Finite Group Databasebeta

← C22×#126⋊C9 #11919705

Subgroups — C22×#126⋊C9 #11919705

Order 1152.

552 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 90 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×#126⋊C9C216yesmaximal, direct product factor
2C22×C42⋊C9C211yesmaximal, characteristic, complemented
3C22×C3×#126C311yesmaximal, characteristic
4C23×C22⋊C9—41nomaximal
5#126⋊C9C2214yesdirect product factor
6C2×C42⋊C9C2213yescomplemented
7C2×C3×#126—36no—
8C2×C3×#126C2×C316yes—
9C22×C3×C4⋊C4—31no—
10C22×C3×C22⋊C4—31no—
11C22×C3×C42C2×C311yescharacteristic
12C22×C22⋊C9—41no—
13C22×C22⋊C9—46no—
14C42⋊C9C2311yescharacteristic, complemented
15C22×#126C911yescharacteristic, complemented, Sylow
16C25×C3A411yescharacteristic
17C23×C3×C4—31no—
18C23×C3×C4—31no—
19C3×#126—38no—
20C3×#126—312no—
21C3×#126C22×C314yes—
22C2×C3×C4⋊C4—36no—
23C2×C3×C4⋊C4—36no—
24C2×C3×C22⋊C4—36no—
25C2×C3×C22⋊C4—36no—
26C2×C3×C42—33no—
27C2×C3×C42C22×C313yes—
28C23×C9—161no—
29C2×C22⋊C9—43no—
30C2×C22⋊C9—44no—
31C2×#126—36no—
32C2×#126C2×C916yescomplemented
33C22×C4⋊C4—31no—
34C22×C22⋊C4—31no—
35C22×C42C2×C911yescharacteristic, complemented
36C24×C3—61no—
37C24×C3—63no—
38C24×C3C2×A411yescharacteristic
39C24×C3C2×A416yes—
40C22×C3×C4—33no—
41C22×C3×C4—33no—
42C22×C3×C4—36no—
43C22×C3×C4—63no—
44C22×C3×C4—63no—
45C22×C3×C4—61no—
46C22×C3×C4—61no—
47C3×C4⋊C4—34no—
48C3×C4⋊C4—312no—
49C3×C22⋊C4—312no—
50C3×C22⋊C4—34no—
51C3×C42—33no—
52C3×C42—32no—
53C3×C42C23×C311yescharacteristic
54C22×C9—166no—
55C22×C9—161no—
56C22⋊C9—41no—
57C25C22⋊C911yescharacteristic
58C23×C4—31no—
59C23×C4—31no—
60#126—38no—
61#126—312no—
62#126C22×C914yescomplemented
63C2×C4⋊C4—36no—
64C2×C4⋊C4—36no—
65C2×C22⋊C4—36no—
66C2×C22⋊C4—36no—
67C2×C42—33no—
68C2×C42C22×C913yescomplemented
69C23×C3—33no—
70C23×C3—31no—
71C23×C3—41no—
72C23×C3—66no—
73C23×C3—66no—
74C23×C3—123no—
75C23×C3—122no—
76C23×C3C22×A413yes—
77C23×C3C22×A414yes—
78C2×C3×C4—31no—
79C2×C3×C4—33no—
80C2×C3×C4—34no—
81C2×C3×C4—63no—
82C2×C3×C4—63no—
83C2×C3×C4—66no—
84C2×C3×C4—63no—
85C2×C3×C4—63no—
86C2×C3×C4—66no—
87C2×C9—164no—
88C2×C9—163no—
89C24—61no—
90C24—63no—
91C24C2×C22⋊C911yescharacteristic
92C24C2×C22⋊C916yes—
93C22×C4—33no—
94C22×C4—33no—
95C22×C4—36no—
96C22×C4—63no—
97C22×C4—63no—
98C22×C4—61no—
99C22×C4—61no—
100C4⋊C4—34no—
101C4⋊C4—312no—
102C22⋊C4—34no—
103C22⋊C4—312no—
104C42—33no—
105C42—32no—
106C42C23×C911yescharacteristic, complemented
107C22×C3—33no—
108C22×C3—32no—
109C22×C3—33no—
110C22×C3—33no—
111C22×C3—46no—
112C22×C3—64no—
113C22×C3—126no—
114C22×C3#126⋊C311yescharacteristic
115C22×C3C23×A411yescharacteristic
116C3×C4—61no—
117C3×C4—63no—
118C3×C4—64no—
119C9—161noSylow
120C23—33no—
121C23—31no—
122C23—41no—
123C23—66no—
124C23—66no—
125C23—123no—
126C23—122no—
127C23C22×C22⋊C913yes—
128C23C22×C22⋊C914yes—
129C2×C4—31no—
130C2×C4—33no—
131C2×C4—34no—
132C2×C4—63no—
133C2×C4—63no—
134C2×C4—66no—
135C2×C4—63no—
136C2×C4—63no—
137C2×C4—66no—
138C2×C3—31no—
139C2×C3—33no—
140C2×C3—44no—
141C2×C3C2×#126⋊C313yes—
142C22—33no—
143C22—32no—
144C22—33no—
145C22—33no—
146C22—46no—
147C22—64no—
148C22—126no—
149C22#126⋊C911yescharacteristic, direct product factor
150C22C23×C22⋊C911yescharacteristic
151C4—61no—
152C4—63no—
153C4—64no—
154C3C22×#126⋊C311yescharacteristic
155C2—31no—
156C2—33no—
157C2—44no—
158C2C2×#126⋊C913yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#126⋊C3
commutator, hypocenterC42C23×C9
FrattiniC22×C3C23×A4
FittingC22×C3×#126C3
socleC24×C3C2×A4
prime radicalC25×C3A4
cosocleC3×C42C23×C3
LCSG ▷ C42
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×#126⋊C9C2direct (N × H)
2. C22×C42⋊C9C2semidirect
5. #126⋊C9C22direct (N × H)
6. C2×C42⋊C9C22semidirect
14. C42⋊C9C23semidirect
15. C22×#126C9semidirect
32. C2×#126C2×C9semidirect
35. C22×C42C2×C9semidirect
62. #126C22×C9semidirect
68. C2×C42C22×C9semidirect
106. C42C23×C9semidirect
149. C22#126⋊C9direct (N × H)
158. C2C2×#126⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#126⋊C92
C22×C42⋊C92
C22×C3×#1263
C23×C22⋊C94