Finite Group Databasebeta

← #31824024

Subgroups — #31824024

Order 17984.

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×D1124C211yesmaximal, characteristic, complemented
2C2×D4×D281C211yesmaximal, characteristic, complemented
3#22582864C212yesmaximal, complemented
4#22582862C211yesmaximal, characteristic, complemented
5#22582854C212yesmaximal, complemented
6#22582852C212yesmaximal, complemented
7#22582849C211yesmaximal, characteristic, complemented
8#22582849C212yesmaximal, complemented
9#22582847C211yesmaximal, characteristic, complemented
10C22⋊C4×D281C211yesmaximal, characteristic, complemented
11#123×C281C211yesmaximal, characteristic, complemented
12C23×D281C2211yescharacteristic
13C23×D281C2211yescharacteristic, complemented
14C23×D281C2211yescharacteristic, complemented
15C2×D1124—24no—
16C2×D1124C2212yes—
17C2×D1124C2212yescomplemented
18C2×D1124C2211yescharacteristic, complemented
19C2×D1124C2212yescomplemented
20C2×C22⋊D281C2211yescharacteristic
21C2×C22⋊D281C2211yescharacteristic, complemented
22C2×C22⋊D281C2212yescomplemented
23D4×D281—24no—
24C2×C4×D281C2211yescharacteristic, complemented
25C2×C4×D281C2212yescomplemented
26#17985117C2212yescomplemented
27C22⋊Dic281C2211yescharacteristic, complemented
28#17985115C2212yescomplemented
29#17985115C2212yescomplemented
30#17985115C2212yescomplemented
31#17985115C2212yescomplemented
32C2×D4×C281C2211yescharacteristic
33C22×C4×C281C2211yescharacteristic, complemented
34C4⋊C4×C281C2212yescomplemented
35C22⋊C4×C281C2211yescharacteristic, complemented
36C22⋊C4×C281C2212yescomplemented
37C22×D281—22no—
38C22×D281—22no—
39C22×D281—42no—
40C22×D281—42no—
41C22×D281—42no—
42C22×D281D414yescomplemented
43C22×D281C2311yescharacteristic
44C22×D281C2312yes—
45C22×D281C2311yescharacteristic
46C22×D281C2311yescharacteristic
47D1124—22no—
48D1124—24no—
49D1124—24no—
50D1124—42no—
51C22⋊D281—22no—
52C22⋊D281—22no—
53C22⋊D281—42no—
54C2×Dic281C2312yes—
55C2×Dic281C2311yescharacteristic
56C4×D281—22no—
57C4×D281—42no—
58C23×C281C2311yescharacteristic
59C23×C281C2311yescharacteristic
60D4×C281—22no—
61C2×C4×C281—22no—
62C2×C4×C281C2312yes—
63C2×C4×C281C2312yes—
64C2×C4×C281C2311yescharacteristic
65C22×C281—22no—
66C22×C281—21no—
67C22×C281—22no—
68C22×C281C2×D412yes—
69C22×C281C2411yescharacteristic
70C2×D281—22no—
71C2×D281—22no—
72C2×D281—21no—
73C2×D281—22no—
74C2×D281—22no—
75C2×D281—22no—
76C2×D281—22no—
77C2×D281—44no—
78C2×D281—44no—
79C2×D281—44no—
80C2×D281C2×D412yes—
81C2×D281C2×D412yes—
82C4×C281—22no—
83C4×C281—21no—
84C4×C281—22no—
85Dic281—21no—
86Dic281—22no—
87C2×C281—22no—
88C2×C281—41no—
89C2×C281C22×D411yescharacteristic
90C2×C281H(5,𝔽2)12yes—
91D281—22no—
92D281—22no—
93D281—41no—
94D281—42no—
95C281#53411yescharacteristic, complemented, Sylow
96#534—2811nomaximal, Sylow
97C22×D4—2811no—
98C22×D4—2811no—
99#123—2811no—
100#123—2812no—
101#123D28111yescharacteristic, complemented
102#121—2812no—
103#121—2812no—
104#120—2811no—
105#120—2812no—
106#120—2811no—
107C2×C22⋊C4—2811no—
108C24—2811no—
109C24—2811no—
110C24—2811no—
111C2×D4—2811no—
112C2×D4—2811no—
113C2×D4—2812no—
114C2×D4—2812no—
115C2×D4—2811no—
116C2×D4—2812no—
117C2×D4—2812no—
118C2×D4—5624no—
119C2×D4—5624no—
120C2×D4C2×D28111yescharacteristic
121C22×C4—2811no—
122C22×C4—2812no—
123C22×C4C2×D28111yescharacteristic, complemented
124C4⋊C4—2812no—
125C4⋊C4C2×D28112yescomplemented
126C22⋊C4—2811no—
127C22⋊C4—2812no—
128C22⋊C4—2812no—
129C22⋊C4—2812no—
130C22⋊C4—2812no—
131C22⋊C4C2×D28111yescharacteristic, complemented
132C22⋊C4C2×D28112yescomplemented
133C23—2811no—
134C23—2811no—
135C23—2814no—
136C23—2811no—
137C23—2812no—
138C23—5622no—
139C23—5622no—
140C23—11242no—
141C23—11242no—
142C23—11242no—
143C23C22×D28111yescharacteristic
144C23C22×D28111yescharacteristic
145D4—22no—
146D4—5624no—
147D4—5624no—
148D4—5622no—
149D4—5622no—
150D4—5622no—
151D4—11242no—
152D4—11242no—
153C2×C4—22no—
154C2×C4—2812no—
155C2×C4—2811no—
156C2×C4—5622no—
157C2×C4—11242no—
158C2×C4C22×D28112yes—
159C2×C4C22×D28111yescharacteristic
160C2×C4C22×D28112yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—2812no—
165C22—2812no—
166C22—5622no—
167C22—5622no—
168C22—5622no—
169C22—5621no—
170C22—5622no—
171C22—5622no—
172C22—5622no—
173C22—11244no—
174C22—11244no—
175C22—11244no—
176C22D4×D28112yes—
177C22C23×D28111yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—5621no—
182C4—5622no—
183C2—22no—
184C2—41no—
185C2—5622no—
186C2—5622no—
187C2—11242no—
188C2—11241no—
189C2C2×D4×D28111yescharacteristic
190C2#2258292512yes—
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
Fitting#123×C281C2
hypercenter#123D281
hypocenterC281#534
LCSG ▷ C22×C281 ▷ C281
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D1124C2semidirect
2. C2×D4×D281C2semidirect
3. #22582864C2semidirect
4. #22582862C2semidirect
5. #22582854C2semidirect
6. #22582852C2semidirect
7. #22582849C2semidirect
8. #22582849C2semidirect
9. #22582847C2semidirect
10. C22⋊C4×D281C2semidirect
11. #123×C281C2semidirect
13. C23×D281C22semidirect
14. C23×D281C22semidirect
17. C2×D1124C22semidirect
18. C2×D1124C22semidirect
19. C2×D1124C22semidirect
21. C2×C22⋊D281C22semidirect
22. C2×C22⋊D281C22semidirect
24. C2×C4×D281C22semidirect
25. C2×C4×D281C22semidirect
26. #17985117C22semidirect
27. C22⋊Dic281C22semidirect
28. #17985115C22semidirect
29. #17985115C22semidirect
30. #17985115C22semidirect
31. #17985115C22semidirect
33. C22×C4×C281C22semidirect
34. C4⋊C4×C281C22semidirect
35. C22⋊C4×C281C22semidirect
36. C22⋊C4×C281C22semidirect
42. C22×D281D4semidirect
95. C281#534semidirect
101. #123D281semidirect
123. C22×C4C2×D281semidirect
125. C4⋊C4C2×D281semidirect
131. C22⋊C4C2×D281semidirect
132. C22⋊C4C2×D281semidirect

Maximal subgroups

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

subgroupindex
C22×D11242
C2×D4×D2812
#225828642
#225828622
#225828542
#225828522
#225828492
#225828472
C22⋊C4×D2812
#123×C2812
#534281