Finite Group Databasebeta

← #10589383⋊C3 #13135515

Subgroups — #10589383⋊C3 #13135515

Order 1728.

215 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 60 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
2H(3,𝔽3)×M32C211yesmaximal, characteristic
3#63646⋊C3C211yesmaximal, characteristic
4#10589383C311yesmaximal, characteristic, complemented
5C3×#4517—31nomaximal
6C2×#634⋊C3—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#3748⋊C3—21no—
9#3746⋊C3—21no—
10#3746⋊C3C412yes—
11C16×H(3,𝔽3)—21no—
12C2×#3746C2×C311yescharacteristic
13C2×C3×Dic12—31no—
14C32×M32—21no—
15C32×M32D311yescharacteristic
16C32×M32C2×C311yescharacteristic
17#63646C2×C311yescharacteristic
18C3×#819—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(C3⋊(C3⋊C8))⋊C3—41no—
26C3×#362—91no—
27#4517—31no—
28#4517—91no—
29C2×#634—21no—
30C2×C3×Dic6—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D311yescharacteristic
33C2×C32×C8C22×C311yescharacteristic
34#3748—21no—
35#3746—21no—
36#3746C3×C412yes—
37C3×#208—61no—
38C3×Dic12—32no—
39C3×Dic12—61no—
40C32×C16—22no—
41C32×C16—21no—
42C32×C16—21no—
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—31no—
51C3×M32—31no—
52C3×M32—61no—
53C3×M32C3×D311yescharacteristic
54C3×#108—91no—
55#819—31no—
56#819—91no—
57C2×C32×C4—21no—
58C2×C32×C4D1211yescharacteristic
59C2×C32×C4C3×D411yescharacteristic
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×C8#6711yescharacteristic
70C32×C8C2×C3×C411yescharacteristic
71C32×C8C3×D411yescharacteristic
72C3⋊(C3⋊C8)—41no—
73C3×C3⋊C8—121no—
74#362—91noSylow
75C2×H(3,𝔽3)—21no—
76C2×H(3,𝔽3)#10211yescharacteristic
77C2×C3×Q8—181no—
78C2×Dic6—61no—
79C2×Dic6—181no—
80C3×Q16—92no—
81C3×Q16—181no—
82C3×M16—181no—
83C2×C3×C8—31no—
84C2×C3×C8—31no—
85C2×C3×C8—61no—
86C2×C3×C8C2×C3×D311yescharacteristic
87#208—61no—
88#208—181no—
89Dic12—32no—
90Dic12—61no—
91Dic12—92no—
92Dic12—181no—
93C3×C16—21no—
94C3×C16—61no—
95C3×C16—61no—
96C3×C16—62no—
97C22×C32—21no—
98C22×C32#20411yescharacteristic
99C22×C32C3×SD1611yescharacteristic
100C32×C4—21no—
101C32×C4—21no—
102C32×C4D2411yescharacteristic
103C32×C4(C2×D3)⋊C411yescharacteristic
104C32×C4C3×C22⋊C411yescharacteristic
105C32×C4C3×D811yescharacteristic
106C3⋊Dic3—42no—
107C3×Dic3—122no—
108C2×Q16—91no—
109M32(C3⋊D3)⋊C311yescharacteristic
110#108—91no—
111H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
112C3×Q8—182no—
113C3×Q8—361no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—61no—
117C2×C3×C4—361no—
118C2×C3×C4C3×D1211yescharacteristic
119C2×Dic3—121no—
120C2×Dic3—361no—
121Dic6—62no—
122Dic6—121no—
123Dic6—182no—
124Dic6—361no—
125C3×C8—31no—
126C3×C8—31no—
127C3×C8—31no—
128C3×C8—31no—
129C3×C8—61no—
130C3×C8—61no—
131C3×C8—361no—
132C3×C8C3×C4×D311yescharacteristic
133C3×C8C3×#6711yescharacteristic
134C3⋊C8—121no—
135C3⋊C8—361no—
136C2×C32—21no—
137C2×C32—21no—
138C2×C32—21no—
139C2×C32—41no—
140C2×C32#82111yescharacteristic
141C2×C32C3×#10211yescharacteristic
142C2×Q8—181no—
143Q16—92no—
144Q16—181no—
145M16—181no—
146C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
147C16—21no—
148C22×C3—31no—
149C22×C3—31no—
150C22×C3—61no—
151C22×C3C3×#20411yescharacteristic
152C3×C4—31no—
153C3×C4—31no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—61no—
157C3×C4—61no—
158C3×C4—362no—
159C3×C4C3×D2411yescharacteristic
160C3×C4C3×(C2×D3)⋊C411yescharacteristic
161Dic3—122no—
162Dic3—362no—
163C32—21no—
164C32#451711yescharacteristic, complemented
165C32C3×#36211yescharacteristic, complemented
166Q8—182no—
167Q8—361no—
168C2×C4—361no—
169C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
170C8—361no—
171C8C4×(C3⋊D3)⋊C311yescharacteristic
172C8#638⋊C311yescharacteristic
173C2×C3—21no—
174C2×C3—31no—
175C2×C3—31no—
176C2×C3—61no—
177C2×C3—61no—
178C2×C3—61no—
179C2×C3—121no—
180C2×C3C3×#82111yescharacteristic
181C22#3744⋊C311yescharacteristic
182C4—362no—
183C4#3745⋊C311yescharacteristic
184C4#3752⋊C311yescharacteristic
185C3—31no—
186C3—31no—
187C3—61no—
188C3C3×#451711yescharacteristic
189C2—21no—
190C2#63648⋊C311yescharacteristic
subgroupG/H
centerC2#63648⋊C3
commutatorC32×C8C2×C3×C4
FrattiniC2×C3×C8C2×C3×D3
FittingH(3,𝔽3)×M32C2
socleC2×C3C3×#821
prime radicalC22×H(3,𝔽3)SD16
cosocleC2×C32×C8C22×C3
hypercenterM32(C3⋊D3)⋊C3
hypocenterC32C3×#362
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32

Complements

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

NH ≅ G/Ntype
4. #10589383C3semidirect
111. H(3,𝔽3)#362semidirect
164. C32#4517semidirect
165. C32C3×#362semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
H(3,𝔽3)×M322
#63646⋊C32
#105893833
C3×#45173