Finite Group Databasebeta

← #29505415

Subgroups — #29505415

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×#16838549C211yesmaximal, characteristic, complemented
2C2×D4×D239C211yesmaximal, characteristic, complemented
3#21439942C212yesmaximal, complemented
4#21439936C211yesmaximal, characteristic, complemented
5#21439933C212yesmaximal, complemented
6#21439931C211yesmaximal, characteristic, complemented
7#21439930C212yesmaximal, complemented
8C4×D956C211yesmaximal, characteristic, complemented
9C4×D956C211yesmaximal, characteristic, complemented
10C22⋊C4×D239C212yesmaximal, complemented
11#124×C239C211yesmaximal, characteristic, complemented
12C23×D239C2212yes—
13#16838549—24no—
14C2×D956C2211yescharacteristic
15C2×D956C2211yescharacteristic
16C2×D956C2212yescomplemented
17C2×C22⋊D239C2212yes—
18C2×C22⋊D239C2212yescomplemented
19D4×D239—24no—
20C2×C4×D239C2211yescharacteristic
21C2×C4×D239C2212yescomplemented
22C2×C4×D239C2211yescharacteristic, complemented
23C2×C4×D239C2212yescomplemented
24C4⋊Dic239C2211yescharacteristic, complemented
25C4⋊Dic239C2211yescharacteristic, complemented
26Dic239⋊C4C2212yescomplemented
27C22⋊Dic239C2212yescomplemented
28(C2×D239)⋊C4C2211yescharacteristic
29(C2×D239)⋊C4C2212yes—
30(C2×D239)⋊C4C2212yescomplemented
31(C2×D239)⋊C4C2211yescharacteristic, complemented
32C2×Q8×C239C2211yescharacteristic, complemented
33C2×D4×C239C2211yescharacteristic
34C22⋊C4×C239C2212yescomplemented
35C22⋊C4×C239C2212yescomplemented
36C42×C239C2211yescharacteristic, complemented
37C22×D239—24no—
38C22×D239—24no—
39C22×D239—42no—
40C22×D239C2311yescharacteristic
41C22×D239C2312yes—
42C22×D239C2311yescharacteristic
43D956—24no—
44D956—22no—
45D956D414yescomplemented
46C22⋊D239—24no—
47C22⋊D239—42no—
48C2×Dic239C2312yes—
49C2×Dic239C2311yescharacteristic
50C2×Dic239C2311yescharacteristic
51C4×D239—24no—
52C4×D239—22no—
53C4×D239—22no—
54C4×D239—42no—
55C23×C239C2312yes—
56Q8×C239—22no—
57D4×C239—22no—
58C2×C4×C239C2311yescharacteristic
59C2×C4×C239C2311yescharacteristic
60C2×C4×C239C2312yes—
61C2×C4×C239C2311yescharacteristic
62C22×C239—22no—
63C22×C239—22no—
64C22×C239—22no—
65C22×C239C2411yescharacteristic
66C2×D239—21no—
67C2×D239—22no—
68C2×D239—22no—
69C2×D239—21no—
70C2×D239—21no—
71C2×D239—21no—
72C2×D239—21no—
73C2×D239—48no—
74C2×D239C2×D414yes—
75C2×D239#4112yescomplemented
76C4×C239—22no—
77C4×C239—21no—
78C4×C239—21no—
79C4×C239C2×D412yes—
80Dic239—21no—
81Dic239—21no—
82Dic239—22no—
83C2×C239—42no—
84C2×C239C22×D411yescharacteristic
85C2×C239C2×#4111yescharacteristic
86C2×C239H(5,𝔽2)11yescharacteristic
87D239—22no—
88D239—24no—
89D239—41no—
90C239#54611yescharacteristic, complemented, Sylow
91#546—2391nomaximal, Sylow
92C2×#41—2391no—
93C22×D4—2391no—
94#124D23911yescharacteristic, complemented
95#123—2392no—
96C22⋊Q8—2391no—
97#121—2391no—
98#121—2392no—
99#120—2392no—
100C4×D4—2391no—
101C4×D4—2391no—
102C2×C22⋊C4—2392no—
103C24—2392no—
104#41—4784no—
105C2×Q8C2×D23911yescharacteristic, complemented
106C2×D4—4784no—
107C2×D4—2392no—
108C2×D4—2392no—
109C2×D4—2391no—
110C2×D4—2392no—
111C2×D4—2391no—
112C2×D4C2×D23911yescharacteristic
113C22×C4—2392no—
114C22×C4—2392no—
115C22×C4—2391no—
116C22×C4—2391no—
117C4⋊C4—2391no—
118C4⋊C4—2391no—
119C4⋊C4—2392no—
120C22⋊C4—2392no—
121C22⋊C4—2391no—
122C22⋊C4—2392no—
123C22⋊C4—2392no—
124C22⋊C4—2391no—
125C22⋊C4C2×D23912yescomplemented
126C22⋊C4C2×D23912yescomplemented
127C42C2×D23911yescharacteristic, complemented
128C23—9562no—
129C23—4784no—
130C23—4784no—
131C23—2391no—
132C23—2391no—
133C23—2392no—
134C23C22×D23912yes—
135Q8—22no—
136D4—22no—
137D4—9562no—
138D4—4784no—
139D4—4782no—
140D4—4784no—
141D4—2394no—
142C2×C4—9562no—
143C2×C4—4784no—
144C2×C4—4782no—
145C2×C4—4782no—
146C2×C4—2392no—
147C2×C4—2391no—
148C2×C4—2391no—
149C2×C4C22×D23911yescharacteristic
150C2×C4C22×D23911yescharacteristic
151C2×C4C22×D23912yes—
152C2×C4C22×D23911yescharacteristic
153C22—22no—
154C22—22no—
155C22—22no—
156C22—9568no—
157C22—4781no—
158C22—4781no—
159C22—4781no—
160C22—4782no—
161C22—4782no—
162C22—4781no—
163C22—4781no—
164C22—2392no—
165C22—2394no—
166C22C23×D23911yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—4782no—
171C4—4781no—
172C4—4781no—
173C4D4×D23912yes—
174C2—42no—
175C2—9561no—
176C2—4782no—
177C2—4784no—
178C2C2×D4×D23911yescharacteristic
179C2#41×D23911yescharacteristic
180C2#2144000611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
Fitting#124×C239C2
hypercenter#124D239
hypocenterC239#546
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#16838549C2semidirect
2. C2×D4×D239C2semidirect
3. #21439942C2semidirect
4. #21439936C2semidirect
5. #21439933C2semidirect
6. #21439931C2semidirect
7. #21439930C2semidirect
8. C4×D956C2semidirect
9. C4×D956C2semidirect
10. C22⋊C4×D239C2semidirect
11. #124×C239C2semidirect
16. C2×D956C22semidirect
18. C2×C22⋊D239C22semidirect
21. C2×C4×D239C22semidirect
22. C2×C4×D239C22semidirect
23. C2×C4×D239C22semidirect
24. C4⋊Dic239C22semidirect
25. C4⋊Dic239C22semidirect
26. Dic239⋊C4C22semidirect
27. C22⋊Dic239C22semidirect
30. (C2×D239)⋊C4C22semidirect
31. (C2×D239)⋊C4C22semidirect
32. C2×Q8×C239C22semidirect
34. C22⋊C4×C239C22semidirect
35. C22⋊C4×C239C22semidirect
36. C42×C239C22semidirect
45. D956D4semidirect
75. C2×D239#41semidirect
90. C239#546semidirect
94. #124D239semidirect
105. C2×Q8C2×D239semidirect
125. C22⋊C4C2×D239semidirect
126. C22⋊C4C2×D239semidirect
127. C42C2×D239semidirect

Maximal subgroups

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

subgroupindex
C2×#168385492
C2×D4×D2392
#214399422
#214399362
#214399332
#214399312
#214399302
C4×D9562
C22⋊C4×D2392
#124×C2392
#546239