Finite Group Databasebeta

← #41001008

Subgroups — #41001008

Order 26816.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#20269411C211yesmaximal, characteristic, complemented
2C2×D4×D419C211yesmaximal, characteristic, complemented
3#27211705C211yesmaximal, characteristic, complemented
4#27211700C212yesmaximal, complemented
5#27211698C211yesmaximal, characteristic, complemented
6#27211689C212yesmaximal, complemented
7#27211686C212yesmaximal, complemented
8#27211680C211yesmaximal, characteristic, complemented
9C4×C22⋊D419C211yesmaximal, characteristic, complemented
10C22⋊C4×D419C212yesmaximal, complemented
11#121×C419C211yesmaximal, characteristic, complemented
12C23×D419C2212yes—
13#20269411—24no—
14C2×Dic838C2211yescharacteristic, complemented
15C2×D1676C2211yescharacteristic
16C2×C22⋊D419C2211yescharacteristic
17C2×C22⋊D419C2212yes—
18C2×C22⋊D419C2212yescomplemented
19D4×D419—24no—
20C2×C4×D419C2211yescharacteristic
21C2×C4×D419C2212yescomplemented
22C2×C4×D419C2212yescomplemented
23C4⋊Dic419C2212yescomplemented
24Dic419⋊C4C2211yescharacteristic, complemented
25C22⋊Dic419C2212yescomplemented
26C22⋊Dic419C2211yescharacteristic, complemented
27C22⋊Dic419C2212yescomplemented
28(C2×D419)⋊C4C2211yescharacteristic
29(C2×D419)⋊C4C2212yes—
30(C2×D419)⋊C4C2212yescomplemented
31C4×Dic419C2211yescharacteristic, complemented
32C2×D4×C419C2211yescharacteristic
33C2×D4×C419C2212yescomplemented
34C22×C4×C419C2211yescharacteristic, complemented
35C4⋊C4×C419C2211yescharacteristic, complemented
36C22⋊C4×C419C2212yescomplemented
37C22×D419—24no—
38C22×D419—24no—
39C22×D419—42no—
40C22×D419C2312yes—
41C22×D419C2311yescharacteristic
42Dic838—22no—
43D1676D414yescomplemented
44C22⋊D419—24no—
45C22⋊D419—24no—
46C22⋊D419—22no—
47C2×Dic419C2311yescharacteristic
48C2×Dic419C2311yescharacteristic
49C2×Dic419C2312yes—
50C2×Dic419C2311yescharacteristic
51C4×D419—24no—
52C4×D419—22no—
53C4×D419—42no—
54C23×C419C2312yes—
55C23×C419C2311yescharacteristic
56D4×C419—22no—
57D4×C419—42no—
58C2×C4×C419—22no—
59C2×C4×C419C2311yescharacteristic
60C2×C4×C419C2312yes—
61C2×C4×C419C2311yescharacteristic
62C22×C419—21no—
63C22×C419—22no—
64C22×C419—21no—
65C22×C419—22no—
66C22×C419—21no—
67C22×C419—22no—
68C22×C419C2411yescharacteristic
69C2×D419—21no—
70C2×D419—22no—
71C2×D419—22no—
72C2×D419—21no—
73C2×D419—48no—
74C2×D419C2×D414yes—
75C2×D419#4112yescomplemented
76C4×C419—21no—
77C4×C419—22no—
78C4×C419C2×D412yes—
79Dic419—21no—
80Dic419—22no—
81Dic419—21no—
82Dic419—21no—
83C2×C419—42no—
84C2×C419—41no—
85C2×C419C22×D411yescharacteristic
86C2×C419C2×#4111yescharacteristic
87C2×C419H(5,𝔽2)11yescharacteristic
88D419—22no—
89D419—24no—
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#121D41911yescharacteristic, complemented
99#120—4192no—
100C4×D4—4191no—
101C4×D4—4191no—
102C2×C22⋊C4—4192no—
103C24—4192no—
104#41—8384no—
105C2×Q8—4191no—
106C2×D4—8384no—
107C2×D4—4191no—
108C2×D4—4192no—
109C2×D4—4192no—
110C2×D4—4191no—
111C2×D4C2×D41911yescharacteristic
112C2×D4C2×D41912yescomplemented
113C22×C4—4191no—
114C22×C4—4192no—
115C22×C4—4192no—
116C22×C4C2×D41911yescharacteristic, complemented
117C4⋊C4—4191no—
118C4⋊C4—4192no—
119C4⋊C4C2×D41911yescharacteristic, complemented
120C22⋊C4—4192no—
121C22⋊C4—4191no—
122C22⋊C4—4192no—
123C22⋊C4—4191no—
124C22⋊C4—4192no—
125C22⋊C4—4192no—
126C22⋊C4C2×D41912yescomplemented
127C42—4191no—
128C23—8384no—
129C23—8384no—
130C23—16762no—
131C23—4191no—
132C23—4192no—
133C23C22×D41912yes—
134C23C22×D41911yescharacteristic
135Q8—8382no—
136D4—22no—
137D4—42no—
138D4—8384no—
139D4—8384no—
140D4—8382no—
141D4—4194no—
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×C4C22×D41912yes—
151C2×C4C22×D41911yescharacteristic
152C2×C4C22×D41911yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—8381no—
160C22—8382no—
161C22—8382no—
162C22—8381no—
163C22—16768no—
164C22—4194no—
165C22—4192no—
166C22C23×D41911yescharacteristic
167C4—21no—
168C4—22no—
169C4—8381no—
170C4—8382no—
171C4—8381no—
172C4—8381no—
173C4D4×D41912yes—
174C2—42no—
175C2—41no—
176C2—8382no—
177C2—8384no—
178C2C2×D4×D41911yescharacteristic
179C2#41×D41911yescharacteristic
180C2#2721176311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D419
commutator, socle, cosocleC22×C419C24
Fitting#121×C419C2
hypercenter#121D419
hypocenterC419#546
LCSG ▷ C22×C419 ▷ C419
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20269411C2semidirect
2. C2×D4×D419C2semidirect
3. #27211705C2semidirect
4. #27211700C2semidirect
5. #27211698C2semidirect
6. #27211689C2semidirect
7. #27211686C2semidirect
8. #27211680C2semidirect
9. C4×C22⋊D419C2semidirect
10. C22⋊C4×D419C2semidirect
11. #121×C419C2semidirect
14. C2×Dic838C22semidirect
18. C2×C22⋊D419C22semidirect
21. C2×C4×D419C22semidirect
22. C2×C4×D419C22semidirect
23. C4⋊Dic419C22semidirect
24. Dic419⋊C4C22semidirect
25. C22⋊Dic419C22semidirect
26. C22⋊Dic419C22semidirect
27. C22⋊Dic419C22semidirect
30. (C2×D419)⋊C4C22semidirect
31. C4×Dic419C22semidirect
33. C2×D4×C419C22semidirect
34. C22×C4×C419C22semidirect
35. C4⋊C4×C419C22semidirect
36. C22⋊C4×C419C22semidirect
43. D1676D4semidirect
75. C2×D419#41semidirect
90. C419#546semidirect
98. #121D419semidirect
112. C2×D4C2×D419semidirect
116. C22×C4C2×D419semidirect
119. C4⋊C4C2×D419semidirect
126. C22⋊C4C2×D419semidirect

Maximal subgroups

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

subgroupindex
C2×#202694112
C2×D4×D4192
#272117052
#272117002
#272116982
#272116892
#272116862
#272116802
C4×C22⋊D4192
C22⋊C4×D4192
#121×C4192
#546419