Finite Group Databasebeta

← C22×#11758485 #16860446

Subgroups — C22×#11758485 #16860446

Order 3968.

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×#92495C211yesmaximal, characteristic, complemented
2C23×D124C211yesmaximal, characteristic, complemented
3C2×#11758485C2124yesmaximal, direct product factor
4C22×#92470C212yesmaximal, complemented
5C22×D248C212yesmaximal, complemented
6C22×M16×C31C211yesmaximal, characteristic, complemented
7C24×D31—21no—
8C23×C4×C31C2211yescharacteristic
9C22×C22⋊D31—21no—
10C2×#92495—26no—
11C2×#92495C22112yescomplemented
12C22×D124—26no—
13C22×D124—21no—
14C22×D124C2212yes—
15C22×D124C22112yescomplemented
16C22×D124C2211yescharacteristic, complemented
17C22×C4×D31—21no—
18C22×Dic62C2211yescharacteristic, complemented
19C2×M16×C31C22112yescomplemented
20C22×C8×C31C2212yescomplemented
21#11758485C22164yesdirect product factor
22C2×D248C22124yescomplemented
23C2×#92470C22124yescomplemented
24C24×C31D411yescharacteristic
25C23×D31—21no—
26C23×D31—22no—
27C23×D31—212no—
28C23×D31—48no—
29C22×C4×C31D411yescharacteristic
30C22×C4×C31D416yes—
31C22×C4×C31C2311yescharacteristic
32C22×C4×C31C2316yes—
33C2×C22⋊D31—212no—
34C22×Dic31—21no—
35#92495—224no—
36#92495C23116yescomplemented
37C2×D124—26no—
38C2×D124—224no—
39C2×D124—212no—
40C2×D124—23no—
41C2×D124C23112yes—
42C2×D124C23116yescomplemented
43C2×D124C2316yescomplemented
44C2×C4×D31—212no—
45C2×Dic62—23no—
46C2×Dic62C2316yescomplemented
47M16×C31C23116yescomplemented
48C2×C8×C31C23112yescomplemented
49D248C23132yescomplemented
50#92470C23132yescomplemented
51C23×C31—24no—
52C23×C31C2×D416yes—
53C23×C31C2×D411yescharacteristic
54C22×D31—26no—
55C22×D31—212no—
56C22×D31—216no—
57C22×D31—448no—
58C22×D31—44no—
59C22×D31—48no—
60C2×C4×C31C2×D413yes—
61C2×C4×C31C2×D416yes—
62C2×C4×C31C2×D4112yes—
63C2×C4×C31C2413yes—
64C2×C4×C31C2414yes—
65C22⋊D31—216no—
66C2×Dic31—26no—
67D124—212no—
68D124—216no—
69D124—26no—
70D124C2418yes—
71D124C2414yescomplemented
72C4×D31—216no—
73Dic62—26no—
74Dic62C2414yescomplemented
75C8×C31C2418yescomplemented
76C22×AGL(1,ℤ/8)—311nomaximal, Sylow
77C22×C31—212no—
78C22×C31AGL(1,ℤ/8)14yescomplemented
79C22×C31C22×D414yes—
80C22×C31C22×D413yes—
81C2×D31—24no—
82C2×D31—28no—
83C2×D31—432no—
84C2×D31—412no—
85C2×D31—424no—
86C4×C31C22×D413yes—
87C4×C31C22×D414yes—
88C4×C31C2511yescharacteristic
89Dic31—24no—
90C22×#41—311no—
91C23×D4—311no—
92C2×AGL(1,ℤ/8)—3124no—
93C22×SD16—312no—
94C22×D8—312no—
95C22×M16D3111yescharacteristic, complemented
96C2×C31—24no—
97C2×C31C2×AGL(1,ℤ/8)16yescomplemented
98C2×C31C23×D411yescharacteristic
99D31—44no—
100D31—48no—
101C25—621no—
102C2×#41—3112no—
103C2×#41—626no—
104C22×Q8—311no—
105C22×D4—311no—
106C22×D4—312no—
107C22×D4—3112no—
108C22×D4—621no—
109C22×D4—626no—
110C22×D4—621no—
111C23×C4—621no—
112C23×C4C2×D3111yescharacteristic
113AGL(1,ℤ/8)—3164no—
114C2×SD16—3124no—
115C2×D8—3124no—
116C2×M16C2×D31112yescomplemented
117C22×C8C2×D3112yescomplemented
118C31C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
119C24—6212no—
120C24—621no—
121C24—622no—
122C24—1248no—
123C24D12411yescharacteristic
124#41—3116no—
125#41—6224no—
126C2×Q8—316no—
127C2×Q8—623no—
128C2×D4—3112no—
129C2×D4—316no—
130C2×D4—3116no—
131C2×D4—6212no—
132C2×D4—6212no—
133C2×D4—626no—
134C2×D4—6224no—
135C2×D4—623no—
136C22×C4—621no—
137C22×C4—6212no—
138C22×C4D12416yes—
139C22×C4D12411yescharacteristic
140C22×C4C22×D3116yes—
141C22×C4C22×D3111yescharacteristic
142SD16—3132no—
143D8—3132no—
144M16C22×D31116yescomplemented
145C2×C8C22×D31112yescomplemented
146C23—24no—
147C23—626no—
148C23—6216no—
149C23—6212no—
150C23—12448no—
151C23—1244no—
152C23—1248no—
153C23C2×D12416yes—
154C23C2×D12411yescharacteristic
155Q8—314no—
156Q8—626no—
157D4—318no—
158D4—314no—
159D4—6216no—
160D4—6216no—
161D4—6212no—
162D4—626no—
163C2×C4—626no—
164C2×C4—6216no—
165C2×C4C2×D12416yes—
166C2×C4C2×D12413yes—
167C2×C4C2×D124112yes—
168C2×C4C23×D3113yes—
169C2×C4C23×D3114yes—
170C8C23×D3118yescomplemented
171C22—212no—
172C22—624no—
173C22—628no—
174C22—12432no—
175C22—12424no—
176C22—12412no—
177C22#1175848514yesdirect product factor
178C22C22×D12414yes—
179C22C22×D12413yes—
180C4—624no—
181C4C22×D12414yes—
182C4C22×D12413yes—
183C4C24×D3111yescharacteristic
184C2—24no—
185C2—1248no—
186C2—1244no—
187C2C2×#1175848516yesdirect product factor
188C2C23×D12411yescharacteristic
subgroupG/H
centerC23C2×D124
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC22×M16×C31C2
socleC23×C31C2×D4
hypercenterC22×M16D31
hypocenterC31C22×AGL(1,ℤ/8)
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C23×D124C2semidirect
3. C2×#11758485C2direct (N × H)
4. C22×#92470C2semidirect
5. C22×D248C2semidirect
6. C22×M16×C31C2semidirect
11. C2×#92495C22semidirect
15. C22×D124C22semidirect
16. C22×D124C22semidirect
18. C22×Dic62C22semidirect
19. C2×M16×C31C22semidirect
20. C22×C8×C31C22semidirect
21. #11758485C22direct (N × H)
22. C2×D248C22semidirect
23. C2×#92470C22semidirect
36. #92495C23semidirect
42. C2×D124C23semidirect
43. C2×D124C23semidirect
46. C2×Dic62C23semidirect
47. M16×C31C23semidirect
48. C2×C8×C31C23semidirect
49. D248C23semidirect
50. #92470C23semidirect
71. D124C24semidirect
74. Dic62C24semidirect
75. C8×C31C24semidirect
78. C22×C31AGL(1,ℤ/8)semidirect
95. C22×M16D31semidirect
97. C2×C31C2×AGL(1,ℤ/8)semidirect
116. C2×M16C2×D31semidirect
117. C22×C8C2×D31semidirect
118. C31C22×AGL(1,ℤ/8)semidirect
144. M16C22×D31semidirect
145. C2×C8C22×D31semidirect
170. C8C23×D31semidirect
177. C22#11758485direct (N × H)
187. C2C2×#11758485direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#924952
C23×D1242
C2×#117584852
C22×#924702
C22×D2482
C22×M16×C312
C22×AGL(1,ℤ/8)31