Finite Group Databasebeta

← C22×#16813648 #28405307

Subgroups — C22×#16813648 #28405307

Order 14464.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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×#14268647C211yesmaximal, characteristic, complemented
2C2×#16813648C2124yesmaximal, direct product factor
3C22×#14268635C212yesmaximal, complemented
4C22×#14268632C212yesmaximal, complemented
5C22×#14268631C211yesmaximal, characteristic, complemented
6C23×D4×C113C211yesmaximal, characteristic, complemented
7C2×#14268647—26no—
8C2×#14268647C22112yescomplemented
9C22×Dic226C2211yescharacteristic, complemented
10C22×D452C2211yescharacteristic, complemented
11C22×C22⋊D113—21no—
12C22×C4×D113—21no—
13#16813648C22164yesdirect product factor
14C2×#14268635C22124yescomplemented
15C2×#14268632C22124yescomplemented
16C2×#14268631C22112yescomplemented
17C22×#11744751C2212yescomplemented
18C25×C113—21no—
19C22×D4×C113—21no—
20C22×D4×C113—26no—
21C22×D4×C113C2212yes—
22C22×D4×C113C22112yescomplemented
23C23×C4×C113C2211yescharacteristic
24C23×D113—21no—
25#14268647—224no—
26#14268647C23116yescomplemented
27C2×Dic226—23no—
28C2×Dic226C2316yescomplemented
29C2×D452—23no—
30C2×D452C2316yescomplemented
31C2×C22⋊D113—212no—
32C22×Dic113—21no—
33C2×C4×D113—212no—
34#14268635C23132yescomplemented
35#14268632C23132yescomplemented
36#14268631C23116yescomplemented
37C2×#11744751C23112yescomplemented
38C24×C113—22no—
39C24×C113—212no—
40C24×C113—48no—
41C24×C113D411yescharacteristic
42C2×D4×C113—26no—
43C2×D4×C113—212no—
44C2×D4×C113—224no—
45C2×D4×C113C23112yes—
46C2×D4×C113C23116yescomplemented
47C22×C4×C113D411yescharacteristic
48C22×C4×C113D416yes—
49C22×C4×C113C2311yescharacteristic
50C22×C4×C113C2316yes—
51C23×C113—24no—
52C23×C113—212no—
53C23×C113—216no—
54C23×C113—448no—
55C23×C113—48no—
56C23×C113C2×D416yes—
57C23×C113C2×D411yescharacteristic
58C22×D113—26no—
59C22×D113—44no—
60D4×C113—212no—
61D4×C113—216no—
62D4×C113C2418yes—
63C2×C4×C113C2×D416yes—
64C2×C4×C113C2×D413yes—
65C2×C4×C113C2×D4112yes—
66C2×C4×C113C2413yes—
67C2×C4×C113C2414yes—
68C22⋊D113—216no—
69C2×Dic113—26no—
70D452—26no—
71D452C2414yescomplemented
72C4×D113—216no—
73Dic226—26no—
74Dic226C2414yescomplemented
75#11744751C2418yescomplemented
76C22×C113—212no—
77C22×C113—28no—
78C22×C113—432no—
79C22×C113—424no—
80C22×C113AGL(1,ℤ/8)14yescomplemented
81C22×C113C22×D414yes—
82C22×C113C22×D413yes—
83C2×D113—24no—
84C2×D113—412no—
85C4×C113C22×D413yes—
86C4×C113C22×D414yes—
87C4×C113C2511yescharacteristic
88Dic113—24no—
89C2×C113—24no—
90C2×C113—48no—
91C2×C113C2×AGL(1,ℤ/8)16yescomplemented
92C2×C113C23×D411yescharacteristic
93D113—44no—
94C22×AGL(1,ℤ/8)—1131nomaximal, Sylow
95C113C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
96C22×#41—1131no—
97C23×D4D11311yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—11324no—
99C22×SD16—1132no—
100C22×D8—1132no—
101C22×M16—1131no—
102C25—21no—
103C2×#41—11312no—
104C2×#41—2266no—
105C22×Q8—1131no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—1131no—
109C22×D4—2261no—
110C22×D4C2×D11312yes—
111C22×D4C2×D113112yescomplemented
112C23×C4—2261no—
113C23×C4C2×D11311yescharacteristic
114AGL(1,ℤ/8)—11364no—
115C2×SD16—11324no—
116C2×D8—11324no—
117C2×M16—11312no—
118C22×C8—1132no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—2261no—
123C24C22⋊D11311yescharacteristic
124#41—11316no—
125#41—22624no—
126C2×Q8—1136no—
127C2×Q8—2263no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—1136no—
132C2×D4—22612no—
133C2×D4—2263no—
134C2×D4C22×D113112yes—
135C2×D4C22×D113116yescomplemented
136C22×C4—2261no—
137C22×C4—22612no—
138C22×C4C22⋊D11316yes—
139C22×C4C22⋊D11311yescharacteristic
140C22×C4C22×D11316yes—
141C22×C4C22×D11311yescharacteristic
142SD16—11332no—
143D8—11332no—
144M16—11316no—
145C2×C8—11312no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—4524no—
152C23—2266no—
153C23C2×C22⋊D11316yes—
154C23C2×C22⋊D11311yescharacteristic
155Q8—1134no—
156Q8—2266no—
157D4—212no—
158D4—216no—
159D4—1134no—
160D4—22616no—
161D4—2266no—
162D4C23×D11318yes—
163C2×C4—2266no—
164C2×C4—22616no—
165C2×C4C2×C22⋊D11316yes—
166C2×C4C2×C22⋊D11313yes—
167C2×C4C2×C22⋊D113112yes—
168C2×C4C23×D11313yes—
169C2×C4C23×D11314yes—
170C8—1138no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—45212no—
176C22—2264no—
177C22#1681364814yesdirect product factor
178C22C22×C22⋊D11314yes—
179C22C22×C22⋊D11313yes—
180C4—2264no—
181C4C22×C22⋊D11314yes—
182C4C22×C22⋊D11313yes—
183C4C24×D11311yescharacteristic
184C2—24no—
185C2—48no—
186C2—4524no—
187C2C2×#1681364816yesdirect product factor
188C2C23×C22⋊D11311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D113
commutator, cosocleC4×C113C25
FrattiniC4C24×D113
FittingC23×D4×C113C2
socleC23×C113C2×D4
hypercenterC23×D4D113
hypocenterC113C22×AGL(1,ℤ/8)
LCSG ▷ C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14268647C2semidirect
2. C2×#16813648C2direct (N × H)
3. C22×#14268635C2semidirect
4. C22×#14268632C2semidirect
5. C22×#14268631C2semidirect
6. C23×D4×C113C2semidirect
8. C2×#14268647C22semidirect
9. C22×Dic226C22semidirect
10. C22×D452C22semidirect
13. #16813648C22direct (N × H)
14. C2×#14268635C22semidirect
15. C2×#14268632C22semidirect
16. C2×#14268631C22semidirect
17. C22×#11744751C22semidirect
22. C22×D4×C113C22semidirect
26. #14268647C23semidirect
28. C2×Dic226C23semidirect
30. C2×D452C23semidirect
34. #14268635C23semidirect
35. #14268632C23semidirect
36. #14268631C23semidirect
37. C2×#11744751C23semidirect
46. C2×D4×C113C23semidirect
71. D452C24semidirect
74. Dic226C24semidirect
75. #11744751C24semidirect
80. C22×C113AGL(1,ℤ/8)semidirect
91. C2×C113C2×AGL(1,ℤ/8)semidirect
95. C113C22×AGL(1,ℤ/8)semidirect
97. C23×D4D113semidirect
111. C22×D4C2×D113semidirect
135. C2×D4C22×D113semidirect
177. C22#16813648direct (N × H)
187. C2C2×#16813648direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142686472
C2×#168136482
C22×#142686352
C22×#142686322
C22×#142686312
C23×D4×C1132
C22×AGL(1,ℤ/8)113