Finite Group Databasebeta

← C42×#1094 #13152160

Subgroups — C42×#1094 #13152160

Order 1728.

678 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 150 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×C4×#1094C213yesmaximal
2C22×C32×C42C311yesmaximal, characteristic
3C42×C22⋊C9C313yesmaximal, complemented
4C22×#1094C2211yescharacteristic
5C4×#1094C416yesdirect product factor
6C42×3−1+2—41nomaximal
7C23×C32×C4C2×C313yes—
8C2×C32×C42—31no—
9C2×C32×C42—33no—
10C2×C4×C22⋊C9C2×C319yes—
11C2×#1094C2×C413yes—
12C2×C4×3−1+2—43no—
13C22×C3×C42—31no—
14C22×C3×C42C3211yescharacteristic
15C24×C32C22×C311yescharacteristic
16C22×C32×C4—33no—
17C22×C32×C4—36no—
18C22×C32×C4—33no—
19C22×C32×C4C3×C416yes—
20C22×C22⋊C9C22×C313yes—
21C32×C42—32no—
22C32×C42—33no—
23C32×C42A411yescharacteristic
24C4×C22⋊C9C3×C4118yescomplemented
25C42×C9—43no—
26C22×3−1+2—41no—
27#1094C4211yescharacteristic, direct product factor
28C4×3−1+2—46no—
29C23×C3×C4—33no—
30C23×C3×C4C2×C3213yes—
31C2×C3×C42—33no—
32C2×C3×C42—39no—
33C2×C3×C42—31no—
34C2×C3×C42—33no—
35C23×C32—33no—
36C23×C32—31no—
37C23×C32C2×C3×C413yes—
38C2×C32×C4—36no—
39C2×C32×C4—33no—
40C2×C32×C4—36no—
41C2×C32×C4—36no—
42C2×C32×C4—36no—
43C2×C32×C4C2×A413yes—
44C2×C22⋊C9C2×C3×C419yes—
45C2×C4×C9—49no—
46C22×C423−1+211yescharacteristic, complemented, Sylow
47C2×3−1+2—43no—
48C24×C3—31no—
49C24×C3C22×C3211yescharacteristic
50C22×C3×C4—39no—
51C22×C3×C4—39no—
52C22×C3×C4—318no—
53C22×C3×C4—33no—
54C22×C3×C4—33no—
55C22×C3×C4—36no—
56C22×C3×C4—36no—
57C22×C3×C4C32×C416yes—
58C3×C42—36no—
59C3×C42—32no—
60C3×C42—39no—
61C3×C42—33no—
62C3×C42—31no—
63C3×C42C3×A411yescharacteristic
64C22×C32—33no—
65C22×C32—32no—
66C22×C32—33no—
67C22×C32—33no—
68C22×C32C3×C4211yescharacteristic
69C22×C32C22×A411yescharacteristic
70C32×C4—36no—
71C32×C4C4×A416yes—
72C22×C9—43no—
73C22⋊C9C3×C4213yescomplemented
74C4×C9—418no—
75C23×C4C2×3−1+213yes—
76C2×C42—31no—
77C2×C42—33no—
783−1+2—41noSylow
79C23×C3—33no—
80C23×C3—33no—
81C23×C3—39no—
82C23×C3—31no—
83C23×C3—33no—
84C23×C3C2×C32×C413yes—
85C2×C3×C4—39no—
86C2×C3×C4—318no—
87C2×C3×C4—36no—
88C2×C3×C4—318no—
89C2×C3×C4—318no—
90C2×C3×C4—318no—
91C2×C3×C4—33no—
92C2×C3×C4—36no—
93C2×C3×C4—36no—
94C2×C3×C4—36no—
95C2×C3×C4—33no—
96C2×C3×C4C2×C3×A413yes—
97C2×C32—33no—
98C2×C32—31no—
99C2×C32C2×C4×A413yes—
100C2×C9—49no—
101C24C22×3−1+211yescharacteristic
102C22×C4—33no—
103C22×C4—33no—
104C22×C4—36no—
105C22×C4C4×3−1+216yescomplemented
106C42—32no—
107C42—33no—
108C42#109411yescharacteristic, direct product factor
109C22×C3—33no—
110C22×C3—39no—
111C22×C3—39no—
112C22×C3—39no—
113C22×C3—36no—
114C22×C3—32no—
115C22×C3—33no—
116C22×C3—33no—
117C22×C3—31no—
118C22×C3—31no—
119C22×C3C32×C4211yescharacteristic
120C22×C3C22×C3×A411yescharacteristic
121C3×C4—318no—
122C3×C4—36no—
123C3×C4—36no—
124C3×C4C3×C4×A416yes—
125C32C42×A411yescharacteristic
126C9—43no—
127C23—33no—
128C23—31no—
129C23C2×C4×3−1+213yes—
130C2×C4—36no—
131C2×C4—33no—
132C2×C4—36no—
133C2×C4—36no—
134C2×C4—36no—
135C2×C4C2×#109413yes—
136C2×C3—33no—
137C2×C3—39no—
138C2×C3—31no—
139C2×C3—33no—
140C2×C3—33no—
141C2×C3C2×C3×C4×A413yes—
142C22—33no—
143C22—32no—
144C22—33no—
145C22—33no—
146C22C42×3−1+211yescharacteristic, complemented
147C22C22×#109411yescharacteristic
148C4—36no—
149C4C4×#109416yesdirect product factor
150C3—31no—
151C3C3×C42×A411yescharacteristic
152C2—31no—
153C2—33no—
154C2C2×C4×#109413yes—
subgroupG/H
centerC3×C42C3×A4
commutatorC22×C3C32×C42
FrattiniC22×C3C22×C3×A4
FittingC22×C32×C42C3
socle, cosocleC24×C3C22×C32
prime radicalC24×C32C22×C3
hypercenterC32×C42A4
hypocenterC22C42×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C3×C42 ◁ C32×C42

Complements

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

NH ≅ G/Ntype
3. C42×C22⋊C9C3semidirect
5. C4×#1094C4direct (N × H)
24. C4×C22⋊C9C3×C4semidirect
27. #1094C42direct (N × H)
46. C22×C423−1+2semidirect
73. C22⋊C9C3×C42semidirect
105. C22×C4C4×3−1+2semidirect
108. C42#1094direct (N × H)
146. C22C42×3−1+2semidirect
149. C4C4×#1094direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#10942
C22×C32×C423
C42×C22⋊C93
C42×3−1+24