Finite Group Databasebeta

← Dic3×C3⋊#226 #13166015

Subgroups — Dic3×C3⋊#226 #13166015

Order 1728.

466 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 64 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×C3×C3⋊#226C211yesmaximal, characteristic
2C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C3⋊(C3⋊#859)C211yesmaximal, characteristic
4C4×C3⋊#226—31nomaximal
5Dic3×#226—33nomaximal
6Dic3×#216—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊#226C412yescomplemented
9Dic3×C3⋊Dic3—41nomaximal
10C2×C3⋊#226D311yescharacteristic
11C2×C3×#226—33no—
12C3×Q8×Dic3D311yescharacteristic
13C2×C3×#216—31no—
14C3×C4×SL(2,𝔽3)—31no—
15Dic3×Dic6—31no—
16Dic3×SL(2,𝔽3)D313yes—
17C3⋊#859—31no—
18C3⋊#859—33no—
19C3⋊#859—61no—
20C3⋊#859—63no—
21#63662—31no—
22#63575—31no—
23#63565—31no—
24Dic3×C3⋊C8—31no—
25C2×C3⋊(C3⋊Dic3)—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C32×Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic
29C4×#226—93no—
30Dic3×Q16—91no—
31C4×#216—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×C3×Dic6—31no—
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)—23no—
36C2×C3×SL(2,𝔽3)C2×D313yes—
37C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
38C3⋊#226Dic312yesdirect product factor
39C3×#226—36no—
40C4⋊(C3⋊Dic3)—31no—
41C4×C3⋊Dic3—121no—
42C2×C3⋊(C3⋊C8)—31no—
43C3×#216—32no—
44C3×#216—32no—
45C3×C4⋊Dic3—31no—
46C3×C4×Dic3—31no—
47C2×C3×C3⋊C8—31no—
48Dic3⋊Dic3—61no—
49Dic32—61no—
50Dic32—123no—
51C22×C33—41no—
52C3⋊(C3⋊Dic3)—42no—
53C3×C3⋊Dic3—42no—
54C32×Dic3—42no—
55C2×#226—33no—
56C2×C3×Q16—91no—
57C3×C4×Q8—31no—
58Q8×Dic3—91no—
59Q8×Dic3C3⋊D311yescharacteristic
60C2×#216—31no—
61C4×Dic6—91no—
62C4×SL(2,𝔽3)—33no—
63#859—93no—
64#859—183no—
65#835—91no—
66#835—91no—
67#835—91no—
68#835—181no—
69#818—91no—
70C8×Dic3—91no—
71#810—91no—
72#807—91no—
73C4×C3⋊C8—91no—
74C32×Q8—31no—
75C32×Q8C4×D311yescharacteristic
76C2×C32×C4—31no—
77C2×C32×C4—121no—
78C2×C3⋊Dic3—41no—
79C2×C3⋊Dic3—61no—
80C2×C3⋊Dic3—121no—
81C2×C3⋊Dic3—123no—
82C2×C3⋊Dic3—241no—
83C2×C3⋊Dic3—243no—
84C2×C3×Dic3—31no—
85C2×C3×Dic3—43no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—123no—
88C2×C3×Dic3S411yescharacteristic
89C3×Dic6—32no—
90C3×Dic6—61no—
91C3×SL(2,𝔽3)—21no—
92C3×SL(2,𝔽3)—23no—
93C3×SL(2,𝔽3)C4×D313yes—
94C3×SL(2,𝔽3)C2×Dic311yescharacteristic
95C3⋊(C3⋊C8)—61no—
96C3×C3⋊C8—32no—
97C4×Q16—271noSylow
98C2×C33—41no—
99C2×C33—41no—
100C2×C33—41no—
101C2×C3×Q8—21no—
102C2×C3×Q8—91no—
103C2×C3×Q8D3211yescharacteristic
104C2×C3×Q8C2×C3⋊D311yescharacteristic
105C2×Dic6—31no—
106C2×SL(2,𝔽3)—23no—
107C2×SL(2,𝔽3)D3213yes—
108#226—36no—
109C3×Q16—92no—
110C3×Q16—92no—
111C2×C3×C8—91no—
112C3×C4⋊C4—91no—
113C3×C42—91no—
114#216—32no—
115#216—32no—
116C4⋊Dic3—31no—
117C4⋊Dic3—91no—
118C4⋊Dic3—91no—
119C4⋊Dic3—181no—
120C4⋊Dic3—181no—
121Dic3⋊C4—181no—
122C4×Dic3—31no—
123C4×Dic3—181no—
124C4×Dic3—181no—
125C4×Dic3—363no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—91no—
128C2×C3⋊C8—91no—
129C2×C3⋊C8—181no—
130C22×C32—41no—
131C22×C32—43no—
132C22×C32—83no—
133C22×C32—81no—
134C22×C32C2×S411yescharacteristic
135C32×C4—31no—
136C32×C4—31no—
137C32×C4—122no—
138C3⋊Dic3—42no—
139C3⋊Dic3—122no—
140C3⋊Dic3—126no—
141C3⋊Dic3—121no—
142C3⋊Dic3—246no—
143C3⋊Dic3—242no—
144C3×Dic3—46no—
145C3×Dic3—62no—
146C3×Dic3—61no—
147C3×Dic3—126no—
148C3×Dic3#22612yescomplemented
149C2×Q16—91no—
150C4×Q8—31no—
151C4×Q8—271no—
152#107—271no—
153#103—271no—
154#103—271no—
155C4×C8—271no—
156C33—41noSylow
157C3×Q8—21no—
158C3×Q8—31no—
159C3×Q8—31no—
160C3×Q8—61no—
161C3×Q8—92no—
162C3×Q8—181no—
163C3×Q8D3×Dic311yescharacteristic
164C3×Q8C4×C3⋊D311yescharacteristic
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—61no—
169C2×C3×C4—91no—
170C2×C3×C4—123no—
171C2×C3×C4—181no—
172C2×Dic3—31no—
173C2×Dic3—61no—
174C2×Dic3—123no—
175C2×Dic3—181no—
176C2×Dic3—181no—
177C2×Dic3—363no—
178C2×Dic3—361no—
179C2×Dic3—723no—
180C2×Dic3C3⋊S411yescharacteristic
181Dic6—32no—
182Dic6—61no—
183SL(2,𝔽3)—23no—
184SL(2,𝔽3)D3×Dic313yes—
185C3×C8—92no—
186C3⋊C8—32no—
187C3⋊C8—181no—
188C3⋊C8—181no—
189C3⋊C8—361no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—43no—
194C2×C32—43no—
195C2×C32—43no—
196C2×C32—83no—
197C2×C32—83no—
198C2×C32—83no—
199C2×C32—81no—
200C2×C32—81no—
201C2×C32—81no—
202C2×C32C4×S411yescharacteristic
203C2×C32C2×#22611yescharacteristic
204C2×C32#98511yescharacteristic
205C2×Q8—91no—
206C2×Q8D3×C3⋊D311yescharacteristic
207Q16—92no—
208Q16—92no—
209C2×C8—91no—
210C2×C8—271no—
211C4⋊C4—91no—
212C4⋊C4—271no—
213C4⋊C4—541no—
214C42—91no—
215C42—541no—
216C22×C3—21no—
217C22×C3—43no—
218C22×C3—83no—
219C22×C3D3×S411yescharacteristic
220C22×C3C2×C3⋊S411yescharacteristic
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—32no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—126no—
229C3×C4—182no—
230C3×C4—181no—
231Dic3—61no—
232Dic3—62no—
233Dic3—126no—
234Dic3—366no—
235Dic3—361no—
236Dic3—361no—
237Dic3—721no—
238Dic3—726no—
239Dic3C3⋊#22612yesdirect product factor
240C32—41no—
241C32—43no—
242C32—83no—
243C32—81no—
244C32C4×#22611yescharacteristic, complemented
245Q8—31no—
246Q8—92no—
247Q8—181no—
248Q8Dic3×C3⋊D311yescharacteristic
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—91no—
252C2×C4—181no—
253C2×C4—541no—
254C8—92no—
255C8—541no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—43no—
260C2×C3—43no—
261C2×C3—43no—
262C2×C3—83no—
263C2×C3—83no—
264C2×C3—83no—
265C2×C3#6419811yescharacteristic
266C2×C3D3×#22611yescharacteristic
267C2×C3Dic3×S411yescharacteristic
268C2×C3C4×C3⋊S411yescharacteristic
269C2×C3C2×C3⋊#22611yescharacteristic
270C2×C3#6426611yescharacteristic
271C22D3×C3⋊S411yescharacteristic
272C4—32no—
273C4—31no—
274C4—31no—
275C4—181no—
276C4—182no—
277C4—1081no—
278C3—21no—
279C3—43no—
280C3—83no—
281C3Dic3×#22611yescharacteristic, complemented
282C3C4×C3⋊#22611yescharacteristic, complemented
283C2#1172419311yescharacteristic
284C2D3×C3⋊#22611yescharacteristic
285C2Dic3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radical, cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. C3×C3⋊#226C4semidirect
38. C3⋊#226Dic3direct (N × H)
148. C3×Dic3#226semidirect
239. Dic3C3⋊#226direct (N × H)
244. C32C4×#226semidirect
281. C3Dic3×#226semidirect
282. C3C4×C3⋊#226semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#2262
C3×Dic3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C4×C3⋊#2263
Dic3×#2263
Dic3×#2163
Dic3×C3⋊Dic34