Finite Group Databasebeta

← C22×#11758488 #16860442

Subgroups — C22×#11758488 #16860442

Order 3968.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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×#92497C211yesmaximal, characteristic, complemented
2C22×D4×D31C211yesmaximal, characteristic, complemented
3C2×#11758488C2124yesmaximal, direct product factor
4C22×#92470C211yesmaximal, characteristic, complemented
5C22×#92480C211yesmaximal, characteristic, complemented
6C22×#92479C211yesmaximal, characteristic, complemented
7C22×#92469C211yesmaximal, characteristic, complemented
8C22×D8×C31C211yesmaximal, characteristic, complemented
9C24×D31—21no—
10C22×D4×C31C2211yescharacteristic
11C22×D4×C31C2211yescharacteristic, complemented
12C22×C22⋊D31—21no—
13C22×C22⋊D31—21no—
14C23×Dic31—21no—
15C2×#92497—26no—
16C2×#92497C22112yescomplemented
17C2×D4×D31—26no—
18C2×D4×D31C22112yescomplemented
19C22×D124C2211yescharacteristic
20C22×C4×D31C2211yescharacteristic
21C22×Dic62C2211yescharacteristic, complemented
22C2×D8×C31C22112yescomplemented
23C22×C8×C31C2211yescharacteristic, complemented
24C2×#92480C22112yescomplemented
25C2×#92479C22112yescomplemented
26C22×C31⋊C8C2211yescharacteristic, complemented
27#11758488C22164yesdirect product factor
28C2×#92470C22112yescomplemented
29C2×#92469C22112yescomplemented
30C24×C31—21no—
31C24×C31—21no—
32C23×D31—21no—
33C23×D31—212no—
34C23×D31—48no—
35C23×D31D411yescharacteristic
36C2×D4×C31—23no—
37C2×D4×C31—23no—
38C2×D4×C31C2316yes—
39C2×D4×C31C2316yescomplemented
40C22×C4×C31C2311yescharacteristic
41C2×C22⋊D31—212no—
42C2×C22⋊D31—212no—
43C22×Dic31—21no—
44C22×Dic31—212no—
45C22×Dic31D411yescharacteristic
46#92497—224no—
47#92497C23116yescomplemented
48D4×D31—224no—
49D4×D31C23116yescomplemented
50C2×D124—23no—
51C2×D124C2316yes—
52C2×C4×D31D416yes—
53C2×C4×D31C2316yes—
54C2×Dic62—23no—
55C2×Dic62C2316yescomplemented
56D8×C31C23116yescomplemented
57C2×C8×C31C2316yescomplemented
58#92480C23116yescomplemented
59#92479C23116yescomplemented
60C2×C31⋊C8C2316yescomplemented
61#92470C23116yescomplemented
62#92469C23116yescomplemented
63C23×C31—26no—
64C23×C31—26no—
65C23×C31—44no—
66C23×C31—44no—
67C23×C31C2×D411yescharacteristic
68C22×D31—24no—
69C22×D31—26no—
70C22×D31—216no—
71C22×D31—448no—
72C22×D31—44no—
73C22×D31C2×D416yes—
74D4×C31—26no—
75D4×C31—26no—
76D4×C31C2414yes—
77D4×C31C2414yescomplemented
78C2×C4×C31C2×D413yes—
79C2×C4×C31C2413yes—
80C22⋊D31—216no—
81C22⋊D31—216no—
82C2×Dic31—26no—
83C2×Dic31—216no—
84C2×Dic31C2×D416yes—
85D124—26no—
86D124C2414yes—
87C4×D31C2×D4112yes—
88C4×D31C2414yes—
89Dic62—26no—
90Dic62C2414yescomplemented
91C8×C31C2414yescomplemented
92C31⋊C8C2414yescomplemented
93C22×AGL(1,ℤ/8)—311nomaximal, Sylow
94C22×C31—24no—
95C22×C31—24no—
96C22×C31—412no—
97C22×C31—412no—
98C22×C31AGL(1,ℤ/8)14yescomplemented
99C22×C31C22×D413yes—
100C2×D31—212no—
101C2×D31—24no—
102C2×D31—432no—
103C2×D31—412no—
104C2×D31C22×D414yes—
105C4×C31C22×D413yes—
106C4×C31C2511yescharacteristic
107Dic31—24no—
108Dic31C22×D414yes—
109C22×#41—311no—
110C23×D4—311no—
111C2×AGL(1,ℤ/8)—3124no—
112C22×SD16—311no—
113C22×SD16—311no—
114C22×D8—311no—
115C22×D8D3111yescharacteristic, complemented
116C22×M16—311no—
117C2×C31—44no—
118C2×C31—44no—
119C2×C31C2×AGL(1,ℤ/8)16yescomplemented
120C2×C31C23×D411yescharacteristic
121D31—24no—
122D31—44no—
123C25—621no—
124C2×#41—3112no—
125C2×#41—626no—
126C22×Q8—311no—
127C22×D4—3112no—
128C22×D4—311no—
129C22×D4—626no—
130C22×D4—621no—
131C22×D4—621no—
132C22×D4C2×D3111yescharacteristic
133C22×D4C2×D3111yescharacteristic, complemented
134C23×C4—311no—
135C23×C4—621no—
136AGL(1,ℤ/8)—3164no—
137C2×SD16—3112no—
138C2×SD16—3112no—
139C2×D8—3112no—
140C2×D8C2×D31112yescomplemented
141C2×M16—3112no—
142C22×C8—311no—
143C22×C8C2×D3111yescharacteristic, complemented
144C31C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
145C24—21no—
146C24—21no—
147C24—311no—
148C24—6212no—
149C24—621no—
150C24—1248no—
151#41—3116no—
152#41—6224no—
153C2×Q8—316no—
154C2×Q8—623no—
155C2×D4—23no—
156C2×D4—23no—
157C2×D4—3116no—
158C2×D4—316no—
159C2×D4—6224no—
160C2×D4—6212no—
161C2×D4—6212no—
162C2×D4—623no—
163C2×D4C22×D3116yes—
164C2×D4C22×D3116yescomplemented
165C22×C4—311no—
166C22×C4—316no—
167C22×C4—316no—
168C22×C4—6212no—
169C22×C4—621no—
170C22×C4C22×D3111yescharacteristic
171SD16—3116no—
172SD16—3116no—
173D8—3116no—
174D8C22×D31116yescomplemented
175M16—3116no—
176C2×C8—316no—
177C2×C8C22×D3116yescomplemented
178C23—26no—
179C23—26no—
180C23—44no—
181C23—44no—
182C23—316no—
183C23—6216no—
184C23—624no—
185C23—626no—
186C23—12448no—
187C23—1244no—
188C23D4×D3111yescharacteristic
189Q8—314no—
190Q8—626no—
191D4—26no—
192D4—26no—
193D4—314no—
194D4—6216no—
195D4—6216no—
196D4—626no—
197D4C23×D3114yes—
198D4C23×D3114yescomplemented
199C2×C4—316no—
200C2×C4—3112no—
201C2×C4—314no—
202C2×C4—6216no—
203C2×C4—626no—
204C2×C4D4×D3113yes—
205C2×C4C23×D3113yes—
206C8—314no—
207C8C23×D3114yescomplemented
208C22—24no—
209C22—24no—
210C22—412no—
211C22—412no—
212C22—314no—
213C22—6212no—
214C22—624no—
215C22—12432no—
216C22—12412no—
217C22#1175848814yesdirect product factor
218C22C2×D4×D3113yes—
219C4—314no—
220C4—624no—
221C4C2×D4×D3113yes—
222C4C24×D3111yescharacteristic
223C2—44no—
224C2—44no—
225C2—624no—
226C2—1244no—
227C2C2×#1175848816yesdirect product factor
228C2C22×D4×D3111yescharacteristic
subgroupG/H
centerC23D4×D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC22×D8×C31C2
socleC23×C31C2×D4
hypercenterC22×D8D31
hypocenterC31C22×AGL(1,ℤ/8)
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#92497C2semidirect
2. C22×D4×D31C2semidirect
3. C2×#11758488C2direct (N × H)
4. C22×#92470C2semidirect
5. C22×#92480C2semidirect
6. C22×#92479C2semidirect
7. C22×#92469C2semidirect
8. C22×D8×C31C2semidirect
11. C22×D4×C31C22semidirect
16. C2×#92497C22semidirect
18. C2×D4×D31C22semidirect
21. C22×Dic62C22semidirect
22. C2×D8×C31C22semidirect
23. C22×C8×C31C22semidirect
24. C2×#92480C22semidirect
25. C2×#92479C22semidirect
26. C22×C31⋊C8C22semidirect
27. #11758488C22direct (N × H)
28. C2×#92470C22semidirect
29. C2×#92469C22semidirect
39. C2×D4×C31C23semidirect
47. #92497C23semidirect
49. D4×D31C23semidirect
55. C2×Dic62C23semidirect
56. D8×C31C23semidirect
57. C2×C8×C31C23semidirect
58. #92480C23semidirect
59. #92479C23semidirect
60. C2×C31⋊C8C23semidirect
61. #92470C23semidirect
62. #92469C23semidirect
77. D4×C31C24semidirect
90. Dic62C24semidirect
91. C8×C31C24semidirect
92. C31⋊C8C24semidirect
98. C22×C31AGL(1,ℤ/8)semidirect
115. C22×D8D31semidirect
119. C2×C31C2×AGL(1,ℤ/8)semidirect
133. C22×D4C2×D31semidirect
140. C2×D8C2×D31semidirect
143. C22×C8C2×D31semidirect
144. C31C22×AGL(1,ℤ/8)semidirect
164. C2×D4C22×D31semidirect
174. D8C22×D31semidirect
177. C2×C8C22×D31semidirect
198. D4C23×D31semidirect
207. C8C23×D31semidirect
217. C22#11758488direct (N × H)
227. C2C2×#11758488direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#924972
C22×D4×D312
C2×#117584882
C22×#924702
C22×#924802
C22×#924792
C22×#924692
C22×D8×C312
C22×AGL(1,ℤ/8)31