Finite Group Databasebeta

← AGL(1,ℤ/8)×D23 #13093201

Subgroups — AGL(1,ℤ/8)×D23 #13093201

Order 1472.

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
1#41×D23C211yesmaximal, characteristic, complemented
2C2×D4×D23C211yesmaximal, characteristic, complemented
3C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#10625120C211yesmaximal, characteristic, complemented
5#10625103C211yesmaximal, characteristic, complemented
6#10625085C212yesmaximal, complemented
7SD16×D23C212yesmaximal, complemented
8#10625082C212yesmaximal, complemented
9D8×D23C212yesmaximal, complemented
10#10625079C211yesmaximal, characteristic, complemented
11M16×D23C211yesmaximal, characteristic, complemented
12C23×D23—21no—
13#41×C23C2211yescharacteristic, complemented
14C2×D4×C23C2211yescharacteristic, complemented
15C2×C22⋊D23—21no—
16#67292C2211yescharacteristic, complemented
17Q8×D23C2211yescharacteristic, complemented
18#67290—21no—
19#67290C2211yescharacteristic, complemented
20D4×D23—21no—
21D4×D23—21no—
22D4×D23—21no—
23D4×D23C2212yes—
24D4×D23C2212yescomplemented
25D4×D23C2211yescharacteristic, complemented
26#67288—21no—
27#67288C2211yescharacteristic, complemented
28C2×D92C2211yescharacteristic, complemented
29C2×C4×D23—21no—
30C2×C4×D23C2211yescharacteristic
31SD16×C23C2212yescomplemented
32D8×C23C2212yescomplemented
33M16×C23C2211yescharacteristic, complemented
34#67274C2212yescomplemented
35#67273C2212yescomplemented
36#67272C2212yescomplemented
37#67272C2212yescomplemented
38#67267C2211yescharacteristic, complemented
39D184C2212yescomplemented
40#67263C2212yescomplemented
41#67262C2212yescomplemented
42C8×D23C2212yescomplemented
43C23×C23—21no—
44C22×D23—21no—
45C22×D23—21no—
46C22×D23—22no—
47C22×D23—22no—
48C22×D23—44no—
49C22×D23D411yescharacteristic
50Q8×C23C2311yescharacteristic, complemented
51D4×C23—21no—
52D4×C23—21no—
53D4×C23C2312yes—
54D4×C23C2311yescharacteristic, complemented
55C2×C4×C23—21no—
56C2×C4×C23C2311yescharacteristic
57C22⋊D23—21no—
58C22⋊D23—21no—
59C22⋊D23—21no—
60C22⋊D23—22no—
61C22⋊D23—22no—
62C2×Dic23—21no—
63C2×Dic23D411yescharacteristic
64D92—21no—
65D92—21no—
66D92C2312yes—
67D92C2311yescharacteristic, complemented
68C4×D23—21no—
69C4×D23—21no—
70C4×D23—21no—
71C4×D23D411yescharacteristic
72C4×D23D411yescharacteristic
73C4×D23C2311yescharacteristic
74C4×D23C2311yescharacteristic
75Dic46—21no—
76Dic46C2311yescharacteristic, complemented
77C8×C23C2312yescomplemented
78C23⋊C8C2312yescomplemented
79C22×C23—21no—
80C22×C23—22no—
81C22×C23—42no—
82C22×C23C2×D411yescharacteristic
83C2×D23—22no—
84C2×D23—21no—
85C2×D23—22no—
86C2×D23—44no—
87C2×D23—42no—
88C2×D23—42no—
89C2×D23—44no—
90C2×D23C2×D411yescharacteristic
91C2×D23C2×D411yescharacteristic
92C4×C23—21no—
93C4×C23C2×D411yescharacteristic
94C4×C23C2411yescharacteristic
95Dic23—21no—
96Dic23C2×D411yescharacteristic
97Dic23C2×D411yescharacteristic
98C2×AGL(1,ℤ/8)—231nomaximal, Sylow
99C2×C23—21no—
100C2×C23—42no—
101C2×C23—41no—
102C2×C23C22×D411yescharacteristic
103D23—21no—
104D23—41no—
105D23—42no—
106D23AGL(1,ℤ/8)12yesdirect product factor
107C2×#41—231no—
108C22×D4—231no—
109AGL(1,ℤ/8)—231no—
110AGL(1,ℤ/8)—231no—
111AGL(1,ℤ/8)—232no—
112AGL(1,ℤ/8)—232no—
113AGL(1,ℤ/8)—231no—
114AGL(1,ℤ/8)D2311yescharacteristic, direct product factor
115C2×SD16—232no—
116C2×D8—232no—
117C2×M16—231no—
118C23C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
119C24—461no—
120#41—231no—
121#41—231no—
122#41—231no—
123#41—461no—
124#41—461no—
125#41C2×D2311yescharacteristic, complemented
126C2×Q8—231no—
127C2×D4—232no—
128C2×D4—232no—
129C2×D4—231no—
130C2×D4—231no—
131C2×D4—461no—
132C2×D4—461no—
133C2×D4—461no—
134C2×D4—461no—
135C2×D4C2×D2311yescharacteristic, complemented
136C22×C4—231no—
137C22×C4—461no—
138SD16—232no—
139SD16—232no—
140SD16—232no—
141SD16C2×D2312yescomplemented
142D8—232no—
143D8—232no—
144D8—232no—
145D8C2×D2312yescomplemented
146M16—231no—
147M16—232no—
148M16C2×D2311yescharacteristic, complemented
149C2×C8—232no—
150C23—21no—
151C23—231no—
152C23—461no—
153C23—462no—
154C23—462no—
155C23—461no—
156C23—924no—
157Q8—231no—
158Q8—461no—
159Q8C22×D2311yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—231no—
163D4—232no—
164D4—461no—
165D4—462no—
166D4—462no—
167D4—461no—
168D4—461no—
169D4—461no—
170D4—461no—
171D4C22×D2312yes—
172D4C22×D2311yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—231no—
175C2×C4—231no—
176C2×C4—231no—
177C2×C4—231no—
178C2×C4—231no—
179C2×C4—461no—
180C2×C4—461no—
181C2×C4—461no—
182C2×C4—461no—
183C2×C4C22×D2311yescharacteristic
184C8—232no—
185C8C22×D2312yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—231no—
190C22—231no—
191C22—462no—
192C22—461no—
193C22—462no—
194C22—922no—
195C22—924no—
196C22—922no—
197C22—924no—
198C22D4×D2311yescharacteristic
199C4—21no—
200C4—231no—
201C4—231no—
202C4—461no—
203C4D4×D2311yescharacteristic
204C4C23×D2311yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—232no—
209C2—461no—
210C2—922no—
211C2—921no—
212C2C2×D4×D2311yescharacteristic
subgroupG/H
centerC2C2×D4×D23
commutator, cosocleC4×C23C24
FrattiniC4C23×D23
FittingC23×AGL(1,ℤ/8)C2
socleC2×C23C22×D4
hypercenterAGL(1,ℤ/8)D23
hypocenterC23C2×AGL(1,ℤ/8)
LCSG ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D23C2semidirect
2. C2×D4×D23C2semidirect
3. C23×AGL(1,ℤ/8)C2semidirect
4. #10625120C2semidirect
5. #10625103C2semidirect
6. #10625085C2semidirect
7. SD16×D23C2semidirect
8. #10625082C2semidirect
9. D8×D23C2semidirect
10. #10625079C2semidirect
11. M16×D23C2semidirect
13. #41×C23C22semidirect
14. C2×D4×C23C22semidirect
16. #67292C22semidirect
17. Q8×D23C22semidirect
19. #67290C22semidirect
24. D4×D23C22semidirect
25. D4×D23C22semidirect
27. #67288C22semidirect
28. C2×D92C22semidirect
31. SD16×C23C22semidirect
32. D8×C23C22semidirect
33. M16×C23C22semidirect
34. #67274C22semidirect
35. #67273C22semidirect
36. #67272C22semidirect
37. #67272C22semidirect
38. #67267C22semidirect
39. D184C22semidirect
40. #67263C22semidirect
41. #67262C22semidirect
42. C8×D23C22semidirect
50. Q8×C23C23semidirect
54. D4×C23C23semidirect
67. D92C23semidirect
76. Dic46C23semidirect
77. C8×C23C23semidirect
78. C23⋊C8C23semidirect
106. D23AGL(1,ℤ/8)direct (N × H)
114. AGL(1,ℤ/8)D23direct (N × H)
118. C23C2×AGL(1,ℤ/8)semidirect
125. #41C2×D23semidirect
135. C2×D4C2×D23semidirect
141. SD16C2×D23semidirect
145. D8C2×D23semidirect
148. M16C2×D23semidirect
159. Q8C22×D23semidirect
172. D4C22×D23semidirect
185. C8C22×D23semidirect

Maximal subgroups

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

subgroupindex
#41×D232
C2×D4×D232
C23×AGL(1,ℤ/8)2
#106251202
#106251032
#106250852
SD16×D232
#106250822
D8×D232
#106250792
M16×D232
C2×AGL(1,ℤ/8)23