Finite Group Databasebeta

← #41001006

Subgroups — #41001006

Order 26816.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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
2C2×D4×D419C211yesmaximal, characteristic, complemented
3#27211706C211yesmaximal, characteristic, complemented
4#27211702C212yesmaximal, complemented
5#27211698C211yesmaximal, characteristic, complemented
6#27211693C211yesmaximal, characteristic, complemented
7#27211692C212yesmaximal, complemented
8#27211688C212yesmaximal, complemented
9C4×D1676C211yesmaximal, characteristic, complemented
10C2×(C2×D419)⋊C4C212yesmaximal, complemented
11C4×D4×C419C211yesmaximal, characteristic, complemented
12C23×D419C2212yes—
13#20269410—24no—
14C2×Dic838C2211yescharacteristic, complemented
15C2×D1676C2211yescharacteristic
16C2×D1676C2211yescharacteristic
17C2×D1676C2212yescomplemented
18C2×C22⋊D419C2212yes—
19C2×C22⋊D419C2212yescomplemented
20D4×D419—24no—
21C22×Dic419C2212yescomplemented
22C2×C4×D419C2211yescharacteristic
23C2×C4×D419C2211yescharacteristic, complemented
24C4⋊Dic419C2211yescharacteristic, complemented
25C4⋊Dic419C2212yescomplemented
26(C2×D419)⋊C4C2211yescharacteristic
27(C2×D419)⋊C4C2212yes—
28(C2×D419)⋊C4C2212yescomplemented
29(C2×D419)⋊C4C2212yescomplemented
30(C2×D419)⋊C4C2211yescharacteristic, complemented
31(C2×D419)⋊C4C2212yescomplemented
32C2×D4×C419C2211yescharacteristic
33C22×C4×C419C2212yescomplemented
34C4⋊C4×C419C2211yescharacteristic, complemented
35C22⋊C4×C419C2212yescomplemented
36C42×C419C2211yescharacteristic, complemented
37C22×D419—24no—
38C22×D419—24no—
39C22×D419—42no—
40C22×D419C2312yes—
41C22×D419C2311yescharacteristic
42C22×D419C2311yescharacteristic
43Dic838—22no—
44D1676—22no—
45D1676—22no—
46D1676—42no—
47C22⋊D419—24no—
48C22⋊D419—24no—
49C2×Dic419—24no—
50C2×Dic419C2311yescharacteristic
51C2×Dic419C2312yes—
52C2×Dic419C2311yescharacteristic
53C4×D419—22no—
54C4×D419—22no—
55C23×C419C2312yes—
56D4×C419D414yescomplemented
57C2×C4×C419—42no—
58C2×C4×C419C2311yescharacteristic
59C2×C4×C419C2311yescharacteristic
60C2×C4×C419C2312yes—
61C2×C4×C419C2311yescharacteristic
62C22×C419—22no—
63C22×C419—22no—
64C22×C419C2×D414yes—
65C22×C419C2411yescharacteristic
66C2×D419—21no—
67C2×D419—21no—
68C2×D419—21no—
69C2×D419—22no—
70C2×D419—21no—
71C2×D419—22no—
72C2×D419—21no—
73C2×D419—22no—
74C2×D419—48no—
75C2×D419#4112yescomplemented
76C4×C419—21no—
77C4×C419—21no—
78C4×C419—22no—
79C4×C419C2×D412yes—
80Dic419—21no—
81Dic419—21no—
82Dic419—22no—
83C2×C419—24no—
84C2×C419C22×D411yescharacteristic
85C2×C419C2×#4111yescharacteristic
86C2×C419H(5,𝔽2)11yescharacteristic
87D419—22no—
88D419—41no—
89D419—42no—
90C419#54611yescharacteristic, complemented, Sylow
91#546—4191nomaximal, Sylow
92C2×#41—4191no—
93C22×D4—4191no—
94#124—4191no—
95#123—4192no—
96C22⋊Q8—4191no—
97#121—4192no—
98#121—4191no—
99#120—4192no—
100C4×D4—4191no—
101C4×D4D41911yescharacteristic, complemented
102C2×C22⋊C4—4192no—
103C24—4192no—
104#41—8384no—
105C2×Q8—4191no—
106C2×D4—8384no—
107C2×D4—4192no—
108C2×D4—4192no—
109C2×D4—4191no—
110C2×D4—4191no—
111C2×D4—4192no—
112C2×D4C2×D41911yescharacteristic
113C22×C4—4192no—
114C22×C4—4191no—
115C22×C4—4191no—
116C22×C4C2×D41912yescomplemented
117C4⋊C4—4191no—
118C4⋊C4—4192no—
119C4⋊C4C2×D41911yescharacteristic, complemented
120C22⋊C4—4191no—
121C22⋊C4—4192no—
122C22⋊C4—4192no—
123C22⋊C4—4192no—
124C22⋊C4—4191no—
125C22⋊C4—4192no—
126C22⋊C4C2×D41912yescomplemented
127C42C2×D41911yescharacteristic, complemented
128C23—8384no—
129C23—8384no—
130C23—16762no—
131C23—4192no—
132C23—4191no—
133C23—4191no—
134C23C22×D41912yes—
135Q8—8382no—
136D4—8382no—
137D4—8384no—
138D4—8384no—
139D4—8382no—
140D4—16762no—
141D4D167614yescomplemented
142C2×C4—42no—
143C2×C4—8384no—
144C2×C4—8382no—
145C2×C4—8382no—
146C2×C4—4191no—
147C2×C4—4192no—
148C2×C4—4191no—
149C2×C4C22×D41911yescharacteristic
150C2×C4C22×D41911yescharacteristic
151C2×C4C22×D41912yes—
152C2×C4C22×D41911yescharacteristic
153C22—22no—
154C22—22no—
155C22—8381no—
156C22—8381no—
157C22—8381no—
158C22—8382no—
159C22—8381no—
160C22—8382no—
161C22—8381no—
162C22—8382no—
163C22—16768no—
164C22—4192no—
165C22C2×D167614yes—
166C22C23×D41911yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—8381no—
171C4—8381no—
172C4—8382no—
173C4C2×D167612yes—
174C2—24no—
175C2—8382no—
176C2—16762no—
177C2—16761no—
178C2C22×D167611yescharacteristic
179C2#41×D41911yescharacteristic
180C2#2721176311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D419
commutator, socle, cosocleC22×C419C24
FittingC4×D4×C419C2
hypercenterC4×D4D419
hypocenterC419#546
LCSG ▷ C22×C419 ▷ C419
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20269410C2semidirect
2. C2×D4×D419C2semidirect
3. #27211706C2semidirect
4. #27211702C2semidirect
5. #27211698C2semidirect
6. #27211693C2semidirect
7. #27211692C2semidirect
8. #27211688C2semidirect
9. C4×D1676C2semidirect
10. C2×(C2×D419)⋊C4C2semidirect
11. C4×D4×C419C2semidirect
14. C2×Dic838C22semidirect
17. C2×D1676C22semidirect
19. C2×C22⋊D419C22semidirect
21. C22×Dic419C22semidirect
23. C2×C4×D419C22semidirect
24. C4⋊Dic419C22semidirect
25. C4⋊Dic419C22semidirect
28. (C2×D419)⋊C4C22semidirect
29. (C2×D419)⋊C4C22semidirect
30. (C2×D419)⋊C4C22semidirect
31. (C2×D419)⋊C4C22semidirect
33. C22×C4×C419C22semidirect
34. C4⋊C4×C419C22semidirect
35. C22⋊C4×C419C22semidirect
36. C42×C419C22semidirect
56. D4×C419D4semidirect
75. C2×D419#41semidirect
90. C419#546semidirect
101. C4×D4D419semidirect
116. C22×C4C2×D419semidirect
119. C4⋊C4C2×D419semidirect
126. C22⋊C4C2×D419semidirect
127. C42C2×D419semidirect
141. D4D1676semidirect

Maximal subgroups

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

subgroupindex
C2×#202694102
C2×D4×D4192
#272117062
#272117022
#272116982
#272116932
#272116922
#272116882
C4×D16762
C2×(C2×D419)⋊C42
C4×D4×C4192
#546419