Finite Group Databasebeta

← C3×#10593658 #13173026

Subgroups — C3×#10593658 #13173026

Order 1728.

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

normal subgroups only

This group has 535 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#10593658C311yesmaximal, characteristic, direct product factor
4C3×#5424—33nomaximal
5C3×#5233—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
14C3×#926—31no—
15C22×C3×C3⋊C8—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
23#5424—33no—
24C3×#416—91no—
25#5233—31no—
26C32×#41—32no—
27C32×#41Dic312yescomplemented
28C2×C32×Q8C2×D311yescharacteristic
29C2×C32×D4—31no—
30C22×C32×C4—31no—
31C3×#41⋊C3—22no—
32C3×#41⋊C3—26no—
33C3×#41⋊C3Dic316yescomplemented
34C3×#41⋊C3C3×C412yescomplemented
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)—23no—
37C2×C3×SL(2,𝔽3)C2×D313yes—
38C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
39C4×C3⋊Dic3—41no—
40C3×#217—61no—
41C3×C4⋊Dic3—32no—
42C3×C4×Dic3—61no—
43C3×C4×Dic3—123no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—62no—
46C22×C33—41no—
47C33×C4—42no—
48C3×C3⋊Dic3—44no—
49C2×C3×#41—21no—
50C2×C3×#41C3×D311yescharacteristic
51C2×C3×#41C3⋊D311yescharacteristic
52C2×#41⋊C3—23no—
53C2×#41⋊C3C3×D313yes—
54C22×C3×C8—91no—
55C3×#117—91no—
56#926—31no—
57C22×C3⋊C8—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—
79C3×SL(2,𝔽3)—21no—
80C3×SL(2,𝔽3)—23no—
81C3×SL(2,𝔽3)C2×Dic313yes—
82C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
83C3×C3⋊C8—62no—
84#416—91noSylow
85C2×C33—42no—
86C2×C33—41no—
87C3×#41—22no—
88C3×#41—32no—
89C3×#41—32no—
90C3×#41—62no—
91C3×#41C3×Dic312yescomplemented
92C3×#41C3⋊Dic312yescomplemented
93C2×C3×Q8—21no—
94C2×C3×Q8C2×C3×D311yescharacteristic
95C2×C3×Q8C2×C3⋊D311yescharacteristic
96C2×C3×D4—31no—
97C2×C3×D4—31no—
98C2×C3×D4—61no—
99C22×C3×C4—31no—
100C22×C3×C4—31no—
101C22×C3×C4—61no—
102#41⋊C3—26no—
103#41⋊C3C3×Dic316yescomplemented
104C2×SL(2,𝔽3)—23no—
105C2×SL(2,𝔽3)C2×C3×D313yes—
106C2×C3×C8—92no—
107C2×C3×C8—182no—
108C3×C4⋊C4—92no—
109C3×C22⋊C4—181no—
110C3×C42—181no—
111#217—61no—
112C4⋊Dic3—32no—
113C4×Dic3—61no—
114C4×Dic3—123no—
115C2×C3⋊C8—32no—
116C2×C3⋊C8—62no—
117C22×C32—32no—
118C22×C32—41no—
119C22×C32—43no—
120C22×C32—62no—
121C22×C32—83no—
122C22×C32—81no—
123C22×C32C2×S411yescharacteristic
124C32×C4—31no—
125C32×C4—31no—
126C32×C4—42no—
127C32×C4—46no—
128C32×C4—86no—
129C32×C4—82no—
130C32×C4SL(2,ℤ/4)12yes—
131C3⋊Dic3—44no—
132C3×Dic3—1212no—
133C3×Dic3—122no—
134C2×#41C3×C3⋊D311yescharacteristic
135C22×C8—91no—
136#117—91no—
137#103—92no—
138#102—92no—
139C33—41noSylow
140C23×C3—31no—
141C23×C3—31no—
142C23×C3—61no—
143C3×Q8—21no—
144C3×Q8—31no—
145C3×Q8—31no—
146C3×Q8—61no—
147C3×Q8C2×C3×Dic311yescharacteristic
148C3×Q8C2×C3⋊Dic311yescharacteristic
149C3×D4—32no—
150C3×D4—32no—
151C3×D4—62no—
152C3×D4—61no—
153C3×D4—61no—
154C3×D4—62no—
155C2×C3×C4—21no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—32no—
159C2×C3×C4—32no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—43no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—61no—
166C2×C3×C4—83no—
167C2×C3×C4—182no—
168C2×C3×C4C3×S411yescharacteristic
169C2×C3×C4C3⋊S411yescharacteristic
170C2×Dic3—62no—
171C2×Dic3—126no—
172SL(2,𝔽3)—23no—
173SL(2,𝔽3)C2×C3×Dic313yes—
174C3×C8—182no—
175C3⋊C8—62no—
176C2×C32—41no—
177C2×C32—42no—
178C2×C32—46no—
179C2×C32—43no—
180C2×C32—62no—
181C2×C32—83no—
182C2×C32—86no—
183C2×C32—81no—
184C2×C32—82no—
185C2×C32#98512yes—
186C2×C32C2×SL(2,ℤ/4)11yescharacteristic
187#41—32no—
188#41C3×C3⋊Dic312yescomplemented
189C2×Q8C2×C3×C3⋊D311yescharacteristic
190C2×D4—31no—
191C22×C4—31no—
192C2×C8—92no—
193C2×C8—182no—
194C4⋊C4—92no—
195C22⋊C4—181no—
196C42—181no—
197C22×C3—21no—
198C22×C3—32no—
199C22×C3—32no—
200C22×C3—43no—
201C22×C3—62no—
202C22×C3—62no—
203C22×C3—62no—
204C22×C3—83no—
205C22×C3—122no—
206C22×C3C2×C3×S411yescharacteristic
207C22×C3C2×C3⋊S411yescharacteristic
208C3×C4—22no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—31no—
213C3×C4—46no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—86no—
217C3×C4—362no—
218C3×C4C3×SL(2,ℤ/4)12yes—
219C3×C4C3⋊SL(2,ℤ/4)12yes—
220Dic3—1212no—
221Dic3—122no—
222C32—41no—
223C32—43no—
224C32—83no—
225C32—81no—
226C32#542411yescharacteristic, complemented
227C23—31no—
228Q8—31no—
229Q8C2×C3×C3⋊Dic311yescharacteristic
230D4—32no—
231D4—61no—
232C2×C4—31no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—182no—
236C2×C4C3×C3⋊S411yescharacteristic
237C8—182no—
238C2×C3—22no—
239C2×C3—21no—
240C2×C3—46no—
241C2×C3—43no—
242C2×C3—62no—
243C2×C3—62no—
244C2×C3—86no—
245C2×C3—83no—
246C2×C3—122no—
247C2×C3C3×#98512yes—
248C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
249C2×C3#6426612yes—
250C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
251C22—32no—
252C22—62no—
253C22C2×C3×C3⋊S411yescharacteristic
254C4—31no—
255C4—31no—
256C4—362no—
257C4C3×C3⋊SL(2,ℤ/4)12yes—
258C3—21no—
259C3—43no—
260C3—83no—
261C3C3×#542411yescharacteristic, complemented
262C3#1059365811yescharacteristic, direct product factor
263C2—62no—
264C2C3×#6426612yes—
265C2C2×C3×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C3⋊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
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
3. #10593658C3direct (N × H)
6. C32×#41⋊C3C4semidirect
16. C3⋊#859C2×C3semidirect
27. C32×#41Dic3semidirect
33. C3×#41⋊C3Dic3semidirect
34. C3×#41⋊C3C3×C4semidirect
91. C3×#41C3×Dic3semidirect
92. C3×#41C3⋊Dic3semidirect
103. #41⋊C3C3×Dic3semidirect
188. #41C3×C3⋊Dic3semidirect
226. C32#5424semidirect
261. C3C3×#5424semidirect
262. C3#10593658direct (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
#105936583
C3×#54243
C3×#52333
C3×C4×C3⋊Dic34