Finite Group Databasebeta

← C3×#10593660 #13173028

Subgroups — C3×#10593660 #13173028

Order 1728.

527 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 98 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×C32×#41⋊C3C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal, complemented
3#10593660C311yesmaximal, characteristic, direct product factor
4C3×#5426—33nomaximal
5C3×#5232—31nomaximal
6C32×#41⋊C3C412yescomplemented
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C4⋊(C3⋊Dic3)—41nomaximal
9C2×C32×#41D311yescharacteristic
10C2×C3×#41⋊C3—21no—
11C2×C3×#41⋊C3—23no—
12C2×C3×#41⋊C3D313yes—
13C2×C3×#41⋊C3C2×C311yescharacteristic
14C2×C3×C4⋊Dic3—31no—
15C2×C3×#208—31no—
16C3⋊#859C2×C312yescomplemented
17C3×#859—36no—
18C3×#835—32no—
19C3×#832—32no—
20C2×C33×C4—41no—
21C2×C3×C3⋊Dic3—42no—
22C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
23#5426—33no—
24C3×#417—91no—
25#5232—31no—
26C32×#41—32no—
27C32×#41Dic312yescomplemented
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C22×C3×Dic3—61no—
32C3×#41⋊C3—22no—
33C3×#41⋊C3—26no—
34C3×#41⋊C3Dic316yescomplemented
35C3×#41⋊C3C3×C412yescomplemented
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—23no—
38C2×C3×SL(2,𝔽3)C2×D313yes—
39C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
40C4⋊(C3⋊Dic3)—41no—
41C3×C4⋊Dic3—32no—
42C3×C4⋊Dic3—61no—
43C3×C4⋊Dic3—123no—
44C3×#208—62no—
45C2×C3×C3⋊C8—32no—
46C22×C33—41no—
47C33×C4—42no—
48C3×C3⋊Dic3—82no—
49C2×C3×#41—21no—
50C2×C3×#41C3×D311yescharacteristic
51C2×C3×#41C3⋊D311yescharacteristic
52C2×#41⋊C3—23no—
53C2×#41⋊C3C3×D313yes—
54C2×C3×M16—91no—
55C2×C3×C4⋊C4—91no—
56C2×C4⋊Dic3—31no—
57C2×#208—31no—
58#859—36no—
59C3×#103—92no—
60C3×#102—92no—
61#835—32no—
62#832—32no—
63C23×C32—31no—
64C32×Q8—31no—
65C32×Q8C2×Dic311yescharacteristic
66C32×D4—32no—
67C32×D4—61no—
68C2×C32×C4—31no—
69C2×C32×C4—32no—
70C2×C32×C4—32no—
71C2×C32×C4—41no—
72C2×C32×C4—43no—
73C2×C32×C4—83no—
74C2×C32×C4—81no—
75C2×C32×C4S411yescharacteristic
76C2×C3⋊Dic3—42no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—126no—
79C2×C3×Dic3—122no—
80C3×SL(2,𝔽3)—21no—
81C3×SL(2,𝔽3)—23no—
82C3×SL(2,𝔽3)C2×Dic313yescomplemented
83C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
84C3×C3⋊C8—62no—
85#417—91noSylow
86C2×C33—41no—
87C2×C33—41no—
88C2×C33—41no—
89C3×#41—22no—
90C3×#41—32no—
91C3×#41—32no—
92C3×#41—62no—
93C3×#41C3×Dic312yescomplemented
94C3×#41C3⋊Dic312yescomplemented
95C2×C3×Q8—21no—
96C2×C3×Q8C2×C3×D311yescharacteristic
97C2×C3×Q8C2×C3⋊D311yescharacteristic
98C2×C3×D4—31no—
99C2×C3×D4—31no—
100C2×C3×D4—61no—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—61no—
104C22×C3×C4—181no—
105C22×Dic3—61no—
106#41⋊C3—26no—
107#41⋊C3C3×Dic316yescomplemented
108C2×SL(2,𝔽3)—23no—
109C2×SL(2,𝔽3)C2×C3×D313yes—
110C3×M16—182no—
111C2×C3×C8—92no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—181no—
114C4⋊Dic3—32no—
115C4⋊Dic3—61no—
116C4⋊Dic3—123no—
117#208—62no—
118C2×C3⋊C8—32no—
119C22×C32—32no—
120C22×C32—41no—
121C22×C32—43no—
122C22×C32—61no—
123C22×C32—61no—
124C22×C32—83no—
125C22×C32—81no—
126C22×C32C2×S411yescharacteristic
127C32×C4—31no—
128C32×C4—31no—
129C32×C4—42no—
130C32×C4—46no—
131C32×C4—86no—
132C32×C4—82no—
133C32×C4SL(2,ℤ/4)12yes—
134C3⋊Dic3—82no—
135C3×Dic3—122no—
136C3×Dic3—246no—
137C2×#41C3×C3⋊D311yescharacteristic
138C2×M16—91no—
139C2×C4⋊C4—91no—
140#103—92no—
141#102—92no—
142C33—41noSylow
143C23×C3—31no—
144C23×C3—31no—
145C23×C3—61no—
146C3×Q8—21no—
147C3×Q8—31no—
148C3×Q8—31no—
149C3×Q8—61no—
150C3×Q8C2×C3×Dic311yescharacteristic
151C3×Q8C2×C3⋊Dic311yescharacteristic
152C3×D4—32no—
153C3×D4—32no—
154C3×D4—62no—
155C3×D4—61no—
156C3×D4—61no—
157C3×D4—62no—
158C2×C3×C4—21no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—32no—
164C2×C3×C4—32no—
165C2×C3×C4—43no—
166C2×C3×C4—62no—
167C2×C3×C4—62no—
168C2×C3×C4—61no—
169C2×C3×C4—83no—
170C2×C3×C4—182no—
171C2×C3×C4—362no—
172C2×C3×C4C3×S411yescharacteristic
173C2×C3×C4C3⋊S411yescharacteristic
174C2×Dic3—62no—
175C2×Dic3—126no—
176C2×Dic3—122no—
177SL(2,𝔽3)—23no—
178SL(2,𝔽3)C2×C3×Dic313yescomplemented
179C3×C8—182no—
180C3⋊C8—62no—
181C2×C32—41no—
182C2×C32—41no—
183C2×C32—41no—
184C2×C32—43no—
185C2×C32—43no—
186C2×C32—43no—
187C2×C32—62no—
188C2×C32—83no—
189C2×C32—83no—
190C2×C32—83no—
191C2×C32—81no—
192C2×C32—81no—
193C2×C32—81no—
194C2×C32#98411yescharacteristic
195C2×C32#98611yescharacteristic
196C2×C32C2×SL(2,ℤ/4)11yescharacteristic
197#41—32no—
198#41C3×C3⋊Dic312yescomplemented
199C2×Q8C2×C3×C3⋊D311yescharacteristic
200C2×D4—31no—
201C22×C4—31no—
202C22×C4—181no—
203M16—182no—
204C2×C8—92no—
205C4⋊C4—92no—
206C4⋊C4—181no—
207C22×C3—21no—
208C22×C3—32no—
209C22×C3—32no—
210C22×C3—43no—
211C22×C3—62no—
212C22×C3—61no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—61no—
216C22×C3—83no—
217C22×C3—121no—
218C22×C3—121no—
219C22×C3C2×C3×S411yescharacteristic
220C22×C3C2×C3⋊S411yescharacteristic
221C3×C4—22no—
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—46no—
227C3×C4—61no—
228C3×C4—61no—
229C3×C4—86no—
230C3×C4—362no—
231C3×C4C3×SL(2,ℤ/4)12yes—
232C3×C4C3⋊SL(2,ℤ/4)12yes—
233Dic3—122no—
234Dic3—246no—
235C32—41no—
236C32—43no—
237C32—83no—
238C32—81no—
239C32#542611yescharacteristic, complemented
240C23—31no—
241Q8—31no—
242Q8C2×C3×C3⋊Dic311yescharacteristic
243D4—32no—
244D4—61no—
245C2×C4—31no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—182no—
249C2×C4—362no—
250C2×C4C3×C3⋊S411yescharacteristic
251C8—182no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—43no—
256C2×C3—43no—
257C2×C3—43no—
258C2×C3—62no—
259C2×C3—62no—
260C2×C3—83no—
261C2×C3—83no—
262C2×C3—83no—
263C2×C3—122no—
264C2×C3C3×#98411yescharacteristic
265C2×C3C3×#98611yescharacteristic
266C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
267C2×C3#6426511yescharacteristic
268C2×C3#6426711yescharacteristic
269C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
270C22—32no—
271C22—61no—
272C22—61no—
273C22C2×C3×C3⋊S411yescharacteristic
274C4—31no—
275C4—31no—
276C4—362no—
277C4C3×C3⋊SL(2,ℤ/4)12yes—
278C3—21no—
279C3—43no—
280C3—83no—
281C3C3×#542611yescharacteristic, complemented
282C3#1059366011yescharacteristic, direct product factor
283C2—62no—
284C2C3×#6426511yescharacteristic
285C2C3×#6426711yescharacteristic
286C2C2×C3×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C2×C3×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
prime radicalC2×C32×#41⋊C3C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
3. #10593660C3direct (N × H)
6. C32×#41⋊C3C4semidirect
16. C3⋊#859C2×C3semidirect
22. C32×SL(2,𝔽3)C2×C4semidirect
27. C32×#41Dic3semidirect
34. C3×#41⋊C3Dic3semidirect
35. C3×#41⋊C3C3×C4semidirect
82. C3×SL(2,𝔽3)C2×Dic3semidirect
83. C3×SL(2,𝔽3)C2×C3×C4semidirect
93. C3×#41C3×Dic3semidirect
94. C3×#41C3⋊Dic3semidirect
107. #41⋊C3C3×Dic3semidirect
178. SL(2,𝔽3)C2×C3×Dic3semidirect
198. #41C3×C3⋊Dic3semidirect
239. C32#5426semidirect
281. C3C3×#5426semidirect
282. C3#10593660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×C3⋊#8592
#105936603
C3×#54263
C3×#52323
C3×C4⋊(C3⋊Dic3)4