Finite Group Databasebeta

← AGL(1,ℤ/8)×D229 #28412265

Subgroups — AGL(1,ℤ/8)×D229 #28412265

Order 14656.

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×D229C211yesmaximal, characteristic, complemented
2C2×D4×D229C211yesmaximal, characteristic, complemented
3#20301016C211yesmaximal, characteristic, complemented
4#20301015C212yesmaximal, complemented
5#20301014C211yesmaximal, characteristic, complemented
6#20301013C211yesmaximal, characteristic, complemented
7#20301012C212yesmaximal, complemented
8SD16×D229C212yesmaximal, complemented
9D8×D229C212yesmaximal, complemented
10M16×D229C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C229C211yesmaximal, characteristic, complemented
12C23×D229—21no—
13#16814667C2211yescharacteristic, complemented
14#16814666—21no—
15#16814666C2211yescharacteristic, complemented
16#16814665—21no—
17#16814665C2211yescharacteristic, complemented
18Q8×D229C2211yescharacteristic, complemented
19C2×D916C2211yescharacteristic, complemented
20C2×C22⋊D229—21no—
21D4×D229—21no—
22D4×D229—21no—
23D4×D229—21no—
24D4×D229C2212yes—
25D4×D229C2212yescomplemented
26D4×D229C2211yescharacteristic, complemented
27C2×C4×D229—21no—
28C2×C4×D229C2211yescharacteristic
29#16814655C2212yescomplemented
30#16814654C2212yescomplemented
31#16814653C2212yescomplemented
32D1832C2212yescomplemented
33#16814651C2212yescomplemented
34#16814651C2212yescomplemented
35#16814650C2211yescharacteristic, complemented
36#16814649C2212yescomplemented
37C8×D229C2212yescomplemented
38#41×C229C2211yescharacteristic, complemented
39C2×D4×C229C2211yescharacteristic, complemented
40SD16×C229C2212yescomplemented
41D8×C229C2212yescomplemented
42M16×C229C2211yescharacteristic, complemented
43C22×D229—21no—
44C22×D229—21no—
45C22×D229—22no—
46C22×D229—22no—
47C22×D229—44no—
48C22×D229D411yescharacteristic
49Dic458—21no—
50Dic458C2311yescharacteristic, complemented
51D916—21no—
52D916—21no—
53D916C2312yes—
54D916C2311yescharacteristic, complemented
55C22⋊D229—21no—
56C22⋊D229—21no—
57C22⋊D229—22no—
58C22⋊D229—22no—
59C22⋊D229—21no—
60C2×Dic229—21no—
61C2×Dic229D411yescharacteristic
62C4×D229—21no—
63C4×D229—21no—
64C4×D229—21no—
65C4×D229D411yescharacteristic
66C4×D229D411yescharacteristic
67C4×D229C2311yescharacteristic
68C4×D229C2311yescharacteristic
69#14270080C2312yescomplemented
70C23×C229—21no—
71Q8×C229C2311yescharacteristic, complemented
72D4×C229—21no—
73D4×C229—21no—
74D4×C229C2312yes—
75D4×C229C2311yescharacteristic, complemented
76C2×C4×C229—21no—
77C2×C4×C229C2311yescharacteristic
78C8×C229C2312yescomplemented
79C22×C229—21no—
80C22×C229—22no—
81C22×C229—42no—
82C22×C229C2×D411yescharacteristic
83C2×D229—22no—
84C2×D229—21no—
85C2×D229—22no—
86C2×D229—44no—
87C2×D229—42no—
88C2×D229—42no—
89C2×D229—44no—
90C2×D229C2×D411yescharacteristic
91C2×D229C2×D411yescharacteristic
92C4×C229—21no—
93C4×C229C2×D411yescharacteristic
94C4×C229C2411yescharacteristic
95Dic229—21no—
96Dic229C2×D411yescharacteristic
97Dic229C2×D411yescharacteristic
98C2×C229—21no—
99C2×C229—42no—
100C2×C229—41no—
101C2×C229C22×D411yescharacteristic
102D229—21no—
103D229—41no—
104D229—42no—
105D229AGL(1,ℤ/8)12yesdirect product factor
106C229C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—2291nomaximal, Sylow
108C2×#41—2291no—
109C22×D4—2291no—
110AGL(1,ℤ/8)—2291no—
111AGL(1,ℤ/8)—2291no—
112AGL(1,ℤ/8)—2292no—
113AGL(1,ℤ/8)—2292no—
114AGL(1,ℤ/8)—2291no—
115AGL(1,ℤ/8)D22911yescharacteristic, direct product factor
116C2×SD16—2292no—
117C2×D8—2292no—
118C2×M16—2291no—
119C24—4581no—
120#41—4581no—
121#41—4581no—
122#41—2291no—
123#41—2291no—
124#41—2291no—
125#41C2×D22911yescharacteristic, complemented
126C2×Q8—2291no—
127C2×D4—4581no—
128C2×D4—4581no—
129C2×D4—4581no—
130C2×D4—4581no—
131C2×D4—2292no—
132C2×D4—2292no—
133C2×D4—2291no—
134C2×D4—2291no—
135C2×D4C2×D22911yescharacteristic, complemented
136C22×C4—4581no—
137C22×C4—2291no—
138SD16—2292no—
139SD16—2292no—
140SD16—2292no—
141SD16C2×D22912yescomplemented
142D8—2292no—
143D8—2292no—
144D8—2292no—
145D8C2×D22912yescomplemented
146M16—2291no—
147M16—2292no—
148M16C2×D22911yescharacteristic, complemented
149C2×C8—2292no—
150C23—21no—
151C23—9164no—
152C23—4581no—
153C23—4582no—
154C23—4582no—
155C23—4581no—
156C23—2291no—
157Q8—4581no—
158Q8—2291no—
159Q8C22×D22911yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—4581no—
163D4—4581no—
164D4—4581no—
165D4—4582no—
166D4—4582no—
167D4—4581no—
168D4—4581no—
169D4—2291no—
170D4—2292no—
171D4C22×D22912yes—
172D4C22×D22911yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—4581no—
175C2×C4—4581no—
176C2×C4—4581no—
177C2×C4—4581no—
178C2×C4—2291no—
179C2×C4—2291no—
180C2×C4—2291no—
181C2×C4—2291no—
182C2×C4—2291no—
183C2×C4C22×D22911yescharacteristic
184C8—2292no—
185C8C22×D22912yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—9162no—
190C22—9164no—
191C22—9164no—
192C22—9162no—
193C22—4582no—
194C22—4581no—
195C22—4582no—
196C22—2291no—
197C22—2291no—
198C22D4×D22911yescharacteristic
199C4—21no—
200C4—4581no—
201C4—2291no—
202C4—2291no—
203C4D4×D22911yescharacteristic
204C4C23×D22911yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—9162no—
209C2—9161no—
210C2—4581no—
211C2—2292no—
212C2C2×D4×D22911yescharacteristic
subgroupG/H
centerC2C2×D4×D229
commutator, cosocleC4×C229C24
FrattiniC4C23×D229
FittingAGL(1,ℤ/8)×C229C2
socleC2×C229C22×D4
hypercenterAGL(1,ℤ/8)D229
hypocenterC229C2×AGL(1,ℤ/8)
LCSG ▷ C4×C229 ▷ C2×C229 ▷ C229
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D229C2semidirect
2. C2×D4×D229C2semidirect
3. #20301016C2semidirect
4. #20301015C2semidirect
5. #20301014C2semidirect
6. #20301013C2semidirect
7. #20301012C2semidirect
8. SD16×D229C2semidirect
9. D8×D229C2semidirect
10. M16×D229C2semidirect
11. AGL(1,ℤ/8)×C229C2semidirect
13. #16814667C22semidirect
15. #16814666C22semidirect
17. #16814665C22semidirect
18. Q8×D229C22semidirect
19. C2×D916C22semidirect
25. D4×D229C22semidirect
26. D4×D229C22semidirect
29. #16814655C22semidirect
30. #16814654C22semidirect
31. #16814653C22semidirect
32. D1832C22semidirect
33. #16814651C22semidirect
34. #16814651C22semidirect
35. #16814650C22semidirect
36. #16814649C22semidirect
37. C8×D229C22semidirect
38. #41×C229C22semidirect
39. C2×D4×C229C22semidirect
40. SD16×C229C22semidirect
41. D8×C229C22semidirect
42. M16×C229C22semidirect
50. Dic458C23semidirect
54. D916C23semidirect
69. #14270080C23semidirect
71. Q8×C229C23semidirect
75. D4×C229C23semidirect
78. C8×C229C23semidirect
105. D229AGL(1,ℤ/8)direct (N × H)
106. C229C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D229direct (N × H)
125. #41C2×D229semidirect
135. C2×D4C2×D229semidirect
141. SD16C2×D229semidirect
145. D8C2×D229semidirect
148. M16C2×D229semidirect
159. Q8C22×D229semidirect
172. D4C22×D229semidirect
185. C8C22×D229semidirect

Maximal subgroups

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

subgroupindex
#41×D2292
C2×D4×D2292
#203010162
#203010152
#203010142
#203010132
#203010122
SD16×D2292
D8×D2292
M16×D2292
AGL(1,ℤ/8)×C2292
C2×AGL(1,ℤ/8)229