Finite Group Databasebeta

← #10589382⋊C3 #13135513

Subgroups — #10589382⋊C3 #13135513

Order 1728.

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

normal subgroups only

This group has 243 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×#3745⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×M32C211yesmaximal, characteristic, complemented
3#63646⋊C3C211yesmaximal, characteristic, complemented
4#10589382C311yesmaximal, characteristic, complemented
5C3×#4516—31nomaximal
6C2×C4⋊((C3⋊D3)⋊C3)—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#3748⋊C3—21no—
9#3745⋊C3—21no—
10#3745⋊C3C412yes—
11C16×H(3,𝔽3)—21no—
12C2×#3745C2×C311yescharacteristic
13C2×C3×D24—31no—
14C32×M32—21no—
15C32×M32D311yescharacteristic, complemented
16C32×M32C2×C311yescharacteristic, complemented
17#63646C2×C311yescharacteristic, complemented
18C3×#819—31no—
19C22×(C3⋊D3)⋊C3—41no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C4⋊((C3⋊D3)⋊C3)—22no—
22C4⋊((C3⋊D3)⋊C3)—41no—
23C8×H(3,𝔽3)C2×C411yescharacteristic
24C8×H(3,𝔽3)D411yescharacteristic
25(C3⋊(C3⋊C8))⋊C3—41no—
26C3×#361—91no—
27#4516—31no—
28#4516—91no—
29C2×C4⋊(C3⋊D3)—21no—
30C2×C3×D12—61no—
31C2×C32×C8—21no—
32C2×C32×C8C2×D311yescharacteristic
33C2×C32×C8C22×C311yescharacteristic
34#3748—21no—
35#3745—21no—
36#3745C3×C412yes—
37C3×#208—61no—
38C3×D24—32no—
39C3×D24—61no—
40C32×C16—22no—
41C32×C16—21no—
42C32×C16—21no—
43C22×H(3,𝔽3)SD1611yescharacteristic
44C2×(C3⋊D3)⋊C3—42no—
45C2×(C3⋊D3)⋊C3—82no—
46C4×H(3,𝔽3)C22⋊C411yescharacteristic
47C4×H(3,𝔽3)D811yescharacteristic
48C2×C3×D8—91no—
49C2×D24—31no—
50C2×D24—91no—
51C3×M32—31no—
52C3×M32—31no—
53C3×M32—61no—
54C3×M32C3×D311yescharacteristic
55C3×#108—91no—
56#819—31no—
57#819—91no—
58C22×C3⋊D3—41no—
59C22×C3×D3—121no—
60C2×C32×C4—21no—
61C2×C32×C4D1211yescharacteristic
62C2×C32×C4C3×D411yescharacteristic
63C4⋊(C3⋊D3)—22no—
64C4⋊(C3⋊D3)—41no—
65C3×D12—62no—
66C3×D12—121no—
67C32×C8—21no—
68C32×C8—21no—
69C32×C8C4×D311yescharacteristic
70C32×C8#6711yescharacteristic
71C32×C8C2×C3×C411yescharacteristic
72C32×C8C3×D411yescharacteristic
73C3⋊(C3⋊C8)—41no—
74C3×C3⋊C8—121no—
75#361—91noSylow
76C2×H(3,𝔽3)—21no—
77C2×H(3,𝔽3)#10211yescharacteristic
78(C3⋊D3)⋊C3—82no—
79C2×C3×D4—181no—
80C2×D12—61no—
81C2×D12—181no—
82C3×D8—92no—
83C3×D8—181no—
84C3×M16—181no—
85C2×C3×C8—31no—
86C2×C3×C8—31no—
87C2×C3×C8—61no—
88C2×C3×C8C2×C3×D311yescharacteristic
89#208—61no—
90#208—181no—
91D24—32no—
92D24—61no—
93D24—92no—
94D24—181no—
95C3×C16—21no—
96C3×C16—61no—
97C3×C16—61no—
98C3×C16—62no—
99C22×C32—21no—
100C22×C32#20411yescharacteristic
101C22×C32C3×SD1611yescharacteristic
102C2×C3⋊D3—42no—
103C2×C3⋊D3—82no—
104C2×C3×D3—122no—
105C2×C3×D3—242no—
106C32×C4—21no—
107C32×C4—21no—
108C32×C4D2411yescharacteristic
109C32×C4(C2×D3)⋊C411yescharacteristic
110C32×C4C3×C22⋊C411yescharacteristic
111C32×C4C3×D811yescharacteristic
112C2×D8—91no—
113M32(C3⋊D3)⋊C311yescharacteristic, complemented
114#108—91no—
115H(3,𝔽3)#36111yescharacteristic, complemented, Sylow
116C23×C3—361no—
117C22×D3—121no—
118C22×D3—361no—
119C3×D4—182no—
120C3×D4—361no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—61no—
124C2×C3×C4C3×D1211yescharacteristic
125D12—62no—
126D12—121no—
127D12—182no—
128D12—361no—
129C3×C8—31no—
130C3×C8—31no—
131C3×C8—31no—
132C3×C8—31no—
133C3×C8—61no—
134C3×C8—61no—
135C3×C8—361no—
136C3×C8C3×C4×D311yescharacteristic
137C3×C8C3×#6711yescharacteristic
138C3⋊C8—121no—
139C3⋊C8—361no—
140C2×C32—21no—
141C2×C32—21no—
142C2×C32—21no—
143C2×C32—41no—
144C2×C32#82111yescharacteristic
145C2×C32C3×#10211yescharacteristic
146C3⋊D3—82no—
147C3×D3—242no—
148C2×D4—181no—
149D8—92no—
150D8—181no—
151M16—181no—
152C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
153C16—21no—
154C22×C3—31no—
155C22×C3—31no—
156C22×C3—61no—
157C22×C3—362no—
158C22×C3—722no—
159C22×C3C3×#20411yescharacteristic
160C2×D3—122no—
161C2×D3—242no—
162C2×D3—362no—
163C2×D3—722no—
164C3×C4—31no—
165C3×C4—31no—
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—61no—
169C3×C4—61no—
170C3×C4C3×D2411yescharacteristic
171C3×C4C3×(C2×D3)⋊C411yescharacteristic
172C32—21no—
173C32#451611yescharacteristic, complemented
174C32C3×#36111yescharacteristic, complemented
175C23—361no—
176D4—182no—
177D4—361no—
178C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
179C8—361no—
180C8C4×(C3⋊D3)⋊C311yescharacteristic
181C8#638⋊C311yescharacteristic
182C2×C3—21no—
183C2×C3—31no—
184C2×C3—31no—
185C2×C3—61no—
186C2×C3—61no—
187C2×C3—61no—
188C2×C3—121no—
189C2×C3—722no—
190C2×C3C3×#82111yescharacteristic
191D3—242no—
192D3—722no—
193C22—362no—
194C22—722no—
195C22#3744⋊C311yescharacteristic
196C4#3745⋊C311yescharacteristic
197C4#3752⋊C311yescharacteristic
198C3—31no—
199C3—31no—
200C3—61no—
201C3C3×#451611yescharacteristic
202C2—21no—
203C2—722no—
204C2#63648⋊C311yescharacteristic
subgroupG/H
centerC2#63648⋊C3
commutatorC32×C8C2×C3×C4
FrattiniC2×C3×C8C2×C3×D3
FittingH(3,𝔽3)×M32C2
socleC2×C3C3×#821
prime radicalC2×#3745⋊C3C2
cosocleC2×C32×C8C22×C3
hypercenterM32(C3⋊D3)⋊C3
hypocenterC32C3×#361
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
2. H(3,𝔽3)×M32C2semidirect
3. #63646⋊C3C2semidirect
4. #10589382C3semidirect
15. C32×M32D3semidirect
16. C32×M32C2×C3semidirect
17. #63646C2×C3semidirect
113. M32(C3⋊D3)⋊C3semidirect
115. H(3,𝔽3)#361semidirect
173. C32#4516semidirect
174. C32C3×#361semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
H(3,𝔽3)×M322
#63646⋊C32
#105893823
C3×#45163