Finite Group Databasebeta

← C2×D4×#10621728 #72259170

Subgroups — C2×D4×#10621728 #72259170

Order 10816.

3408 conjugacy classes of subgroups; only the 422 normal subgroups are shown.

normal subgroups only

This group has 3408 conjugacy classes of subgroups — too many to draw. Only its 422 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×#72229039C212yesmaximal, complemented
2C2×C4×#10621728C211yesmaximal, characteristic, complemented
3C2×#72229144C211yesmaximal, characteristic, complemented
4C2×D4×C13⋊D13C211yesmaximal, characteristic
5C2×C22⋊(C13⋊D13)C412yescomplemented
6C2×C4⋊(C13⋊D13)C411yescharacteristic, complemented
7C2×D4×C132C411yescharacteristic, complemented
8D4×#10621728C218yesmaximal, direct product factor
9C23×#10621728C212yesmaximal, complemented
10#72229144C2214yescomplemented
11C2×C4×C13⋊D13C2211yescharacteristic
12C4×#10621728C2214yescomplemented
13#72229039C2218yescomplemented
14D4×C13⋊D13C414yescomplemented
15D4×C13⋊D13C2214yes—
16C23×C13⋊D13C2212yes—
17C22×#10621728—28no—
18C22×#10621728C2212yes—
19C22×#10621728C2211yescharacteristic
20C22×#10621728C2211yescharacteristic
21C22×#10621728C2218yescomplemented
22C23×C132C2×C412yescomplemented
23C22×C13⋊D13—24no—
24C22×C13⋊D13—24no—
25C22×C13⋊D13C2×C414yescomplemented
26C22×C13⋊D13C2×C412yescomplemented
27C22×C13⋊D13C2314yes—
28C22×C13⋊D13C2311yescharacteristic
29C2×#10621728—24no—
30C2×#10621728—22no—
31C2×#10621728—216no—
32C2×#10621728D414yesdirect product factor
33C2×#10621728C2314yes—
34C2×#10621728C2312yes—
35C2×#10621728C2312yes—
36D4×C132C2×C414yescomplemented
37C2×C4×C132C2×C411yescharacteristic, complemented
38C22⋊(C13⋊D13)C2×C418yescomplemented
39C2×#10621725C2×C411yescharacteristic, complemented
40C4⋊(C13⋊D13)C2×C414yescomplemented
41C4×C13⋊D13C2×C412yescomplemented
42C4×C13⋊D13C2312yes—
43C2×D4×#261—1314nomaximal
44C22×C132—24no—
45C22×C132C22×C411yescharacteristic
46C22×C132C22×C414yescomplemented
47C2×C13⋊D13—24no—
48C2×C13⋊D13—28no—
49C2×C13⋊D13—24no—
50C2×C13⋊D13—24no—
51C2×C13⋊D13C22×C411yescharacteristic
52C2×C13⋊D13C22×C414yescomplemented
53C2×C13⋊D13C2×D412yes—
54C2×C13⋊D13#4112yes—
55C2×C13⋊D13C2411yescharacteristic
56#10621728—24no—
57#10621728—22no—
58#10621728C2×D414yesdirect product factor
59C4×C132C22×C412yescomplemented
60#10621725C22×C412yescomplemented
61C23×#261—1328no—
62C2×D4×D13—1314no—
63C2×C22⋊#261—1328no—
64D4×#261—13112no—
65C2×C4⋊#261—1314no—
66C2×C4×#261—1314no—
67C2×C132—24no—
68C2×C132C4×D412yescomplemented
69C2×C132C23×C411yescharacteristic
70C13⋊D13—24no—
71C13⋊D13C4×D412yescomplemented
72C13⋊D13C22×D411yescharacteristic
73C13⋊D13C2×#4111yescharacteristic
74C23×D13—1328no—
75C22×#261—1314no—
76C22×#261—1314no—
77C22×#261—1328no—
78C22×#261—13112no—
79C22×#261—26112no—
80C2×D4×C13#261114yescomplemented
81C2×C22⋊D13—1328no—
82D4×D13—1356no—
83D4×D13—1356no—
84C2×D52—1314no—
85C2×C4×D13—1314no—
86C22⋊#261—13112no—
87C4⋊#261—1356no—
88C4×#261—1356no—
89C132C2×C4×D411yescharacteristic, complemented, Sylow
90C23×C13C2×#261128yescomplemented
91C22×D13—1314no—
92C22×D13—1328no—
93C22×D13—1356no—
94C22×D13—1356no—
95C22×D13—2656no—
96C22×D13—2656no—
97C2×#261—1356no—
98C2×#261—1328no—
99C2×#261—1328no—
100C2×#261—1356no—
101C2×#261—26224no—
102C2×#261—2628no—
103C2×#261—2656no—
104D4×C13C2×#261156yescomplemented
105C2×C4×C13C2×#261114yescomplemented
106C22⋊D13—13112no—
107C2×Dic13—1314no—
108D52—1356no—
109C4×D13—1328no—
110C4×D13—1328no—
111C2×C4×D4—1691noSylow
112C22×C13—256no—
113C22×C13C22×#261114yes—
114C22×C13C22×#261156yescomplemented
115C2×D13—1328no—
116C2×D13—1328no—
117C2×D13—1314no—
118C2×D13—1314no—
119C2×D13—1356no—
120C2×D13—2656no—
121C2×D13—2656no—
122C2×D13—26112no—
123C2×D13—2656no—
124#261—1356no—
125#261—2628no—
126#261—2656no—
127C4×C13C22×#261128yescomplemented
128Dic13—1328no—
129C22×D4—1691no—
130C23×C4—1692no—
131C4×D4—1698no—
132C2×C4⋊C4—1691no—
133C2×C22⋊C4—1692no—
134C2×C42—1691no—
135C2×C13—256no—
136C2×C13D4×#261128yescomplemented
137C2×C13C23×#261114yes—
138D13—1314no—
139D13—1314no—
140D13—1328no—
141D13—2656no—
142C24—1692no—
143C2×D4—1692no—
144C2×D4—1694no—
145C2×D4—1694no—
146C2×D4—1691no—
147C2×D4#1062172811yescharacteristic, direct product factor
148C22×C4—3388no—
149C22×C4—1691no—
150C22×C4—1692no—
151C22×C4—1698no—
152C22×C4—1691no—
153C22×C4—1691no—
154C4⋊C4—1694no—
155C22⋊C4—1698no—
156C42—1694no—
157C13C2×D4×#261114yescomplemented
158C23—3384no—
159C23—3384no—
160C23—1692no—
161C23—1694no—
162C23—1694no—
163C23—1691no—
164C23C2×#1062172812yescomplemented
165D4—1698no—
166D4—1694no—
167D4C2×#1062172814yesdirect product factor
168C2×C4—3384no—
169C2×C4—3382no—
170C2×C4—33816no—
171C2×C4—1692no—
172C2×C4—1694no—
173C2×C4—1692no—
174C2×C4—1694no—
175C2×C4—1691no—
176C2×C4—1692no—
177C2×C4—1692no—
178C2×C4C2×#1062172811yescharacteristic, complemented
179C22—24no—
180C22—3384no—
181C22—3384no—
182C22—3388no—
183C22—3384no—
184C22—1694no—
185C22—1691no—
186C22—1691no—
187C22—1692no—
188C22—1692no—
189C22C22×#1062172811yescharacteristic
190C22C22×#1062172814yescomplemented
191C4—3384no—
192C4—3382no—
193C4—1694no—
194C4—1692no—
195C4C22×#1062172812yescomplemented
196C2—24no—
197C2—3384no—
198C2—1691no—
199C2—1691no—
200C2—1692no—
201C2C23×#1062172811yescharacteristic
202C2D4×#1062172812yesdirect product factor
subgroupG/H
centerC22C22×#10621728
commutatorC2×C132C23×C4
FrattiniC2C23×#10621728
FittingC2×D4×C132C4
socleC22×C132C22×C4
prime radicalC2×D4×C13⋊D13C2
cosocleC2×C13⋊D13C24
hypercenterC2×D4#10621728
hypocenterC132C2×C4×D4
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#72229039C2semidirect
2. C2×C4×#10621728C2semidirect
3. C2×#72229144C2semidirect
5. C2×C22⋊(C13⋊D13)C4semidirect
6. C2×C4⋊(C13⋊D13)C4semidirect
7. C2×D4×C132C4semidirect
8. D4×#10621728C2direct (N × H)
9. C23×#10621728C2semidirect
10. #72229144C22semidirect
12. C4×#10621728C22semidirect
13. #72229039C22semidirect
14. D4×C13⋊D13C4semidirect
21. C22×#10621728C22semidirect
22. C23×C132C2×C4semidirect
25. C22×C13⋊D13C2×C4semidirect
26. C22×C13⋊D13C2×C4semidirect
32. C2×#10621728D4direct (N × H)
36. D4×C132C2×C4semidirect
37. C2×C4×C132C2×C4semidirect
38. C22⋊(C13⋊D13)C2×C4semidirect
39. C2×#10621725C2×C4semidirect
40. C4⋊(C13⋊D13)C2×C4semidirect
41. C4×C13⋊D13C2×C4semidirect
46. C22×C132C22×C4semidirect
52. C2×C13⋊D13C22×C4semidirect
58. #10621728C2×D4direct (N × H)
59. C4×C132C22×C4semidirect
60. #10621725C22×C4semidirect
68. C2×C132C4×D4semidirect
71. C13⋊D13C4×D4semidirect
80. C2×D4×C13#261semidirect
89. C132C2×C4×D4semidirect
90. C23×C13C2×#261semidirect
104. D4×C13C2×#261semidirect
105. C2×C4×C13C2×#261semidirect
114. C22×C13C22×#261semidirect
127. C4×C13C22×#261semidirect
136. C2×C13D4×#261semidirect
147. C2×D4#10621728direct (N × H)
157. C13C2×D4×#261semidirect
164. C23C2×#10621728semidirect
167. D4C2×#10621728direct (N × H)
178. C2×C4C2×#10621728semidirect
190. C22C22×#10621728semidirect
195. C4C22×#10621728semidirect
202. C2D4×#10621728direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722290392
C2×C4×#106217282
C2×#722291442
C2×D4×C13⋊D132
D4×#106217282
C23×#106217282
C2×D4×#26113