Finite Group Databasebeta

← #41001005

Subgroups — #41001005

Order 26816.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#20269410C211yesmaximal, characteristic, complemented
2#27211705C211yesmaximal, characteristic, complemented
3#27211699C212yesmaximal, complemented
4#27211697C211yesmaximal, characteristic, complemented
5#27211690C212yesmaximal, complemented
6#27211689C211yesmaximal, characteristic, complemented
7#27211687C212yesmaximal, complemented
8D4×Dic419C211yesmaximal, characteristic, complemented
9C4×C22⋊D419C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic419C212yesmaximal, complemented
11C22×D4×C419C211yesmaximal, characteristic, complemented
12#20269410—24no—
13C2×Dic838C2211yescharacteristic, complemented
14C2×C22⋊D419C2211yescharacteristic
15C2×C22⋊D419C2212yescomplemented
16C2×C22⋊D419C2212yescomplemented
17C22×Dic419C2212yescomplemented
18C22×Dic419C2212yescomplemented
19C2×C4×D419C2211yescharacteristic, complemented
20C4⋊Dic419C2211yescharacteristic, complemented
21Dic419⋊C4C2211yescharacteristic, complemented
22Dic419⋊C4C2212yescomplemented
23C22⋊Dic419C2211yescharacteristic
24C22⋊Dic419C2212yes—
25C22⋊Dic419C2212yescomplemented
26C22⋊Dic419C2212yescomplemented
27C22⋊Dic419C2212yescomplemented
28C22⋊Dic419C2212yescomplemented
29(C2×D419)⋊C4C2211yescharacteristic, complemented
30C4×Dic419C2211yescharacteristic, complemented
31C24×C419C2212yes—
32C2×D4×C419—24no—
33C2×D4×C419C2211yescharacteristic
34C2×D4×C419C2212yes—
35C2×D4×C419C2211yescharacteristic
36C22×C4×C419C2211yescharacteristic
37C22×D419C2311yescharacteristic
38Dic838—22no—
39C22⋊D419—24no—
40C22⋊D419—22no—
41C22⋊D419—42no—
42C2×Dic419—24no—
43C2×Dic419—42no—
44C2×Dic419C2311yescharacteristic
45C2×Dic419C2311yescharacteristic
46C2×Dic419C2312yes—
47C2×Dic419C2311yescharacteristic
48C2×Dic419C2312yes—
49C4×D419—22no—
50C23×C419—24no—
51C23×C419—24no—
52C23×C419—42no—
53C23×C419C2312yes—
54C23×C419C2312yes—
55C23×C419C2311yescharacteristic
56D4×C419—24no—
57D4×C419—22no—
58D4×C419D414yescomplemented
59C2×C4×C419—22no—
60C2×C4×C419C2311yescharacteristic
61C2×C4×C419C2311yescharacteristic
62C22×C419—21no—
63C22×C419—22no—
64C22×C419—21no—
65C22×C419—22no—
66C22×C419—22no—
67C22×C419—22no—
68C22×C419—22no—
69C22×C419—48no—
70C22×C419C2×D414yes—
71C22×C419#4112yescomplemented
72C22×C419C2411yescharacteristic
73C2×D419—21no—
74C2×D419—21no—
75C2×D419—21no—
76C4×C419—21no—
77C4×C419C2×D412yes—
78Dic419—21no—
79Dic419—22no—
80Dic419—21no—
81Dic419—21no—
82Dic419—22no—
83C2×C419—22no—
84C2×C419—24no—
85C2×C419—42no—
86C2×C419C22×D411yescharacteristic
87C2×C419C2×#4111yescharacteristic
88C2×C419H(5,𝔽2)11yescharacteristic
89D419—41no—
90C419#54611yescharacteristic, complemented, Sylow
91#546—4191nomaximal, Sylow
92C2×#41—4191no—
93C22×D4D41911yescharacteristic, complemented
94#124—4191no—
95#123—4192no—
96C22⋊Q8—4191no—
97#121—4191no—
98#121—4192no—
99#120—4192no—
100C4×D4—4191no—
101C4×D4—4191no—
102C2×C22⋊C4—4192no—
103C24C2×D41912yes—
104#41—8384no—
105C2×Q8—4191no—
106C2×D4—24no—
107C2×D4—4191no—
108C2×D4—4192no—
109C2×D4—4192no—
110C2×D4C2×D41912yes—
111C2×D4C2×D41911yescharacteristic
112C2×D4C2×D41911yescharacteristic
113C22×C4—4192no—
114C22×C4—4191no—
115C22×C4—4192no—
116C22×C4C2×D41911yescharacteristic
117C4⋊C4—4191no—
118C4⋊C4—4192no—
119C4⋊C4—4191no—
120C22⋊C4—4191no—
121C22⋊C4—4192no—
122C22⋊C4—4192no—
123C22⋊C4—4192no—
124C22⋊C4—4192no—
125C22⋊C4—4192no—
126C22⋊C4—4191no—
127C42—4191no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—4191no—
132C23C22×D41912yes—
133C23C22×D41912yes—
134C23C22×D41911yescharacteristic
135Q8—8382no—
136D4—24no—
137D4—22no—
138D4—8384no—
139D4—8382no—
140D4—16762no—
141D4C22⋊D41914yescomplemented
142C2×C4—22no—
143C2×C4—8384no—
144C2×C4—8382no—
145C2×C4—16762no—
146C2×C4—4191no—
147C2×C4—4191no—
148C2×C4—4192no—
149C2×C4—4191no—
150C2×C4—4192no—
151C2×C4C22×D41911yescharacteristic
152C2×C4C22×D41911yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—8381no—
162C22—8381no—
163C22—8381no—
164C22C2×C22⋊D41914yes—
165C22#2026941012yescomplemented
166C22C23×D41911yescharacteristic
167C4—21no—
168C4—8381no—
169C4—8382no—
170C4—8381no—
171C4—8381no—
172C4—8382no—
173C4C2×C22⋊D41912yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—16761no—
178C2C22×C22⋊D41911yescharacteristic
179C2C2×#2026941011yescharacteristic
180C2#2721176311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D419
commutator, socle, cosocleC22×C419C24
FittingC22×D4×C419C2
hypercenterC22×D4D419
hypocenterC419#546
LCSG ▷ C22×C419 ▷ C419
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20269410C2semidirect
2. #27211705C2semidirect
3. #27211699C2semidirect
4. #27211697C2semidirect
5. #27211690C2semidirect
6. #27211689C2semidirect
7. #27211687C2semidirect
8. D4×Dic419C2semidirect
9. C4×C22⋊D419C2semidirect
10. C2×C22⋊Dic419C2semidirect
11. C22×D4×C419C2semidirect
13. C2×Dic838C22semidirect
15. C2×C22⋊D419C22semidirect
16. C2×C22⋊D419C22semidirect
17. C22×Dic419C22semidirect
18. C22×Dic419C22semidirect
19. C2×C4×D419C22semidirect
20. C4⋊Dic419C22semidirect
21. Dic419⋊C4C22semidirect
22. Dic419⋊C4C22semidirect
25. C22⋊Dic419C22semidirect
26. C22⋊Dic419C22semidirect
27. C22⋊Dic419C22semidirect
28. C22⋊Dic419C22semidirect
29. (C2×D419)⋊C4C22semidirect
30. C4×Dic419C22semidirect
58. D4×C419D4semidirect
71. C22×C419#41semidirect
90. C419#546semidirect
93. C22×D4D419semidirect
141. D4C22⋊D419semidirect
165. C22#20269410semidirect

Maximal subgroups

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

subgroupindex
C2×#202694102
#272117052
#272116992
#272116972
#272116902
#272116892
#272116872
D4×Dic4192
C4×C22⋊D4192
C2×C22⋊Dic4192
C22×D4×C4192
#546419