Finite Group Databasebeta

← D3×AGL(1,ℤ/8) #5772

Subgroups — D3×AGL(1,ℤ/8) #5772

Order 192.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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
1D3×#41C211yesmaximal, characteristic, complemented
2C2×D3×D4C211yesmaximal, characteristic, complemented
3C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#949C211yesmaximal, characteristic, complemented
5#932C211yesmaximal, characteristic, complemented
6#914C212yesmaximal, complemented
7D3×SD16C212yesmaximal, complemented
8#911C212yesmaximal, complemented
9D3×D8C212yesmaximal, complemented
10#908C211yesmaximal, characteristic, complemented
11D3×M16C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)—31nomaximal, Sylow
13C23×D3—21no—
14C3×#41C2211yescharacteristic, complemented
15C2×C3×D4C2211yescharacteristic, complemented
16C2×#67—21no—
17#239C2211yescharacteristic, complemented
18D3×Q8C2211yescharacteristic, complemented
19#237—21no—
20#237C2211yescharacteristic, complemented
21D3×D4—21no—
22D3×D4—21no—
23D3×D4—21no—
24D3×D4C2212yes—
25D3×D4C2212yescomplemented
26D3×D4C2211yescharacteristic, complemented
27#235—21no—
28#235C2211yescharacteristic, complemented
29C2×D12C2211yescharacteristic, complemented
30C2×C4×D3—21no—
31C2×C4×D3C2211yescharacteristic
32C3×SD16C2212yescomplemented
33C3×D8C2212yescomplemented
34C3×M16C2211yescharacteristic, complemented
35#215C2212yescomplemented
36#214C2212yescomplemented
37#213C2212yescomplemented
38#213C2212yescomplemented
39#208C2211yescharacteristic, complemented
40D24C2212yescomplemented
41#204C2212yescomplemented
42ΓL(1,𝔽25)C2212yescomplemented
43D3×C8C2212yescomplemented
44C2×#41—31no—
45C22×D4—31no—
46AGL(1,ℤ/8)—31no—
47AGL(1,ℤ/8)—31no—
48AGL(1,ℤ/8)—32no—
49AGL(1,ℤ/8)—32no—
50AGL(1,ℤ/8)—31no—
51AGL(1,ℤ/8)D311yescharacteristic, direct product factor
52C2×SD16—32no—
53C2×D8—32no—
54C2×M16—31no—
55C23×C3—21no—
56C22×D3—21no—
57C22×D3—21no—
58C22×D3—22no—
59C22×D3—22no—
60C22×D3—44no—
61C22×D3D411yescharacteristic
62C3×Q8C2311yescharacteristic, complemented
63C3×D4—21no—
64C3×D4—21no—
65C3×D4C2312yes—
66C3×D4C2311yescharacteristic, complemented
67C2×C3×C4—21no—
68C2×C3×C4C2311yescharacteristic
69#67—21no—
70#67—21no—
71#67—21no—
72#67—22no—
73#67—22no—
74C2×Dic3—21no—
75C2×Dic3D411yescharacteristic
76D12—21no—
77D12—21no—
78D12C2312yes—
79D12C2311yescharacteristic, complemented
80C4×D3—21no—
81C4×D3—21no—
82C4×D3—21no—
83C4×D3D411yescharacteristic
84C4×D3D411yescharacteristic
85C4×D3C2311yescharacteristic
86C4×D3C2311yescharacteristic
87Dic6—21no—
88Dic6C2311yescharacteristic, complemented
89C3×C8C2312yescomplemented
90C3⋊C8C2312yescomplemented
91C24—61no—
92#41—31no—
93#41—31no—
94#41—31no—
95#41—61no—
96#41—61no—
97#41C2×D311yescharacteristic, complemented
98C2×Q8—31no—
99C2×D4—32no—
100C2×D4—32no—
101C2×D4—31no—
102C2×D4—31no—
103C2×D4—61no—
104C2×D4—61no—
105C2×D4—61no—
106C2×D4—61no—
107C2×D4C2×D311yescharacteristic, complemented
108C22×C4—31no—
109C22×C4—61no—
110SD16—32no—
111SD16—32no—
112SD16—32no—
113SD16C2×D312yescomplemented
114D8—32no—
115D8—32no—
116D8—32no—
117D8C2×D312yescomplemented
118M16—31no—
119M16—32no—
120M16C2×D311yescharacteristic, complemented
121C2×C8—32no—
122C22×C3—21no—
123C22×C3—22no—
124C22×C3—42no—
125C22×C3C2×D411yescharacteristic
126C2×D3—22no—
127C2×D3—21no—
128C2×D3—22no—
129C2×D3—44no—
130C2×D3—42no—
131C2×D3—42no—
132C2×D3—44no—
133C2×D3C2×D411yescharacteristic
134C2×D3C2×D411yescharacteristic
135C3×C4—21no—
136C3×C4C2×D411yescharacteristic
137C3×C4C2411yescharacteristic
138Dic3—21no—
139Dic3C2×D411yescharacteristic
140Dic3C2×D411yescharacteristic
141C23—21no—
142C23—31no—
143C23—61no—
144C23—62no—
145C23—62no—
146C23—61no—
147C23—124no—
148Q8—31no—
149Q8—61no—
150Q8C22×D311yescharacteristic, complemented
151D4—21no—
152D4—21no—
153D4—31no—
154D4—32no—
155D4—61no—
156D4—62no—
157D4—62no—
158D4—61no—
159D4—61no—
160D4—61no—
161D4—61no—
162D4C22×D312yes—
163D4C22×D311yescharacteristic, complemented
164C2×C4—21no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—31no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4C22×D311yescharacteristic
175C8—32no—
176C8C22×D312yescomplemented
177C2×C3—21no—
178C2×C3—42no—
179C2×C3—41no—
180C2×C3C22×D411yescharacteristic
181D3—21no—
182D3—41no—
183D3—42no—
184D3AGL(1,ℤ/8)12yesdirect product factor
185C22—21no—
186C22—22no—
187C22—31no—
188C22—31no—
189C22—42no—
190C22—62no—
191C22—61no—
192C22—62no—
193C22—122no—
194C22—124no—
195C22—122no—
196C22—124no—
197C22D3×D411yescharacteristic
198C4—21no—
199C4—31no—
200C4—31no—
201C4—61no—
202C4D3×D411yescharacteristic
203C4C23×D311yescharacteristic
204C3C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
205C2—21no—
206C2—32no—
207C2—42no—
208C2—41no—
209C2—61no—
210C2—122no—
211C2—121no—
212C2C2×D3×D411yescharacteristic
subgroupG/H
centerC2C2×D3×D4
commutator, cosocleC3×C4C24
FrattiniC4C23×D3
FittingC3×AGL(1,ℤ/8)C2
socleC2×C3C22×D4
hypercenterAGL(1,ℤ/8)D3
hypocenterC3C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. D3×#41C2semidirect
2. C2×D3×D4C2semidirect
3. C3×AGL(1,ℤ/8)C2semidirect
4. #949C2semidirect
5. #932C2semidirect
6. #914C2semidirect
7. D3×SD16C2semidirect
8. #911C2semidirect
9. D3×D8C2semidirect
10. #908C2semidirect
11. D3×M16C2semidirect
14. C3×#41C22semidirect
15. C2×C3×D4C22semidirect
17. #239C22semidirect
18. D3×Q8C22semidirect
20. #237C22semidirect
25. D3×D4C22semidirect
26. D3×D4C22semidirect
28. #235C22semidirect
29. C2×D12C22semidirect
32. C3×SD16C22semidirect
33. C3×D8C22semidirect
34. C3×M16C22semidirect
35. #215C22semidirect
36. #214C22semidirect
37. #213C22semidirect
38. #213C22semidirect
39. #208C22semidirect
40. D24C22semidirect
41. #204C22semidirect
42. ΓL(1,𝔽25)C22semidirect
43. D3×C8C22semidirect
51. AGL(1,ℤ/8)D3direct (N × H)
62. C3×Q8C23semidirect
66. C3×D4C23semidirect
79. D12C23semidirect
88. Dic6C23semidirect
89. C3×C8C23semidirect
90. C3⋊C8C23semidirect
97. #41C2×D3semidirect
107. C2×D4C2×D3semidirect
113. SD16C2×D3semidirect
117. D8C2×D3semidirect
120. M16C2×D3semidirect
150. Q8C22×D3semidirect
163. D4C22×D3semidirect
176. C8C22×D3semidirect
184. D3AGL(1,ℤ/8)direct (N × H)
204. C3C2×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
D3×#412
C2×D3×D42
C3×AGL(1,ℤ/8)2
#9492
#9322
#9142
D3×SD162
#9112
D3×D82
#9082
D3×M162
C2×AGL(1,ℤ/8)3