Finite Group Databasebeta

← #13141242

Subgroups — #13141242

Order 1728.

242 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 29 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
1#11722586C211yesmaximal, characteristic
2C32×#395C311yesmaximal, characteristic
3#395⋊C9C313yesmaximal, complemented
4C4×#1094—41nomaximal
5C22×C32×Q8C2×C311yescharacteristic
6C2×C32×C4⋊C4—31no—
7C22⋊(Q8⋊C9)C2×C313yes—
8C32×#95—31no—
9C2×#1094—41no—
10#6204—82no—
11C3×#395—31no—
12C3×#395C3211yescharacteristic
13C2×C32×Q8—62no—
14C22×C32×C4—31no—
15C22×C32×C4—31no—
16C22×C32×C4A411yescharacteristic
17C32×C4⋊C4—62no—
18C4×C22⋊C9—43no—
19#1094—41no—
20C4×3−1+2—161no—
21C22×C3×Q8—31no—
22C22×C3×Q8C2×C3211yescharacteristic
23C2×C3×C4⋊C4—33no—
24C2×C3×C4⋊C4—31no—
25C3×#95—31no—
26C3×#95—33no—
27C23×C32C2×A411yescharacteristic
28C32×Q8—22no—
29C32×Q8—62no—
30C2×C32×C4—32no—
31C2×C32×C4—32no—
32C2×C32×C4—61no—
33C2×C32×C4—61no—
34C2×C32×C4—62no—
35C2×C32×C4—61no—
36C2×C22⋊C9—43no—
37Q8⋊C9—86no—
38#3953−1+211yescharacteristic, complemented, Sylow
39C2×3−1+2—161no—
40C2×C3×Q8—62no—
41C2×C3×Q8—66no—
42C22×C3×C4—33no—
43C22×C3×C4—33no—
44C22×C3×C4—31no—
45C22×C3×C4—31no—
46C22×C3×C4—31no—
47C22×C3×C4C3×A411yescharacteristic
48C3×C4⋊C4—62no—
49C3×C4⋊C4—66no—
50C22×C32—31no—
51C22×C32—31no—
52C22×C32#41⋊C311yescharacteristic
53C32×C4—41no—
54C32×C4—62no—
55C32×C4—62no—
56C22⋊C9—43no—
57C4×C9—163no—
58C22×Q8C2×3−1+211yescharacteristic
59C2×C4⋊C4—31no—
60#95—31no—
613−1+2—161noSylow
62C23×C3—31no—
63C23×C3C2×C3×A411yescharacteristic
64C3×Q8—22no—
65C3×Q8—62no—
66C3×Q8—62no—
67C3×Q8—66no—
68C2×C3×C4—36no—
69C2×C3×C4—32no—
70C2×C3×C4—32no—
71C2×C3×C4—36no—
72C2×C3×C4—61no—
73C2×C3×C4—63no—
74C2×C3×C4—61no—
75C2×C3×C4—63no—
76C2×C3×C4—66no—
77C2×C3×C4—62no—
78C2×C3×C4—61no—
79C2×C3×C4—63no—
80C2×C32—31no—
81C2×C32—31no—
82C2×C32#120⋊C311yescharacteristic
83C2×C9—163no—
84C2×Q8—62no—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4#109411yescharacteristic
88C4⋊C4—62no—
89C22×C3—31no—
90C22×C3—33no—
91C22×C3—33no—
92C22×C3—31no—
93C22×C3—31no—
94C22×C3C3×#41⋊C311yescharacteristic
95C3×C4—41no—
96C3×C4—66no—
97C3×C4—66no—
98C3×C4—62no—
99C3×C4—62no—
100C3×C4—121no—
101C32#395⋊C311yescharacteristic
102C9—163no—
103C23C2×#109411yescharacteristic
104Q8—22no—
105Q8—62no—
106C2×C4—32no—
107C2×C4—32no—
108C2×C4—61no—
109C2×C4—61no—
110C2×C4—62no—
111C2×C4—61no—
112C2×C3—33no—
113C2×C3—33no—
114C2×C3—31no—
115C2×C3—31no—
116C2×C3—31no—
117C2×C3C3×#120⋊C311yescharacteristic
118C22—31no—
119C22—31no—
120C22#8869111yescharacteristic
121C4—41no—
122C4—62no—
123C4—62no—
124C3—31no—
125C3C3×#395⋊C311yescharacteristic
126C2—31no—
127C2—31no—
128C2#1172134711yescharacteristic
subgroupG/H
centerC2×C3C3×#120⋊C3
commutator, cosocleC22×C3×Q8C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC32×#395C3
prime radicalC23×C32C2×A4
hypercenterC2×C32#120⋊C3
hypocenterC22×Q8C2×3−1+2
LCSG ▷ C22×C3×Q8 ▷ C22×Q8
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
3. #395⋊C9C3semidirect
38. #3953−1+2semidirect

Maximal subgroups

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

subgroupindex
#117225862
C32×#3953
#395⋊C93
C4×#10944