Finite Group Databasebeta

← C2×#20260907 #27151071

Subgroups — C2×#20260907 #27151071

Order 12416.

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

normal subgroups only

This group has 728 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
1C2×#15690858C211yesmaximal, characteristic, complemented
2C2×#15690836C212yesmaximal, complemented
3#20260907C218yesmaximal, direct product factor
4C2×#15690738C212yesmaximal, complemented
5C2×#15690729C211yesmaximal, characteristic, complemented
6C2×#124×C97C211yesmaximal, characteristic, complemented
7#15690858—24no—
8#15690858—22no—
9#15690858C2214yescomplemented
10C2×#13095512—22no—
11C2×#13095511—22no—
12C2×#13095511C2212yes—
13C2×Q8×D97—22no—
14#15690836—24no—
15#15690836C2218yescomplemented
16C2×#13095499—22no—
17C2×#13095496—22no—
18C2×#13095495C2212yes—
19#15690738C2218yescomplemented
20#15690729C2214yescomplemented
21C22×Q8×C97C2211yescharacteristic
22C22×D4×C97C2211yescharacteristic
23#124×C97—22no—
24#124×C97C2214yescomplemented
25C2×C22⋊C4×C97—22no—
26C2×C42×C97C2211yescharacteristic
27#13095512—28no—
28#13095512—28no—
29#13095511—28no—
30#13095511—28no—
31#13095511—22no—
32#13095511—22no—
33#13095511D414yes—
34#13095511C2314yes—
35C2×Dic194—22no—
36C2×Dic194D412yes—
37Q8×D97—28no—
38Q8×D97—28no—
39C2×D388—22no—
40C2×D388D412yes—
41C2×C22⋊D97—22no—
42C2×C4×D97—24no—
43C2×C4×D97—22no—
44#13095499—28no—
45#13095496—28no—
46#13095495—22no—
47#13095495C2314yes—
48C2×#11715895—22no—
49C24×C97—21no—
50C2×Q8×C97—22no—
51C2×Q8×C97—22no—
52C2×Q8×C97D412yes—
53C2×Q8×C97C2312yes—
54C2×D4×C97—22no—
55C2×D4×C97—22no—
56C2×D4×C97—21no—
57C2×D4×C97C2312yes—
58C22×C4×C97—21no—
59C22×C4×C97D412yes—
60C22×C4×C97C2311yescharacteristic
61C22⋊C4×C97—28no—
62C42×C97—21no—
63C42×C97C2312yes—
64C23×C97—22no—
65C23×C97—44no—
66C23×C97—41no—
67C23×C97—41no—
68C23×C97C2×D411yescharacteristic
69C22×D97—22no—
70Q8×C97—24no—
71Q8×C97—24no—
72D4×C97—24no—
73D4×C97—42no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—42no—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D414yes—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2411yescharacteristic
83C22⋊D97—24no—
84C22⋊D97—24no—
85C2×Dic97—22no—
86D388—28no—
87D388—22no—
88D388C2×D414yes—
89C4×D97—216no—
90C4×D97—24no—
91C4×D97—24no—
92Dic194—28no—
93Dic194—22no—
94Dic194C2×D414yes—
95#11715895—24no—
96C22×C97—22no—
97C22×C97—44no—
98C22×C97—42no—
99C22×C97—82no—
100C22×C97—82no—
101C22×C97#12011yescharacteristic
102C22×C97#12011yescharacteristic
103C22×C97C22×D411yescharacteristic
104C2×D97—24no—
105C2×D97—44no—
106C4×C97—24no—
107C4×C97—42no—
108C4×C97#12012yes—
109C4×C97C22×D412yes—
110Dic97—24no—
111C2×C97—21no—
112C2×C97—21no—
113C2×C97—82no—
114C2×C97#45412yescomplemented
115C2×C97C2×#12011yescharacteristic
116D97—44no—
117C2×#454—971nomaximal, Sylow
118C97C2×#45411yescharacteristic, complemented, Sylow
119C2×2−1+4—971no—
120C2×AGL(1,ℤ/8)—972no—
121C2×#124D9711yescharacteristic, complemented
122#454—978no—
123C2×#104—972no—
124C2×#101—971no—
1252−1+4—974no—
1262−1+4—1944no—
1272−1+4—1942no—
128C2×#41—972no—
129C2×#41—1942no—
130C2×#41—1942no—
131C22×Q8—1942no—
132C22×Q8C2×D9711yescharacteristic
133C22×D4C2×D9711yescharacteristic
134AGL(1,ℤ/8)—978no—
135AGL(1,ℤ/8)—1944no—
136C2×SD16—1942no—
137C2×D8—1942no—
138C2×M16—972no—
139#124—22no—
140#124C2×D9714yescomplemented
141C2×C22⋊C4—22no—
142C2×C42C2×D9711yescharacteristic
143#104—978no—
144#101—974no—
145C24—21no—
146#41—974no—
147#41—974no—
148#41—1948no—
149#41—1948no—
150#41—1948no—
151#41—1942no—
152#41—1942no—
153#41—1948no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—972no—
157C2×Q8—1942no—
158C2×Q8—1948no—
159C2×Q8—1948no—
160C2×Q8C22⋊D9712yes—
161C2×Q8C22×D9712yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—972no—
166C2×D4—1942no—
167C2×D4—1942no—
168C2×D4C22×D9712yes—
169C22×C4—21no—
170C22×C4—1944no—
171C22×C4—1942no—
172C22×C4C22⋊D9712yes—
173C22×C4C22×D9711yescharacteristic
174SD16—1948no—
175D8—1948no—
176M16—974no—
177M16—1942no—
178C2×C8—1942no—
179C22⋊C4—28no—
180C42—21no—
181C42C22×D9712yes—
182C23—22no—
183C23—44no—
184C23—41no—
185C23—41no—
186C23—1942no—
187C23C2×C22⋊D9711yescharacteristic
188Q8—24no—
189Q8—24no—
190Q8—974no—
191Q8—1948no—
192Q8—1942no—
193D4—24no—
194D4—42no—
195D4—974no—
196D4—1948no—
197D4—1944no—
198D4—1944no—
199D4—1942no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—19416no—
205C2×C4—1942no—
206C2×C4—1944no—
207C2×C4—1944no—
208C2×C4D4×D9712yes—
209C2×C4D4×D9712yes—
210C2×C4C2×C22⋊D9714yes—
211C2×C4C2×C22⋊D9711yescharacteristic
212C2×C4C23×D9711yescharacteristic
213C8—1944no—
214C22—22no—
215C22—44no—
216C22—42no—
217C22—82no—
218C22—82no—
219C22—3884no—
220C22—1944no—
221C22#1569077811yescharacteristic
222C22#1569077811yescharacteristic
223C22C22×C22⋊D9711yescharacteristic
224C4—24no—
225C4—42no—
226C4—1944no—
227C4#1569077812yes—
228C4C2×D4×D9712yes—
229C2—21no—
230C2—21no—
231C2—82no—
232C2—3884no—
233C2#2026090712yesdirect product factor
234C2C2×#1569077811yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#124×C97C2
socleC22×C97#120
hypercenterC2×#124D97
hypocenterC97C2×#454
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#15690858C2semidirect
2. C2×#15690836C2semidirect
3. #20260907C2direct (N × H)
4. C2×#15690738C2semidirect
5. C2×#15690729C2semidirect
6. C2×#124×C97C2semidirect
9. #15690858C22semidirect
15. #15690836C22semidirect
19. #15690738C22semidirect
20. #15690729C22semidirect
24. #124×C97C22semidirect
114. C2×C97#454semidirect
118. C97C2×#454semidirect
121. C2×#124D97semidirect
140. #124C2×D97semidirect
233. C2#20260907direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908582
C2×#156908362
#202609072
C2×#156907382
C2×#156907292
C2×#124×C972
C2×#45497