077911yescharacteristic
223C22C22×C22⋊D9711yescharacteristic
224C4—24no—
225C4—24no—
226C4—3882no—
227C4#1569077912yes—
228C4C22×C22⋊D9712yes—
229C2—21no—
230C2—21no—
231C2—44no—
232C2—7762no—
233C2#2026090812yesdirect product factor
234C2C2×#1569077911yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×2−1+4×C97C2
socleC22×C97#120
hypercenterC2×2−1+4D97
hypocenterC97C2×#454
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. C2×#15690799C2semidirect
3. #20260908C2direct (N × H)
4. C2×#15690737C2semidirect
5. C2×#15690729C2semidirect
6. C2×2−1+4×C97C2semidirect
9. #15690835C22semidirect
14. #15690799C22semidirect
17. #15690737C22semidirect
18. #15690729C22semidirect
21. 2−1+4×C97C22semidirect
114. C2×C97#454semidirect
118. C97C2×#454semidirect
119. C2×2−1+4D97semidirect
127. 2−1+4C2×D97semidirect
233. C2#20260908direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908352
C2×#156907992
#202609082
C2×#156907372
C2×#156907292
C2×2−1+4×C972
C2×#45497