Finite Group Databasebeta

← #31824104

Subgroups — #31824104

Order 17984.

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×#17985139C211yesmaximal, characteristic, complemented
2C22×D1124C211yesmaximal, characteristic, complemented
3#22582868C211yesmaximal, characteristic, complemented
4#22582864C212yesmaximal, complemented
5#22582854C211yesmaximal, characteristic, complemented
6#22582854C212yesmaximal, complemented
7#22582849C212yesmaximal, complemented
8#22582841C211yesmaximal, characteristic, complemented
9C4×C22⋊D281C211yesmaximal, characteristic, complemented
10C22⋊C4×D281C212yesmaximal, complemented
11C22⋊Q8×C281C211yesmaximal, characteristic, complemented
12C23×D281C2212yes—
13#17985139—24no—
14C2×D1124—24no—
15C2×D1124C2212yes—
16C2×D1124C2211yescharacteristic
17C2×D1124C2211yescharacteristic
18C2×D1124C2212yescomplemented
19C2×D1124C2212yescomplemented
20C2×C22⋊D281C2211yescharacteristic
21C2×C4×D281C2212yescomplemented
22C2×C4×D281C2211yescharacteristic, complemented
23C2×C4×D281C2212yescomplemented
24#17985117C2211yescharacteristic, complemented
25C22⋊Dic281C2211yescharacteristic
26#17985115C2212yescomplemented
27#17985115C2212yescomplemented
28#17985115C2211yescharacteristic, complemented
29#17985115C2212yescomplemented
30#17985115C2212yescomplemented
31C4×Dic281C2211yescharacteristic, complemented
32C2×Q8×C281C2211yescharacteristic, complemented
33C22×C4×C281C2211yescharacteristic
34C4⋊C4×C281C2211yescharacteristic, complemented
35C4⋊C4×C281C2212yescomplemented
36C22⋊C4×C281C2212yes—
37C22×D281—24no—
38C22×D281—24no—
39C22×D281—42no—
40C22×D281C2312yes—
41C22×D281C2311yescharacteristic
42C22×D281C2312yes—
43D1124—24no—
44D1124—24no—
45D1124—22no—
46D1124—42no—
47D1124D414yescomplemented
48C22⋊D281—22no—
49C2×Dic281C2311yescharacteristic
50C2×Dic281C2311yescharacteristic
51C2×Dic281C2311yescharacteristic
52C4×D281—24no—
53C4×D281—22no—
54C4×D281—42no—
55C23×C281C2311yescharacteristic
56Q8×C281—22no—
57C2×C4×C281—22no—
58C2×C4×C281C2312yes—
59C2×C4×C281C2312yes—
60C2×C4×C281C2311yescharacteristic
61C2×C4×C281C2311yescharacteristic
62C22×C281—21no—
63C22×C281—21no—
64C22×C281#4112yescomplemented
65C22×C281C2411yescharacteristic
66C2×D281—22no—
67C2×D281—22no—
68C2×D281—21no—
69C2×D281—22no—
70C2×D281—21no—
71C2×D281—22no—
72C2×D281—21no—
73C2×D281—22no—
74C2×D281—48no—
75C2×D281C2×D414yes—
76C4×C281—21no—
77C4×C281—22no—
78C4×C281—22no—
79C4×C281C2×D412yes—
80Dic281—21no—
81Dic281—21no—
82Dic281—21no—
83C2×C281—22no—
84C2×C281C22×D411yescharacteristic
85C2×C281C2×#4111yescharacteristic
86C2×C281H(5,𝔽2)11yescharacteristic
87D281—24no—
88D281—41no—
89D281—42no—
90C281#54611yescharacteristic, complemented, Sylow
91#546—2811nomaximal, Sylow
92C2×#41—2811no—
93C22×D4—2811no—
94#124—2811no—
95#123—2812no—
96C22⋊Q8D28111yescharacteristic, complemented
97#121—2811no—
98#121—2812no—
99#120—2812no—
100C4×D4—2811no—
101C4×D4—2811no—
102C2×C22⋊C4—2812no—
103C24—2812no—
104#41—5624no—
105C2×Q8C2×D28111yescharacteristic, complemented
106C2×D4—2811no—
107C2×D4—2811no—
108C2×D4—2812no—
109C2×D4—2811no—
110C2×D4—2812no—
111C2×D4—2812no—
112C2×D4—5624no—
113C22×C4—2812no—
114C22×C4—2811no—
115C22×C4—2812no—
116C22×C4C2×D28111yescharacteristic
117C4⋊C4—2811no—
118C4⋊C4C2×D28111yescharacteristic, complemented
119C4⋊C4C2×D28112yescomplemented
120C22⋊C4—2811no—
121C22⋊C4—2812no—
122C22⋊C4—2812no—
123C22⋊C4—2812no—
124C22⋊C4—2811no—
125C22⋊C4—2812no—
126C22⋊C4C2×D28112yes—
127C42—2811no—
128C23—2812no—
129C23—2812no—
130C23—2811no—
131C23—5624no—
132C23—5624no—
133C23—11242no—
134C23C22×D28111yescharacteristic
135Q8—22no—
136D4—2814no—
137D4—5624no—
138D4—5624no—
139D4—5622no—
140D4—5622no—
141D4—11242no—
142C2×C4—22no—
143C2×C4—2811no—
144C2×C4—2811no—
145C2×C4—2811no—
146C2×C4—5624no—
147C2×C4—5622no—
148C2×C4—11242no—
149C2×C4C22×D28112yes—
150C2×C4C22×D28112yes—
151C2×C4C22×D28111yescharacteristic
152C2×C4C22×D28111yescharacteristic
153C22—21no—
154C22—21no—
155C22—2814no—
156C22—5622no—
157C22—5622no—
158C22—5621no—
159C22—5622no—
160C22—5621no—
161C22—5622no—
162C22—5621no—
163C22—5622no—
164C22—11248no—
165C22#1798513912yescomplemented
166C22C23×D28111yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—5621no—
171C4—5621no—
172C4—5621no—
173C4D4×D28112yes—
174C2—22no—
175C2—5624no—
176C2—11242no—
177C2—11241no—
178C2C2×D4×D28111yescharacteristic
179C2C2×#1798513911yescharacteristic
180C2#2258292511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
FittingC22⋊Q8×C281C2
hypercenterC22⋊Q8D281
hypocenterC281#546
LCSG ▷ C22×C281 ▷ C281
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17985139C2semidirect
2. C22×D1124C2semidirect
3. #22582868C2semidirect
4. #22582864C2semidirect
5. #22582854C2semidirect
6. #22582854C2semidirect
7. #22582849C2semidirect
8. #22582841C2semidirect
9. C4×C22⋊D281C2semidirect
10. C22⋊C4×D281C2semidirect
11. C22⋊Q8×C281C2semidirect
18. C2×D1124C22semidirect
19. C2×D1124C22semidirect
21. C2×C4×D281C22semidirect
22. C2×C4×D281C22semidirect
23. C2×C4×D281C22semidirect
24. #17985117C22semidirect
26. #17985115C22semidirect
27. #17985115C22semidirect
28. #17985115C22semidirect
29. #17985115C22semidirect
30. #17985115C22semidirect
31. C4×Dic281C22semidirect
32. C2×Q8×C281C22semidirect
34. C4⋊C4×C281C22semidirect
35. C4⋊C4×C281C22semidirect
47. D1124D4semidirect
64. C22×C281#41semidirect
90. C281#546semidirect
96. C22⋊Q8D281semidirect
105. C2×Q8C2×D281semidirect
118. C4⋊C4C2×D281semidirect
119. C4⋊C4C2×D281semidirect
165. C22#17985139semidirect

Maximal subgroups

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

subgroupindex
C2×#179851392
C22×D11242
#225828682
#225828642
#225828542
#225828492
#225828412
C4×C22⋊D2812
C22⋊C4×D2812
C22⋊Q8×C2812
#546281