Finite Group Databasebeta

← #13135076

Subgroups — #13135076

Order 1728.

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

normal subgroups only

This group has 219 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
1C2×#3746⋊C3C211yesmaximal, characteristic
2#11720552C211yesmaximal, characteristic
3#63639⋊C3C211yesmaximal, characteristic
4#10589131—31nomaximal
5#10589129—31nomaximal
6C2×#634⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88538—21no—
9#3746⋊C3—21no—
10#3746⋊C3C412yes—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×#3746D311yescharacteristic
13C2×C3×Dic12—31no—
14#63639D311yescharacteristic
15C3×#819—31no—
16C3×#819—31no—
17C3×#819—61no—
18C3×#812—31no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×(C3⋊Dic3)⋊C3—41no—
21#634⋊C3—22no—
22#634⋊C3—41no—
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25#6180—41no—
26#4565—91no—
27#4517—91no—
28#4496—91no—
29C2×#634—21no—
30C2×C3×Dic6—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D311yescharacteristic
33C2×C32×C8C2×D311yescharacteristic
34#3746—21no—
35#3746Dic312yes—
36C3×#208—62no—
37C3×#208—61no—
38C3×#208—61no—
39C3×Dic12—32no—
40C3×Dic12—61no—
41C3⋊(C3⋊C16)—21no—
42C3×C3⋊C16—61no—
43C22×H(3,𝔽3)SD1611yescharacteristic
44C4×H(3,𝔽3)C22⋊C411yescharacteristic
45C4×H(3,𝔽3)D811yescharacteristic
46(C3⋊Dic3)⋊C3—42no—
47C2×C3×Q16—91no—
48C2×Dic12—31no—
49C2×Dic12—91no—
50C3×M32—91no—
51C3×#108—91no—
52#819—31no—
53#819—31no—
54#819—61no—
55#812—31no—
56#812—91no—
57C2×C32×C4—21no—
58C2×C32×C4D1211yescharacteristic
59C2×C32×C4#6711yescharacteristic
60C2×C3⋊Dic3—41no—
61#634—22no—
62#634—41no—
63C2×C3×Dic3—121no—
64C3×Dic6—62no—
65C3×Dic6—121no—
66C32×C8—21no—
67C32×C8—21no—
68C32×C8C4×D311yescharacteristic
69C32×C8C2×Dic311yescharacteristic
70C32×C8#6711yescharacteristic
71C32×C8#6711yescharacteristic
72C3×C3⋊C8—122no—
73C3×C3⋊C8—121no—
74C3×C3⋊C8—121no—
75#362—271noSylow
76C2×H(3,𝔽3)—21no—
77C2×H(3,𝔽3)#10211yescharacteristic
78C2×C3×Q8—181no—
79C2×Dic6—61no—
80C2×Dic6—181no—
81C3×Q16—92no—
82C3×Q16—181no—
83C3×M16—181no—
84C2×C3×C8—31no—
85C2×C3×C8—31no—
86C2×C3×C8—61no—
87C2×C3×C8D3211yescharacteristic
88#208—61no—
89#208—61no—
90#208—62no—
91Dic12—32no—
92Dic12—61no—
93Dic12—92no—
94Dic12—181no—
95C3×C16—181no—
96C3⋊C16—61no—
97C3⋊C16—181no—
98C22×C32—21no—
99C22×C32#20411yescharacteristic
100C22×C32#21411yescharacteristic
101C32×C4—21no—
102C32×C4—21no—
103C32×C4D2411yescharacteristic
104C32×C4(C2×D3)⋊C411yescharacteristic
105C32×C4#21311yescharacteristic
106C32×C4#21711yescharacteristic
107C3⋊Dic3—42no—
108C3×Dic3—122no—
109C2×Q16—91no—
110M32—91no—
111#108—91no—
112H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
113C3×Q8—182no—
114C3×Q8—361no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—61no—
118C2×C3×C4—361no—
119C2×C3×C4#62611yescharacteristic
120C2×Dic3—121no—
121C2×Dic3—361no—
122Dic6—62no—
123Dic6—121no—
124Dic6—182no—
125Dic6—361no—
126C3×C8—31no—
127C3×C8—31no—
128C3×C8—31no—
129C3×C8—31no—
130C3×C8—61no—
131C3×C8—61no—
132C3×C8—361no—
133C3×C8D3×Dic311yescharacteristic
134C3×C8#62511yescharacteristic
135C3⋊C8—121no—
136C3⋊C8—121no—
137C3⋊C8—122no—
138C2×C32—21no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32—41no—
142C2×C32#82111yescharacteristic
143C2×C32#83211yescharacteristic
144C2×Q8—181no—
145Q16—92no—
146Q16—181no—
147M16—181no—
148C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
149C16—181no—
150C22×C3—31no—
151C22×C3—31no—
152C22×C3—61no—
153C22×C3#371611yescharacteristic
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—31no—
157C3×C4—31no—
158C3×C4—61no—
159C3×C4—61no—
160C3×C4—362no—
161C3×C4#371411yescharacteristic
162C3×C4(C2×D3)⋊Dic311yescharacteristic
163Dic3—122no—
164Dic3—362no—
165C32—21no—
166C32#451711yescharacteristic, complemented
167C32#456511yescharacteristic, complemented
168Q8—182no—
169Q8—361no—
170C2×C4—361no—
171C2×C4#620011yescharacteristic
172C8—361no—
173C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
174C8#619811yescharacteristic
175C2×C3—21no—
176C2×C3—31no—
177C2×C3—31no—
178C2×C3—61no—
179C2×C3—61no—
180C2×C3—61no—
181C2×C3—121no—
182C2×C3#6356311yescharacteristic
183C22#8843811yescharacteristic
184C4—362no—
185C4#8843611yescharacteristic
186C4#8845011yescharacteristic
187C3—31no—
188C3—31no—
189C3—61no—
190C3#1058913011yescharacteristic
191C2—21no—
192C2#1172030811yescharacteristic
subgroupG/H
centerC2#11720308
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3#63563
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
112. H(3,𝔽3)#362semidirect
166. C32#4517semidirect
167. C32#4565semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
#117205522
#63639⋊C32
#105891313
#105891293