Finite Group Databasebeta

← #10593640⋊C3 #13146863

Subgroups — #10593640⋊C3 #13146863

Order 1728.

252 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 33 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#64267⋊C3C211yesmaximal, characteristic
2#64265⋊C3C211yesmaximal, characteristic, complemented
3#131⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#10593640C311yesmaximal, characteristic, complemented
5C3×#5141—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7(C3⋊GL(2,𝔽3))⋊C3—21no—
8(C3⋊#226)⋊C3—21no—
9#3744⋊C3—41nomaximal
10#64267C2×C311yescharacteristic
11#64265C2×C311yescharacteristic, complemented
12C32×#131D311yescharacteristic, complemented
13C3×#951—31no—
14C3×#949—31no—
15C2×C3×#204—31no—
16C3×#131⋊C3—21no—
17C3×#131⋊C3C2×C311yescharacteristic, complemented
18C3×#824—31no—
19C3×#823—31no—
20C3×#819—31no—
21C4⋊((C3⋊D3)⋊C3)—41no—
22#634⋊C3—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24C8×H(3,𝔽3)—41no—
25#5406—91no—
26C3×#471—91no—
27#5141—31no—
28C32×#41C2×D311yescharacteristic
29C2×C3×D12—31no—
30C2×C3×Dic6—31no—
31C3×#41⋊C3—21no—
32C3×#41⋊C3C22×C311yescharacteristic
33C3⋊GL(2,𝔽3)—21no—
34C3⋊#226—21no—
35C32×M16—31no—
36C2×C32×C8—31no—
37#3744—41no—
38C3×#216—61no—
39C3×#215—61no—
40C3×#214—61no—
41C3×#213—61no—
42C3×#208—31no—
43C3×#208—31no—
44C3×#204—61no—
45C3×#204—61no—
46C2×(C3⋊D3)⋊C3—81no—
47C4×H(3,𝔽3)—41no—
48(C3⋊Dic3)⋊C3—81no—
49#986—91no—
50#984—91no—
51C3×#137—91no—
52C3×AGL(1,ℤ/8)—91no—
53C2×C3×SD16—91no—
54C3×#131—31no—
55C3×#131C3×D311yescharacteristic
56#951—31no—
57#949—31no—
58C2×#204—31no—
59#131⋊C3—31no—
60#131⋊C3—61no—
61C3×#108—91no—
62C3×#101—91no—
63C3×#100—91no—
64#824—31no—
65#823—31no—
66#819—31no—
67C22×C3×D3—61no—
68C32×Q8D1211yescharacteristic
69C32×D4—31no—
70C2×C32×C4—31no—
71C4⋊(C3⋊D3)—41no—
72#634—41no—
73C2×C3×Dic3—61no—
74C3×D12—61no—
75C3×D12—61no—
76C3×Dic6—61no—
77C3×Dic6—61no—
78C3×SL(2,𝔽3)—21no—
79C3×SL(2,𝔽3)C3×D411yescharacteristic
80C32×C8—31no—
81C32×C8—41no—
82C32×C8—81no—
83C32×C8S411yescharacteristic
84C3×C3⋊C8—61no—
85C3×C3⋊C8—61no—
86#471—91noSylow
87C2×H(3,𝔽3)—41no—
88(C3⋊D3)⋊C3—161no—
89C3×#41—31no—
90C3×#41C2×C3×D311yescharacteristic
91C2×C3×Q8—91no—
92C2×C3×D4—91no—
93C2×D12—31no—
94C2×Dic6—31no—
95#41⋊C3—31no—
96#41⋊C3—61no—
97GL(2,𝔽3)—181no—
98#226—181no—
99C3×Q16—181no—
100C3×SD16—181no—
101C3×SD16—181no—
102C3×SD16—181no—
103C3×SD16—181no—
104C3×D8—181no—
105C3×M16—31no—
106C3×M16—31no—
107C3×M16—61no—
108C3×M16—91no—
109C3×M16—91no—
110C2×C3×C8—31no—
111C2×C3×C8—31no—
112C2×C3×C8—61no—
113#216—61no—
114#215—61no—
115#214—61no—
116#213—61no—
117#208—31no—
118#208—31no—
119#204—61no—
120#204—61no—
121#204—361no—
122C22×C32—31no—
123C2×C3⋊D3—81no—
124C2×C3×D3—121no—
125C2×C3×D3—241no—
126C32×C4—31no—
127C32×C4—41no—
128C32×C4—81no—
129C32×C4C2×S411yescharacteristic
130C3⋊Dic3—81no—
131C3×Dic3—121no—
132#137—91no—
133AGL(1,ℤ/8)—91no—
134C2×SD16—91no—
135#131(C3⋊D3)⋊C311yescharacteristic, complemented
136#108—91no—
137#101—91no—
138#100—91no—
139H(3,𝔽3)—41noSylow
140C23×C3—181no—
141C22×D3—61no—
142C3×Q8—31no—
143C3×Q8—181no—
144C3×Q8—181no—
145C3×Q8C3×D1211yescharacteristic
146C3×D4—31no—
147C3×D4—31no—
148C3×D4—61no—
149C3×D4—181no—
150C3×D4—181no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—61no—
154C2×C3×C4—181no—
155C2×Dic3—61no—
156D12—61no—
157D12—61no—
158D12—361no—
159Dic6—61no—
160Dic6—61no—
161Dic6—361no—
162SL(2,𝔽3)—31no—
163SL(2,𝔽3)—61no—
164C3×C8—31no—
165C3×C8—31no—
166C3×C8—31no—
167C3×C8—61no—
168C3×C8—121no—
169C3×C8—181no—
170C3×C8—181no—
171C3×C8—241no—
172C3×C8C3×S411yescharacteristic
173C3⋊C8—61no—
174C3⋊C8—61no—
175C2×C32—41no—
176C2×C32—61no—
177C2×C32—81no—
178C2×C32C4⋊S411yescharacteristic
179C3⋊D3—161no—
180C3×D3—241no—
181#41C2×(C3⋊D3)⋊C311yescharacteristic
182C2×Q8—91no—
183C2×D4—91no—
184Q16—181no—
185SD16—181no—
186SD16—181no—
187SD16—181no—
188SD16—181no—
189D8—181no—
190M16—31no—
191M16—91no—
192M16—91no—
193C2×C8—31no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—61no—
197C22×C3—361no—
198C22×C3—721no—
199C2×D3—121no—
200C2×D3—241no—
201C2×D3—721no—
202C3×C4—31no—
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—61no—
206C3×C4—121no—
207C3×C4—241no—
208C3×C4—361no—
209C3×C4C2×C3×S411yescharacteristic
210Dic3—121no—
211Dic3—721no—
212C32—41no—
213C32—81no—
214C32#540611yescharacteristic, complemented
215C23—181no—
216Q8—181no—
217Q8—181no—
218Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
219D4—31no—
220D4—181no—
221D4—181no—
222C2×C4—31no—
223C2×C4—181no—
224C8—31no—
225C8—181no—
226C8—181no—
227C8(C3⋊S4)⋊C311yescharacteristic
228C2×C3—31no—
229C2×C3—61no—
230C2×C3—61no—
231C2×C3—121no—
232C2×C3—121no—
233C2×C3—241no—
234C2×C3—721no—
235C2×C3C3×C4⋊S411yescharacteristic
236D3—241no—
237D3—1441no—
238C22—31no—
239C22—361no—
240C22—721no—
241C4—31no—
242C4—361no—
243C4C2×(C3⋊S4)⋊C311yescharacteristic
244C3—31no—
245C3—121no—
246C3—241no—
247C3C3×#540611yescharacteristic
248C2—61no—
249C2—721no—
250C2C4⋊((C3⋊S4)⋊C3)11yescharacteristic
subgroupG/H
centerC2C4⋊((C3⋊S4)⋊C3)
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C3C3×C4⋊S4
prime radical#64267⋊C3C2
hypercenterC8(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #64265⋊C3C2semidirect
3. #131⋊H(3,𝔽3)C2semidirect
4. #10593640C3semidirect
11. #64265C2×C3semidirect
12. C32×#131D3semidirect
17. C3×#131⋊C3C2×C3semidirect
135. #131(C3⋊D3)⋊C3semidirect
214. C32#5406semidirect

Maximal subgroups

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

subgroupindex
#64267⋊C32
#64265⋊C32
#131⋊H(3,𝔽3)2
#105936403
C3×#51413
#3744⋊C34