Finite Group Databasebeta

← #123×D43 #14583399

Subgroups — #123×D43 #14583399

Order 2752.

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×D43C211yesmaximal, characteristic
2C22×C4×D43C211yesmaximal, characteristic, complemented
3#13063310C212yesmaximal, complemented
4#13063309C211yesmaximal, characteristic, complemented
5#13063308C211yesmaximal, characteristic, complemented
6#13063307C212yesmaximal, complemented
7#13063306C211yesmaximal, characteristic, complemented
8C4⋊C4×D43C212yesmaximal, complemented
9C22⋊C4×D43C211yesmaximal, characteristic, complemented
10C22⋊C4×D43C212yesmaximal, complemented
11#123×C43C211yesmaximal, characteristic, complemented
12C23×D43C2211yescharacteristic
13C23×D43C2211yescharacteristic
14C2×D4×C43C2211yescharacteristic
15C22×C4×C43C2211yescharacteristic, complemented
16C2×C22⋊D43C2211yescharacteristic
17C2×C22⋊D43C2211yescharacteristic
18C22×Dic43C2211yescharacteristic, complemented
19D4×D43—24no—
20C2×D172C2211yescharacteristic
21C2×C4×D43—24no—
22C2×C4×D43C2212yes—
23C2×C4×D43C2212yes—
24C2×C4×D43C2211yescharacteristic
25C2×C4×D43C2212yescomplemented
26C4⋊C4×C43C2212yescomplemented
27C22⋊C4×C43C2212yescomplemented
28C22⋊C4×C43C2211yescharacteristic, complemented
29C22⋊Dic43C2212yescomplemented
30C22⋊Dic43C2211yescharacteristic, complemented
31(C2×D43)⋊C4C2212yescomplemented
32(C2×D43)⋊C4C2212yescomplemented
33(C2×D43)⋊C4C2212yescomplemented
34C4⋊Dic43C2212yescomplemented
35Dic43⋊C4C2212yescomplemented
36Dic43⋊C4C2212yescomplemented
37C23×C43C2311yescharacteristic
38C23×C43C2311yescharacteristic
39C22×D43—24no—
40C22×D43—22no—
41C22×D43—24no—
42C22×D43D414yes—
43C22×D43C2311yescharacteristic
44C22×D43C2311yescharacteristic
45C22×D43C2311yescharacteristic
46D4×C43—22no—
47C2×C4×C43—22no—
48C2×C4×C43C2311yescharacteristic
49C2×C4×C43C2312yes—
50C2×C4×C43C2312yes—
51C22⋊D43—22no—
52C22⋊D43—22no—
53C2×Dic43—22no—
54C2×Dic43C2311yescharacteristic
55C2×Dic43C2312yes—
56C2×Dic43C2312yes—
57D172—22no—
58C4×D43—24no—
59C4×D43—24no—
60C4×D43—22no—
61C4×D43—24no—
62C22×C43—22no—
63C22×C43—21no—
64C22×C43—22no—
65C22×C43C2×D412yes—
66C22×C43C2411yescharacteristic
67C2×D43—22no—
68C2×D43—28no—
69C2×D43—21no—
70C2×D43—22no—
71C2×D43—44no—
72C2×D43C2×D412yes—
73C2×D43C2×D412yes—
74C2×D43#4114yes—
75C4×C43—22no—
76C4×C43—21no—
77C4×C43—22no—
78Dic43—22no—
79Dic43—21no—
80Dic43—22no—
81C2×C43—22no—
82C2×C43—41no—
83C2×C43C22×D411yescharacteristic
84C2×C43C2×#4112yes—
85D43—22no—
86D43—41no—
87D43#12314yesdirect product factor
88C2×#123—431nomaximal, Sylow
89C43C2×#12311yescharacteristic, complemented, Sylow
90C22×D4—431no—
91C23×C4—431no—
92#123—431no—
93#123—431no—
94#123—432no—
95#123—432no—
96#123—431no—
97#123D4311yescharacteristic, direct product factor
98C2×C4⋊C4—432no—
99C2×C22⋊C4—431no—
100C2×C22⋊C4—432no—
101C24—431no—
102C24—431no—
103C2×D4—431no—
104C2×D4—431no—
105C2×D4—431no—
106C2×D4—864no—
107C2×D4C2×D4311yescharacteristic
108C22×C4—431no—
109C22×C4—432no—
110C22×C4—432no—
111C22×C4—431no—
112C22×C4—432no—
113C22×C4—864no—
114C22×C4C2×D4311yescharacteristic, complemented
115C4⋊C4—432no—
116C4⋊C4—432no—
117C4⋊C4—432no—
118C4⋊C4C2×D4312yescomplemented
119C22⋊C4—431no—
120C22⋊C4—432no—
121C22⋊C4—432no—
122C22⋊C4—432no—
123C22⋊C4—432no—
124C22⋊C4C2×D4311yescharacteristic, complemented
125C22⋊C4C2×D4312yescomplemented
126C23—431no—
127C23—434no—
128C23—431no—
129C23—431no—
130C23—864no—
131C23—862no—
132C23—864no—
133C23C22×D4311yescharacteristic
134C23C22×D4311yescharacteristic
135D4—22no—
136D4—862no—
137D4—862no—
138D4—862no—
139C2×C4—22no—
140C2×C4—432no—
141C2×C4—432no—
142C2×C4—431no—
143C2×C4—864no—
144C2×C4—864no—
145C2×C4—862no—
146C2×C4—864no—
147C2×C4—862no—
148C2×C4C22×D4312yes—
149C2×C4C22×D4312yes—
150C2×C4C22×D4311yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—432no—
155C22—434no—
156C22—432no—
157C22—868no—
158C22—862no—
159C22—861no—
160C22—862no—
161C22—1724no—
162C22D4×D4312yes—
163C22C23×D4311yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—862no—
168C4—861no—
169C4—862no—
170C2—22no—
171C2—41no—
172C2—434no—
173C2—862no—
174C2—1721no—
175C2C2×D4×D4311yescharacteristic
176C2#41×D4312yes—
subgroupG/H
center, FrattiniC22C23×D43
commutator, socle, cosocleC22×C43C24
Fitting#123×C43C2
prime radicalC2×D4×D43C2
hypercenter#123D43
hypocenterC43C2×#123
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D43C2semidirect
3. #13063310C2semidirect
4. #13063309C2semidirect
5. #13063308C2semidirect
6. #13063307C2semidirect
7. #13063306C2semidirect
8. C4⋊C4×D43C2semidirect
9. C22⋊C4×D43C2semidirect
10. C22⋊C4×D43C2semidirect
11. #123×C43C2semidirect
15. C22×C4×C43C22semidirect
18. C22×Dic43C22semidirect
25. C2×C4×D43C22semidirect
26. C4⋊C4×C43C22semidirect
27. C22⋊C4×C43C22semidirect
28. C22⋊C4×C43C22semidirect
29. C22⋊Dic43C22semidirect
30. C22⋊Dic43C22semidirect
31. (C2×D43)⋊C4C22semidirect
32. (C2×D43)⋊C4C22semidirect
33. (C2×D43)⋊C4C22semidirect
34. C4⋊Dic43C22semidirect
35. Dic43⋊C4C22semidirect
36. Dic43⋊C4C22semidirect
87. D43#123direct (N × H)
89. C43C2×#123semidirect
97. #123D43direct (N × H)
114. C22×C4C2×D43semidirect
118. C4⋊C4C2×D43semidirect
124. C22⋊C4C2×D43semidirect
125. C22⋊C4C2×D43semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D432
C22×C4×D432
#130633102
#130633092
#130633082
#130633072
#130633062
C4⋊C4×D432
C22⋊C4×D432
#123×C432
C2×#12343