Finite Group Databasebeta

← C22×#949 #87255

Subgroups — C22×#949 #87255

Order 384.

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×C3×#41C211yesmaximal, characteristic, complemented
2C23×D12C211yesmaximal, characteristic, complemented
3C2×#949C2124yesmaximal, direct product factor
4C22×#215C212yesmaximal, complemented
5C22×#213C212yesmaximal, complemented
6C22×#208C211yesmaximal, characteristic, complemented
7C22×AGL(1,ℤ/8)—31nomaximal, Sylow
8C24×D3—21no—
9C2×C3×#41—26no—
10C2×C3×#41C22112yescomplemented
11C22×C3×Q8C2211yescharacteristic, complemented
12C22×C3×D4—21no—
13C22×C3×D4C2211yescharacteristic, complemented
14C23×C3×C4—21no—
15C23×C3×C4C2211yescharacteristic
16C22×D12—26no—
17C22×D12—21no—
18C22×D12C2212yes—
19C22×D12C22112yescomplemented
20#949C22164yesdirect product factor
21C2×#215C22124yescomplemented
22C2×#213C22124yescomplemented
23C2×#208C22112yescomplemented
24C22×C3⋊C8C2212yescomplemented
25C22×#41D311yescharacteristic, complemented
26C23×D4—31no—
27C2×AGL(1,ℤ/8)—324no—
28C22×SD16—32no—
29C22×D8—32no—
30C22×M16—31no—
31C24×C3—21no—
32C24×C3D411yescharacteristic
33C23×D3—22no—
34C23×D3—212no—
35C23×D3—48no—
36C3×#41—224no—
37C3×#41C23116yescomplemented
38C2×C3×Q8—23no—
39C2×C3×Q8C2316yescomplemented
40C2×C3×D4—212no—
41C2×C3×D4—23no—
42C2×C3×D4C2316yescomplemented
43C22×C3×C4—21no—
44C22×C3×C4—212no—
45C22×C3×C4D411yescharacteristic
46C22×C3×C4D416yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2316yes—
49C2×D12—26no—
50C2×D12—224no—
51C2×D12—212no—
52C2×D12C23112yes—
53C2×D12C23116yescomplemented
54#215C23132yescomplemented
55#213C23132yescomplemented
56#208C23116yescomplemented
57C2×C3⋊C8C23112yescomplemented
58C25—61no—
59C2×#41—26no—
60C2×#41C2×D3112yescomplemented
61C22×Q8C2×D311yescharacteristic, complemented
62C22×D4—21no—
63C22×D4—32no—
64C22×D4—312no—
65C22×D4—66no—
66C22×D4—61no—
67C22×D4C2×D311yescharacteristic, complemented
68C23×C4—21no—
69C23×C4C2×D311yescharacteristic
70AGL(1,ℤ/8)—364no—
71C2×SD16—324no—
72C2×D8—324no—
73C2×M16—312no—
74C22×C8—32no—
75C23×C3—24no—
76C23×C3—26no—
77C23×C3—44no—
78C23×C3C2×D416yes—
79C23×C3C2×D411yescharacteristic
80C22×D3—212no—
81C22×D3—216no—
82C22×D3—448no—
83C22×D3—48no—
84C3×Q8—26no—
85C3×Q8C2414yescomplemented
86C3×D4—216no—
87C3×D4—26no—
88C3×D4C2414yescomplemented
89C2×C3×C4—26no—
90C2×C3×C4—216no—
91C2×C3×C4C2×D416yes—
92C2×C3×C4C2×D413yes—
93C2×C3×C4C2×D4112yes—
94C2×C3×C4C2413yes—
95C2×C3×C4C2414yes—
96D12—212no—
97D12—216no—
98D12C2418yes—
99C3⋊C8C2418yescomplemented
100C24—21no—
101C24—612no—
102C24—62no—
103C24—128no—
104C24#6711yescharacteristic
105#41—224no—
106#41C22×D3116yescomplemented
107C2×Q8—23no—
108C2×Q8C22×D316yescomplemented
109C2×D4—212no—
110C2×D4—23no—
111C2×D4—312no—
112C2×D4—316no—
113C2×D4—612no—
114C2×D4—66no—
115C2×D4—624no—
116C2×D4C22×D316yescomplemented
117C22×C4—21no—
118C22×C4—212no—
119C22×C4#6716yes—
120C22×C4#6711yescharacteristic
121C22×C4C22×D316yes—
122C22×C4C22×D311yescharacteristic
123SD16—332no—
124D8—332no—
125M16—316no—
126C2×C8—312no—
127C22×C3—212no—
128C22×C3—24no—
129C22×C3—412no—
130C22×C3AGL(1,ℤ/8)14yescomplemented
131C22×C3C22×D414yes—
132C22×C3C22×D413yes—
133C2×D3—28no—
134C2×D3—432no—
135C2×D3—424no—
136C3×C4—24no—
137C3×C4C22×D414yes—
138C3×C4C22×D413yes—
139C3×C4C2511yescharacteristic
140C23—24no—
141C23—26no—
142C23—44no—
143C23—616no—
144C23—612no—
145C23—1248no—
146C23—128no—
147C23C2×#6716yes—
148C23C2×#6711yescharacteristic
149Q8—26no—
150Q8C23×D314yescomplemented
151D4—216no—
152D4—26no—
153D4—38no—
154D4—616no—
155D4—612no—
156D4C23×D314yescomplemented
157C2×C4—26no—
158C2×C4—216no—
159C2×C4C2×#6716yes—
160C2×C4C2×#6713yes—
161C2×C4C2×#67112yes—
162C2×C4C23×D313yes—
163C2×C4C23×D314yes—
164C8—38no—
165C2×C3—24no—
166C2×C3—44no—
167C2×C3C2×AGL(1,ℤ/8)16yescomplemented
168C2×C3C23×D411yescharacteristic
169D3—48no—
170C22—212no—
171C22—24no—
172C22—412no—
173C22—68no—
174C22—1232no—
175C22—1224no—
176C22#94914yesdirect product factor
177C22C22×#6714yes—
178C22C22×#6713yes—
179C4—24no—
180C4C22×#6714yes—
181C4C22×#6713yes—
182C4C24×D311yescharacteristic
183C3C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
184C2—24no—
185C2—44no—
186C2—128no—
187C2C2×#94916yesdirect product factor
188C2C23×#6711yescharacteristic
subgroupG/H
centerC23C2×#67
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC22×C3×#41C2
socleC23×C3C2×D4
hypercenterC22×#41D3
hypocenterC3C22×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41C2semidirect
2. C23×D12C2semidirect
3. C2×#949C2direct (N × H)
4. C22×#215C2semidirect
5. C22×#213C2semidirect
6. C22×#208C2semidirect
10. C2×C3×#41C22semidirect
11. C22×C3×Q8C22semidirect
13. C22×C3×D4C22semidirect
19. C22×D12C22semidirect
20. #949C22direct (N × H)
21. C2×#215C22semidirect
22. C2×#213C22semidirect
23. C2×#208C22semidirect
24. C22×C3⋊C8C22semidirect
25. C22×#41D3semidirect
37. C3×#41C23semidirect
39. C2×C3×Q8C23semidirect
42. C2×C3×D4C23semidirect
53. C2×D12C23semidirect
54. #215C23semidirect
55. #213C23semidirect
56. #208C23semidirect
57. C2×C3⋊C8C23semidirect
60. C2×#41C2×D3semidirect
61. C22×Q8C2×D3semidirect
67. C22×D4C2×D3semidirect
85. C3×Q8C24semidirect
88. C3×D4C24semidirect
99. C3⋊C8C24semidirect
106. #41C22×D3semidirect
108. C2×Q8C22×D3semidirect
116. C2×D4C22×D3semidirect
130. C22×C3AGL(1,ℤ/8)semidirect
150. Q8C23×D3semidirect
156. D4C23×D3semidirect
167. C2×C3C2×AGL(1,ℤ/8)semidirect
176. C22#949direct (N × H)
183. C3C22×AGL(1,ℤ/8)semidirect
187. C2C2×#949direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#412
C23×D122
C2×#9492
C22×#2152
C22×#2132
C22×#2082
C22×AGL(1,ℤ/8)3