Finite Group Databasebeta

← #43244165

Subgroups — #43244165

Order 28096.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×D1756C211yesmaximal, characteristic, complemented
2C2×D4×D439C211yesmaximal, characteristic, complemented
3#28380728C212yesmaximal, complemented
4#28380726C211yesmaximal, characteristic, complemented
5#28380718C212yesmaximal, complemented
6#28380716C212yesmaximal, complemented
7#28380713C211yesmaximal, characteristic, complemented
8#28380713C212yesmaximal, complemented
9#28380711C211yesmaximal, characteristic, complemented
10C22⋊C4×D439C211yesmaximal, characteristic, complemented
11#123×C439C211yesmaximal, characteristic, complemented
12C23×D439C2211yescharacteristic
13C23×D439C2211yescharacteristic, complemented
14C23×D439C2211yescharacteristic, complemented
15C2×D1756—24no—
16C2×D1756C2212yes—
17C2×D1756C2212yescomplemented
18C2×D1756C2211yescharacteristic, complemented
19C2×D1756C2212yescomplemented
20C2×C22⋊D439C2211yescharacteristic
21C2×C22⋊D439C2211yescharacteristic, complemented
22C2×C22⋊D439C2212yescomplemented
23D4×D439—24no—
24C2×C4×D439C2211yescharacteristic, complemented
25C2×C4×D439C2212yescomplemented
26Dic439⋊C4C2212yescomplemented
27C22⋊Dic439C2211yescharacteristic, complemented
28(C2×D439)⋊C4C2212yescomplemented
29(C2×D439)⋊C4C2212yescomplemented
30(C2×D439)⋊C4C2212yescomplemented
31(C2×D439)⋊C4C2212yescomplemented
32C2×D4×C439C2211yescharacteristic
33C22×C4×C439C2211yescharacteristic, complemented
34C4⋊C4×C439C2212yescomplemented
35C22⋊C4×C439C2211yescharacteristic, complemented
36C22⋊C4×C439C2212yescomplemented
37C22×D439—22no—
38C22×D439—22no—
39C22×D439—42no—
40C22×D439—42no—
41C22×D439—42no—
42C22×D439D414yescomplemented
43C22×D439C2311yescharacteristic
44C22×D439C2312yes—
45C22×D439C2311yescharacteristic
46C22×D439C2311yescharacteristic
47D1756—22no—
48D1756—24no—
49D1756—24no—
50D1756—42no—
51C22⋊D439—22no—
52C22⋊D439—22no—
53C22⋊D439—42no—
54C2×Dic439C2312yes—
55C2×Dic439C2311yescharacteristic
56C4×D439—22no—
57C4×D439—42no—
58C23×C439C2311yescharacteristic
59C23×C439C2311yescharacteristic
60D4×C439—22no—
61C2×C4×C439—22no—
62C2×C4×C439C2312yes—
63C2×C4×C439C2312yes—
64C2×C4×C439C2311yescharacteristic
65C22×C439—22no—
66C22×C439—21no—
67C22×C439—22no—
68C22×C439C2×D412yes—
69C22×C439C2411yescharacteristic
70C2×D439—22no—
71C2×D439—22no—
72C2×D439—21no—
73C2×D439—22no—
74C2×D439—22no—
75C2×D439—22no—
76C2×D439—22no—
77C2×D439—44no—
78C2×D439—44no—
79C2×D439—44no—
80C2×D439C2×D412yes—
81C2×D439C2×D412yes—
82C4×C439—22no—
83C4×C439—21no—
84C4×C439—22no—
85Dic439—21no—
86Dic439—22no—
87C2×C439—22no—
88C2×C439—41no—
89C2×C439C22×D411yescharacteristic
90C2×C439H(5,𝔽2)12yes—
91D439—22no—
92D439—22no—
93D439—41no—
94D439—42no—
95C439#53411yescharacteristic, complemented, Sylow
96#534—4391nomaximal, Sylow
97C22×D4—4391no—
98C22×D4—4391no—
99#123—4391no—
100#123—4392no—
101#123D43911yescharacteristic, complemented
102#121—4392no—
103#121—4392no—
104#120—4391no—
105#120—4392no—
106#120—4391no—
107C2×C22⋊C4—4391no—
108C24—4391no—
109C24—4391no—
110C24—4391no—
111C2×D4—8784no—
112C2×D4—8784no—
113C2×D4—4391no—
114C2×D4—4391no—
115C2×D4—4392no—
116C2×D4—4392no—
117C2×D4—4391no—
118C2×D4—4392no—
119C2×D4—4392no—
120C2×D4C2×D43911yescharacteristic
121C22×C4—4391no—
122C22×C4—4392no—
123C22×C4C2×D43911yescharacteristic, complemented
124C4⋊C4—4392no—
125C4⋊C4C2×D43912yescomplemented
126C22⋊C4—4391no—
127C22⋊C4—4392no—
128C22⋊C4—4392no—
129C22⋊C4—4392no—
130C22⋊C4—4392no—
131C22⋊C4C2×D43911yescharacteristic, complemented
132C22⋊C4C2×D43912yescomplemented
133C23—8782no—
134C23—8782no—
135C23—4391no—
136C23—4392no—
137C23—4391no—
138C23—4391no—
139C23—4394no—
140C23—17562no—
141C23—17562no—
142C23—17562no—
143C23C22×D43911yescharacteristic
144C23C22×D43911yescharacteristic
145D4—22no—
146D4—8784no—
147D4—8784no—
148D4—8782no—
149D4—8782no—
150D4—8782no—
151D4—17562no—
152D4—17562no—
153C2×C4—22no—
154C2×C4—8782no—
155C2×C4—4392no—
156C2×C4—4391no—
157C2×C4—17562no—
158C2×C4C22×D43912yes—
159C2×C4C22×D43911yescharacteristic
160C2×C4C22×D43912yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—8782no—
165C22—8782no—
166C22—8782no—
167C22—8781no—
168C22—8782no—
169C22—8782no—
170C22—8782no—
171C22—4392no—
172C22—4392no—
173C22—17564no—
174C22—17564no—
175C22—17564no—
176C22D4×D43912yes—
177C22C23×D43911yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—8781no—
182C4—8782no—
183C2—22no—
184C2—41no—
185C2—8782no—
186C2—8782no—
187C2—17562no—
188C2—17561no—
189C2C2×D4×D43911yescharacteristic
190C2#2838078912yes—
subgroupG/H
center, FrattiniC22C23×D439
commutator, socle, cosocleC22×C439C24
Fitting#123×C439C2
hypercenter#123D439
hypocenterC439#534
LCSG ▷ C22×C439 ▷ C439
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D1756C2semidirect
2. C2×D4×D439C2semidirect
3. #28380728C2semidirect
4. #28380726C2semidirect
5. #28380718C2semidirect
6. #28380716C2semidirect
7. #28380713C2semidirect
8. #28380713C2semidirect
9. #28380711C2semidirect
10. C22⋊C4×D439C2semidirect
11. #123×C439C2semidirect
13. C23×D439C22semidirect
14. C23×D439C22semidirect
17. C2×D1756C22semidirect
18. C2×D1756C22semidirect
19. C2×D1756C22semidirect
21. C2×C22⋊D439C22semidirect
22. C2×C22⋊D439C22semidirect
24. C2×C4×D439C22semidirect
25. C2×C4×D439C22semidirect
26. Dic439⋊C4C22semidirect
27. C22⋊Dic439C22semidirect
28. (C2×D439)⋊C4C22semidirect
29. (C2×D439)⋊C4C22semidirect
30. (C2×D439)⋊C4C22semidirect
31. (C2×D439)⋊C4C22semidirect
33. C22×C4×C439C22semidirect
34. C4⋊C4×C439C22semidirect
35. C22⋊C4×C439C22semidirect
36. C22⋊C4×C439C22semidirect
42. C22×D439D4semidirect
95. C439#534semidirect
101. #123D439semidirect
123. C22×C4C2×D439semidirect
125. C4⋊C4C2×D439semidirect
131. C22⋊C4C2×D439semidirect
132. C22⋊C4C2×D439semidirect

Maximal subgroups

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

subgroupindex
C22×D17562
C2×D4×D4392
#283807282
#283807262
#283807182
#283807162
#283807132
#283807112
C22⋊C4×D4392
#123×C4392
#534439