Finite Group Databasebeta

← #10593638⋊C3 #13146861

Subgroups — #10593638⋊C3 #13146861

Order 1728.

259 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 37 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#64266⋊C3C211yesmaximal, characteristic
2#131⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3(C3⋊GU(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
4#10593638C311yesmaximal, characteristic, complemented
5C3×#5144—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7(C3⋊GL(2,𝔽3))⋊C3C411yescharacteristic, complemented
8(C3⋊#226)⋊C3C411yescharacteristic, complemented
9C8×(C3⋊D3)⋊C3—41nomaximal
10#64266C2×C311yescharacteristic
11C32×#131D311yescharacteristic, complemented
12C3×#950—31no—
13C3×#901—31no—
14C3×#131⋊C3—21no—
15C3×#131⋊C3C2×C311yescharacteristic, complemented
16C3⋊GU(2,𝔽3)C2×C311yescharacteristic, complemented
17C3×#837—31no—
18C3×#825—31no—
19C3×#822—31no—
20C3×C8×Dic3—31no—
21C4×(C3⋊D3)⋊C3—41no—
22Q8⋊H(3,𝔽3)C2×C411yescharacteristic
23C8×H(3,𝔽3)—41no—
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5404—91no—
26C3×#443—91no—
27#5144—31no—
28C32×#41C2×D311yescharacteristic
29C3×#235—31no—
30C3×#41⋊C3—21no—
31C3×#41⋊C3C22×C311yescharacteristic
32C3⋊GL(2,𝔽3)C3×C411yescharacteristic, complemented
33C3⋊#226C3×C411yescharacteristic, complemented
34C32×M16—31no—
35C2×C32×C8—31no—
36C8×C3⋊D3—41no—
37C3×#216—31no—
38C3×#215—31no—
39C3×#214—31no—
40C3×#213—31no—
41C3×C4×Dic3—31no—
42C3×#208—31no—
43C2×C3×C3⋊C8—31no—
44C3×ΓL(1,𝔽25)—61no—
45C3×D3×C8—61no—
46C2×(C3⋊D3)⋊C3—41no—
47C4×H(3,𝔽3)—41no—
48(C3⋊Dic3)⋊C3—41no—
49#985—91no—
50C3×#135—91no—
51C3×#131—31no—
52C3×#131—91no—
53C3×#131C3×D311yescharacteristic
54#950—31no—
55#901—31no—
56#131⋊C3—31no—
57#131⋊C3—61no—
58GU(2,𝔽3)—91no—
59C3×#108—91no—
60C3×#104—91no—
61C3×#104—91no—
62C3×C4×C8—91no—
63#837—31no—
64#825—31no—
65#822—31no—
66C8×Dic3—31no—
67C32×Q8C4×D311yescharacteristic
68C32×D4—31no—
69C2×C32×C4—31no—
70C4×C3⋊D3—41no—
71C3×#67—61no—
72C2×C3×Dic3—61no—
73C3×D12—31no—
74C3×C4×D3—61no—
75C3×Dic6—31no—
76C3×SL(2,𝔽3)—21no—
77C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
78C32×C8—31no—
79C32×C8—41no—
80C32×C8—81no—
81C32×C8S411yescharacteristic
82C3⋊(C3⋊C8)—41no—
83C3×C3⋊C8—31no—
84C3×C3⋊C8—31no—
85C3×C3⋊C8—61no—
86#443—91noSylow
87C2×H(3,𝔽3)—41no—
88(C3⋊D3)⋊C3—42no—
89C3×#41—31no—
90C3×#41—91no—
91C3×#41C2×C3×D311yescharacteristic
92#235—31no—
93#41⋊C3—31no—
94#41⋊C3—61no—
95GL(2,𝔽3)—91no—
96#226—91no—
97C3×Q16—91no—
98C3×SD16—91no—
99C3×SD16—91no—
100C3×D8—91no—
101C3×M16—31no—
102C3×M16—31no—
103C3×M16—61no—
104C3×M16—91no—
105C3×M16—181no—
106C2×C3×C8—31no—
107C2×C3×C8—31no—
108C2×C3×C8—61no—
109C2×C3×C8—91no—
110C2×C3×C8—181no—
111C3×C42—91no—
112#216—31no—
113#215—31no—
114#214—31no—
115#213—31no—
116C4×Dic3—31no—
117#208—31no—
118C2×C3⋊C8—31no—
119ΓL(1,𝔽25)—61no—
120D3×C8—61no—
121D3×C8—361no—
122C22×C32—31no—
123C2×C3⋊D3—41no—
124C2×C3×D3—61no—
125C32×C4—31no—
126C32×C4—41no—
127C32×C4—81no—
128C32×C4C2×S411yescharacteristic
129C3⋊Dic3—41no—
130C3×Dic3—62no—
131C3×Dic3—61no—
132#135—91no—
133#131—91no—
134#131(C3⋊D3)⋊C311yescharacteristic, complemented
135#108—91no—
136#104—91no—
137#104—91no—
138C4×C8—91no—
139H(3,𝔽3)—41noSylow
140C3×Q8—31no—
141C3×Q8—91no—
142C3×Q8C3×C4×D311yescharacteristic
143C3×D4—31no—
144C3×D4—31no—
145C3×D4—61no—
146C3×D4—91no—
147C3×D4—181no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—61no—
151C2×C3×C4—181no—
152C2×C3×C4—181no—
153#67—61no—
154C2×Dic3—61no—
155D12—31no—
156C4×D3—61no—
157C4×D3—361no—
158Dic6—31no—
159SL(2,𝔽3)—31no—
160SL(2,𝔽3)—61no—
161C3×C8—31no—
162C3×C8—31no—
163C3×C8—31no—
164C3×C8—61no—
165C3×C8—91no—
166C3×C8—91no—
167C3×C8—121no—
168C3×C8—181no—
169C3×C8—241no—
170C3×C8C3×S411yescharacteristic
171C3⋊C8—31no—
172C3⋊C8—31no—
173C3⋊C8—61no—
174C3⋊C8—361no—
175C2×C32—41no—
176C2×C32—61no—
177C2×C32—81no—
178C2×C32C4×S411yescharacteristic
179C3⋊D3—42no—
180C3×D3—121no—
181#41—91no—
182#41C2×(C3⋊D3)⋊C311yescharacteristic
183Q16—91no—
184SD16—91no—
185SD16—91no—
186D8—91no—
187M16—31no—
188M16—91no—
189M16—181no—
190C2×C8—31no—
191C2×C8—91no—
192C2×C8—181no—
193C42—91no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—61no—
197C22×C3—181no—
198C2×D3—61no—
199C2×D3—361no—
200C3×C4—31no—
201C3×C4—31no—
202C3×C4—31no—
203C3×C4—61no—
204C3×C4—121no—
205C3×C4—181no—
206C3×C4—182no—
207C3×C4—241no—
208C3×C4C2×C3×S411yescharacteristic
209Dic3—62no—
210Dic3—61no—
211Dic3—361no—
212C32—41no—
213C32—81no—
214C32#540411yescharacteristic, complemented
215Q8—91no—
216Q8C4×(C3⋊D3)⋊C311yescharacteristic
217D4—31no—
218D4—91no—
219D4—181no—
220C2×C4—31no—
221C2×C4—181no—
222C2×C4—181no—
223C8—31no—
224C8—91no—
225C8—91no—
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—361no—
235C2×C3C3×C4×S411yescharacteristic
236D3—121no—
237D3—362no—
238C22—31no—
239C22—181no—
240C4—31no—
241C4—182no—
242C4—181no—
243C4C2×(C3⋊S4)⋊C311yescharacteristic
244C3—31no—
245C3—121no—
246C3—241no—
247C3C3×#540411yescharacteristic
248C2—61no—
249C2—361no—
250C2C4×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC8(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C3C3×C4×S4
prime radical#64266⋊C3C2
cosocleC3×#41⋊C3C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #131⋊H(3,𝔽3)C2semidirect
3. (C3⋊GU(2,𝔽3))⋊C3C2semidirect
4. #10593638C3semidirect
7. (C3⋊GL(2,𝔽3))⋊C3C4semidirect
8. (C3⋊#226)⋊C3C4semidirect
11. C32×#131D3semidirect
15. C3×#131⋊C3C2×C3semidirect
16. C3⋊GU(2,𝔽3)C2×C3semidirect
32. C3⋊GL(2,𝔽3)C3×C4semidirect
33. C3⋊#226C3×C4semidirect
134. #131(C3⋊D3)⋊C3semidirect
214. C32#5404semidirect

Maximal subgroups

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

subgroupindex
#64266⋊C32
#131⋊H(3,𝔽3)2
(C3⋊GU(2,𝔽3))⋊C32
#105936383
C3×#51443
C8×(C3⋊D3)⋊C34