Finite Group Databasebeta

← C22×#11758526 #16860443

Subgroups — C22×#11758526 #16860443

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
1C23×D124C211yesmaximal, characteristic, complemented
2C2×#11758526C2124yesmaximal, direct product factor
3C22×#92481C212yesmaximal, complemented
4C22×#92479C212yesmaximal, complemented
5C22×#92474C211yesmaximal, characteristic, complemented
6C22×#41×C31C211yesmaximal, characteristic, complemented
7C24×D31—21no—
8C2×#41×C31—26no—
9C2×#41×C31C22112yescomplemented
10C22×Q8×C31C2211yescharacteristic, complemented
11C22×D4×C31—21no—
12C22×D4×C31C2211yescharacteristic, complemented
13C23×C4×C31—21no—
14C23×C4×C31C2211yescharacteristic
15C22×D124—26no—
16C22×D124—21no—
17C22×D124C2212yes—
18C22×D124C22112yescomplemented
19#11758526C22164yesdirect product factor
20C2×#92481C22124yescomplemented
21C2×#92479C22124yescomplemented
22C2×#92474C22112yescomplemented
23C22×C31⋊C8C2212yescomplemented
24C24×C31—21no—
25C24×C31D411yescharacteristic
26C23×D31—22no—
27C23×D31—212no—
28C23×D31—48no—
29#41×C31—224no—
30#41×C31C23116yescomplemented
31C2×Q8×C31—23no—
32C2×Q8×C31C2316yescomplemented
33C2×D4×C31—212no—
34C2×D4×C31—23no—
35C2×D4×C31C2316yescomplemented
36C22×C4×C31—21no—
37C22×C4×C31—212no—
38C22×C4×C31D411yescharacteristic
39C22×C4×C31D416yes—
40C22×C4×C31C2311yescharacteristic
41C22×C4×C31C2316yes—
42C2×D124—26no—
43C2×D124—224no—
44C2×D124—212no—
45C2×D124C23112yes—
46C2×D124C23116yescomplemented
47#92481C23132yescomplemented
48#92479C23132yescomplemented
49#92474C23116yescomplemented
50C2×C31⋊C8C23112yescomplemented
51C23×C31—24no—
52C23×C31—26no—
53C23×C31—44no—
54C23×C31C2×D416yes—
55C23×C31C2×D411yescharacteristic
56C22×D31—212no—
57C22×D31—216no—
58C22×D31—448no—
59C22×D31—48no—
60Q8×C31—26no—
61Q8×C31C2414yescomplemented
62D4×C31—216no—
63D4×C31—26no—
64D4×C31C2414yescomplemented
65C2×C4×C31—26no—
66C2×C4×C31—216no—
67C2×C4×C31C2×D416yes—
68C2×C4×C31C2×D413yes—
69C2×C4×C31C2×D4112yes—
70C2×C4×C31C2413yes—
71C2×C4×C31C2414yes—
72D124—212no—
73D124—216no—
74D124C2418yes—
75C31⋊C8C2418yescomplemented
76C22×AGL(1,ℤ/8)—311nomaximal, Sylow
77C22×C31—212no—
78C22×C31—24no—
79C22×C31—412no—
80C22×C31AGL(1,ℤ/8)14yescomplemented
81C22×C31C22×D414yes—
82C22×C31C22×D413yes—
83C2×D31—28no—
84C2×D31—432no—
85C2×D31—424no—
86C4×C31—24no—
87C4×C31C22×D414yes—
88C4×C31C22×D413yes—
89C4×C31C2511yescharacteristic
90C22×#41D3111yescharacteristic, complemented
91C23×D4—311no—
92C2×AGL(1,ℤ/8)—3124no—
93C22×SD16—312no—
94C22×D8—312no—
95C22×M16—311no—
96C2×C31—24no—
97C2×C31—44no—
98C2×C31C2×AGL(1,ℤ/8)16yescomplemented
99C2×C31C23×D411yescharacteristic
100D31—48no—
101C25—621no—
102C2×#41—26no—
103C2×#41C2×D31112yescomplemented
104C22×Q8C2×D3111yescharacteristic, complemented
105C22×D4—21no—
106C22×D4—312no—
107C22×D4—3112no—
108C22×D4—626no—
109C22×D4—621no—
110C22×D4C2×D3111yescharacteristic, complemented
111C23×C4—21no—
112C23×C4C2×D3111yescharacteristic
113AGL(1,ℤ/8)—3164no—
114C2×SD16—3124no—
115C2×D8—3124no—
116C2×M16—3112no—
117C22×C8—312no—
118C31C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
119C24—21no—
120C24—6212no—
121C24—622no—
122C24—1248no—
123C24C22⋊D3111yescharacteristic
124#41—224no—
125#41C22×D31116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D3116yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—3112no—
131C2×D4—3116no—
132C2×D4—6212no—
133C2×D4—626no—
134C2×D4—6224no—
135C2×D4C22×D3116yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D3116yes—
139C22×C4C22⋊D3111yescharacteristic
140C22×C4C22×D3116yes—
141C22×C4C22×D3111yescharacteristic
142SD16—3132no—
143D8—3132no—
144M16—3116no—
145C2×C8—3112no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—6216no—
150C23—6212no—
151C23—12448no—
152C23—1248no—
153C23C2×C22⋊D3116yes—
154C23C2×C22⋊D3111yescharacteristic
155Q8—26no—
156Q8C23×D3114yescomplemented
157D4—216no—
158D4—26no—
159D4—318no—
160D4—6216no—
161D4—6212no—
162D4C23×D3114yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D3116yes—
166C2×C4C2×C22⋊D3113yes—
167C2×C4C2×C22⋊D31112yes—
168C2×C4C23×D3113yes—
169C2×C4C23×D3114yes—
170C8—318no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—628no—
175C22—12432no—
176C22—12424no—
177C22#1175852614yesdirect product factor
178C22C22×C22⋊D3114yes—
179C22C22×C22⋊D3113yes—
180C4—24no—
181C4C22×C22⋊D3114yes—
182C4C22×C22⋊D3113yes—
183C4C24×D3111yescharacteristic
184C2—24no—
185C2—44no—
186C2—1248no—
187C2C2×#1175852616yesdirect product factor
188C2C23×C22⋊D3111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC22×#41×C31C2
socleC23×C31C2×D4
hypercenterC22×#41D31
hypocenterC31C22×AGL(1,ℤ/8)
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D124C2semidirect
2. C2×#11758526C2direct (N × H)
3. C22×#92481C2semidirect
4. C22×#92479C2semidirect
5. C22×#92474C2semidirect
6. C22×#41×C31C2semidirect
9. C2×#41×C31C22semidirect
10. C22×Q8×C31C22semidirect
12. C22×D4×C31C22semidirect
18. C22×D124C22semidirect
19. #11758526C22direct (N × H)
20. C2×#92481C22semidirect
21. C2×#92479C22semidirect
22. C2×#92474C22semidirect
23. C22×C31⋊C8C22semidirect
30. #41×C31C23semidirect
32. C2×Q8×C31C23semidirect
35. C2×D4×C31C23semidirect
46. C2×D124C23semidirect
47. #92481C23semidirect
48. #92479C23semidirect
49. #92474C23semidirect
50. C2×C31⋊C8C23semidirect
61. Q8×C31C24semidirect
64. D4×C31C24semidirect
75. C31⋊C8C24semidirect
80. C22×C31AGL(1,ℤ/8)semidirect
90. C22×#41D31semidirect
98. C2×C31C2×AGL(1,ℤ/8)semidirect
103. C2×#41C2×D31semidirect
104. C22×Q8C2×D31semidirect
110. C22×D4C2×D31semidirect
118. C31C22×AGL(1,ℤ/8)semidirect
125. #41C22×D31semidirect
127. C2×Q8C22×D31semidirect
135. C2×D4C22×D31semidirect
156. Q8C23×D31semidirect
162. D4C23×D31semidirect
177. C22#11758526direct (N × H)
187. C2C2×#11758526direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D1242
C2×#117585262
C22×#924812
C22×#924792
C22×#924742
C22×#41×C312
C22×AGL(1,ℤ/8)31