Finite Group Databasebeta

← #13135103

Subgroups — #13135103

Order 1728.

249 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 53 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×#88530C211yesmaximal, characteristic
2#63646⋊C3C211yesmaximal, characteristic
3#63639⋊C3C211yesmaximal, characteristic
4#10589130—31nomaximal
5#10589129—31nomaximal
6C2×#6229—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88530—21no—
9#88530C412yes—
10#3748⋊C3—21no—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×C3×Dic12—31no—
13C2×C3×Dic12—31no—
14C2×C3×Dic12—61no—
15#63646D311yescharacteristic
16#63639D311yescharacteristic
17C3×#819—31no—
18C3×#812—31no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×#1084—41no—
21#6229—22no—
22#6229—41no—
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25(C3⋊(C3⋊C8))⋊C3—41no—
26#4565—91no—
27#4517—91no—
28#4496—91no—
29C2×C3×Dic6—62no—
30C2×C3×Dic6—61no—
31C2×C3×Dic6—61no—
32C2×C32×C8—21no—
33C2×C32×C8C2×D311yescharacteristic
34C2×C32×C8C2×D311yescharacteristic
35#3748—21no—
36C3×#208—61no—
37C3×Dic12—32no—
38C3×Dic12—32no—
39C3×Dic12—62no—
40C3×Dic12—62no—
41C3×Dic12—61no—
42C3×Dic12—61no—
43C3⋊(C3⋊C16)—21no—
44C3×C3⋊C16—61no—
45C22×H(3,𝔽3)SD1611yescharacteristic
46C4×H(3,𝔽3)C22⋊C411yescharacteristic
47C4×H(3,𝔽3)D811yescharacteristic
48#1084—42no—
49C2×C3×Q16—91no—
50C2×Dic12—31no—
51C2×Dic12—31no—
52C2×Dic12—61no—
53C3×M32—91no—
54C3×#108—91no—
55#819—31no—
56#819—91no—
57#812—31no—
58#812—91no—
59C2×C32×C4—21no—
60C2×C32×C4D1211yescharacteristic
61C2×C32×C4#6711yescharacteristic
62C2×C3×Dic3—122no—
63C2×C3×Dic3—121no—
64C2×C3×Dic3—121no—
65C3×Dic6—64no—
66C3×Dic6—62no—
67C3×Dic6—62no—
68C3×Dic6—122no—
69C3×Dic6—121no—
70C3×Dic6—121no—
71C32×C8—21no—
72C32×C8—21no—
73C32×C8C4×D311yescharacteristic
74C32×C8C4×D311yescharacteristic
75C32×C8D1211yescharacteristic
76C32×C8#6711yescharacteristic
77C3⋊(C3⋊C8)—41no—
78C3×C3⋊C8—121no—
79#362—271noSylow
80C2×H(3,𝔽3)—21no—
81C2×H(3,𝔽3)#10211yescharacteristic
82C2×C3×Q8—181no—
83C2×Dic6—61no—
84C2×Dic6—61no—
85C2×Dic6—62no—
86C3×Q16—92no—
87C3×Q16—181no—
88C3×M16—181no—
89C2×C3×C8—31no—
90C2×C3×C8—31no—
91C2×C3×C8—61no—
92C2×C3×C8D3211yescharacteristic
93#208—61no—
94#208—181no—
95Dic12—32no—
96Dic12—32no—
97Dic12—62no—
98Dic12—61no—
99Dic12—61no—
100Dic12—62no—
101C3×C16—181no—
102C3⋊C16—61no—
103C3⋊C16—181no—
104C22×C32—21no—
105C22×C32#20411yescharacteristic
106C22×C32#21511yescharacteristic
107C32×C4—21no—
108C32×C4—21no—
109C32×C4D2411yescharacteristic
110C32×C4(C2×D3)⋊C411yescharacteristic
111C32×C4(C2×D3)⋊C411yescharacteristic
112C32×C4#21311yescharacteristic
113C3×Dic3—124no—
114C3×Dic3—122no—
115C3×Dic3—122no—
116C2×Q16—91no—
117M32—91no—
118#108—91no—
119H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
120C3×Q8—182no—
121C3×Q8—361no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—61no—
125C2×C3×C4—361no—
126C2×C3×C4#62611yescharacteristic
127C2×Dic3—121no—
128C2×Dic3—121no—
129C2×Dic3—122no—
130Dic6—62no—
131Dic6—62no—
132Dic6—64no—
133Dic6—121no—
134Dic6—121no—
135Dic6—122no—
136C3×C8—31no—
137C3×C8—31no—
138C3×C8—31no—
139C3×C8—31no—
140C3×C8—61no—
141C3×C8—61no—
142C3×C8—361no—
143C3×C8#62411yescharacteristic
144C3×C8#62611yescharacteristic
145C3⋊C8—121no—
146C3⋊C8—361no—
147C2×C32—22no—
148C2×C32—21no—
149C2×C32—21no—
150C2×C32—21no—
151C2×C32#80911yescharacteristic
152C2×C32#82111yescharacteristic
153C2×Q8—181no—
154Q16—92no—
155Q16—181no—
156M16—181no—
157C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
158C16—181no—
159C22×C3—31no—
160C22×C3—31no—
161C22×C3—61no—
162C22×C3#371711yescharacteristic
163C3×C4—31no—
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—31no—
167C3×C4—61no—
168C3×C4—61no—
169C3×C4—362no—
170C3×C4#371411yescharacteristic
171C3×C4#372211yescharacteristic
172Dic3—122no—
173Dic3—122no—
174Dic3—124no—
175C32—21no—
176C32#449611yescharacteristic, complemented
177C32#451711yescharacteristic, complemented
178Q8—182no—
179Q8—361no—
180C2×C4—361no—
181C2×C4#620011yescharacteristic
182C8—361no—
183C8#619911yescharacteristic
184C8#620011yescharacteristic
185C2×C3—21no—
186C2×C3—31no—
187C2×C3—31no—
188C2×C3—61no—
189C2×C3—61no—
190C2×C3—62no—
191C2×C3—61no—
192C2×C3#6356411yescharacteristic
193C22#8843711yescharacteristic
194C4—362no—
195C4#8843611yescharacteristic
196C4#8845111yescharacteristic
197C3—31no—
198C3—31no—
199C3—61no—
200C3#1058913111yescharacteristic
201C2—21no—
202C2#1172032411yescharacteristic
subgroupG/H
centerC2#11720324
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63564
prime radicalC22×H(3,𝔽3)SD16
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#362
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
119. H(3,𝔽3)#362semidirect
176. C32#4496semidirect
177. C32#4517semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#63646⋊C32
#63639⋊C32
#105891303
#105891293