Finite Group Databasebeta

← C2×#20260986 #27151330

Subgroups — C2×#20260986 #27151330

Order 12416.

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

normal subgroups only

This group has 596 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×#13095496C211yesmaximal, characteristic, complemented
2C22×#13095495C211yesmaximal, characteristic, complemented
3C2×#15690784C211yesmaximal, characteristic, complemented
4#20260986C218yesmaximal, direct product factor
5C2×#15690741C211yesmaximal, characteristic, complemented
6C2×#15690731C211yesmaximal, characteristic, complemented
7C2×#15690730C211yesmaximal, characteristic, complemented
8C2×#121×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×C22⋊D97—21no—
11C2×#13095496—24no—
12C2×#13095496C2214yescomplemented
13C2×#13095495—24no—
14C2×#13095495C2214yescomplemented
15C22×#11715895C2211yescharacteristic
16C22×#11715895C2211yescharacteristic
17#15690784—22no—
18#15690784C2214yescomplemented
19C2×C4⋊Dic97C2211yescharacteristic
20C2×#13095488—21no—
21#15690741C2214yescomplemented
22#15690731C2214yescomplemented
23#15690730C2214yescomplemented
24C22×D4×C97—21no—
25C22×D4×C97C2211yescharacteristic
26C23×C4×C97C2211yescharacteristic
27#121×C97—22no—
28#121×C97C2214yescomplemented
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#13095496—28no—
40#13095495—28no—
41C2×#11715895—22no—
42C2×#11715895D414yescomplemented
43C2×#11715895C2312yes—
44C2×#11715895C2312yes—
45C4⋊Dic97—21no—
46C4⋊Dic97C2312yes—
47#13095488—24no—
48C24×C97—21no—
49C24×C97D411yescharacteristic
50C2×D4×C97—24no—
51C2×D4×C97—22no—
52C2×D4×C97—21no—
53C2×D4×C97—22no—
54C2×D4×C97—44no—
55C2×D4×C97C2312yes—
56C22×C4×C97—22no—
57C22×C4×C97—22no—
58C22×C4×C97—21no—
59C22×C4×C97D411yescharacteristic
60C22×C4×C97D412yes—
61C22×C4×C97C2312yes—
62C22×C4×C97C2311yescharacteristic
63C4⋊C4×C97—21no—
64C4⋊C4×C97C2312yes—
65C22⋊C4×C97—24no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—22no—
69C23×C97—22no—
70C23×C97—42no—
71C23×C97—42no—
72C23×C97—42no—
73C23×C97C2×D412yes—
74C23×C97C2×D411yescharacteristic
75C22×D97—22no—
76C22×D97—42no—
77C22×D97—42no—
78C22×D97—42no—
79D4×C97—22no—
80D4×C97—22no—
81D4×C97—48no—
82D4×C97—42no—
83C2×C4×C97—22no—
84C2×C4×C97—24no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—42no—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D412yes—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97#4112yes—
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—48no—
95C2×Dic97—22no—
96C2×Dic97—42no—
97D388—22no—
98D388—22no—
99D388—42no—
100#11715895—22no—
101#11715895C2×D414yescomplemented
102C22×C97—22no—
103C22×C97—22no—
104C22×C97—22no—
105C22×C97—44no—
106C22×C97—42no—
107C22×C97—42no—
108C22×C97—42no—
109C22×C97—84no—
110C22×C97#12112yes—
111C22×C97AGL(1,ℤ/8)12yes—
112C22×C97AGL(1,ℤ/8)12yes—
113C22×C97C22×D411yescharacteristic
114C2×D97—42no—
115C2×D97—42no—
116C2×D97—42no—
117C2×D97—84no—
118C4×C97—22no—
119C4×C97—42no—
120C4×C97#12112yes—
121C4×C97C22×D411yescharacteristic
122C4×C97C2×#4111yescharacteristic
123Dic97—42no—
124C2×C97—42no—
125C2×C97—82no—
126C2×C97#46914yescomplemented
127C2×C97C2×#12111yescharacteristic
128C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
129C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
130D97—82no—
131C2×#469—971nomaximal, Sylow
132C97C2×#46911yescharacteristic, complemented, Sylow
133C22×D8—971no—
134C22×M16—971no—
135C2×#121—971no—
136C2×#121D9711yescharacteristic, complemented
137#469—978no—
138C2×#106—971no—
139C2×#102—971no—
140C2×#102—971no—
141C22×D4—21no—
142C22×D4—971no—
143C22×D4—1941no—
144C22×D4C2×D9711yescharacteristic
145C23×C4C2×D9711yescharacteristic
146C2×D8—974no—
147C2×D8—1944no—
148C2×M16—974no—
149C2×M16—1944no—
150C22×C8—971no—
151C22×C8—971no—
152#121—22no—
153#121—974no—
154#121—1942no—
155#121C2×D9714yescomplemented
156C2×C4⋊C4—971no—
157C2×C4⋊C4C2×D9711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—1941no—
160#106—974no—
161#102—974no—
162#102—974no—
163C24—21no—
164C24—1941no—
165C24C22⋊D9711yescharacteristic
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—972no—
172C2×D4—3884no—
173C2×D4—1944no—
174C2×D4—1942no—
175C2×D4—1941no—
176C2×D4—1942no—
177C2×D4C22×D9712yes—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—1941no—
182C22×C4C22⋊D9712yes—
183C22×C4C22⋊D9711yescharacteristic
184C22×C4C22×D9712yes—
185C22×C4C22×D9711yescharacteristic
186D8—1948no—
187M16—1948no—
188C2×C8—974no—
189C2×C8—972no—
190C2×C8—972no—
191C2×C8—1942no—
192C4⋊C4—21no—
193C4⋊C4—972no—
194C4⋊C4—1941no—
195C4⋊C4C22×D9712yes—
196C22⋊C4—24no—
197C22⋊C4—1944no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—3882no—
206C23—3882no—
207C23—3882no—
208C23—1942no—
209C23C2×C22⋊D9712yes—
210C23C2×C22⋊D9711yescharacteristic
211D4—22no—
212D4—22no—
213D4—48no—
214D4—42no—
215D4—3888no—
216D4—3882no—
217D4—1942no—
218D4—1942no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—3882no—
226C2×C4—1942no—
227C2×C4D4×D9712yes—
228C2×C4C2×C22⋊D9712yes—
229C2×C4C2×C22⋊D9711yescharacteristic
230C2×C4#1309551012yes—
231C2×C4C23×D9711yescharacteristic
232C8—974no—
233C8—1942no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22—7764no—
243C22—3882no—
244C22—3882no—
245C22—3882no—
246C22#1569078212yes—
247C22#1569083512yes—
248C22#1569083612yes—
249C22C22×C22⋊D9711yescharacteristic
250C4—22no—
251C4—42no—
252C4—3882no—
253C4#1569078212yes—
254C4C2×D4×D9711yescharacteristic
255C4C2×#1309551011yescharacteristic
256C2—42no—
257C2—82no—
258C2—7762no—
259C2#2026098614yesdirect product factor
260C2C2×#1569078211yescharacteristic
261C2C2×#1569083511yescharacteristic
262C2C2×#1569083611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#121×C97C2
socleC23×C97C2×D4
hypercenterC2×#121D97
hypocenterC97C2×#469
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13095496C2semidirect
2. C22×#13095495C2semidirect
3. C2×#15690784C2semidirect
4. #20260986C2direct (N × H)
5. C2×#15690741C2semidirect
6. C2×#15690731C2semidirect
7. C2×#15690730C2semidirect
8. C2×#121×C97C2semidirect
12. C2×#13095496C22semidirect
14. C2×#13095495C22semidirect
18. #15690784C22semidirect
21. #15690741C22semidirect
22. #15690731C22semidirect
23. #15690730C22semidirect
28. #121×C97C22semidirect
42. C2×#11715895D4semidirect
101. #11715895C2×D4semidirect
126. C2×C97#469semidirect
132. C97C2×#469semidirect
136. C2×#121D97semidirect
155. #121C2×D97semidirect
259. C2#20260986direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954962
C22×#130954952
C2×#156907842
#202609862
C2×#156907412
C2×#156907312
C2×#156907302
C2×#121×C972
C2×#46997