Finite Group Databasebeta

← C22×#11758491 #16860445

Subgroups — C22×#11758491 #16860445

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×#92499C211yesmaximal, characteristic, complemented
2C22×D4×D31C211yesmaximal, characteristic, complemented
3C2×#11758491C2124yesmaximal, direct product factor
4C22×#92481C211yesmaximal, characteristic, complemented
5C22×D248C211yesmaximal, characteristic, complemented
6C22×#92479C211yesmaximal, characteristic, complemented
7C22×#92469C211yesmaximal, characteristic, complemented
8C22×SD16×C31C211yesmaximal, characteristic, complemented
9C24×D31—21no—
10C22×Q8×C31C2211yescharacteristic, complemented
11C22×D4×C31C2211yescharacteristic
12C22×C22⋊D31—21no—
13C2×#92499—26no—
14C2×#92499C22112yescomplemented
15C2×D4×D31—26no—
16C2×D4×D31C22112yescomplemented
17C22×D124—21no—
18C22×D124C2211yescharacteristic
19C22×D124C2211yescharacteristic, complemented
20C22×C4×D31—21no—
21C22×C4×D31C2211yescharacteristic
22C2×SD16×C31C22112yescomplemented
23C22×C8×C31C2211yescharacteristic, complemented
24C2×#92481C22112yescomplemented
25C2×#92479C22112yescomplemented
26C22×C31⋊C8C2211yescharacteristic, complemented
27#11758491C22164yesdirect product factor
28C2×D248C22112yescomplemented
29C2×#92469C22112yescomplemented
30C24×C31—21no—
31C23×D31—21no—
32C23×D31—21no—
33C23×D31—212no—
34C23×D31—48no—
35C23×D31D411yescharacteristic
36C2×Q8×C31—23no—
37C2×Q8×C31C2316yescomplemented
38C2×D4×C31—23no—
39C2×D4×C31C2316yes—
40C22×C4×C31—21no—
41C22×C4×C31C2311yescharacteristic
42C2×C22⋊D31—212no—
43C22×Dic31D411yescharacteristic
44#92499—224no—
45#92499C23116yescomplemented
46D4×D31—224no—
47D4×D31C23116yescomplemented
48C2×D124—212no—
49C2×D124—23no—
50C2×D124—23no—
51C2×D124C2316yes—
52C2×D124C2316yescomplemented
53C2×C4×D31—212no—
54C2×C4×D31D416yes—
55C2×C4×D31C2316yes—
56SD16×C31C23116yescomplemented
57C2×C8×C31C2316yescomplemented
58#92481C23116yescomplemented
59#92479C23116yescomplemented
60C2×C31⋊C8C2316yescomplemented
61D248C23116yescomplemented
62#92469C23116yescomplemented
63C23×C31—26no—
64C23×C31—44no—
65C23×C31C2×D411yescharacteristic
66C22×D31—24no—
67C22×D31—26no—
68C22×D31—26no—
69C22×D31—216no—
70C22×D31—448no—
71C22×D31—44no—
72C22×D31—44no—
73C22×D31C2×D416yes—
74Q8×C31—26no—
75Q8×C31C2414yescomplemented
76D4×C31—26no—
77D4×C31C2414yes—
78C2×C4×C31—26no—
79C2×C4×C31C2×D413yes—
80C2×C4×C31C2413yes—
81C22⋊D31—216no—
82C2×Dic31C2×D416yes—
83D124—216no—
84D124—26no—
85D124—26no—
86D124C2414yes—
87D124C2414yescomplemented
88C4×D31—216no—
89C4×D31C2×D4112yes—
90C4×D31C2414yes—
91C8×C31C2414yescomplemented
92C31⋊C8C2414yescomplemented
93C22×AGL(1,ℤ/8)—311nomaximal, Sylow
94C22×C31—24no—
95C22×C31—412no—
96C22×C31AGL(1,ℤ/8)14yescomplemented
97C22×C31C22×D413yes—
98C2×D31—212no—
99C2×D31—24no—
100C2×D31—24no—
101C2×D31—432no—
102C2×D31—412no—
103C2×D31—412no—
104C2×D31C22×D414yes—
105C4×C31—24no—
106C4×C31C22×D413yes—
107C4×C31C2511yescharacteristic
108Dic31C22×D414yes—
109C22×#41—311no—
110C23×D4—311no—
111C2×AGL(1,ℤ/8)—3124no—
112C22×SD16—311no—
113C22×SD16D3111yescharacteristic, complemented
114C22×D8—311no—
115C22×D8—311no—
116C22×M16—311no—
117C2×C31—44no—
118C2×C31C2×AGL(1,ℤ/8)16yescomplemented
119C2×C31C23×D411yescharacteristic
120D31—24no—
121D31—44no—
122D31—44no—
123C25—621no—
124C2×#41—3112no—
125C2×#41—626no—
126C22×Q8C2×D3111yescharacteristic, complemented
127C22×D4—3112no—
128C22×D4—311no—
129C22×D4—311no—
130C22×D4—621no—
131C22×D4—626no—
132C22×D4—621no—
133C22×D4C2×D3111yescharacteristic
134C23×C4—311no—
135C23×C4—621no—
136AGL(1,ℤ/8)—3164no—
137C2×SD16—3112no—
138C2×SD16C2×D31112yescomplemented
139C2×D8—3112no—
140C2×D8—3112no—
141C2×M16—3112no—
142C22×C8—311no—
143C22×C8C2×D3111yescharacteristic, complemented
144C31C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
145C24—21no—
146C24—311no—
147C24—6212no—
148C24—621no—
149C24—621no—
150C24—1248no—
151#41—3116no—
152#41—6224no—
153C2×Q8—23no—
154C2×Q8C22×D3116yescomplemented
155C2×D4—23no—
156C2×D4—3116no—
157C2×D4—316no—
158C2×D4—316no—
159C2×D4—6212no—
160C2×D4—6224no—
161C2×D4—6212no—
162C2×D4—623no—
163C2×D4—623no—
164C2×D4C22×D3116yes—
165C22×C4—21no—
166C22×C4—311no—
167C22×C4—316no—
168C22×C4—316no—
169C22×C4—6212no—
170C22×C4C22×D3111yescharacteristic
171SD16—3116no—
172SD16C22×D31116yescomplemented
173D8—3116no—
174D8—3116no—
175M16—3116no—
176C2×C8—316no—
177C2×C8C22×D3116yescomplemented
178C23—26no—
179C23—44no—
180C23—316no—
181C23—6216no—
182C23—624no—
183C23—626no—
184C23—626no—
185C23—12448no—
186C23—1244no—
187C23—1244no—
188C23D4×D3111yescharacteristic
189Q8—26no—
190Q8C23×D3114yescomplemented
191D4—26no—
192D4—314no—
193D4—314no—
194D4—6216no—
195D4—6216no—
196D4—626no—
197D4—626no—
198D4C23×D3114yes—
199C2×C4—26no—
200C2×C4—316no—
201C2×C4—3112no—
202C2×C4—314no—
203C2×C4—6216no—
204C2×C4D4×D3113yes—
205C2×C4C23×D3113yes—
206C8—314no—
207C8C23×D3114yescomplemented
208C22—24no—
209C22—412no—
210C22—314no—
211C22—6212no—
212C22—624no—
213C22—624no—
214C22—12432no—
215C22—12412no—
216C22—12412no—
217C22#1175849114yesdirect product factor
218C22C2×D4×D3113yes—
219C4—24no—
220C4—314no—
221C4C2×D4×D3113yes—
222C4C24×D3111yescharacteristic
223C2—44no—
224C2—624no—
225C2—1244no—
226C2—1244no—
227C2C2×#1175849116yesdirect product factor
228C2C22×D4×D3111yescharacteristic
subgroupG/H
centerC23D4×D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC22×SD16×C31C2
socleC23×C31C2×D4
hypercenterC22×SD16D31
hypocenterC31C22×AGL(1,ℤ/8)
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#92499C2semidirect
2. C22×D4×D31C2semidirect
3. C2×#11758491C2direct (N × H)
4. C22×#92481C2semidirect
5. C22×D248C2semidirect
6. C22×#92479C2semidirect
7. C22×#92469C2semidirect
8. C22×SD16×C31C2semidirect
10. C22×Q8×C31C22semidirect
14. C2×#92499C22semidirect
16. C2×D4×D31C22semidirect
19. C22×D124C22semidirect
22. C2×SD16×C31C22semidirect
23. C22×C8×C31C22semidirect
24. C2×#92481C22semidirect
25. C2×#92479C22semidirect
26. C22×C31⋊C8C22semidirect
27. #11758491C22direct (N × H)
28. C2×D248C22semidirect
29. C2×#92469C22semidirect
37. C2×Q8×C31C23semidirect
45. #92499C23semidirect
47. D4×D31C23semidirect
52. C2×D124C23semidirect
56. SD16×C31C23semidirect
57. C2×C8×C31C23semidirect
58. #92481C23semidirect
59. #92479C23semidirect
60. C2×C31⋊C8C23semidirect
61. D248C23semidirect
62. #92469C23semidirect
75. Q8×C31C24semidirect
87. D124C24semidirect
91. C8×C31C24semidirect
92. C31⋊C8C24semidirect
96. C22×C31AGL(1,ℤ/8)semidirect
113. C22×SD16D31semidirect
118. C2×C31C2×AGL(1,ℤ/8)semidirect
126. C22×Q8C2×D31semidirect
138. C2×SD16C2×D31semidirect
143. C22×C8C2×D31semidirect
144. C31C22×AGL(1,ℤ/8)semidirect
154. C2×Q8C22×D31semidirect
172. SD16C22×D31semidirect
177. C2×C8C22×D31semidirect
190. Q8C23×D31semidirect
207. C8C23×D31semidirect
217. C22#11758491direct (N × H)
227. C2C2×#11758491direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#924992
C22×D4×D312
C2×#117584912
C22×#924812
C22×D2482
C22×#924792
C22×#924692
C22×SD16×C312
C22×AGL(1,ℤ/8)31