Finite Group Databasebeta

← #29505419

Subgroups — #29505419

Order 15296.

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×#16838548C211yesmaximal, characteristic, complemented
2C2×D4×D239C211yesmaximal, characteristic, complemented
3#21439947C211yesmaximal, characteristic, complemented
4#21439942C212yesmaximal, complemented
5#21439940C211yesmaximal, characteristic, complemented
6#21439931C212yesmaximal, complemented
7#21439928C212yesmaximal, complemented
8#21439922C211yesmaximal, characteristic, complemented
9C4×C22⋊D239C211yesmaximal, characteristic, complemented
10C22⋊C4×D239C212yesmaximal, complemented
11#121×C239C211yesmaximal, characteristic, complemented
12C23×D239C2212yes—
13#16838548—24no—
14C2×Dic478C2211yescharacteristic, complemented
15C2×D956C2211yescharacteristic
16C2×C22⋊D239C2211yescharacteristic
17C2×C22⋊D239C2212yes—
18C2×C22⋊D239C2212yescomplemented
19D4×D239—24no—
20C2×C4×D239C2211yescharacteristic
21C2×C4×D239C2212yescomplemented
22C2×C4×D239C2212yescomplemented
23C4⋊Dic239C2212yescomplemented
24Dic239⋊C4C2211yescharacteristic, complemented
25C22⋊Dic239C2212yescomplemented
26C22⋊Dic239C2211yescharacteristic, complemented
27C22⋊Dic239C2212yescomplemented
28(C2×D239)⋊C4C2211yescharacteristic
29(C2×D239)⋊C4C2212yes—
30(C2×D239)⋊C4C2212yescomplemented
31C4×Dic239C2211yescharacteristic, complemented
32C2×D4×C239C2211yescharacteristic
33C2×D4×C239C2212yescomplemented
34C22×C4×C239C2211yescharacteristic, complemented
35C4⋊C4×C239C2211yescharacteristic, complemented
36C22⋊C4×C239C2212yescomplemented
37C22×D239—24no—
38C22×D239—24no—
39C22×D239—42no—
40C22×D239C2312yes—
41C22×D239C2311yescharacteristic
42Dic478—22no—
43D956D414yescomplemented
44C22⋊D239—24no—
45C22⋊D239—24no—
46C22⋊D239—22no—
47C2×Dic239C2311yescharacteristic
48C2×Dic239C2311yescharacteristic
49C2×Dic239C2312yes—
50C2×Dic239C2311yescharacteristic
51C4×D239—24no—
52C4×D239—22no—
53C4×D239—42no—
54C23×C239C2312yes—
55C23×C239C2311yescharacteristic
56D4×C239—22no—
57D4×C239—42no—
58C2×C4×C239—22no—
59C2×C4×C239C2311yescharacteristic
60C2×C4×C239C2312yes—
61C2×C4×C239C2311yescharacteristic
62C22×C239—21no—
63C22×C239—22no—
64C22×C239—21no—
65C22×C239—22no—
66C22×C239—21no—
67C22×C239—22no—
68C22×C239C2411yescharacteristic
69C2×D239—21no—
70C2×D239—22no—
71C2×D239—22no—
72C2×D239—21no—
73C2×D239—48no—
74C2×D239C2×D414yes—
75C2×D239#4112yescomplemented
76C4×C239—21no—
77C4×C239—22no—
78C4×C239C2×D412yes—
79Dic239—21no—
80Dic239—22no—
81Dic239—21no—
82Dic239—21no—
83C2×C239—42no—
84C2×C239—41no—
85C2×C239C22×D411yescharacteristic
86C2×C239C2×#4111yescharacteristic
87C2×C239H(5,𝔽2)11yescharacteristic
88D239—22no—
89D239—24no—
90C239#54611yescharacteristic, complemented, Sylow
91#546—2391nomaximal, Sylow
92C2×#41—2391no—
93C22×D4—2391no—
94#124—2391no—
95#123—2392no—
96C22⋊Q8—2391no—
97#121—2392no—
98#121D23911yescharacteristic, complemented
99#120—2392no—
100C4×D4—2391no—
101C4×D4—2391no—
102C2×C22⋊C4—2392no—
103C24—2392no—
104#41—4784no—
105C2×Q8—2391no—
106C2×D4—4784no—
107C2×D4—2391no—
108C2×D4—2392no—
109C2×D4—2392no—
110C2×D4—2391no—
111C2×D4C2×D23911yescharacteristic
112C2×D4C2×D23912yescomplemented
113C22×C4—2392no—
114C22×C4—2392no—
115C22×C4—2391no—
116C22×C4C2×D23911yescharacteristic, complemented
117C4⋊C4—2391no—
118C4⋊C4—2392no—
119C4⋊C4C2×D23911yescharacteristic, complemented
120C22⋊C4—2392no—
121C22⋊C4—2391no—
122C22⋊C4—2392no—
123C22⋊C4—2391no—
124C22⋊C4—2392no—
125C22⋊C4—2392no—
126C22⋊C4C2×D23912yescomplemented
127C42—2391no—
128C23—9562no—
129C23—4784no—
130C23—4784no—
131C23—2391no—
132C23—2392no—
133C23C22×D23912yes—
134C23C22×D23911yescharacteristic
135Q8—4782no—
136D4—22no—
137D4—42no—
138D4—4784no—
139D4—4784no—
140D4—4782no—
141D4—2394no—
142C2×C4—22no—
143C2×C4—9562no—
144C2×C4—4784no—
145C2×C4—4782no—
146C2×C4—2391no—
147C2×C4—2391no—
148C2×C4—2392no—
149C2×C4—2391no—
150C2×C4C22×D23912yes—
151C2×C4C22×D23911yescharacteristic
152C2×C4C22×D23911yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—9568no—
160C22—4782no—
161C22—4782no—
162C22—4781no—
163C22—4781no—
164C22—2392no—
165C22—2394no—
166C22C23×D23911yescharacteristic
167C4—22no—
168C4—21no—
169C4—4781no—
170C4—4781no—
171C4—4781no—
172C4—4782no—
173C4D4×D23912yes—
174C2—42no—
175C2—41no—
176C2—4782no—
177C2—4784no—
178C2C2×D4×D23911yescharacteristic
179C2#41×D23911yescharacteristic
180C2#2144000511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
Fitting#121×C239C2
hypercenter#121D239
hypocenterC239#546
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16838548C2semidirect
2. C2×D4×D239C2semidirect
3. #21439947C2semidirect
4. #21439942C2semidirect
5. #21439940C2semidirect
6. #21439931C2semidirect
7. #21439928C2semidirect
8. #21439922C2semidirect
9. C4×C22⋊D239C2semidirect
10. C22⋊C4×D239C2semidirect
11. #121×C239C2semidirect
14. C2×Dic478C22semidirect
18. C2×C22⋊D239C22semidirect
21. C2×C4×D239C22semidirect
22. C2×C4×D239C22semidirect
23. C4⋊Dic239C22semidirect
24. Dic239⋊C4C22semidirect
25. C22⋊Dic239C22semidirect
26. C22⋊Dic239C22semidirect
27. C22⋊Dic239C22semidirect
30. (C2×D239)⋊C4C22semidirect
31. C4×Dic239C22semidirect
33. C2×D4×C239C22semidirect
34. C22×C4×C239C22semidirect
35. C4⋊C4×C239C22semidirect
36. C22⋊C4×C239C22semidirect
43. D956D4semidirect
75. C2×D239#41semidirect
90. C239#546semidirect
98. #121D239semidirect
112. C2×D4C2×D239semidirect
116. C22×C4C2×D239semidirect
119. C4⋊C4C2×D239semidirect
126. C22⋊C4C2×D239semidirect

Maximal subgroups

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

subgroupindex
C2×#168385482
C2×D4×D2392
#214399472
#214399422
#214399402
#214399312
#214399282
#214399222
C4×C22⋊D2392
C22⋊C4×D2392
#121×C2392
#546239