Finite Group Databasebeta

← #31823976

Subgroups — #31823976

Order 17984.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4×D281C211yesmaximal, characteristic, complemented
3#22582864C212yesmaximal, complemented
4#22582855C211yesmaximal, characteristic, complemented
5#22582854C211yesmaximal, characteristic, complemented
6#22582853C211yesmaximal, characteristic, complemented
7#22582849C212yesmaximal, complemented
8#22582841C212yesmaximal, complemented
9#22582839C212yesmaximal, complemented
10#22582837C211yesmaximal, characteristic, complemented
11C22⋊Q8×C281C211yesmaximal, characteristic, complemented
12C23×D281C2211yescharacteristic
13#17985139—24no—
14C2×D1124C2212yes—
15C2×D1124C2212yes—
16C2×D1124C2211yescharacteristic, complemented
17C2×C22⋊D281C2212yes—
18C22×Dic281C2211yescharacteristic
19C2×C4×D281—24no—
20C2×C4×D281C2211yescharacteristic
21C2×C4×D281C2211yescharacteristic
22C2×C4×D281C2211yescharacteristic
23C2×C4×D281C2211yescharacteristic
24C2×C4×D281C2212yescomplemented
25C4⋊Dic281C2211yescharacteristic, complemented
26#17985117C2212yescomplemented
27#17985115C2212yes—
28#17985115C2212yes—
29#17985115C2212yescomplemented
30#17985115C2212yescomplemented
31C4×Dic281C2212yescomplemented
32C2×Q8×C281C2211yescharacteristic, complemented
33C22×C4×C281C2211yescharacteristic
34C4⋊C4×C281C2211yescharacteristic, complemented
35C4⋊C4×C281C2212yescomplemented
36C22⋊C4×C281C2212yes—
37C22×D281—22no—
38C22×D281—22no—
39C22×D281—22no—
40C22×D281C2312yes—
41C22×D281C2311yescharacteristic
42C22×D281C2311yescharacteristic
43D1124—24no—
44D1124—24no—
45D1124—22no—
46C22⋊D281—24no—
47C2×Dic281—22no—
48C2×Dic281C2312yes—
49C2×Dic281C2311yescharacteristic
50C2×Dic281C2311yescharacteristic
51C4×D281—22no—
52C4×D281—22no—
53C4×D281—22no—
54C4×D281—24no—
55C4×D281D414yescomplemented
56C23×C281C2311yescharacteristic
57Q8×C281—22no—
58C2×C4×C281—22no—
59C2×C4×C281C2312yes—
60C2×C4×C281C2312yes—
61C2×C4×C281C2311yescharacteristic
62C2×C4×C281C2311yescharacteristic
63C22×C281—21no—
64C22×C281—21no—
65C22×C281#4112yescomplemented
66C22×C281C2411yescharacteristic
67C2×D281—28no—
68C2×D281—21no—
69C2×D281—21no—
70C2×D281—21no—
71C2×D281—21no—
72C2×D281—22no—
73C2×D281—22no—
74C2×D281—22no—
75C2×D281C2×D412yes—
76C2×D281#4112yescomplemented
77C4×C281—21no—
78C4×C281—22no—
79C4×C281—22no—
80C4×C281C2×D412yes—
81Dic281—21no—
82Dic281—22no—
83Dic281C2×D412yes—
84C2×C281—22no—
85C2×C281C22×D411yescharacteristic
86C2×C281C2×#4111yescharacteristic
87C2×C281C2×#4111yescharacteristic
88D281—22no—
89D281—22no—
90D281—42no—
91C281#52511yescharacteristic, complemented, Sylow
92#525—2811nomaximal, Sylow
93C2×#41—2811no—
94C23×C4—2811no—
95#123—2812no—
96C22⋊Q8—2811no—
97C22⋊Q8D28111yescharacteristic, complemented
98#121—2811no—
99#121—2811no—
100#120—2812no—
101C4×D4—2812no—
102C4×D4—2812no—
103#117—2811no—
104C24—2811no—
105#41—5624no—
106C2×Q8C2×D28111yescharacteristic, complemented
107C2×D4—2812no—
108C2×D4—2812no—
109C2×D4—2812no—
110C2×D4—2811no—
111C22×C4—2811no—
112C22×C4—2811no—
113C22×C4—2811no—
114C22×C4—2811no—
115C22×C4—2811no—
116C22×C4—2812no—
117C22×C4—5624no—
118C22×C4C2×D28111yescharacteristic
119C4⋊C4—2812no—
120C4⋊C4—2811no—
121C4⋊C4C2×D28111yescharacteristic, complemented
122C4⋊C4C2×D28112yescomplemented
123C22⋊C4—2812no—
124C22⋊C4—2812no—
125C22⋊C4—2812no—
126C22⋊C4—2812no—
127C22⋊C4C2×D28112yes—
128C42—2812no—
129C23—2811no—
130C23—2811no—
131C23—2812no—
132C23—5622no—
133C23—5622no—
134C23—5622no—
135C23C22×D28111yescharacteristic
136Q8—22no—
137D4—5624no—
138D4—5624no—
139D4—5624no—
140D4—5622no—
141C2×C4—22no—
142C2×C4—2811no—
143C2×C4—2814no—
144C2×C4—2812no—
145C2×C4—2811no—
146C2×C4—5622no—
147C2×C4—5624no—
148C2×C4—5622no—
149C2×C4—5622no—
150C2×C4—5622no—
151C2×C4C22×D28112yes—
152C2×C4C22×D28111yescharacteristic
153C2×C4C22×D28112yes—
154C2×C4C22×D28111yescharacteristic
155C22—21no—
156C22—21no—
157C22—2812no—
158C22—2812no—
159C22—5628no—
160C22—5621no—
161C22—5621no—
162C22—5621no—
163C22—5621no—
164C22—5622no—
165C22—5622no—
166C22—5622no—
167C22#1798513912yescomplemented
168C22C23×D28111yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—2812no—
173C4—5621no—
174C4—5622no—
175C4D4×D28112yes—
176C2—22no—
177C2—5622no—
178C2—5622no—
179C2—11242no—
180C2C2×D4×D28111yescharacteristic
181C2#41×D28111yescharacteristic
182C2C2×#1798513911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
FittingC22⋊Q8×C281C2
hypercenterC22⋊Q8D281
hypocenterC281#525
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×C4×D281C2semidirect
3. #22582864C2semidirect
4. #22582855C2semidirect
5. #22582854C2semidirect
6. #22582853C2semidirect
7. #22582849C2semidirect
8. #22582841C2semidirect
9. #22582839C2semidirect
10. #22582837C2semidirect
11. C22⋊Q8×C281C2semidirect
16. C2×D1124C22semidirect
24. C2×C4×D281C22semidirect
25. C4⋊Dic281C22semidirect
26. #17985117C22semidirect
29. #17985115C22semidirect
30. #17985115C22semidirect
31. C4×Dic281C22semidirect
32. C2×Q8×C281C22semidirect
34. C4⋊C4×C281C22semidirect
35. C4⋊C4×C281C22semidirect
55. C4×D281D4semidirect
65. C22×C281#41semidirect
76. C2×D281#41semidirect
91. C281#525semidirect
97. C22⋊Q8D281semidirect
106. C2×Q8C2×D281semidirect
121. C4⋊C4C2×D281semidirect
122. C4⋊C4C2×D281semidirect
167. 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×C4×D2812
#225828642
#225828552
#225828542
#225828532
#225828492
#225828412
#225828392
#225828372
C22⋊Q8×C2812
#525281