Finite Group Databasebeta

← #29505413

Subgroups — #29505413

Order 15296.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C22×D956C211yesmaximal, characteristic, complemented
3#21439949C211yesmaximal, characteristic, complemented
4#21439945C212yesmaximal, complemented
5#21439935C211yesmaximal, characteristic, complemented
6#21439935C212yesmaximal, complemented
7#21439930C212yesmaximal, complemented
8#21439922C211yesmaximal, characteristic, complemented
9C4×C22⋊D239C211yesmaximal, characteristic, complemented
10C22⋊C4×D239C212yesmaximal, complemented
11C22⋊Q8×C239C211yesmaximal, characteristic, complemented
12C23×D239C2212yes—
13#16838549—24no—
14C2×D956—24no—
15C2×D956C2212yes—
16C2×D956C2211yescharacteristic
17C2×D956C2211yescharacteristic
18C2×D956C2212yescomplemented
19C2×D956C2212yescomplemented
20C2×C22⋊D239C2211yescharacteristic
21C2×C4×D239C2212yescomplemented
22C2×C4×D239C2211yescharacteristic, complemented
23C2×C4×D239C2212yescomplemented
24Dic239⋊C4C2211yescharacteristic, complemented
25C22⋊Dic239C2211yescharacteristic
26(C2×D239)⋊C4C2212yescomplemented
27(C2×D239)⋊C4C2212yescomplemented
28(C2×D239)⋊C4C2211yescharacteristic, complemented
29(C2×D239)⋊C4C2212yescomplemented
30(C2×D239)⋊C4C2212yescomplemented
31C4×Dic239C2211yescharacteristic, complemented
32C2×Q8×C239C2211yescharacteristic, complemented
33C22×C4×C239C2211yescharacteristic
34C4⋊C4×C239C2211yescharacteristic, complemented
35C4⋊C4×C239C2212yescomplemented
36C22⋊C4×C239C2212yes—
37C22×D239—24no—
38C22×D239—24no—
39C22×D239—42no—
40C22×D239C2312yes—
41C22×D239C2311yescharacteristic
42C22×D239C2312yes—
43D956—24no—
44D956—24no—
45D956—22no—
46D956—42no—
47D956D414yescomplemented
48C22⋊D239—22no—
49C2×Dic239C2311yescharacteristic
50C2×Dic239C2311yescharacteristic
51C2×Dic239C2311yescharacteristic
52C4×D239—24no—
53C4×D239—22no—
54C4×D239—42no—
55C23×C239C2311yescharacteristic
56Q8×C239—22no—
57C2×C4×C239—22no—
58C2×C4×C239C2312yes—
59C2×C4×C239C2312yes—
60C2×C4×C239C2311yescharacteristic
61C2×C4×C239C2311yescharacteristic
62C22×C239—21no—
63C22×C239—21no—
64C22×C239#4112yescomplemented
65C22×C239C2411yescharacteristic
66C2×D239—22no—
67C2×D239—22no—
68C2×D239—21no—
69C2×D239—22no—
70C2×D239—21no—
71C2×D239—22no—
72C2×D239—21no—
73C2×D239—22no—
74C2×D239—48no—
75C2×D239C2×D414yes—
76C4×C239—21no—
77C4×C239—22no—
78C4×C239—22no—
79C4×C239C2×D412yes—
80Dic239—21no—
81Dic239—21no—
82Dic239—21no—
83C2×C239—22no—
84C2×C239C22×D411yescharacteristic
85C2×C239C2×#4111yescharacteristic
86C2×C239H(5,𝔽2)11yescharacteristic
87D239—24no—
88D239—41no—
89D239—42no—
90C239#54611yescharacteristic, complemented, Sylow
91#546—2391nomaximal, Sylow
92C2×#41—2391no—
93C22×D4—2391no—
94#124—2391no—
95#123—2392no—
96C22⋊Q8D23911yescharacteristic, complemented
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—2391no—
108C2×D4—2392no—
109C2×D4—2392no—
110C2×D4—2392no—
111C2×D4—2391no—
112C2×D4—2391no—
113C22×C4—2392no—
114C22×C4—2392no—
115C22×C4—2391no—
116C22×C4C2×D23911yescharacteristic
117C4⋊C4—2391no—
118C4⋊C4C2×D23911yescharacteristic, complemented
119C4⋊C4C2×D23912yescomplemented
120C22⋊C4—2391no—
121C22⋊C4—2392no—
122C22⋊C4—2392no—
123C22⋊C4—2392no—
124C22⋊C4—2391no—
125C22⋊C4—2392no—
126C22⋊C4C2×D23912yes—
127C42—2391no—
128C23—9562no—
129C23—4784no—
130C23—4784no—
131C23—2392no—
132C23—2391no—
133C23—2392no—
134C23C22×D23911yescharacteristic
135Q8—22no—
136D4—9562no—
137D4—4784no—
138D4—4784no—
139D4—4782no—
140D4—4782no—
141D4—2394no—
142C2×C4—22no—
143C2×C4—9562no—
144C2×C4—4784no—
145C2×C4—4782no—
146C2×C4—2391no—
147C2×C4—2391no—
148C2×C4—2391no—
149C2×C4C22×D23912yes—
150C2×C4C22×D23912yes—
151C2×C4C22×D23911yescharacteristic
152C2×C4C22×D23911yescharacteristic
153C22—21no—
154C22—21no—
155C22—9568no—
156C22—4781no—
157C22—4782no—
158C22—4781no—
159C22—4782no—
160C22—4781no—
161C22—4782no—
162C22—4782no—
163C22—4782no—
164C22—2394no—
165C22#1683854912yescomplemented
166C22C23×D23911yescharacteristic
167C4—22no—
168C4—21no—
169C4—22no—
170C4—4781no—
171C4—4781no—
172C4—4781no—
173C4D4×D23912yes—
174C2—22no—
175C2—9562no—
176C2—9561no—
177C2—4784no—
178C2C2×D4×D23911yescharacteristic
179C2C2×#1683854911yescharacteristic
180C2#2144000611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D239
commutator, socle, cosocleC22×C239C24
FittingC22⋊Q8×C239C2
hypercenterC22⋊Q8D239
hypocenterC239#546
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16838549C2semidirect
2. C22×D956C2semidirect
3. #21439949C2semidirect
4. #21439945C2semidirect
5. #21439935C2semidirect
6. #21439935C2semidirect
7. #21439930C2semidirect
8. #21439922C2semidirect
9. C4×C22⋊D239C2semidirect
10. C22⋊C4×D239C2semidirect
11. C22⋊Q8×C239C2semidirect
18. C2×D956C22semidirect
19. C2×D956C22semidirect
21. C2×C4×D239C22semidirect
22. C2×C4×D239C22semidirect
23. C2×C4×D239C22semidirect
24. Dic239⋊C4C22semidirect
26. (C2×D239)⋊C4C22semidirect
27. (C2×D239)⋊C4C22semidirect
28. (C2×D239)⋊C4C22semidirect
29. (C2×D239)⋊C4C22semidirect
30. (C2×D239)⋊C4C22semidirect
31. C4×Dic239C22semidirect
32. C2×Q8×C239C22semidirect
34. C4⋊C4×C239C22semidirect
35. C4⋊C4×C239C22semidirect
47. D956D4semidirect
64. C22×C239#41semidirect
90. C239#546semidirect
96. C22⋊Q8D239semidirect
105. C2×Q8C2×D239semidirect
118. C4⋊C4C2×D239semidirect
119. C4⋊C4C2×D239semidirect
165. C22#16838549semidirect

Maximal subgroups

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

subgroupindex
C2×#168385492
C22×D9562
#214399492
#214399452
#214399352
#214399302
#214399222
C4×C22⋊D2392
C22⋊C4×D2392
C22⋊Q8×C2392
#546239