Finite Group Databasebeta

← AGL(1,ℤ/8)×D211 #27213930

Subgroups — AGL(1,ℤ/8)×D211 #27213930

Order 13504.

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×D211C211yesmaximal, characteristic, complemented
2C2×D4×D211C211yesmaximal, characteristic, complemented
3#20270005C211yesmaximal, characteristic, complemented
4#20270004C212yesmaximal, complemented
5#20270003C211yesmaximal, characteristic, complemented
6#20270002C211yesmaximal, characteristic, complemented
7#20270001C212yesmaximal, complemented
8SD16×D211C212yesmaximal, complemented
9D8×D211C212yesmaximal, complemented
10M16×D211C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
12C23×D211—21no—
13#16809963C2211yescharacteristic, complemented
14#16809962—21no—
15#16809962C2211yescharacteristic, complemented
16#16809961—21no—
17#16809961C2211yescharacteristic, complemented
18Q8×D211C2211yescharacteristic, complemented
19C2×D844C2211yescharacteristic, complemented
20C2×C22⋊D211—21no—
21D4×D211—21no—
22D4×D211—21no—
23D4×D211—21no—
24D4×D211C2212yes—
25D4×D211C2212yescomplemented
26D4×D211C2211yescharacteristic, complemented
27C2×C4×D211—21no—
28C2×C4×D211C2211yescharacteristic
29#16809951C2212yescomplemented
30#16809950C2212yescomplemented
31#16809949C2212yescomplemented
32D1688C2212yescomplemented
33#16809947C2212yescomplemented
34#16809947C2212yescomplemented
35#16809946C2211yescharacteristic, complemented
36#16809945C2212yescomplemented
37C8×D211C2212yescomplemented
38#41×C211C2211yescharacteristic, complemented
39C2×D4×C211C2211yescharacteristic, complemented
40SD16×C211C2212yescomplemented
41D8×C211C2212yescomplemented
42M16×C211C2211yescharacteristic, complemented
43C22×D211—21no—
44C22×D211—21no—
45C22×D211—22no—
46C22×D211—22no—
47C22×D211—44no—
48C22×D211D411yescharacteristic
49Dic422—21no—
50Dic422C2311yescharacteristic, complemented
51D844—21no—
52D844—21no—
53D844C2312yes—
54D844C2311yescharacteristic, complemented
55C22⋊D211—21no—
56C22⋊D211—21no—
57C22⋊D211—22no—
58C22⋊D211—22no—
59C22⋊D211—21no—
60C2×Dic211—21no—
61C2×Dic211D411yescharacteristic
62C4×D211—21no—
63C4×D211—21no—
64C4×D211—21no—
65C4×D211D411yescharacteristic
66C4×D211D411yescharacteristic
67C4×D211C2311yescharacteristic
68C4×D211C2311yescharacteristic
69C211⋊C8C2312yescomplemented
70C23×C211—21no—
71Q8×C211C2311yescharacteristic, complemented
72D4×C211—21no—
73D4×C211—21no—
74D4×C211C2312yes—
75D4×C211C2311yescharacteristic, complemented
76C2×C4×C211—21no—
77C2×C4×C211C2311yescharacteristic
78C8×C211C2312yescomplemented
79C22×C211—21no—
80C22×C211—22no—
81C22×C211—42no—
82C22×C211C2×D411yescharacteristic
83C2×D211—22no—
84C2×D211—21no—
85C2×D211—22no—
86C2×D211—44no—
87C2×D211—42no—
88C2×D211—42no—
89C2×D211—44no—
90C2×D211C2×D411yescharacteristic
91C2×D211C2×D411yescharacteristic
92C4×C211—21no—
93C4×C211C2×D411yescharacteristic
94C4×C211C2411yescharacteristic
95Dic211—21no—
96Dic211C2×D411yescharacteristic
97Dic211C2×D411yescharacteristic
98C2×C211—21no—
99C2×C211—42no—
100C2×C211—41no—
101C2×C211C22×D411yescharacteristic
102D211—21no—
103D211—41no—
104D211—42no—
105D211AGL(1,ℤ/8)12yesdirect product factor
106C211C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—2111nomaximal, Sylow
108C2×#41—2111no—
109C22×D4—2111no—
110AGL(1,ℤ/8)—2111no—
111AGL(1,ℤ/8)—2111no—
112AGL(1,ℤ/8)—2112no—
113AGL(1,ℤ/8)—2112no—
114AGL(1,ℤ/8)—2111no—
115AGL(1,ℤ/8)D21111yescharacteristic, direct product factor
116C2×SD16—2112no—
117C2×D8—2112no—
118C2×M16—2111no—
119C24—4221no—
120#41—4221no—
121#41—4221no—
122#41—2111no—
123#41—2111no—
124#41—2111no—
125#41C2×D21111yescharacteristic, complemented
126C2×Q8—2111no—
127C2×D4—4221no—
128C2×D4—4221no—
129C2×D4—4221no—
130C2×D4—4221no—
131C2×D4—2112no—
132C2×D4—2112no—
133C2×D4—2111no—
134C2×D4—2111no—
135C2×D4C2×D21111yescharacteristic, complemented
136C22×C4—4221no—
137C22×C4—2111no—
138SD16—2112no—
139SD16—2112no—
140SD16—2112no—
141SD16C2×D21112yescomplemented
142D8—2112no—
143D8—2112no—
144D8—2112no—
145D8C2×D21112yescomplemented
146M16—2111no—
147M16—2112no—
148M16C2×D21111yescharacteristic, complemented
149C2×C8—2112no—
150C23—21no—
151C23—8444no—
152C23—4221no—
153C23—4222no—
154C23—4222no—
155C23—4221no—
156C23—2111no—
157Q8—4221no—
158Q8—2111no—
159Q8C22×D21111yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—4221no—
163D4—4221no—
164D4—4221no—
165D4—4222no—
166D4—4222no—
167D4—4221no—
168D4—4221no—
169D4—2111no—
170D4—2112no—
171D4C22×D21112yes—
172D4C22×D21111yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4—2111no—
179C2×C4—2111no—
180C2×C4—2111no—
181C2×C4—2111no—
182C2×C4—2111no—
183C2×C4C22×D21111yescharacteristic
184C8—2112no—
185C8C22×D21112yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—8442no—
190C22—8444no—
191C22—8444no—
192C22—8442no—
193C22—4222no—
194C22—4221no—
195C22—4222no—
196C22—2111no—
197C22—2111no—
198C22D4×D21111yescharacteristic
199C4—21no—
200C4—4221no—
201C4—2111no—
202C4—2111no—
203C4D4×D21111yescharacteristic
204C4C23×D21111yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—8442no—
209C2—8441no—
210C2—4221no—
211C2—2112no—
212C2C2×D4×D21111yescharacteristic
subgroupG/H
centerC2C2×D4×D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
FittingAGL(1,ℤ/8)×C211C2
socleC2×C211C22×D4
hypercenterAGL(1,ℤ/8)D211
hypocenterC211C2×AGL(1,ℤ/8)
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D211C2semidirect
2. C2×D4×D211C2semidirect
3. #20270005C2semidirect
4. #20270004C2semidirect
5. #20270003C2semidirect
6. #20270002C2semidirect
7. #20270001C2semidirect
8. SD16×D211C2semidirect
9. D8×D211C2semidirect
10. M16×D211C2semidirect
11. AGL(1,ℤ/8)×C211C2semidirect
13. #16809963C22semidirect
15. #16809962C22semidirect
17. #16809961C22semidirect
18. Q8×D211C22semidirect
19. C2×D844C22semidirect
25. D4×D211C22semidirect
26. D4×D211C22semidirect
29. #16809951C22semidirect
30. #16809950C22semidirect
31. #16809949C22semidirect
32. D1688C22semidirect
33. #16809947C22semidirect
34. #16809947C22semidirect
35. #16809946C22semidirect
36. #16809945C22semidirect
37. C8×D211C22semidirect
38. #41×C211C22semidirect
39. C2×D4×C211C22semidirect
40. SD16×C211C22semidirect
41. D8×C211C22semidirect
42. M16×C211C22semidirect
50. Dic422C23semidirect
54. D844C23semidirect
69. C211⋊C8C23semidirect
71. Q8×C211C23semidirect
75. D4×C211C23semidirect
78. C8×C211C23semidirect
105. D211AGL(1,ℤ/8)direct (N × H)
106. C211C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D211direct (N × H)
125. #41C2×D211semidirect
135. C2×D4C2×D211semidirect
141. SD16C2×D211semidirect
145. D8C2×D211semidirect
148. M16C2×D211semidirect
159. Q8C22×D211semidirect
172. D4C22×D211semidirect
185. C8C22×D211semidirect

Maximal subgroups

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

subgroupindex
#41×D2112
C2×D4×D2112
#202700052
#202700042
#202700032
#202700022
#202700012
SD16×D2112
D8×D2112
M16×D2112
AGL(1,ℤ/8)×C2112
C2×AGL(1,ℤ/8)211