Finite Group Databasebeta

← #13136932

Subgroups — #13136932

Order 1728.

208 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 25 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#11721808C211yesmaximal, characteristic
2#11720777C211yesmaximal, characteristic, complemented
3#11720770C211yesmaximal, characteristic, complemented
4#88748—21no—
5#88627—21no—
6Q16×H(3,𝔽3)C411yescharacteristic
7#88549—21no—
8#88529C411yescharacteristic
9C8×#272C2211yescharacteristic
10#88410—21no—
11C3×#919—62no—
12Q8×H(3,𝔽3)—21no—
13C4⋊#272—21no—
14C4⋊#272—41no—
15C4×#272—41no—
16C4×#272D411yescharacteristic
17#6188—41no—
18C8×H(3,𝔽3)C2×C411yescharacteristic
19#6180D411yescharacteristic
20C3×#359—91nomaximal
21C3×#239—64no—
22C32×Q16—22no—
23C3×#215—64no—
24C3×D24—62no—
25C3×D3×C8—62no—
26C2×#272—41no—
27C2×#272SD1611yescharacteristic
28C4×H(3,𝔽3)—41no—
29C4×H(3,𝔽3)C22⋊C411yescharacteristic
30#1084D811yescharacteristic
31C3×#135—91no—
32#919—62no—
33C2×C3×C16—91no—
34C3×#108—91no—
35C32×Q8—24no—
36C3×D12—64no—
37C3×D12—124no—
38C3×C4×D3—62no—
39C3×C4×D3—124no—
40C32×C8—22no—
41C3×C3⋊C8—62no—
42#359—91noSylow
43C2×H(3,𝔽3)#10211yescharacteristic
44#272—21no—
45#272—81no—
46C3×#41—181no—
47#239—64no—
48C3×Q16—62no—
49C3×Q16#15711yescharacteristic
50C3×SD16—181no—
51C3×D8—91no—
52C3×M16—181no—
53C2×C3×C8—91no—
54#215—64no—
55D24—62no—
56D3×C8—62no—
57C3×C16—181no—
58C2×C3×D3—62no—
59C2×C3×D3—124no—
60C32×C4—22no—
61C32×C4—44no—
62C3×Dic3—62no—
63#135—91no—
64C2×C16—91no—
65#108—91no—
66H(3,𝔽3)#35911yescharacteristic, complemented, Sylow
67C3×Q8—21no—
68C3×Q8—64no—
69C3×D4—181no—
70C3×D4—361no—
71C2×C3×C4—91no—
72C2×C3×C4—361no—
73D12—64no—
74D12—124no—
75C4×D3—62no—
76C4×D3—124no—
77C3×C8—62no—
78C3×C8—91no—
79C3×C8—361no—
80C3×C8C2×#15711yescharacteristic
81C3⋊C8—62no—
82C2×C32—22no—
83C3×D3—122no—
84C3×D3—244no—
85#41—181no—
86Q16#108811yescharacteristic
87SD16—181no—
88D8—91no—
89M16—181no—
90C2×C8—91no—
91C16—181no—
92C22×C3—91no—
93C22×C3—361no—
94C2×D3—62no—
95C2×D3—124no—
96C3×C4—41no—
97C3×C4—62no—
98C3×C4—91no—
99C3×C4—124no—
100C3×C4C22⋊#15711yescharacteristic
101Dic3—62no—
102C32—22no—
103Q8—21no—
104D4—181no—
105D4—361no—
106C2×C4—91no—
107C2×C4—361no—
108C8—91no—
109C8—361no—
110C8C2×#108811yescharacteristic
111C2×C3—62no—
112C2×C3—181no—
113C2×C3—721no—
114C2×C3#6378211yescharacteristic
115D3—122no—
116D3—244no—
117C22—91no—
118C22—361no—
119C4—41no—
120C4—91no—
121C4C22⋊#108811yescharacteristic
122C3—62no—
123C3#1059019811yescharacteristic
124C2—181no—
125C2—721no—
126C2#1172078811yescharacteristic
subgroupG/H
center, socleC2×C3#63782
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC3×C8C2×#157
FittingQ16×H(3,𝔽3)C4
prime radical#11721808C2
cosocleC8×#272C22
hypercenterC3×Q16#157
hypocenterH(3,𝔽3)#359
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×Q16

Complements

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

NH ≅ G/Ntype
2. #11720777C2semidirect
3. #11720770C2semidirect
66. H(3,𝔽3)#359semidirect

Maximal subgroups

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

subgroupindex
#117218082
#117207772
#117207702
C3×#3599