Finite Group Databasebeta

← 3−1+2×#384 #13144912

Subgroups — 3−1+2×#384 #13144912

Order 1728.

744 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 455 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⋊C4×3−1+2C216yesmaximal
2C2×C42×3−1+2C211yesmaximal, characteristic
3C32×#384C311yesmaximal, characteristic
4C9×#384C313yesmaximal, complemented
5C22×C4×3−1+2C2214yes—
6C22×C4×3−1+2C2213yes—
7C4⋊C4×3−1+2—212no—
8C42×3−1+2C414yescomplemented
9C2×C32×C4⋊C4C2×C316yes—
10C2×C32×C42C2×C311yescharacteristic
11C2×C9×C4⋊C4C2×C3118yes—
12C2×C42×C9C2×C313yes—
13C23×3−1+2C2311yescharacteristic
14C2×C4×3−1+2—212no—
15C2×C4×3−1+2C2×C416yes—
16C2×C4×3−1+2D416yes—
17C2×C4×3−1+2Q816yes—
18C3×#384—31no—
19C3×#384C3211yescharacteristic
20C22×C32×C4C22×C314yes—
21C22×C32×C4C22×C313yes—
22C32×C4⋊C4—212no—
23C32×C42C3×C414yes—
24C22×C4×C9C22×C3112yes—
25C22×C4×C9C22×C319yes—
26C9×C4⋊C4—236no—
27C42×C9C3×C4112yescomplemented
28C22×3−1+2C22×C411yescharacteristic
29C22×3−1+2C2×D413yes—
30C22×3−1+2C2×Q813yes—
31C4×3−1+2—44no—
32C4×3−1+2C4⋊C4112yescomplemented
33C2×C3×C4⋊C4—36no—
34C2×C3×C4⋊C4C2×C3216yes—
35C2×C3×C42—31no—
36C2×C3×C42C2×C3211yescharacteristic
37C23×C32C23×C311yescharacteristic
38C2×C32×C4—212no—
39C2×C32×C4C2×C3×C416yes—
40C2×C32×C4C3×D416yes—
41C2×C32×C4C3×Q816yes—
42C23×C9C23×C313yes—
43C2×C4×C9—236no—
44C2×C4×C9C2×C3×C4118yes—
45C2×C4×C9C3×D4118yes—
46C2×C4×C9C3×Q8118yes—
47#3843−1+211yescharacteristic, direct product factor, Sylow
48C2×3−1+2C2×C4⋊C413yes—
49C2×3−1+2#12711yescharacteristic
50C2×3−1+2C4⋊Q813yes—
51C22×C3×C4—33no—
52C22×C3×C4—34no—
53C22×C3×C4C22×C3214yes—
54C22×C3×C4C22×C3213yes—
55C3×C4⋊C4—212no—
56C3×C4⋊C4—612no—
57C3×C42—34no—
58C3×C42C32×C414yes—
59C22×C32C22×C3×C411yescharacteristic
60C22×C32C2×C3×D413yes—
61C22×C32C2×C3×Q813yes—
62C32×C4—44no—
63C32×C4C3×C4⋊C4112yes—
64C22×C9C22×C3×C413yes—
65C22×C9C2×C3×D419yes—
66C22×C9C2×C3×Q819yes—
67C4×C9—412no—
68C4×C9C3×C4⋊C4136yescomplemented
69C2×C4⋊C4C2×3−1+216yes—
70C2×C42C2×3−1+211yescharacteristic
713−1+2#38411yescharacteristic, direct product factor, Sylow
72C23×C3—31no—
73C23×C3C23×C3211yescharacteristic
74C2×C3×C4—212no—
75C2×C3×C4—36no—
76C2×C3×C4—36no—
77C2×C3×C4—36no—
78C2×C3×C4—612no—
79C2×C3×C4C2×C32×C416yes—
80C2×C3×C4C32×D416yes—
81C2×C3×C4C32×Q816yes—
82C2×C32C2×C3×C4⋊C413yes—
83C2×C32C3×#12711yescharacteristic
84C2×C32C3×C4⋊Q813yes—
85C2×C9C2×C3×C4⋊C419yes—
86C2×C9C3×#12713yes—
87C2×C9C3×C4⋊Q819yes—
88C22×C4C22×3−1+214yes—
89C22×C4C22×3−1+213yes—
90C4⋊C4—212no—
91C42C4×3−1+214yescomplemented
92C22×C3—31no—
93C22×C3—33no—
94C22×C3—33no—
95C22×C3C22×C32×C411yescharacteristic
96C22×C3C2×C32×D413yes—
97C22×C3C2×C32×Q813yes—
98C3×C4—312no—
99C3×C4—44no—
100C3×C4—124no—
101C3×C4C32×C4⋊C4112yes—
102C32C3×#38411yescharacteristic
103C9C3×#38413yescomplemented
104C23C23×3−1+211yescharacteristic
105C2×C4—212no—
106C2×C4C2×C4×3−1+216yes—
107C2×C4D4×3−1+216yes—
108C2×C4Q8×3−1+216yes—
109C2×C3—33no—
110C2×C3—31no—
111C2×C3—33no—
112C2×C3C2×C32×C4⋊C413yes—
113C2×C3C32×#12711yescharacteristic
114C2×C3C32×C4⋊Q813yes—
115C22C22×C4×3−1+211yescharacteristic
116C22C2×D4×3−1+213yes—
117C22C2×Q8×3−1+213yes—
118C4—44no—
119C4C4⋊C4×3−1+2112yescomplemented
120C3—31no—
121C3C32×#38411yescharacteristic
122C2C2×C4⋊C4×3−1+213yes—
123C23−1+2×#12711yescharacteristic
124C23−1+2×C4⋊Q813yes—
subgroupG/H
center, Frattini, socle, cosocleC23×C3C23×C32
commutatorC22×C3C22×C32×C4
prime radicalC23×C32C23×C3
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
4. C9×#384C3semidirect
8. C42×3−1+2C4semidirect
27. C42×C9C3×C4semidirect
32. C4×3−1+2C4⋊C4semidirect
47. #3843−1+2direct (N × H)
68. C4×C9C3×C4⋊C4semidirect
71. 3−1+2#384direct (N × H)
91. C42C4×3−1+2semidirect
103. C9C3×#384semidirect
119. C4C4⋊C4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×C42×3−1+22
C32×#3843
C9×#3843