Finite Group Databasebeta

← #13135067

Subgroups — #13135067

Order 1728.

183 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 55 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#11720550C211yesmaximal, characteristic
2#63646⋊C3C211yesmaximal, characteristic
3#63639⋊C3C211yesmaximal, characteristic
4#10589138—31nomaximal
5#10589136—31nomaximal
6C2×C8×H(3,𝔽3)C2211yescharacteristic
7C4⋊#1084—21no—
8#3748⋊C3—21no—
9(C3⋊(C3⋊C16))⋊C3C412yescomplemented
10#63646D311yescharacteristic
11#63639D311yescharacteristic
12C3×#819—31no—
13C3×#817—31no—
14C3×#817—31no—
15C3×#817—61no—
16C3×#812—31no—
17C2×C4×H(3,𝔽3)D411yescharacteristic
18C2×#1084—41no—
19C8×H(3,𝔽3)C2×C411yescharacteristic
20C8×H(3,𝔽3)Q811yescharacteristic
21(C3⋊(C3⋊C8))⋊C3—41no—
22#4513—91no—
23#4494—91no—
24#4487—91no—
25C2×C32×C8—21no—
26C2×C32×C8C2×D311yescharacteristic
27C2×C32×C8C2×D311yescharacteristic
28#3748—21no—
29C3×C4⋊Dic3—62no—
30C3×C4⋊Dic3—61no—
31C3×C4⋊Dic3—61no—
32C3×#208—61no—
33C3⋊(C3⋊C16)Dic312yescomplemented
34C3×C3⋊C16—32no—
35C22×H(3,𝔽3)SD1611yescharacteristic
36C4×H(3,𝔽3)C4⋊C411yescharacteristic
37C4×H(3,𝔽3)SD1611yescharacteristic
38#1084—81no—
39C3×M32—91no—
40C3×#108—91no—
41C3×#106—91no—
42#819—31no—
43#819—91no—
44#817—31no—
45#817—31no—
46#817—61no—
47#812—31no—
48#812—91no—
49C2×C32×C4—21no—
50C2×C32×C4D1211yescharacteristic
51C2×C32×C4#6711yescharacteristic
52C2×C3×Dic3—122no—
53C2×C3×Dic3—121no—
54C2×C3×Dic3—121no—
55C32×C8—21no—
56C32×C8—21no—
57C32×C8Dic611yescharacteristic
58C32×C8Dic611yescharacteristic
59C32×C8C4×D311yescharacteristic
60C32×C8C2×Dic311yescharacteristic
61C3⋊(C3⋊C8)—41no—
62C3×C3⋊C8—121no—
63#365—271noSylow
64C2×H(3,𝔽3)—21no—
65C2×H(3,𝔽3)#10611yescharacteristic
66C3×M16—181no—
67C2×C3×C8—31no—
68C2×C3×C8—31no—
69C2×C3×C8—61no—
70C2×C3×C8D3211yescharacteristic
71C3×C4⋊C4—181no—
72C4⋊Dic3—61no—
73C4⋊Dic3—61no—
74C4⋊Dic3—62no—
75#208—61no—
76#208—181no—
77C3×C16—92no—
78C3⋊C16—32no—
79C3⋊C16—92no—
80C22×C32—21no—
81C22×C32#20411yescharacteristic
82C22×C32#21511yescharacteristic
83C32×C4—21no—
84C32×C4—21no—
85C32×C4#20411yescharacteristic
86C32×C4Dic3⋊C411yescharacteristic
87C32×C4C4⋊Dic311yescharacteristic
88C32×C4#21411yescharacteristic
89C3×Dic3—242no—
90C3×Dic3—241no—
91C3×Dic3—241no—
92M32—91no—
93#108—91no—
94#106—91no—
95H(3,𝔽3)#36511yescharacteristic, complemented, Sylow
96C2×C3×C4—31no—
97C2×C3×C4—31no—
98C2×C3×C4—61no—
99C2×C3×C4—361no—
100C2×C3×C4#62611yescharacteristic
101C2×Dic3—121no—
102C2×Dic3—121no—
103C2×Dic3—122no—
104C3×C8—31no—
105C3×C8—31no—
106C3×C8—31no—
107C3×C8—31no—
108C3×C8—61no—
109C3×C8—61no—
110C3×C8—361no—
111C3×C8D3×Dic311yescharacteristic
112C3×C8#62711yescharacteristic
113C3⋊C8—121no—
114C3⋊C8—361no—
115C2×C32—22no—
116C2×C32—21no—
117C2×C32—21no—
118C2×C32—21no—
119C2×C32#80811yescharacteristic
120C2×C32#81711yescharacteristic
121M16—181no—
122C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
123C4⋊C4—181no—
124C16—92no—
125C22×C3—31no—
126C22×C3—31no—
127C22×C3—61no—
128C22×C3#371711yescharacteristic
129C3×C4—31no—
130C3×C4—31no—
131C3×C4—31no—
132C3×C4—31no—
133C3×C4—61no—
134C3×C4—61no—
135C3×C4—721no—
136C3×C4#371611yescharacteristic
137C3×C4Dic3⋊Dic311yescharacteristic
138Dic3—241no—
139Dic3—241no—
140Dic3—242no—
141C32—21no—
142C32#449411yescharacteristic, complemented
143C32#451311yescharacteristic, complemented
144C2×C4—361no—
145C2×C4#620011yescharacteristic
146C8—361no—
147C8#619611yescharacteristic
148C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
149C2×C3—21no—
150C2×C3—31no—
151C2×C3—31no—
152C2×C3—61no—
153C2×C3—61no—
154C2×C3—62no—
155C2×C3—61no—
156C2×C3#6357311yescharacteristic
157C22#8843711yescharacteristic
158C4—721no—
159C4#8843811yescharacteristic
160C4#8844811yescharacteristic
161C3—31no—
162C3—31no—
163C3—61no—
164C3#1058913711yescharacteristic
165C2—21no—
166C2#1172030711yescharacteristic
subgroupG/H
centerC2#11720307
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63573
prime radicalC22×H(3,𝔽3)SD16
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#365
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
9. (C3⋊(C3⋊C16))⋊C3C4semidirect
33. C3⋊(C3⋊C16)Dic3semidirect
95. H(3,𝔽3)#365semidirect
142. C32#4494semidirect
143. C32#4513semidirect

Maximal subgroups

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

subgroupindex
#117205502
#63646⋊C32
#63639⋊C32
#105891383
#105891363