Finite Group Databasebeta

← C3×#10593654 #13173022

Subgroups — C3×#10593654 #13173022

Order 1728.

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

normal subgroups only

This group has 611 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
1C22×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal, complemented
3#10593654C311yesmaximal, characteristic, direct product factor
4C3×#5420—33nomaximal
5C3×#5225—31nomaximal
6C2×C32×SL(2,𝔽3)C412yescomplemented
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×#3757—41nomaximal
9C22×C32×Q8D311yescharacteristic
10C22×C3×SL(2,𝔽3)—21no—
11C22×C3×SL(2,𝔽3)—23no—
12C22×C3×SL(2,𝔽3)D313yes—
13C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
14C3×#926—31no—
15C2×C3×#208—31no—
16C3⋊#859C2×C312yescomplemented
17C3×#859—36no—
18C3×#835—32no—
19C3×#835—32no—
20C23×C33—41no—
21C2×C3×C3⋊Dic3—42no—
22C32×SL(2,𝔽3)C2×C411yescharacteristic
23#5420—33no—
24C3×#419—91no—
25#5225—31no—
26C2×C32×Q8—32no—
27C2×C32×Q8—31no—
28C2×C32×Q8Dic312yescomplemented
29C2×C32×Q8C2×D311yescharacteristic
30C22×C32×C4—31no—
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)—26no—
34C2×C3×SL(2,𝔽3)—23no—
35C2×C3×SL(2,𝔽3)Dic316yescomplemented
36C2×C3×SL(2,𝔽3)C3×C412yescomplemented
37C2×C3×SL(2,𝔽3)C2×D313yes—
38C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
39#3757—41no—
40C3×#217—61no—
41C3×#217—123no—
42C3×C4⋊Dic3—32no—
43C3×C4×Dic3—61no—
44C3×#208—62no—
45C2×C3×C3⋊C8—32no—
46C22×C33—41no—
47C22×C33—42no—
48C22×C33—82no—
49C3×C3⋊Dic3—82no—
50C22×C3×Q8—21no—
51C22×C3×Q8C3×D311yescharacteristic
52C22×C3×Q8C3⋊D311yescharacteristic
53C22×SL(2,𝔽3)—23no—
54C22×SL(2,𝔽3)C3×D313yes—
55C2×C3×M16—91no—
56C3×#117—91no—
57#926—31no—
58C2×#208—31no—
59#859—36no—
60C3×#103—92no—
61C3×#103—92no—
62#835—32no—
63#835—32no—
64C23×C32—41no—
65C23×C32—43no—
66C23×C32—83no—
67C23×C32—81no—
68C23×C32S411yescharacteristic
69C32×Q8—32no—
70C32×Q8—31no—
71C32×Q8—61no—
72C32×Q8C2×Dic311yescharacteristic
73C2×C32×C4—32no—
74C2×C32×C4—32no—
75C2×C32×C4—31no—
76C2×C32×C4—31no—
77C2×C3⋊Dic3—42no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—126no—
80C3×SL(2,𝔽3)—21no—
81C3×SL(2,𝔽3)—23no—
82C3×SL(2,𝔽3)C2×Dic313yes—
83C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
84C3×C3⋊C8—62no—
85#419—91noSylow
86C2×C33—42no—
87C2×C33—41no—
88C2×C33—82no—
89C2×C3×Q8—22no—
90C2×C3×Q8—21no—
91C2×C3×Q8—31no—
92C2×C3×Q8—31no—
93C2×C3×Q8—32no—
94C2×C3×Q8—32no—
95C2×C3×Q8—61no—
96C2×C3×Q8—62no—
97C2×C3×Q8C3×Dic312yescomplemented
98C2×C3×Q8C3⋊Dic312yescomplemented
99C2×C3×Q8C2×C3×D311yescharacteristic
100C2×C3×Q8C2×C3⋊D311yescharacteristic
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—61no—
104C2×SL(2,𝔽3)—26no—
105C2×SL(2,𝔽3)—23no—
106C2×SL(2,𝔽3)C3×Dic316yescomplemented
107C2×SL(2,𝔽3)C2×C3×D313yes—
108C3×M16—182no—
109C2×C3×C8—92no—
110C3×C4⋊C4—92no—
111C3×C22⋊C4—181no—
112C3×C42—181no—
113#217—61no—
114#217—123no—
115C4⋊Dic3—32no—
116C4×Dic3—61no—
117#208—62no—
118C2×C3⋊C8—32no—
119C22×C32—22no—
120C22×C32—41no—
121C22×C32—42no—
122C22×C32—43no—
123C22×C32—46no—
124C22×C32—83no—
125C22×C32—86no—
126C22×C32—86no—
127C22×C32—82no—
128C22×C32—812no—
129C22×C32—84no—
130C22×C32—81no—
131C22×C32—82no—
132C22×C32SL(2,ℤ/4)12yes—
133C22×C32C2×S411yescharacteristic
134C32×C4—31no—
135C32×C4—31no—
136C32×C4—32no—
137C3⋊Dic3—82no—
138C3×Dic3—122no—
139C3×Dic3—246no—
140C22×Q8C3×C3⋊D311yescharacteristic
141C2×M16—91no—
142#117—91no—
143#103—92no—
144#103—92no—
145C33—41noSylow
146C23×C3—21no—
147C23×C3—43no—
148C23×C3—83no—
149C23×C3C3×S411yescharacteristic
150C23×C3C3⋊S411yescharacteristic
151C3×Q8—21no—
152C3×Q8—32no—
153C3×Q8—32no—
154C3×Q8—31no—
155C3×Q8—31no—
156C3×Q8—62no—
157C3×Q8—61no—
158C3×Q8—61no—
159C3×Q8—61no—
160C3×Q8—62no—
161C3×Q8C2×C3×Dic311yescharacteristic
162C3×Q8C2×C3⋊Dic311yescharacteristic
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—61no—
172C2×C3×C4—62no—
173C2×C3×C4—61no—
174C2×C3×C4—62no—
175C2×C3×C4—182no—
176C2×Dic3—62no—
177C2×Dic3—126no—
178SL(2,𝔽3)—23no—
179SL(2,𝔽3)C2×C3×Dic313yes—
180C3×C8—182no—
181C3⋊C8—62no—
182C2×C32—22no—
183C2×C32—41no—
184C2×C32—42no—
185C2×C32—46no—
186C2×C32—43no—
187C2×C32—83no—
188C2×C32—86no—
189C2×C32—82no—
190C2×C32—86no—
191C2×C32—812no—
192C2×C32—84no—
193C2×C32—81no—
194C2×C32—82no—
195C2×C32#98312yes—
196C2×C32C2×SL(2,ℤ/4)11yescharacteristic
197C2×Q8—31no—
198C2×Q8—32no—
199C2×Q8C3×C3⋊Dic312yescomplemented
200C2×Q8C2×C3×C3⋊D311yescharacteristic
201C22×C4—31no—
202M16—182no—
203C2×C8—92no—
204C4⋊C4—92no—
205C22⋊C4—181no—
206C42—181no—
207C22×C3—24no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—21no—
211C22×C3—22no—
212C22×C3—43no—
213C22×C3—46no—
214C22×C3—86no—
215C22×C3—812no—
216C22×C3—83no—
217C22×C3—86no—
218C22×C3C3×SL(2,ℤ/4)12yes—
219C22×C3C3⋊SL(2,ℤ/4)12yes—
220C22×C3C2×C3×S411yescharacteristic
221C22×C3C2×C3⋊S411yescharacteristic
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—32no—
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—32no—
228C3×C4—61no—
229C3×C4—61no—
230C3×C4—62no—
231C3×C4—362no—
232Dic3—122no—
233Dic3—246no—
234C32—41no—
235C32—43no—
236C32—83no—
237C32—81no—
238C32#542011yescharacteristic, complemented
239C23C3×C3⋊S411yescharacteristic
240Q8—32no—
241Q8—31no—
242Q8—61no—
243Q8C2×C3×C3⋊Dic311yescharacteristic
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—182no—
249C8—182no—
250C2×C3—24no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—21no—
255C2×C3—46no—
256C2×C3—43no—
257C2×C3—86no—
258C2×C3—812no—
259C2×C3—86no—
260C2×C3—83no—
261C2×C3C3×#98312yes—
262C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
263C2×C3#6426412yes—
264C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
265C22—22no—
266C22C3×C3⋊SL(2,ℤ/4)12yes—
267C22C2×C3×C3⋊S411yescharacteristic
268C4—31no—
269C4—31no—
270C4—32no—
271C4—362no—
272C3—21no—
273C3—43no—
274C3—83no—
275C3C3×#542011yescharacteristic, complemented
276C3#1059365411yescharacteristic, direct product factor
277C2—22no—
278C2C3×#6426412yes—
279C2C2×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
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC22×C32×SL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
hypercenterC23×C3C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
3. #10593654C3direct (N × H)
6. C2×C32×SL(2,𝔽3)C4semidirect
16. C3⋊#859C2×C3semidirect
28. C2×C32×Q8Dic3semidirect
35. C2×C3×SL(2,𝔽3)Dic3semidirect
36. C2×C3×SL(2,𝔽3)C3×C4semidirect
97. C2×C3×Q8C3×Dic3semidirect
98. C2×C3×Q8C3⋊Dic3semidirect
106. C2×SL(2,𝔽3)C3×Dic3semidirect
199. C2×Q8C3×C3⋊Dic3semidirect
238. C32#5420semidirect
275. C3C3×#5420semidirect
276. C3#10593654direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C3×C3⋊#8592
#105936543
C3×#54203
C3×#52253
C3×#37574