Finite Group Databasebeta

← C22×C3×#126⋊C3 #11919704

Subgroups — C22×C3×#126⋊C3 #11919704

Order 1152.

696 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 138 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×C3×#126⋊C3C216yesmaximal, direct product factor
2C22×C3×C42⋊C3C211yesmaximal, characteristic, complemented
3C22×C3×#126C311yesmaximal, characteristic, complemented
4C22×#126⋊C3C313yesmaximal, direct product factor
5C23×C3×A4—41nomaximal
6C3×#126⋊C3C2214yesdirect product factor
7C2×C3×C42⋊C3C2213yescomplemented
8C2×C3×#126—36no—
9C2×C3×#126C2×C316yescomplemented
10C22×C3×C4⋊C4—31no—
11C22×C3×C22⋊C4—31no—
12C22×C3×C42C2×C311yescharacteristic, complemented
13C2×#126⋊C3C2×C3118yesdirect product factor
14C22×C42⋊C3C2×C313yescomplemented
15C22×C3×A4—41no—
16C22×C3×A4—46no—
17C3×C42⋊C3C2311yescharacteristic, complemented
18C22×#126C3211yescharacteristic, complemented, Sylow
19C25×C3A411yescharacteristic
20C23×A4—43no—
21C23×C3×C4—31no—
22C23×C3×C4—31no—
23C3×#126—38no—
24C3×#126—312no—
25C3×#126C22×C314yescomplemented
26C2×C3×C4⋊C4—36no—
27C2×C3×C4⋊C4—36no—
28C2×C3×C22⋊C4—36no—
29C2×C3×C22⋊C4—36no—
30C2×C3×C42—33no—
31C2×C3×C42C22×C313yescomplemented
32#126⋊C3C22×C3112yesdirect product factor
33C2×C42⋊C3C22×C319yescomplemented
34C23×C32—161no—
35C2×C3×A4—43no—
36C2×C3×A4—44no—
37C2×#126—36no—
38C2×#126C2×C3216yescomplemented
39C22×C4⋊C4—31no—
40C22×C22⋊C4—31no—
41C22×C42C2×C3211yescharacteristic, complemented
42C24×C3—61no—
43C24×C3—63no—
44C24×C3C2×A411yescharacteristic
45C24×C3C2×A416yes—
46C22×A4—43no—
47C22×A4—418no—
48C22×C3×C4—33no—
49C22×C3×C4—33no—
50C22×C3×C4—36no—
51C22×C3×C4—63no—
52C22×C3×C4—63no—
53C22×C3×C4—61no—
54C22×C3×C4—61no—
55C3×C4⋊C4—34no—
56C3×C4⋊C4—312no—
57C3×C22⋊C4—312no—
58C3×C22⋊C4—34no—
59C3×C42—33no—
60C3×C42—32no—
61C3×C42C23×C311yescharacteristic, complemented
62C42⋊C3C23×C313yescomplemented
63C22×C32—166no—
64C22×C32—161no—
65C3×A4—41no—
66C25C3×A411yescharacteristic
67C23×C4—31no—
68C23×C4—31no—
69#126—38no—
70#126—312no—
71#126C22×C3214yescomplemented
72C2×C4⋊C4—36no—
73C2×C4⋊C4—36no—
74C2×C22⋊C4—36no—
75C2×C22⋊C4—36no—
76C2×C42—33no—
77C2×C42C22×C3213yescomplemented
78C23×C3—33no—
79C23×C3—31no—
80C23×C3—41no—
81C23×C3—66no—
82C23×C3—66no—
83C23×C3—123no—
84C23×C3—122no—
85C23×C3—163no—
86C23×C3C22×A413yes—
87C23×C3C22×A414yes—
88C2×A4—49no—
89C2×A4—412no—
90C2×C3×C4—31no—
91C2×C3×C4—33no—
92C2×C3×C4—34no—
93C2×C3×C4—63no—
94C2×C3×C4—63no—
95C2×C3×C4—66no—
96C2×C3×C4—63no—
97C2×C3×C4—63no—
98C2×C3×C4—66no—
99C2×C32—164no—
100C2×C32—163no—
101C24—61no—
102C24—63no—
103C24C2×C3×A411yescharacteristic
104C24C2×C3×A416yes—
105C22×C4—33no—
106C22×C4—33no—
107C22×C4—36no—
108C22×C4—63no—
109C22×C4—63no—
110C22×C4—61no—
111C22×C4—61no—
112C4⋊C4—34no—
113C4⋊C4—312no—
114C22⋊C4—34no—
115C22⋊C4—312no—
116C42—33no—
117C42—32no—
118C42C23×C3211yescharacteristic, complemented
119C22×C3—33no—
120C22×C3—32no—
121C22×C3—33no—
122C22×C3—33no—
123C22×C3—46no—
124C22×C3—64no—
125C22×C3—126no—
126C22×C3—1618no—
127C22×C3—163no—
128C22×C3#126⋊C311yescharacteristic, direct product factor
129C22×C3C23×A411yescharacteristic
130A4—43no—
131C3×C4—61no—
132C3×C4—63no—
133C3×C4—64no—
134C32—161noSylow
135C23—33no—
136C23—31no—
137C23—41no—
138C23—66no—
139C23—66no—
140C23—123no—
141C23—122no—
142C23C22×C3×A413yes—
143C23C22×C3×A414yes—
144C2×C4—31no—
145C2×C4—33no—
146C2×C4—34no—
147C2×C4—63no—
148C2×C4—63no—
149C2×C4—66no—
150C2×C4—63no—
151C2×C4—63no—
152C2×C4—66no—
153C2×C3—31no—
154C2×C3—33no—
155C2×C3—44no—
156C2×C3—1612no—
157C2×C3—169no—
158C2×C3C2×#126⋊C313yesdirect product factor
159C22—33no—
160C22—32no—
161C22—33no—
162C22—33no—
163C22—46no—
164C22—64no—
165C22—126no—
166C22C3×#126⋊C311yescharacteristic, direct product factor
167C22C23×C3×A411yescharacteristic
168C4—61no—
169C4—63no—
170C4—64no—
171C3—163no—
172C3C22×#126⋊C311yescharacteristic, direct product factor
173C2—31no—
174C2—33no—
175C2—44no—
176C2C2×C3×#126⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#126⋊C3
commutator, cosocle, hypocenterC42C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#126C3
socleC24×C3C2×A4
LCSG ▷ C42
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126⋊C3C2direct (N × H)
2. C22×C3×C42⋊C3C2semidirect
3. C22×C3×#126C3semidirect
4. C22×#126⋊C3C3direct (N × H)
6. C3×#126⋊C3C22direct (N × H)
7. C2×C3×C42⋊C3C22semidirect
9. C2×C3×#126C2×C3semidirect
12. C22×C3×C42C2×C3semidirect
13. C2×#126⋊C3C2×C3direct (N × H)
14. C22×C42⋊C3C2×C3semidirect
17. C3×C42⋊C3C23semidirect
18. C22×#126C32semidirect
25. C3×#126C22×C3semidirect
31. C2×C3×C42C22×C3semidirect
32. #126⋊C3C22×C3direct (N × H)
33. C2×C42⋊C3C22×C3semidirect
38. C2×#126C2×C32semidirect
41. C22×C42C2×C32semidirect
61. C3×C42C23×C3semidirect
62. C42⋊C3C23×C3semidirect
71. #126C22×C32semidirect
77. C2×C42C22×C32semidirect
118. C42C23×C32semidirect
128. C22×C3#126⋊C3direct (N × H)
158. C2×C3C2×#126⋊C3direct (N × H)
166. C22C3×#126⋊C3direct (N × H)
172. C3C22×#126⋊C3direct (N × H)
176. C2C2×C3×#126⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#126⋊C32
C22×C3×C42⋊C32
C22×C3×#1263
C22×#126⋊C33
C23×C3×A44