Finite Group Databasebeta

← C22×#66985 #13083349

Subgroups — C22×#66985 #13083349

Order 1408.

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
1C23×D4×C11C211yesmaximal, characteristic, complemented
2C22×#4339C211yesmaximal, characteristic, complemented
3C22×#4324C212yesmaximal, complemented
4C2×#66985C2124yesmaximal, direct product factor
5C22×#4323C212yesmaximal, complemented
6C22×#4318C211yesmaximal, characteristic, complemented
7C25×C11—21no—
8C22×D4×C11—21no—
9C22×D4×C11—26no—
10C22×D4×C11C2212yes—
11C22×D4×C11C22112yescomplemented
12C23×C4×C11C2211yescharacteristic
13C22×C22⋊D11—21no—
14C2×#4339—26no—
15C2×#4339C22112yescomplemented
16C22×D44C2211yescharacteristic, complemented
17C22×C4×D11—21no—
18C22×Dic22C2211yescharacteristic, complemented
19C2×#4324C22124yescomplemented
20#66985C22164yesdirect product factor
21C2×#4323C22124yescomplemented
22C2×#4318C22112yescomplemented
23C22×C11⋊C8C2212yescomplemented
24C24×C11—22no—
25C24×C11—212no—
26C24×C11—48no—
27C24×C11D411yescharacteristic
28C23×D11—21no—
29C2×D4×C11—26no—
30C2×D4×C11—212no—
31C2×D4×C11—224no—
32C2×D4×C11C23112yes—
33C2×D4×C11C23116yescomplemented
34C22×C4×C11D411yescharacteristic
35C22×C4×C11D416yes—
36C22×C4×C11C2311yescharacteristic
37C22×C4×C11C2316yes—
38C2×C22⋊D11—212no—
39C22×Dic11—21no—
40#4339—224no—
41#4339C23116yescomplemented
42C2×D44—23no—
43C2×D44C2316yescomplemented
44C2×C4×D11—212no—
45C2×Dic22—23no—
46C2×Dic22C2316yescomplemented
47#4324C23132yescomplemented
48#4323C23132yescomplemented
49#4318C23116yescomplemented
50C2×C11⋊C8C23112yescomplemented
51C22×AGL(1,ℤ/8)—111nomaximal, Sylow
52C23×C11—24no—
53C23×C11—212no—
54C23×C11—216no—
55C23×C11—448no—
56C23×C11—48no—
57C23×C11C2×D416yes—
58C23×C11C2×D411yescharacteristic
59C22×D11—26no—
60C22×D11—44no—
61D4×C11—212no—
62D4×C11—216no—
63D4×C11C2418yes—
64C2×C4×C11C2×D413yes—
65C2×C4×C11C2×D416yes—
66C2×C4×C11C2×D4112yes—
67C2×C4×C11C2413yes—
68C2×C4×C11C2414yes—
69C22⋊D11—216no—
70C2×Dic11—26no—
71D44—26no—
72D44C2414yescomplemented
73C4×D11—216no—
74Dic22—26no—
75Dic22C2414yescomplemented
76C11⋊C8C2418yescomplemented
77C22×#41—111no—
78C23×D4D1111yescharacteristic, complemented
79C2×AGL(1,ℤ/8)—1124no—
80C22×SD16—112no—
81C22×D8—112no—
82C22×M16—111no—
83C22×C11—212no—
84C22×C11—28no—
85C22×C11—432no—
86C22×C11—424no—
87C22×C11AGL(1,ℤ/8)14yescomplemented
88C22×C11C22×D414yes—
89C22×C11C22×D413yes—
90C2×D11—24no—
91C2×D11—412no—
92C4×C11C22×D413yes—
93C4×C11C22×D414yes—
94C4×C11C2511yescharacteristic
95Dic11—24no—
96C25—21no—
97C2×#41—1112no—
98C2×#41—226no—
99C22×Q8—111no—
100C22×D4—26no—
101C22×D4—21no—
102C22×D4—111no—
103C22×D4—221no—
104C22×D4C2×D1112yes—
105C22×D4C2×D11112yescomplemented
106C23×C4—221no—
107C23×C4C2×D1111yescharacteristic
108AGL(1,ℤ/8)—1164no—
109C2×SD16—1124no—
110C2×D8—1124no—
111C2×M16—1112no—
112C22×C8—112no—
113C2×C11—24no—
114C2×C11—48no—
115C2×C11C2×AGL(1,ℤ/8)16yescomplemented
116C2×C11C23×D411yescharacteristic
117D11—44no—
118C24—212no—
119C24—22no—
120C24—48no—
121C24—221no—
122C24C22⋊D1111yescharacteristic
123#41—1116no—
124#41—2224no—
125C2×Q8—116no—
126C2×Q8—223no—
127C2×D4—26no—
128C2×D4—212no—
129C2×D4—224no—
130C2×D4—116no—
131C2×D4—2212no—
132C2×D4—223no—
133C2×D4C22×D11112yes—
134C2×D4C22×D11116yescomplemented
135C22×C4—221no—
136C22×C4—2212no—
137C22×C4C22⋊D1116yes—
138C22×C4C22⋊D1111yescharacteristic
139C22×C4C22×D1116yes—
140C22×C4C22×D1111yescharacteristic
141SD16—1132no—
142D8—1132no—
143M16—1116no—
144C2×C8—1112no—
145C11C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—226no—
152C23—444no—
153C23C2×C22⋊D1116yes—
154C23C2×C22⋊D1111yescharacteristic
155Q8—114no—
156Q8—226no—
157D4—212no—
158D4—216no—
159D4—114no—
160D4—2216no—
161D4—226no—
162D4C23×D1118yes—
163C2×C4—226no—
164C2×C4—2216no—
165C2×C4C2×C22⋊D1116yes—
166C2×C4C2×C22⋊D1113yes—
167C2×C4C2×C22⋊D11112yes—
168C2×C4C23×D1113yes—
169C2×C4C23×D1114yes—
170C8—118no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—224no—
176C22—4412no—
177C22#6698514yesdirect product factor
178C22C22×C22⋊D1114yes—
179C22C22×C22⋊D1113yes—
180C4—224no—
181C4C22×C22⋊D1114yes—
182C4C22×C22⋊D1113yes—
183C4C24×D1111yescharacteristic
184C2—24no—
185C2—48no—
186C2—444no—
187C2C2×#6698516yesdirect product factor
188C2C23×C22⋊D1111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D11
commutator, cosocleC4×C11C25
FrattiniC4C24×D11
FittingC23×D4×C11C2
socleC23×C11C2×D4
hypercenterC23×D4D11
hypocenterC11C22×AGL(1,ℤ/8)
LCSG ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×D4×C11C2semidirect
2. C22×#4339C2semidirect
3. C22×#4324C2semidirect
4. C2×#66985C2direct (N × H)
5. C22×#4323C2semidirect
6. C22×#4318C2semidirect
11. C22×D4×C11C22semidirect
15. C2×#4339C22semidirect
16. C22×D44C22semidirect
18. C22×Dic22C22semidirect
19. C2×#4324C22semidirect
20. #66985C22direct (N × H)
21. C2×#4323C22semidirect
22. C2×#4318C22semidirect
23. C22×C11⋊C8C22semidirect
33. C2×D4×C11C23semidirect
41. #4339C23semidirect
43. C2×D44C23semidirect
46. C2×Dic22C23semidirect
47. #4324C23semidirect
48. #4323C23semidirect
49. #4318C23semidirect
50. C2×C11⋊C8C23semidirect
72. D44C24semidirect
75. Dic22C24semidirect
76. C11⋊C8C24semidirect
78. C23×D4D11semidirect
87. C22×C11AGL(1,ℤ/8)semidirect
105. C22×D4C2×D11semidirect
115. C2×C11C2×AGL(1,ℤ/8)semidirect
134. C2×D4C22×D11semidirect
145. C11C22×AGL(1,ℤ/8)semidirect
177. C22#66985direct (N × H)
187. C2C2×#66985direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4×C112
C22×#43392
C22×#43242
C2×#669852
C22×#43232
C22×#43182
C22×AGL(1,ℤ/8)11