Finite Group Databasebeta

← C2×#20265286 #27179922

Subgroups — C2×#20265286 #27179922

Order 12928.

608 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13108059C211yesmaximal, characteristic
2C2×D4×Dic101C211yesmaximal, characteristic, complemented
3C2×#15692217C211yesmaximal, characteristic, complemented
4#20265286C218yesmaximal, direct product factor
5C2×#15692186C211yesmaximal, characteristic, complemented
6C2×#15692177C211yesmaximal, characteristic, complemented
7C2×#15692165C211yesmaximal, characteristic, complemented
8C2×#102×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic
10C23×Dic101—21no—
11C22×C4×D101—21no—
12C2×#13108059C418yescomplemented
13C2×#13108059C2214yes—
14C22×#11717579C2211yescharacteristic
15D4×Dic101—22no—
16D4×Dic101C2214yescomplemented
17#15692217—22no—
18#15692217C2214yescomplemented
19C2×C4⋊Dic101C2211yescharacteristic
20C2×C22⋊Dic101—21no—
21C2×#13108051—21no—
22C2×C4×Dic101C2211yescharacteristic
23#15692186C2214yescomplemented
24#15692177C2214yescomplemented
25#15692165C2214yescomplemented
26C22×D4×C101C2211yescharacteristic
27C22×C8×C101C2211yescharacteristic
28C2×C4⋊C4×C101C2211yescharacteristic
29#102×C101C2214yescomplemented
30C23×D101—21no—
31C2×D404—21no—
32C2×D404—22no—
33C2×D404C2×C414yescomplemented
34C2×D404C2312yes—
35C22×Dic101—21no—
36C22×Dic101—24no—
37C22×Dic101—44no—
38C22×Dic101D411yescharacteristic
39C22×Dic101D411yescharacteristic
40C2×C4×D101—24no—
41C2×C4×D101—44no—
42#13108059C2×C4116yescomplemented
43C2×#11717579C2×C414yescomplemented
44C2×#11717579C2312yes—
45C4⋊Dic101—21no—
46C4⋊Dic101C2312yes—
47C22⋊Dic101—24no—
48#13108051—24no—
49C4×Dic101D412yes—
50C4×Dic101C2312yes—
51C24×C101—21no—
52C2×D4×C101—21no—
53C2×D4×C101—22no—
54C2×D4×C101C2×C414yescomplemented
55C2×D4×C101C2312yes—
56C22×C4×C101—21no—
57C22×C4×C101C2311yescharacteristic
58C2×C8×C101—22no—
59C2×C8×C101C2312yes—
60C4⋊C4×C101—21no—
61C4⋊C4×C101C2312yes—
62C23×C101—24no—
63C23×C101—22no—
64C23×C101—42no—
65C23×C101—42no—
66C23×C101C2×D411yescharacteristic
67C22×D101—24no—
68C22×D101—22no—
69C22×D101—42no—
70C22×D101—42no—
71D4×C101—24no—
72D4×C101—22no—
73D4×C101C22×C414yescomplemented
74C2×C4×C101—22no—
75C2×C4×C101—42no—
76C2×C4×C101C22×C412yes—
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101#4112yes—
79C2×C4×C101C2411yescharacteristic
80C2×Dic101—22no—
81C2×Dic101—22no—
82C2×Dic101—22no—
83C2×Dic101—48no—
84C2×Dic101—42no—
85C2×Dic101C2×D412yes—
86C2×Dic101C2×D412yes—
87D404—24no—
88D404—22no—
89D404C22×C414yescomplemented
90C4×D101—48no—
91C8×C101—22no—
92#11717579C22×C414yescomplemented
93C22×C101—24no—
94C22×C101—42no—
95C22×C101—42no—
96C22×C101—48no—
97C22×C101C4×D412yes—
98C22×C101AGL(1,ℤ/8)12yes—
99C22×C101AGL(1,ℤ/8)12yes—
100C22×C101C22×D411yescharacteristic
101C2×D101—24no—
102C2×D101—42no—
103C2×D101—42no—
104C2×D101—48no—
105C4×C101—42no—
106C4×C101C4×D412yes—
107C4×C101C23×C411yescharacteristic
108C4×C101C2×#4111yescharacteristic
109Dic101—22no—
110Dic101—22no—
111Dic101—42no—
112C2×C101—44no—
113C2×C101#44214yescomplemented
114C2×C101C2×C4×D411yescharacteristic
115C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
116C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
117D101—44no—
118C2×#442—1011nomaximal, Sylow
119C101C2×#44211yescharacteristic, complemented, Sylow
120C22×D8—1011no—
121C2×C4×D4—1011no—
122C2×C4×D4—1011no—
123#442—1018no—
124C2×#106—1011no—
125C2×#102—1011no—
126C2×#102D10111yescharacteristic, complemented
127C2×#97—1011no—
128C22×D4—1011no—
129C22×D4C2×D10111yescharacteristic
130C23×C4—2021no—
131C23×C4—2021no—
132C2×D8—1018no—
133C2×D8—1014no—
134C22×C8—1011no—
135C22×C8C2×D10111yescharacteristic
136C4×D4—1014no—
137C4×D4—1014no—
138C4×D4—2022no—
139C4×D4—2022no—
140C2×C4⋊C4—1011no—
141C2×C4⋊C4C2×D10111yescharacteristic
142C2×C22⋊C4—2021no—
143C2×C22⋊C4—2021no—
144C2×C42—1011no—
145#106—1014no—
146#102—1014no—
147#102C2×D10114yescomplemented
148#97—1014no—
149C24—21no—
150C24—2021no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4—1014no—
154C2×D4—1012no—
155C2×D4—2021no—
156C2×D4—2022no—
157C2×D4C4×D10114yescomplemented
158C2×D4C22×D10112yes—
159C22×C4—21no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—4044no—
163C22×C4—4044no—
164C22×C4—2024no—
165C22×C4—2024no—
166C22×C4—2021no—
167C22×C4C22×D10111yescharacteristic
168D8—10116no—
169C2×C8—22no—
170C2×C8—1014no—
171C2×C8—1012no—
172C2×C8C22×D10112yes—
173C4⋊C4—21no—
174C4⋊C4—1012no—
175C4⋊C4—2021no—
176C4⋊C4C22×D10112yes—
177C22⋊C4—2024no—
178C22⋊C4—2024no—
179C42—1012no—
180C42—1012no—
181C23—22no—
182C23—24no—
183C23—42no—
184C23—42no—
185C23—4042no—
186C23—4042no—
187C23—2022no—
188C23—2024no—
189C23D4×D10111yescharacteristic
190D4—22no—
191D4—24no—
192D4—1014no—
193D4—2022no—
194D4—2024no—
195D4C2×C4×D10114yescomplemented
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—1012no—
199C2×C4—1012no—
200C2×C4—4048no—
201C2×C4—4042no—
202C2×C4—4048no—
203C2×C4—2022no—
204C2×C4—2022no—
205C2×C4—2022no—
206C2×C4C2×C4×D10112yes—
207C2×C4D4×D10111yescharacteristic
208C2×C4#1310807312yes—
209C2×C4C23×D10111yescharacteristic
210C8—22no—
211C8—1014no—
212C22—24no—
213C22—48no—
214C22—42no—
215C22—42no—
216C22—4048no—
217C22—4042no—
218C22—4042no—
219C22—2024no—
220C22#1569221512yes—
221C22#1569227912yes—
222C22#1569228212yes—
223C22C2×D4×D10111yescharacteristic
224C4—42no—
225C4—4042no—
226C4—2022no—
227C4—2022no—
228C4#1569221512yes—
229C4C22×C4×D10111yescharacteristic
230C4C2×#1310807311yescharacteristic
231C2—44no—
232C2—4044no—
233C2#2026528614yesdirect product factor
234C2C2×#1569221511yescharacteristic
235C2C2×#1569227911yescharacteristic
236C2C2×#1569228211yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
FittingC2×#102×C101C2
socleC23×C101C2×D4
prime radicalC22×#13108059C2
cosocleC2×C4×C101C24
hypercenterC2×#102D101
hypocenterC101C2×#442
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic101C2semidirect
3. C2×#15692217C2semidirect
4. #20265286C2direct (N × H)
5. C2×#15692186C2semidirect
6. C2×#15692177C2semidirect
7. C2×#15692165C2semidirect
8. C2×#102×C101C2semidirect
12. C2×#13108059C4semidirect
16. D4×Dic101C22semidirect
18. #15692217C22semidirect
23. #15692186C22semidirect
24. #15692177C22semidirect
25. #15692165C22semidirect
29. #102×C101C22semidirect
33. C2×D404C2×C4semidirect
42. #13108059C2×C4semidirect
43. C2×#11717579C2×C4semidirect
54. C2×D4×C101C2×C4semidirect
73. D4×C101C22×C4semidirect
89. D404C22×C4semidirect
92. #11717579C22×C4semidirect
113. C2×C101#442semidirect
119. C101C2×#442semidirect
126. C2×#102D101semidirect
147. #102C2×D101semidirect
157. C2×D4C4×D101semidirect
195. D4C2×C4×D101semidirect
233. C2#20265286direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080592
C2×D4×Dic1012
C2×#156922172
#202652862
C2×#156921862
C2×#156921772
C2×#156921652
C2×#102×C1012
C2×#442101