Finite Group Databasebeta

← #41001165

Subgroups — #41001165

Order 26816.

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#27211764C211yesmaximal, characteristic, complemented
2#27211763C211yesmaximal, characteristic, complemented
3#27211746C211yesmaximal, characteristic, complemented
4#27211745C212yesmaximal, complemented
5#27211742C212yesmaximal, complemented
6#27211739C212yesmaximal, complemented
7#27211738C211yesmaximal, characteristic, complemented
8C2×#20269396C211yesmaximal, characteristic, complemented
9D8×D419C212yesmaximal, complemented
10#27211726C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C419C211yesmaximal, characteristic, complemented
12#20269412C2212yescomplemented
13#20269411—21no—
14#20269411C2211yescharacteristic
15#20269411C2211yescharacteristic, complemented
16#20269410—21no—
17#20269410C2212yescomplemented
18C2×D1676—21no—
19C2×D1676C2211yescharacteristic, complemented
20C2×C22⋊D419—22no—
21D4×D419—21no—
22D4×D419—22no—
23D4×D419C2212yescomplemented
24D4×D419C2212yescomplemented
25#20269401C2211yescharacteristic, complemented
26#20269400C2212yescomplemented
27#20269399C2211yescharacteristic, complemented
28#20269398C2211yescharacteristic, complemented
29D3352C2212yescomplemented
30#20269396C2211yescharacteristic, complemented
31#20269396C2212yescomplemented
32#20269396C2212yescomplemented
33#20269394C2212yescomplemented
34C2×C419⋊C8C2211yescharacteristic, complemented
35C8×D419C2212yescomplemented
36#41×C419C2211yescharacteristic, complemented
37C2×D4×C419C2211yescharacteristic, complemented
38SD16×C419C2212yescomplemented
39D8×C419C2212yescomplemented
40M16×C419C2211yescharacteristic, complemented
41C22×D419—21no—
42C22×D419—22no—
43C22×D419—22no—
44Dic838D411yescharacteristic
45Dic838C2311yescharacteristic
46D1676—22no—
47D1676—21no—
48D1676D411yescharacteristic
49D1676C2312yes—
50D1676C2311yescharacteristic
51C22⋊D419—21no—
52C22⋊D419—22no—
53C22⋊D419—22no—
54C22⋊D419—22no—
55C22⋊D419D412yes—
56C2×Dic419—22no—
57C4×D419—21no—
58C4×D419—22no—
59C4×D419C2312yes—
60C419⋊C8C2311yescharacteristic
61C419⋊C8C2311yescharacteristic
62C23×C419—21no—
63Q8×C419C2311yescharacteristic
64D4×C419—21no—
65D4×C419—21no—
66D4×C419C2311yescharacteristic
67D4×C419C2312yes—
68C2×C4×C419—21no—
69C2×C4×C419C2311yescharacteristic
70C8×C419C2312yes—
71C22×C419—21no—
72C22×C419—22no—
73C22×C419—81no—
74C22×C419C2×D411yescharacteristic
75C2×D419—21no—
76C2×D419—22no—
77C2×D419—82no—
78C2×D419—82no—
79C2×D419—81no—
80C2×D419C2×D412yes—
81C4×C419—21no—
82C4×C419C2×D411yescharacteristic
83C4×C419C2411yescharacteristic
84Dic419—21no—
85Dic419C2×D412yes—
86C2×C419—21no—
87C2×C419—41no—
88C2×C419—42no—
89C2×C419C22×D411yescharacteristic
90D419—22no—
91D419—41no—
92D419—42no—
93C419#57611yescharacteristic, complemented, Sylow
94#576—4191nomaximal, Sylow
95H(5,𝔽2)—4191no—
96H(5,𝔽2)—4191no—
97AGL(1,ℤ/8)—4192no—
98AGL(1,ℤ/8)—4192no—
99AGL(1,ℤ/8)—4191no—
100AGL(1,ℤ/8)D41911yescharacteristic, complemented
101#135—4191no—
102#135—4192no—
103C2×D8—4191no—
104C2×D8—4192no—
105#131—4191no—
106#41—8381no—
107#41—8381no—
108#41—4192no—
109#41—4192no—
110#41—4191no—
111#41—4191no—
112#41C2×D41911yescharacteristic, complemented
113C2×D4—8381no—
114C2×D4—8382no—
115C2×D4—8381no—
116C2×D4—8382no—
117C2×D4—4192no—
118C2×D4—4192no—
119C2×D4—4191no—
120C2×D4C2×D41911yescharacteristic, complemented
121Q16—4191no—
122SD16—4191no—
123SD16—4191no—
124SD16—4192no—
125SD16C2×D41912yescomplemented
126D8—4192no—
127D8—4191no—
128D8—4192no—
129D8—4192no—
130D8C2×D41912yescomplemented
131M16—4192no—
132M16C2×D41911yescharacteristic, complemented
133C2×C8—4191no—
134C2×C8—4192no—
135C23—21no—
136C23—8382no—
137C23—8382no—
138C23—8381no—
139Q8—4191no—
140Q8—4191no—
141Q8C22×D41911yescharacteristic
142D4—21no—
143D4—21no—
144D4—8382no—
145D4—8382no—
146D4—8381no—
147D4—8382no—
148D4—8382no—
149D4—8381no—
150D4—4192no—
151D4—4191no—
152D4—4192no—
153D4—4191no—
154D4C22×D41912yes—
155D4C22×D41911yescharacteristic
156C2×C4—21no—
157C2×C4—8382no—
158C2×C4—8382no—
159C2×C4—8381no—
160C2×C4—4192no—
161C2×C4C22×D41911yescharacteristic
162C8—4191no—
163C8—4191no—
164C8C22×D41912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—33521no—
169C22—33522no—
170C22—33522no—
171C22—8382no—
172C22—8381no—
173C22—4192no—
174C22D4×D41911yescharacteristic
175C4—21no—
176C4—8381no—
177C4—4192no—
178C4D4×D41911yescharacteristic
179C4C23×D41911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8382no—
184C2—16762no—
185C2—16761no—
186C2C2×D4×D41911yescharacteristic
subgroupG/H
centerC2C2×D4×D419
commutator, cosocleC4×C419C24
FrattiniC4C23×D419
FittingAGL(1,ℤ/8)×C419C2
socleC2×C419C22×D4
hypercenterAGL(1,ℤ/8)D419
hypocenterC419#576
LCSG ▷ C4×C419 ▷ C2×C419 ▷ C419
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27211764C2semidirect
2. #27211763C2semidirect
3. #27211746C2semidirect
4. #27211745C2semidirect
5. #27211742C2semidirect
6. #27211739C2semidirect
7. #27211738C2semidirect
8. C2×#20269396C2semidirect
9. D8×D419C2semidirect
10. #27211726C2semidirect
11. AGL(1,ℤ/8)×C419C2semidirect
12. #20269412C22semidirect
15. #20269411C22semidirect
17. #20269410C22semidirect
19. C2×D1676C22semidirect
23. D4×D419C22semidirect
24. D4×D419C22semidirect
25. #20269401C22semidirect
26. #20269400C22semidirect
27. #20269399C22semidirect
28. #20269398C22semidirect
29. D3352C22semidirect
30. #20269396C22semidirect
31. #20269396C22semidirect
32. #20269396C22semidirect
33. #20269394C22semidirect
34. C2×C419⋊C8C22semidirect
35. C8×D419C22semidirect
36. #41×C419C22semidirect
37. C2×D4×C419C22semidirect
38. SD16×C419C22semidirect
39. D8×C419C22semidirect
40. M16×C419C22semidirect
93. C419#576semidirect
100. AGL(1,ℤ/8)D419semidirect
112. #41C2×D419semidirect
120. C2×D4C2×D419semidirect
125. SD16C2×D419semidirect
130. D8C2×D419semidirect
132. M16C2×D419semidirect

Maximal subgroups

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

subgroupindex
#272117642
#272117632
#272117462
#272117452
#272117422
#272117392
#272117382
C2×#202693962
D8×D4192
#272117262
AGL(1,ℤ/8)×C4192
#576419