Finite Group Databasebeta

← #123×D281 #31823963

Subgroups — #123×D281 #31823963

Order 17984.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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×D4×D281C211yesmaximal, characteristic
2C22×C4×D281C211yesmaximal, characteristic, complemented
3#22582864C212yesmaximal, complemented
4#22582863C211yesmaximal, characteristic, complemented
5#22582862C211yesmaximal, characteristic, complemented
6#22582861C212yesmaximal, complemented
7#22582860C211yesmaximal, characteristic, complemented
8C4⋊C4×D281C212yesmaximal, complemented
9C22⋊C4×D281C212yesmaximal, complemented
10C22⋊C4×D281C211yesmaximal, characteristic, complemented
11#123×C281C211yesmaximal, characteristic, complemented
12C23×D281C2211yescharacteristic
13C23×D281C2211yescharacteristic
14C2×D1124C2211yescharacteristic
15C2×C22⋊D281C2211yescharacteristic
16C2×C22⋊D281C2211yescharacteristic
17D4×D281—24no—
18C22×Dic281C2211yescharacteristic, complemented
19C2×C4×D281—24no—
20C2×C4×D281C2212yes—
21C2×C4×D281C2212yes—
22C2×C4×D281C2211yescharacteristic
23C2×C4×D281C2212yescomplemented
24C4⋊Dic281C2212yescomplemented
25#17985117C2212yescomplemented
26#17985117C2212yescomplemented
27C22⋊Dic281C2212yescomplemented
28C22⋊Dic281C2211yescharacteristic, complemented
29#17985115C2212yescomplemented
30#17985115C2212yescomplemented
31#17985115C2212yescomplemented
32C2×D4×C281C2211yescharacteristic
33C22×C4×C281C2211yescharacteristic, complemented
34C4⋊C4×C281C2212yescomplemented
35C22⋊C4×C281C2212yescomplemented
36C22⋊C4×C281C2211yescharacteristic, complemented
37C22×D281—24no—
38C22×D281—22no—
39C22×D281—24no—
40C22×D281D414yes—
41C22×D281C2311yescharacteristic
42C22×D281C2311yescharacteristic
43C22×D281C2311yescharacteristic
44D1124—22no—
45C22⋊D281—22no—
46C22⋊D281—22no—
47C2×Dic281—22no—
48C2×Dic281C2312yes—
49C2×Dic281C2312yes—
50C2×Dic281C2311yescharacteristic
51C4×D281—24no—
52C4×D281—24no—
53C4×D281—22no—
54C4×D281—24no—
55C23×C281C2311yescharacteristic
56C23×C281C2311yescharacteristic
57D4×C281—22no—
58C2×C4×C281—22no—
59C2×C4×C281C2312yes—
60C2×C4×C281C2312yes—
61C2×C4×C281C2311yescharacteristic
62C22×C281—22no—
63C22×C281—21no—
64C22×C281—22no—
65C22×C281C2×D412yes—
66C22×C281C2411yescharacteristic
67C2×D281—22no—
68C2×D281—28no—
69C2×D281—21no—
70C2×D281—22no—
71C2×D281—44no—
72C2×D281C2×D412yes—
73C2×D281C2×D412yes—
74C2×D281#4114yes—
75C4×C281—22no—
76C4×C281—21no—
77C4×C281—22no—
78Dic281—22no—
79Dic281—21no—
80Dic281—22no—
81C2×C281—22no—
82C2×C281—41no—
83C2×C281C22×D411yescharacteristic
84C2×C281C2×#4112yes—
85D281—22no—
86D281—41no—
87D281#12314yesdirect product factor
88C281C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—2811nomaximal, Sylow
90C22×D4—2811no—
91C23×C4—2811no—
92#123—2811no—
93#123—2811no—
94#123—2812no—
95#123—2812no—
96#123—2811no—
97#123D28111yescharacteristic, direct product factor
98C2×C4⋊C4—2812no—
99C2×C22⋊C4—2811no—
100C2×C22⋊C4—2812no—
101C24—2811no—
102C24—2811no—
103C2×D4—2811no—
104C2×D4—2811no—
105C2×D4—2811no—
106C2×D4—5624no—
107C2×D4C2×D28111yescharacteristic
108C22×C4—2811no—
109C22×C4—2812no—
110C22×C4—2812no—
111C22×C4—2811no—
112C22×C4—2812no—
113C22×C4—5624no—
114C22×C4C2×D28111yescharacteristic, complemented
115C4⋊C4—2812no—
116C4⋊C4—2812no—
117C4⋊C4—2812no—
118C4⋊C4C2×D28112yescomplemented
119C22⋊C4—2811no—
120C22⋊C4—2812no—
121C22⋊C4—2812no—
122C22⋊C4—2812no—
123C22⋊C4—2812no—
124C22⋊C4C2×D28111yescharacteristic, complemented
125C22⋊C4C2×D28112yescomplemented
126C23—2811no—
127C23—2814no—
128C23—2811no—
129C23—2811no—
130C23—5624no—
131C23—5622no—
132C23—5624no—
133C23C22×D28111yescharacteristic
134C23C22×D28111yescharacteristic
135D4—22no—
136D4—5622no—
137D4—5622no—
138D4—5622no—
139C2×C4—22no—
140C2×C4—2812no—
141C2×C4—2812no—
142C2×C4—2811no—
143C2×C4—5624no—
144C2×C4—5624no—
145C2×C4—5622no—
146C2×C4—5624no—
147C2×C4—5622no—
148C2×C4C22×D28112yes—
149C2×C4C22×D28112yes—
150C2×C4C22×D28111yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—2812no—
155C22—2814no—
156C22—2812no—
157C22—5628no—
158C22—5622no—
159C22—5621no—
160C22—5622no—
161C22—11244no—
162C22D4×D28112yes—
163C22C23×D28111yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—5622no—
168C4—5621no—
169C4—5622no—
170C2—22no—
171C2—41no—
172C2—2814no—
173C2—5622no—
174C2—11241no—
175C2C2×D4×D28111yescharacteristic
176C2#41×D28112yes—
subgroupG/H
center, FrattiniC22C23×D281
commutator, socle, cosocleC22×C281C24
Fitting#123×C281C2
prime radicalC2×D4×D281C2
hypercenter#123D281
hypocenterC281C2×#123
LCSG ▷ C22×C281 ▷ C281
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D281C2semidirect
3. #22582864C2semidirect
4. #22582863C2semidirect
5. #22582862C2semidirect
6. #22582861C2semidirect
7. #22582860C2semidirect
8. C4⋊C4×D281C2semidirect
9. C22⋊C4×D281C2semidirect
10. C22⋊C4×D281C2semidirect
11. #123×C281C2semidirect
18. C22×Dic281C22semidirect
23. C2×C4×D281C22semidirect
24. C4⋊Dic281C22semidirect
25. #17985117C22semidirect
26. #17985117C22semidirect
27. C22⋊Dic281C22semidirect
28. C22⋊Dic281C22semidirect
29. #17985115C22semidirect
30. #17985115C22semidirect
31. #17985115C22semidirect
33. C22×C4×C281C22semidirect
34. C4⋊C4×C281C22semidirect
35. C22⋊C4×C281C22semidirect
36. C22⋊C4×C281C22semidirect
87. D281#123direct (N × H)
88. C281C2×#123semidirect
97. #123D281direct (N × H)
114. C22×C4C2×D281semidirect
118. C4⋊C4C2×D281semidirect
124. C22⋊C4C2×D281semidirect
125. C22⋊C4C2×D281semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2812
C22×C4×D2812
#225828642
#225828632
#225828622
#225828612
#225828602
C4⋊C4×D2812
C22⋊C4×D2812
#123×C2812
C2×#123281