Finite Group Databasebeta

← #10593660⋊C3 #13146924

Subgroups — #10593660⋊C3 #13146924

Order 1728.

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

normal subgroups only

This group has 344 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2(C3⋊#859)⋊C3C212yesmaximal, complemented
3#10593660C311yesmaximal, characteristic, complemented
4C3×#5232—31nomaximal
5#41⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7C4⋊((C3⋊Dic3)⋊C3)—41nomaximal
8C2×C32×#41D311yescharacteristic
9C2×C3×#41⋊C3—21no—
10C2×C3×#41⋊C3C2×C311yescharacteristic
11C2×C3×C4⋊Dic3—31no—
12C2×C3×#208—31no—
13C3⋊#859C2×C312yescomplemented
14C3×#835—32no—
15C3×#832—32no—
16C2×C4×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
19#5426—91no—
20C3×#417—91no—
21#5232—31no—
22C32×#41—32no—
23C32×#41Dic312yescomplemented
24C2×C32×Q8C2×D311yescharacteristic
25C2×C32×D4—31no—
26C22×C32×C4—31no—
27C22×C3×Dic3—61no—
28C3×#41⋊C3—22no—
29C3×#41⋊C3C3×C412yescomplemented
30C2×C3×SL(2,𝔽3)—21no—
31C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
32C4⋊(C3⋊Dic3)—41no—
33C3×C4⋊Dic3—32no—
34C3×C4⋊Dic3—61no—
35C3×#208—62no—
36C2×C3×C3⋊C8—32no—
37C22×H(3,𝔽3)—41no—
38C4×H(3,𝔽3)—42no—
39(C3⋊Dic3)⋊C3—82no—
40C2×C3×#41—31no—
41C2×C3×#41C3×D311yescharacteristic
42C2×#41⋊C3—31no—
43C2×#41⋊C3—61no—
44C2×C3×M16—91no—
45C2×C3×C4⋊C4—91no—
46C2×C4⋊Dic3—31no—
47C2×#208—31no—
48#859—92no—
49C3×#103—92no—
50C3×#102—92no—
51#835—32no—
52#832—32no—
53C23×C32—31no—
54C32×Q8—31no—
55C32×Q8C2×Dic311yescharacteristic
56C32×D4—32no—
57C32×D4—61no—
58C2×C32×C4—31no—
59C2×C32×C4—32no—
60C2×C32×C4—32no—
61C2×C32×C4—41no—
62C2×C32×C4—81no—
63C2×C32×C4S411yescharacteristic
64C2×C3⋊Dic3—42no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—122no—
67C3×SL(2,𝔽3)—21no—
68C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
69C3×C3⋊C8—62no—
70#417—91noSylow
71C2×H(3,𝔽3)—41no—
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—41no—
74C3×#41—32no—
75C3×#41—32no—
76C3×#41—32no—
77C3×#41—62no—
78C3×#41C3×Dic312yes—
79C2×C3×Q8—31no—
80C2×C3×Q8C2×C3×D311yescharacteristic
81C2×C3×D4—31no—
82C2×C3×D4—31no—
83C2×C3×D4—61no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C22×C3×C4—181no—
88C22×Dic3—61no—
89#41⋊C3—32no—
90#41⋊C3—62no—
91C2×SL(2,𝔽3)—31no—
92C2×SL(2,𝔽3)—61no—
93C3×M16—182no—
94C2×C3×C8—92no—
95C3×C4⋊C4—92no—
96C3×C4⋊C4—181no—
97C4⋊Dic3—32no—
98C4⋊Dic3—61no—
99C4⋊Dic3—361no—
100#208—62no—
101C2×C3⋊C8—32no—
102C22×C32—32no—
103C22×C32—41no—
104C22×C32—61no—
105C22×C32—61no—
106C22×C32—81no—
107C22×C32C2×S411yescharacteristic
108C32×C4—31no—
109C32×C4—31no—
110C32×C4—42no—
111C32×C4—82no—
112C32×C4SL(2,ℤ/4)12yes—
113C3⋊Dic3—82no—
114C3×Dic3—122no—
115C2×#41(C3⋊D3)⋊C311yescharacteristic
116C2×M16—91no—
117C2×C4⋊C4—91no—
118#103—92no—
119#102—92no—
120H(3,𝔽3)—41noSylow
121C23×C3—31no—
122C23×C3—31no—
123C23×C3—61no—
124C3×Q8—31no—
125C3×Q8—31no—
126C3×Q8—31no—
127C3×Q8—61no—
128C3×Q8C2×C3×Dic311yescharacteristic
129C3×D4—32no—
130C3×D4—32no—
131C3×D4—62no—
132C3×D4—61no—
133C3×D4—61no—
134C3×D4—62no—
135C2×C3×C4—31no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—31no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—31no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—61no—
145C2×C3×C4—121no—
146C2×C3×C4—182no—
147C2×C3×C4—241no—
148C2×C3×C4—362no—
149C2×C3×C4C3×S411yescharacteristic
150C2×Dic3—62no—
151C2×Dic3—122no—
152C2×Dic3—362no—
153SL(2,𝔽3)—31no—
154SL(2,𝔽3)—61no—
155C3×C8—182no—
156C3⋊C8—62no—
157C2×C32—41no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32—62no—
161C2×C32—81no—
162C2×C32—81no—
163C2×C32—81no—
164C2×C32#98411yescharacteristic
165C2×C32#98611yescharacteristic
166C2×C32C2×SL(2,ℤ/4)11yescharacteristic
167#41—32no—
168#41(C3⋊Dic3)⋊C312yescomplemented
169C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
170C2×D4—31no—
171C22×C4—31no—
172C22×C4—181no—
173M16—182no—
174C2×C8—92no—
175C4⋊C4—92no—
176C4⋊C4—181no—
177C22×C3—32no—
178C22×C3—32no—
179C22×C3—31no—
180C22×C3—62no—
181C22×C3—61no—
182C22×C3—61no—
183C22×C3—61no—
184C22×C3—61no—
185C22×C3—121no—
186C22×C3—121no—
187C22×C3—121no—
188C22×C3—241no—
189C22×C3C2×C3×S411yescharacteristic
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—32no—
195C3×C4—61no—
196C3×C4—61no—
197C3×C4—122no—
198C3×C4—242no—
199C3×C4—362no—
200C3×C4C3×SL(2,ℤ/4)12yes—
201Dic3—122no—
202Dic3—722no—
203C32—41no—
204C32—81no—
205C32#542611yescharacteristic, complemented
206C23—31no—
207Q8—31no—
208Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
209D4—32no—
210D4—61no—
211C2×C4—31no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—182no—
215C2×C4—362no—
216C2×C4(C3⋊S4)⋊C311yescharacteristic
217C8—182no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—31no—
221C2×C3—62no—
222C2×C3—62no—
223C2×C3—122no—
224C2×C3—121no—
225C2×C3—121no—
226C2×C3—121no—
227C2×C3—241no—
228C2×C3—241no—
229C2×C3—241no—
230C2×C3C3×#98411yescharacteristic
231C2×C3C3×#98611yescharacteristic
232C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
233C22—32no—
234C22—61no—
235C22—61no—
236C22C2×(C3⋊S4)⋊C311yescharacteristic
237C4—31no—
238C4—31no—
239C4—362no—
240C4(C3⋊SL(2,ℤ/4))⋊C312yes—
241C3—31no—
242C3—121no—
243C3—241no—
244C3C3×#542611yescharacteristic
245C2—62no—
246C2#64265⋊C311yescharacteristic
247C2#64267⋊C311yescharacteristic
248C2C2×(C3⋊SL(2,ℤ/4))⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
Frattini, socleC22×C3C2×C3×S4
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC2×C4(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. (C3⋊#859)⋊C3C2semidirect
3. #10593660C3semidirect
5. #41⋊H(3,𝔽3)C4semidirect
13. C3⋊#859C2×C3semidirect
18. Q8⋊H(3,𝔽3)C2×C4semidirect
23. C32×#41Dic3semidirect
29. C3×#41⋊C3C3×C4semidirect
68. C3×SL(2,𝔽3)C2×C3×C4semidirect
168. #41(C3⋊Dic3)⋊C3semidirect
205. C32#5426semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
(C3⋊#859)⋊C32
#105936603
C3×#52323
C4⋊((C3⋊Dic3)⋊C3)4