Finite Group Databasebeta

← #41001171

Subgroups — #41001171

Order 26816.

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

normal subgroups only

This group has 258 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#27211765C211yesmaximal, characteristic, complemented
2#27211763C211yesmaximal, characteristic, complemented
3#27211751C211yesmaximal, characteristic, complemented
4#27211747C212yesmaximal, complemented
5#27211742C212yesmaximal, complemented
6#27211738C211yesmaximal, characteristic, complemented
7#27211737C212yesmaximal, complemented
8C2×#20269399C211yesmaximal, characteristic, complemented
9SD16×D419C212yesmaximal, complemented
10#27211726C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C419C211yesmaximal, characteristic, complemented
12#20269411—21no—
13#20269411C2211yescharacteristic
14#20269411C2211yescharacteristic
15#20269410—21no—
16#20269410—22no—
17#20269410C2212yes—
18#20269410C2212yescomplemented
19C2×Dic838—21no—
20C2×Dic838C2211yescharacteristic, complemented
21Q8×D419C2212yescomplemented
22C2×C22⋊D419—22no—
23D4×D419—21no—
24D4×D419C2212yes—
25Dic1676C2212yescomplemented
26#20269401C2211yescharacteristic, complemented
27#20269400C2212yescomplemented
28#20269399C2212yescomplemented
29#20269399C2212yescomplemented
30#20269399C2211yescharacteristic, complemented
31#20269398C2211yescharacteristic, complemented
32#20269396C2211yescharacteristic, complemented
33#20269394C2212yescomplemented
34C2×C419⋊C8C2211yescharacteristic, complemented
35C8×D419C2212yescomplemented
36#41×C419C2211yescharacteristic, complemented
37C2×D4×C419C2211yescharacteristic
38SD16×C419C2212yescomplemented
39D8×C419C2212yescomplemented
40M16×C419C2211yescharacteristic, complemented
41C22×D419—22no—
42Dic838—21no—
43Dic838—22no—
44Dic838D411yescharacteristic
45Dic838C2312yes—
46Dic838C2311yescharacteristic
47D1676D411yescharacteristic
48D1676C2311yescharacteristic
49C22⋊D419—21no—
50C22⋊D419—22no—
51C22⋊D419—22no—
52C22⋊D419—22no—
53C22⋊D419D412yes—
54C2×Dic419—21no—
55C2×Dic419—22no—
56C2×Dic419—22no—
57C4×D419—22no—
58C4×D419—21no—
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—82no—
77C2×D419C2×D412yes—
78C4×C419—21no—
79C4×C419C2×D411yescharacteristic
80C4×C419C2411yescharacteristic
81Dic419—21no—
82Dic419—22no—
83Dic419C2×D412yes—
84C2×C419—21no—
85C2×C419—42no—
86C2×C419—41no—
87C2×C419C22×D411yescharacteristic
88D419—22no—
89D419—41no—
90C419#57711yescharacteristic, complemented, Sylow
91#577—4191nomaximal, Sylow
922−1+4—4191no—
93H(5,𝔽2)—4191no—
94#137—4192no—
95#137—4191no—
96AGL(1,ℤ/8)—4192no—
97AGL(1,ℤ/8)D41911yescharacteristic, complemented
98#135—4191no—
99#135—4192no—
100C2×SD16—4191no—
101C2×SD16—4192no—
102#131—4191no—
103#41—8381no—
104#41—8381no—
105#41—8382no—
106#41—4192no—
107#41—4192no—
108#41—4191no—
109#41—4191no—
110#41C2×D41911yescharacteristic, complemented
111C2×Q8—8381no—
112C2×Q8—4192no—
113C2×Q8—4191no—
114C2×D4—8381no—
115C2×D4—8382no—
116C2×D4—4192no—
117C2×D4C2×D41911yescharacteristic
118Q16—4191no—
119Q16—4192no—
120SD16—4191no—
121SD16—4192no—
122SD16—4192no—
123SD16—4191no—
124SD16—4192no—
125SD16C2×D41912yescomplemented
126D8—4191no—
127D8C2×D41912yescomplemented
128M16—4192no—
129M16C2×D41911yescharacteristic, complemented
130C2×C8—4191no—
131C2×C8—4192no—
132C23—21no—
133C23—8382no—
134Q8—8382no—
135Q8—8381no—
136Q8—4192no—
137Q8—4191no—
138Q8—4191no—
139Q8C22×D41911yescharacteristic
140D4—21no—
141D4—21no—
142D4—8382no—
143D4—8382no—
144D4—8381no—
145D4—8382no—
146D4—4192no—
147D4—4191no—
148D4—4191no—
149D4C22×D41911yescharacteristic
150D4C22×D41912yes—
151C2×C4—21no—
152C2×C4—8382no—
153C2×C4—8381no—
154C2×C4—8382no—
155C2×C4—8382no—
156C2×C4—8381no—
157C2×C4—4192no—
158C2×C4C22×D41911yescharacteristic
159C8—4191no—
160C8—4191no—
161C8C22×D41912yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—33522no—
166C22—8381no—
167C22—4192no—
168C22D4×D41911yescharacteristic
169C4—21no—
170C4—8381no—
171C4—8382no—
172C4—4192no—
173C4D4×D41911yescharacteristic
174C4C23×D41911yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—8382no—
179C2—16761no—
180C2C2×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#577
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. #27211765C2semidirect
2. #27211763C2semidirect
3. #27211751C2semidirect
4. #27211747C2semidirect
5. #27211742C2semidirect
6. #27211738C2semidirect
7. #27211737C2semidirect
8. C2×#20269399C2semidirect
9. SD16×D419C2semidirect
10. #27211726C2semidirect
11. AGL(1,ℤ/8)×C419C2semidirect
18. #20269410C22semidirect
20. C2×Dic838C22semidirect
21. Q8×D419C22semidirect
25. Dic1676C22semidirect
26. #20269401C22semidirect
27. #20269400C22semidirect
28. #20269399C22semidirect
29. #20269399C22semidirect
30. #20269399C22semidirect
31. #20269398C22semidirect
32. #20269396C22semidirect
33. #20269394C22semidirect
34. C2×C419⋊C8C22semidirect
35. C8×D419C22semidirect
36. #41×C419C22semidirect
38. SD16×C419C22semidirect
39. D8×C419C22semidirect
40. M16×C419C22semidirect
90. C419#577semidirect
97. AGL(1,ℤ/8)D419semidirect
110. #41C2×D419semidirect
125. SD16C2×D419semidirect
127. D8C2×D419semidirect
129. M16C2×D419semidirect

Maximal subgroups

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

subgroupindex
#272117652
#272117632
#272117512
#272117472
#272117422
#272117382
#272117372
C2×#202693992
SD16×D4192
#272117262
AGL(1,ℤ/8)×C4192
#577419