Finite Group Databasebeta

← AGL(1,ℤ/8)×D421 #41003624

Subgroups — AGL(1,ℤ/8)×D421 #41003624

Order 26944.

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×D421C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212394C211yesmaximal, characteristic, complemented
4#27212393C212yesmaximal, complemented
5#27212392C211yesmaximal, characteristic, complemented
6#27212391C211yesmaximal, characteristic, complemented
7#27212390C212yesmaximal, complemented
8SD16×D421C212yesmaximal, complemented
9D8×D421C212yesmaximal, complemented
10M16×D421C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C421C211yesmaximal, characteristic, complemented
12C23×D421—21no—
13#20269772C2211yescharacteristic, complemented
14#20269771—21no—
15#20269771C2211yescharacteristic, complemented
16#20269770—21no—
17#20269770C2211yescharacteristic, complemented
18Q8×D421C2211yescharacteristic, complemented
19C2×D1684C2211yescharacteristic, complemented
20C2×C22⋊D421—21no—
21D4×D421—21no—
22D4×D421—21no—
23D4×D421—21no—
24D4×D421C2212yes—
25D4×D421C2212yescomplemented
26D4×D421C2211yescharacteristic, complemented
27C2×C4×D421—21no—
28C2×C4×D421C2211yescharacteristic
29#20269760C2212yescomplemented
30#20269759C2212yescomplemented
31#20269758C2212yescomplemented
32D3368C2212yescomplemented
33#20269756C2212yescomplemented
34#20269756C2212yescomplemented
35#20269755C2211yescharacteristic, complemented
36#20269754C2212yescomplemented
37C8×D421C2212yescomplemented
38#41×C421C2211yescharacteristic, complemented
39C2×D4×C421C2211yescharacteristic, complemented
40SD16×C421C2212yescomplemented
41D8×C421C2212yescomplemented
42M16×C421C2211yescharacteristic, complemented
43C22×D421—21no—
44C22×D421—21no—
45C22×D421—22no—
46C22×D421—22no—
47C22×D421—44no—
48C22×D421D411yescharacteristic
49Dic842—21no—
50Dic842C2311yescharacteristic, complemented
51D1684—21no—
52D1684—21no—
53D1684C2312yes—
54D1684C2311yescharacteristic, complemented
55C22⋊D421—21no—
56C22⋊D421—21no—
57C22⋊D421—22no—
58C22⋊D421—22no—
59C22⋊D421—21no—
60C2×Dic421—21no—
61C2×Dic421D411yescharacteristic
62C4×D421—21no—
63C4×D421—21no—
64C4×D421—21no—
65C4×D421D411yescharacteristic
66C4×D421D411yescharacteristic
67C4×D421C2311yescharacteristic
68C4×D421C2311yescharacteristic
69#16809890C2312yescomplemented
70C23×C421—21no—
71Q8×C421C2311yescharacteristic, complemented
72D4×C421—21no—
73D4×C421—21no—
74D4×C421C2312yes—
75D4×C421C2311yescharacteristic, complemented
76C2×C4×C421—21no—
77C2×C4×C421C2311yescharacteristic
78C8×C421C2312yescomplemented
79C22×C421—21no—
80C22×C421—22no—
81C22×C421—42no—
82C22×C421C2×D411yescharacteristic
83C2×D421—22no—
84C2×D421—21no—
85C2×D421—22no—
86C2×D421—44no—
87C2×D421—42no—
88C2×D421—42no—
89C2×D421—44no—
90C2×D421C2×D411yescharacteristic
91C2×D421C2×D411yescharacteristic
92C4×C421—21no—
93C4×C421C2×D411yescharacteristic
94C4×C421C2411yescharacteristic
95Dic421—21no—
96Dic421C2×D411yescharacteristic
97Dic421C2×D411yescharacteristic
98C2×C421—21no—
99C2×C421—42no—
100C2×C421—41no—
101C2×C421C22×D411yescharacteristic
102D421—21no—
103D421—41no—
104D421—42no—
105D421AGL(1,ℤ/8)12yesdirect product factor
106C421C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
107C2×AGL(1,ℤ/8)—4211nomaximal, Sylow
108C2×#41—4211no—
109C22×D4—4211no—
110AGL(1,ℤ/8)—4211no—
111AGL(1,ℤ/8)—4211no—
112AGL(1,ℤ/8)—4212no—
113AGL(1,ℤ/8)—4212no—
114AGL(1,ℤ/8)—4211no—
115AGL(1,ℤ/8)D42111yescharacteristic, direct product factor
116C2×SD16—4212no—
117C2×D8—4212no—
118C2×M16—4211no—
119C24—8421no—
120#41—8421no—
121#41—8421no—
122#41—4211no—
123#41—4211no—
124#41—4211no—
125#41C2×D42111yescharacteristic, complemented
126C2×Q8—4211no—
127C2×D4—8421no—
128C2×D4—8421no—
129C2×D4—8421no—
130C2×D4—8421no—
131C2×D4—4212no—
132C2×D4—4212no—
133C2×D4—4211no—
134C2×D4—4211no—
135C2×D4C2×D42111yescharacteristic, complemented
136C22×C4—8421no—
137C22×C4—4211no—
138SD16—4212no—
139SD16—4212no—
140SD16—4212no—
141SD16C2×D42112yescomplemented
142D8—4212no—
143D8—4212no—
144D8—4212no—
145D8C2×D42112yescomplemented
146M16—4211no—
147M16—4212no—
148M16C2×D42111yescharacteristic, complemented
149C2×C8—4212no—
150C23—21no—
151C23—8421no—
152C23—8422no—
153C23—8422no—
154C23—8421no—
155C23—16844no—
156C23—4211no—
157Q8—8421no—
158Q8—4211no—
159Q8C22×D42111yescharacteristic, complemented
160D4—21no—
161D4—21no—
162D4—8421no—
163D4—8421no—
164D4—8421no—
165D4—8422no—
166D4—8422no—
167D4—8421no—
168D4—8421no—
169D4—4211no—
170D4—4212no—
171D4C22×D42112yes—
172D4C22×D42111yescharacteristic, complemented
173C2×C4—21no—
174C2×C4—8421no—
175C2×C4—8421no—
176C2×C4—8421no—
177C2×C4—8421no—
178C2×C4—4211no—
179C2×C4—4211no—
180C2×C4—4211no—
181C2×C4—4211no—
182C2×C4—4211no—
183C2×C4C22×D42111yescharacteristic
184C8—4212no—
185C8C22×D42112yescomplemented
186C22—21no—
187C22—22no—
188C22—42no—
189C22—8422no—
190C22—8421no—
191C22—8422no—
192C22—16842no—
193C22—16844no—
194C22—16844no—
195C22—16842no—
196C22—4211no—
197C22—4211no—
198C22D4×D42111yescharacteristic
199C4—21no—
200C4—8421no—
201C4—4211no—
202C4—4211no—
203C4D4×D42111yescharacteristic
204C4C23×D42111yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2—8421no—
209C2—16842no—
210C2—16841no—
211C2—4212no—
212C2C2×D4×D42111yescharacteristic
subgroupG/H
centerC2C2×D4×D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
FittingAGL(1,ℤ/8)×C421C2
socleC2×C421C22×D4
hypercenterAGL(1,ℤ/8)D421
hypocenterC421C2×AGL(1,ℤ/8)
LCSG ▷ C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D421C2semidirect
2. C2×D4×D421C2semidirect
3. #27212394C2semidirect
4. #27212393C2semidirect
5. #27212392C2semidirect
6. #27212391C2semidirect
7. #27212390C2semidirect
8. SD16×D421C2semidirect
9. D8×D421C2semidirect
10. M16×D421C2semidirect
11. AGL(1,ℤ/8)×C421C2semidirect
13. #20269772C22semidirect
15. #20269771C22semidirect
17. #20269770C22semidirect
18. Q8×D421C22semidirect
19. C2×D1684C22semidirect
25. D4×D421C22semidirect
26. D4×D421C22semidirect
29. #20269760C22semidirect
30. #20269759C22semidirect
31. #20269758C22semidirect
32. D3368C22semidirect
33. #20269756C22semidirect
34. #20269756C22semidirect
35. #20269755C22semidirect
36. #20269754C22semidirect
37. C8×D421C22semidirect
38. #41×C421C22semidirect
39. C2×D4×C421C22semidirect
40. SD16×C421C22semidirect
41. D8×C421C22semidirect
42. M16×C421C22semidirect
50. Dic842C23semidirect
54. D1684C23semidirect
69. #16809890C23semidirect
71. Q8×C421C23semidirect
75. D4×C421C23semidirect
78. C8×C421C23semidirect
105. D421AGL(1,ℤ/8)direct (N × H)
106. C421C2×AGL(1,ℤ/8)semidirect
115. AGL(1,ℤ/8)D421direct (N × H)
125. #41C2×D421semidirect
135. C2×D4C2×D421semidirect
141. SD16C2×D421semidirect
145. D8C2×D421semidirect
148. M16C2×D421semidirect
159. Q8C22×D421semidirect
172. D4C22×D421semidirect
185. C8C22×D421semidirect

Maximal subgroups

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

subgroupindex
#41×D4212
C2×D4×D4212
#272123942
#272123932
#272123922
#272123912
#272123902
SD16×D4212
D8×D4212
M16×D4212
AGL(1,ℤ/8)×C4212
C2×AGL(1,ℤ/8)421