Finite Group Databasebeta

← C22×#914 #87175

Subgroups — C22×#914 #87175

Order 384.

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×#239C211yesmaximal, characteristic, complemented
2C22×D3×D4C211yesmaximal, characteristic, complemented
3C22×C3×SD16C211yesmaximal, characteristic, complemented
4C22×#215C211yesmaximal, characteristic, complemented
5C22×#213C211yesmaximal, characteristic, complemented
6C2×#914C2124yesmaximal, direct product factor
7C22×D24C211yesmaximal, characteristic, complemented
8C22×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)—31nomaximal, Sylow
10C24×D3—21no—
11C22×C3×Q8C2211yescharacteristic, complemented
12C22×C3×D4C2211yescharacteristic
13C22×#67—21no—
14C2×#239—26no—
15C2×#239C22112yescomplemented
16C2×D3×D4—26no—
17C2×D3×D4C22112yescomplemented
18C22×D12—21no—
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic, complemented
21C22×C4×D3—21no—
22C22×C4×D3C2211yescharacteristic
23C2×C3×SD16C22112yescomplemented
24C22×C3×C8C2211yescharacteristic, complemented
25C2×#215C22112yescomplemented
26C2×#213C22112yescomplemented
27C22×C3⋊C8C2211yescharacteristic, complemented
28#914C22164yesdirect product factor
29C2×D24C22112yescomplemented
30C2×ΓL(1,𝔽25)C22112yescomplemented
31C22×#41—31no—
32C23×D4—31no—
33C2×AGL(1,ℤ/8)—324no—
34C22×SD16—31no—
35C22×SD16D311yescharacteristic, complemented
36C22×D8—31no—
37C22×D8—31no—
38C22×M16—31no—
39C24×C3—21no—
40C23×D3—21no—
41C23×D3—21no—
42C23×D3—212no—
43C23×D3—48no—
44C23×D3D411yescharacteristic
45C2×C3×Q8—23no—
46C2×C3×Q8C2316yescomplemented
47C2×C3×D4—23no—
48C2×C3×D4C2316yes—
49C22×C3×C4—21no—
50C22×C3×C4C2311yescharacteristic
51C2×#67—212no—
52C22×Dic3D411yescharacteristic
53#239—224no—
54#239C23116yescomplemented
55D3×D4—224no—
56D3×D4C23116yescomplemented
57C2×D12—212no—
58C2×D12—23no—
59C2×D12—23no—
60C2×D12C2316yes—
61C2×D12C2316yescomplemented
62C2×C4×D3—212no—
63C2×C4×D3D416yes—
64C2×C4×D3C2316yes—
65C3×SD16C23116yescomplemented
66C2×C3×C8C2316yescomplemented
67#215C23116yescomplemented
68#213C23116yescomplemented
69C2×C3⋊C8C2316yescomplemented
70D24C23116yescomplemented
71ΓL(1,𝔽25)C23116yescomplemented
72C25—61no—
73C2×#41—312no—
74C2×#41—66no—
75C22×Q8C2×D311yescharacteristic, complemented
76C22×D4—312no—
77C22×D4—31no—
78C22×D4—31no—
79C22×D4—61no—
80C22×D4—66no—
81C22×D4—61no—
82C22×D4C2×D311yescharacteristic
83C23×C4—31no—
84C23×C4—61no—
85AGL(1,ℤ/8)—364no—
86C2×SD16—312no—
87C2×SD16C2×D3112yescomplemented
88C2×D8—312no—
89C2×D8—312no—
90C2×M16—312no—
91C22×C8—31no—
92C22×C8C2×D311yescharacteristic, complemented
93C23×C3—26no—
94C23×C3—44no—
95C23×C3C2×D411yescharacteristic
96C22×D3—24no—
97C22×D3—26no—
98C22×D3—26no—
99C22×D3—216no—
100C22×D3—448no—
101C22×D3—44no—
102C22×D3—44no—
103C22×D3C2×D416yes—
104C3×Q8—26no—
105C3×Q8C2414yescomplemented
106C3×D4—26no—
107C3×D4C2414yes—
108C2×C3×C4—26no—
109C2×C3×C4C2×D413yes—
110C2×C3×C4C2413yes—
111#67—216no—
112C2×Dic3C2×D416yes—
113D12—216no—
114D12—26no—
115D12—26no—
116D12C2414yes—
117D12C2414yescomplemented
118C4×D3—216no—
119C4×D3C2×D4112yes—
120C4×D3C2414yes—
121C3×C8C2414yescomplemented
122C3⋊C8C2414yescomplemented
123C24—21no—
124C24—31no—
125C24—612no—
126C24—61no—
127C24—61no—
128C24—128no—
129#41—316no—
130#41—624no—
131C2×Q8—23no—
132C2×Q8C22×D316yescomplemented
133C2×D4—23no—
134C2×D4—316no—
135C2×D4—36no—
136C2×D4—36no—
137C2×D4—612no—
138C2×D4—624no—
139C2×D4—612no—
140C2×D4—63no—
141C2×D4—63no—
142C2×D4C22×D316yes—
143C22×C4—21no—
144C22×C4—31no—
145C22×C4—36no—
146C22×C4—36no—
147C22×C4—612no—
148C22×C4C22×D311yescharacteristic
149SD16—316no—
150SD16C22×D3116yescomplemented
151D8—316no—
152D8—316no—
153M16—316no—
154C2×C8—36no—
155C2×C8C22×D316yescomplemented
156C22×C3—24no—
157C22×C3—412no—
158C22×C3AGL(1,ℤ/8)14yescomplemented
159C22×C3C22×D413yes—
160C2×D3—212no—
161C2×D3—24no—
162C2×D3—24no—
163C2×D3—432no—
164C2×D3—412no—
165C2×D3—412no—
166C2×D3C22×D414yes—
167C3×C4—24no—
168C3×C4C22×D413yes—
169C3×C4C2511yescharacteristic
170Dic3C22×D414yes—
171C23—26no—
172C23—36no—
173C23—44no—
174C23—616no—
175C23—64no—
176C23—66no—
177C23—66no—
178C23—1248no—
179C23—124no—
180C23—124no—
181C23D3×D411yescharacteristic
182Q8—26no—
183Q8C23×D314yescomplemented
184D4—26no—
185D4—34no—
186D4—34no—
187D4—616no—
188D4—616no—
189D4—66no—
190D4—66no—
191D4C23×D314yes—
192C2×C4—26no—
193C2×C4—36no—
194C2×C4—312no—
195C2×C4—34no—
196C2×C4—616no—
197C2×C4D3×D413yes—
198C2×C4C23×D313yes—
199C8—34no—
200C8C23×D314yescomplemented
201C2×C3—44no—
202C2×C3C2×AGL(1,ℤ/8)16yescomplemented
203C2×C3C23×D411yescharacteristic
204D3—24no—
205D3—44no—
206D3—44no—
207C22—24no—
208C22—34no—
209C22—412no—
210C22—612no—
211C22—64no—
212C22—64no—
213C22—1232no—
214C22—1212no—
215C22—1212no—
216C22#91414yesdirect product factor
217C22C2×D3×D413yes—
218C4—24no—
219C4—34no—
220C4C2×D3×D413yes—
221C4C24×D311yescharacteristic
222C3C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
223C2—44no—
224C2—64no—
225C2—124no—
226C2—124no—
227C2C2×#91416yesdirect product factor
228C2C22×D3×D411yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC22×C3×SD16C2
socleC23×C3C2×D4
hypercenterC22×SD16D3
hypocenterC3C22×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C22×D3×D4C2semidirect
3. C22×C3×SD16C2semidirect
4. C22×#215C2semidirect
5. C22×#213C2semidirect
6. C2×#914C2direct (N × H)
7. C22×D24C2semidirect
8. C22×ΓL(1,𝔽25)C2semidirect
11. C22×C3×Q8C22semidirect
15. C2×#239C22semidirect
17. C2×D3×D4C22semidirect
20. C22×D12C22semidirect
23. C2×C3×SD16C22semidirect
24. C22×C3×C8C22semidirect
25. C2×#215C22semidirect
26. C2×#213C22semidirect
27. C22×C3⋊C8C22semidirect
28. #914C22direct (N × H)
29. C2×D24C22semidirect
30. C2×ΓL(1,𝔽25)C22semidirect
35. C22×SD16D3semidirect
46. C2×C3×Q8C23semidirect
54. #239C23semidirect
56. D3×D4C23semidirect
61. C2×D12C23semidirect
65. C3×SD16C23semidirect
66. C2×C3×C8C23semidirect
67. #215C23semidirect
68. #213C23semidirect
69. C2×C3⋊C8C23semidirect
70. D24C23semidirect
71. ΓL(1,𝔽25)C23semidirect
75. C22×Q8C2×D3semidirect
87. C2×SD16C2×D3semidirect
92. C22×C8C2×D3semidirect
105. C3×Q8C24semidirect
117. D12C24semidirect
121. C3×C8C24semidirect
122. C3⋊C8C24semidirect
132. C2×Q8C22×D3semidirect
150. SD16C22×D3semidirect
155. C2×C8C22×D3semidirect
158. C22×C3AGL(1,ℤ/8)semidirect
183. Q8C23×D3semidirect
200. C8C23×D3semidirect
202. C2×C3C2×AGL(1,ℤ/8)semidirect
216. C22#914direct (N × H)
222. C3C22×AGL(1,ℤ/8)semidirect
227. C2C2×#914direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2392
C22×D3×D42
C22×C3×SD162
C22×#2152
C22×#2132
C2×#9142
C22×D242
C22×ΓL(1,𝔽25)2
C22×AGL(1,ℤ/8)3