Finite Group Databasebeta

← #27213950

Subgroups — #27213950

Order 13504.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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#20270023C211yesmaximal, characteristic, complemented
2#20270022C211yesmaximal, characteristic, complemented
3#20270005C211yesmaximal, characteristic, complemented
4#20270004C212yesmaximal, complemented
5#20270001C212yesmaximal, complemented
6#20269998C212yesmaximal, complemented
7#20269997C211yesmaximal, characteristic, complemented
8C2×#16809947C211yesmaximal, characteristic, complemented
9D8×D211C212yesmaximal, complemented
10#20269985C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
12#16809963C2212yescomplemented
13#16809962—21no—
14#16809962C2211yescharacteristic
15#16809962C2211yescharacteristic, complemented
16#16809961—21no—
17#16809961C2212yescomplemented
18C2×D844—21no—
19C2×D844C2211yescharacteristic, complemented
20C2×C22⋊D211—22no—
21D4×D211—21no—
22D4×D211—22no—
23D4×D211C2212yescomplemented
24D4×D211C2212yescomplemented
25#16809952C2211yescharacteristic, complemented
26#16809951C2212yescomplemented
27#16809950C2211yescharacteristic, complemented
28#16809949C2211yescharacteristic, complemented
29D1688C2212yescomplemented
30#16809947C2211yescharacteristic, complemented
31#16809947C2212yescomplemented
32#16809947C2212yescomplemented
33#16809945C2212yescomplemented
34C2×C211⋊C8C2211yescharacteristic, complemented
35C8×D211C2212yescomplemented
36#41×C211C2211yescharacteristic, complemented
37C2×D4×C211C2211yescharacteristic, complemented
38SD16×C211C2212yescomplemented
39D8×C211C2212yescomplemented
40M16×C211C2211yescharacteristic, complemented
41C22×D211—21no—
42C22×D211—22no—
43C22×D211—22no—
44Dic422D411yescharacteristic
45Dic422C2311yescharacteristic
46D844—22no—
47D844—21no—
48D844D411yescharacteristic
49D844C2312yes—
50D844C2311yescharacteristic
51C22⋊D211—21no—
52C22⋊D211—22no—
53C22⋊D211—22no—
54C22⋊D211—22no—
55C22⋊D211D412yes—
56C2×Dic211—22no—
57C4×D211—21no—
58C4×D211—22no—
59C4×D211C2312yes—
60C211⋊C8C2311yescharacteristic
61C211⋊C8C2311yescharacteristic
62C23×C211—21no—
63Q8×C211C2311yescharacteristic
64D4×C211—21no—
65D4×C211—21no—
66D4×C211C2311yescharacteristic
67D4×C211C2312yes—
68C2×C4×C211—21no—
69C2×C4×C211C2311yescharacteristic
70C8×C211C2312yes—
71C22×C211—21no—
72C22×C211—22no—
73C22×C211—81no—
74C22×C211C2×D411yescharacteristic
75C2×D211—21no—
76C2×D211—22no—
77C2×D211—82no—
78C2×D211—82no—
79C2×D211—81no—
80C2×D211C2×D412yes—
81C4×C211—21no—
82C4×C211C2×D411yescharacteristic
83C4×C211C2411yescharacteristic
84Dic211—21no—
85Dic211C2×D412yes—
86C2×C211—21no—
87C2×C211—41no—
88C2×C211—42no—
89C2×C211C22×D411yescharacteristic
90D211—22no—
91D211—41no—
92D211—42no—
93C211#57611yescharacteristic, complemented, Sylow
94#576—2111nomaximal, Sylow
95H(5,𝔽2)—2111no—
96H(5,𝔽2)—2111no—
97AGL(1,ℤ/8)—2112no—
98AGL(1,ℤ/8)—2112no—
99AGL(1,ℤ/8)—2111no—
100AGL(1,ℤ/8)D21111yescharacteristic, complemented
101#135—2111no—
102#135—2112no—
103C2×D8—2111no—
104C2×D8—2112no—
105#131—2111no—
106#41—4221no—
107#41—4221no—
108#41—2112no—
109#41—2112no—
110#41—2111no—
111#41—2111no—
112#41C2×D21111yescharacteristic, complemented
113C2×D4—4221no—
114C2×D4—4222no—
115C2×D4—4221no—
116C2×D4—4222no—
117C2×D4—2112no—
118C2×D4—2112no—
119C2×D4—2111no—
120C2×D4C2×D21111yescharacteristic, complemented
121Q16—2111no—
122SD16—2111no—
123SD16—2111no—
124SD16—2112no—
125SD16C2×D21112yescomplemented
126D8—2112no—
127D8—2111no—
128D8—2112no—
129D8—2112no—
130D8C2×D21112yescomplemented
131M16—2112no—
132M16C2×D21111yescharacteristic, complemented
133C2×C8—2111no—
134C2×C8—2112no—
135C23—21no—
136C23—4222no—
137C23—4222no—
138C23—4221no—
139Q8—2111no—
140Q8—2111no—
141Q8C22×D21111yescharacteristic
142D4—21no—
143D4—21no—
144D4—4222no—
145D4—4222no—
146D4—4221no—
147D4—4222no—
148D4—4222no—
149D4—4221no—
150D4—2112no—
151D4—2111no—
152D4—2112no—
153D4—2111no—
154D4C22×D21112yes—
155D4C22×D21111yescharacteristic
156C2×C4—21no—
157C2×C4—4222no—
158C2×C4—4222no—
159C2×C4—4221no—
160C2×C4—2112no—
161C2×C4C22×D21111yescharacteristic
162C8—2111no—
163C8—2111no—
164C8C22×D21112yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—16881no—
169C22—16882no—
170C22—16882no—
171C22—4222no—
172C22—4221no—
173C22—2112no—
174C22D4×D21111yescharacteristic
175C4—21no—
176C4—4221no—
177C4—2112no—
178C4D4×D21111yescharacteristic
179C4C23×D21111yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8442no—
184C2—8441no—
185C2—4222no—
186C2C2×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
hypocenterC211#576
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. #20270023C2semidirect
2. #20270022C2semidirect
3. #20270005C2semidirect
4. #20270004C2semidirect
5. #20270001C2semidirect
6. #20269998C2semidirect
7. #20269997C2semidirect
8. C2×#16809947C2semidirect
9. D8×D211C2semidirect
10. #20269985C2semidirect
11. AGL(1,ℤ/8)×C211C2semidirect
12. #16809963C22semidirect
15. #16809962C22semidirect
17. #16809961C22semidirect
19. C2×D844C22semidirect
23. D4×D211C22semidirect
24. D4×D211C22semidirect
25. #16809952C22semidirect
26. #16809951C22semidirect
27. #16809950C22semidirect
28. #16809949C22semidirect
29. D1688C22semidirect
30. #16809947C22semidirect
31. #16809947C22semidirect
32. #16809947C22semidirect
33. #16809945C22semidirect
34. C2×C211⋊C8C22semidirect
35. C8×D211C22semidirect
36. #41×C211C22semidirect
37. C2×D4×C211C22semidirect
38. SD16×C211C22semidirect
39. D8×C211C22semidirect
40. M16×C211C22semidirect
93. C211#576semidirect
100. AGL(1,ℤ/8)D211semidirect
112. #41C2×D211semidirect
120. C2×D4C2×D211semidirect
125. SD16C2×D211semidirect
130. D8C2×D211semidirect
132. M16C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202700232
#202700222
#202700052
#202700042
#202700012
#202699982
#202699972
C2×#168099472
D8×D2112
#202699852
AGL(1,ℤ/8)×C2112
#576211