Finite Group Databasebeta

← #31824107

Subgroups — #31824107

Order 17984.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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×#17985137C211yesmaximal, characteristic, complemented
2#22582866C211yesmaximal, characteristic, complemented
3#22582860C212yesmaximal, complemented
4#22582858C211yesmaximal, characteristic, complemented
5#22582851C212yesmaximal, complemented
6#22582850C211yesmaximal, characteristic, complemented
7#22582848C212yesmaximal, complemented
8D4×Dic281C211yesmaximal, characteristic, complemented
9C4×C22⋊D281C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic281C212yesmaximal, complemented
11C22×D4×C281C211yesmaximal, characteristic, complemented
12#17985137—24no—
13C2×Dic562C2211yescharacteristic, complemented
14C2×C22⋊D281C2211yescharacteristic
15C2×C22⋊D281C2212yescomplemented
16C2×C22⋊D281C2212yescomplemented
17C22×Dic281C2212yescomplemented
18C22×Dic281C2212yescomplemented
19C2×C4×D281C2211yescharacteristic, complemented
20C4⋊Dic281C2211yescharacteristic, complemented
21#17985117C2211yescharacteristic, complemented
22#17985117C2212yescomplemented
23C22⋊Dic281C2211yescharacteristic
24C22⋊Dic281C2212yes—
25C22⋊Dic281C2212yescomplemented
26C22⋊Dic281C2212yescomplemented
27C22⋊Dic281C2212yescomplemented
28C22⋊Dic281C2212yescomplemented
29#17985115C2211yescharacteristic, complemented
30C4×Dic281C2211yescharacteristic, complemented
31C24×C281C2212yes—
32C2×D4×C281—24no—
33C2×D4×C281C2211yescharacteristic
34C2×D4×C281C2212yes—
35C2×D4×C281C2211yescharacteristic
36C22×C4×C281C2211yescharacteristic
37C22×D281C2311yescharacteristic
38Dic562—22no—
39C22⋊D281—24no—
40C22⋊D281—22no—
41C22⋊D281—42no—
42C2×Dic281—24no—
43C2×Dic281—42no—
44C2×Dic281C2311yescharacteristic
45C2×Dic281C2311yescharacteristic
46C2×Dic281C2312yes—
47C2×Dic281C2311yescharacteristic
48C2×Dic281C2312yes—
49C4×D281—22no—
50C23×C281—24no—
51C23×C281—24no—
52C23×C281—42no—
53C23×C281C2312yes—
54C23×C281C2312yes—
55C23×C281C2311yescharacteristic
56D4×C281—24no—
57D4×C281—22no—
58D4×C281D414yescomplemented
59C2×C4×C281—22no—
60C2×C4×C281C2311yescharacteristic
61C2×C4×C281C2311yescharacteristic
62C22×C281—21no—
63C22×C281—22no—
64C22×C281—21no—
65C22×C281—22no—
66C22×C281—22no—
67C22×C281—22no—
68C22×C281—22no—
69C22×C281—48no—
70C22×C281C2×D414yes—
71C22×C281#4112yescomplemented
72C22×C281C2411yescharacteristic
73C2×D281—21no—
74C2×D281—21no—
75C2×D281—21no—
76C4×C281—21no—
77C4×C281C2×D412yes—
78Dic281—21no—
79Dic281—22no—
80Dic281—21no—
81Dic281—21no—
82Dic281—22no—
83C2×C281—22no—
84C2×C281—24no—
85C2×C281—42no—
86C2×C281C22×D411yescharacteristic
87C2×C281C2×#4111yescharacteristic
88C2×C281H(5,𝔽2)11yescharacteristic
89D281—41no—
90C281#54611yescharacteristic, complemented, Sylow
91#546—2811nomaximal, Sylow
92C2×#41—2811no—
93C22×D4D28111yescharacteristic, complemented
94#124—2811no—
95#123—2812no—
96C22⋊Q8—2811no—
97#121—2811no—
98#121—2812no—
99#120—2812no—
100C4×D4—2811no—
101C4×D4—2811no—
102C2×C22⋊C4—2812no—
103C24C2×D28112yes—
104#41—5624no—
105C2×Q8—2811no—
106C2×D4—24no—
107C2×D4—2811no—
108C2×D4—2812no—
109C2×D4—2812no—
110C2×D4C2×D28112yes—
111C2×D4C2×D28111yescharacteristic
112C2×D4C2×D28111yescharacteristic
113C22×C4—2812no—
114C22×C4—2812no—
115C22×C4—2811no—
116C22×C4C2×D28111yescharacteristic
117C4⋊C4—2811no—
118C4⋊C4—2812no—
119C4⋊C4—2811no—
120C22⋊C4—2811no—
121C22⋊C4—2812no—
122C22⋊C4—2812no—
123C22⋊C4—2812no—
124C22⋊C4—2812no—
125C22⋊C4—2812no—
126C22⋊C4—2811no—
127C42—2811no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—2811no—
132C23C22×D28112yes—
133C23C22×D28112yes—
134C23C22×D28111yescharacteristic
135Q8—5622no—
136D4—24no—
137D4—22no—
138D4—5624no—
139D4—5622no—
140D4—11242no—
141D4C22⋊D28114yescomplemented
142C2×C4—22no—
143C2×C4—2811no—
144C2×C4—2811no—
145C2×C4—2812no—
146C2×C4—2811no—
147C2×C4—2812no—
148C2×C4—5624no—
149C2×C4—5622no—
150C2×C4—11242no—
151C2×C4C22×D28111yescharacteristic
152C2×C4C22×D28111yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—5621no—
162C22—5621no—
163C22—5621no—
164C22C2×C22⋊D28114yes—
165C22#1798513712yescomplemented
166C22C23×D28111yescharacteristic
167C4—21no—
168C4—5621no—
169C4—5622no—
170C4—5621no—
171C4—5621no—
172C4—5622no—
173C4C2×C22⋊D28112yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—11241no—
178C2C22×C22⋊D28111yescharacteristic
179C2C2×#1798513711yescharacteristic
180C2#2258292411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
FittingC22×D4×C281C2
hypercenterC22×D4D281
hypocenterC281#546
LCSG ▷ C22×C281 ▷ C281
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#17985137C2semidirect
2. #22582866C2semidirect
3. #22582860C2semidirect
4. #22582858C2semidirect
5. #22582851C2semidirect
6. #22582850C2semidirect
7. #22582848C2semidirect
8. D4×Dic281C2semidirect
9. C4×C22⋊D281C2semidirect
10. C2×C22⋊Dic281C2semidirect
11. C22×D4×C281C2semidirect
13. C2×Dic562C22semidirect
15. C2×C22⋊D281C22semidirect
16. C2×C22⋊D281C22semidirect
17. C22×Dic281C22semidirect
18. C22×Dic281C22semidirect
19. C2×C4×D281C22semidirect
20. C4⋊Dic281C22semidirect
21. #17985117C22semidirect
22. #17985117C22semidirect
25. C22⋊Dic281C22semidirect
26. C22⋊Dic281C22semidirect
27. C22⋊Dic281C22semidirect
28. C22⋊Dic281C22semidirect
29. #17985115C22semidirect
30. C4×Dic281C22semidirect
58. D4×C281D4semidirect
71. C22×C281#41semidirect
90. C281#546semidirect
93. C22×D4D281semidirect
141. D4C22⋊D281semidirect
165. C22#17985137semidirect

Maximal subgroups

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

subgroupindex
C2×#179851372
#225828662
#225828602
#225828582
#225828512
#225828502
#225828482
D4×Dic2812
C4×C22⋊D2812
C2×C22⋊Dic2812
C22×D4×C2812
#546281