Finite Group Databasebeta

← #10593665⋊C3 #13146937

Subgroups — #10593665⋊C3 #13146937

Order 1728.

315 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 33 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64267⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊GU(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
4#10593665C311yesmaximal, characteristic, complemented
5C3×#5245—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)—21no—
8(C3⋊GL(2,𝔽3))⋊C3—21no—
9#3753⋊C3—41nomaximal
10C32×2−1+4D311yescharacteristic, complemented
11C3×2−1+4⋊C3—21no—
12C3×2−1+4⋊C3C2×C311yescharacteristic, complemented
13#64267C2×C311yescharacteristic, complemented
14C3×#949—31no—
15C3×#949—31no—
16C3×#947—31no—
17C3⋊GU(2,𝔽3)C2×C311yescharacteristic, complemented
18C3×#837—31no—
19C3×#837—31no—
20C3×#836—31no—
21D4×H(3,𝔽3)—41no—
22C4⋊((C3⋊D3)⋊C3)—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5431—91no—
26C3×#454—91no—
27#5245—31no—
28C32×#41—31no—
29C32×#41—61no—
30C32×#41C2×D311yescharacteristic
31C2×C32×Q8—21no—
32C2×C32×Q8—31no—
33C2×C3×D12—31no—
34C3×#41⋊C3—21no—
35C3×#41⋊C3C22×C311yescharacteristic
36C2×C3×SL(2,𝔽3)—22no—
37C2×C3×SL(2,𝔽3)—21no—
38C3⋊GL(2,𝔽3)—21no—
39#3753—41no—
40C3×#215—61no—
41C3×#215—61no—
42C3×#213—61no—
43C3×#213—61no—
44C3×(C2×D3)⋊C4—62no—
45C3×C4×Dic3—31no—
46C3×#208—31no—
47C3×#208—31no—
48C22×H(3,𝔽3)—81no—
49C2×(C3⋊D3)⋊C3—81no—
50C4×H(3,𝔽3)—41no—
51C3×2−1+4—31no—
52C3×2−1+4C3×D311yescharacteristic
532−1+4⋊C3—31no—
542−1+4⋊C3—61no—
55#986—91no—
56C3×AGL(1,ℤ/8)—91no—
57C3×AGL(1,ℤ/8)—91no—
58C3×#124—91no—
59#949—31no—
60#949—31no—
61#947—31no—
62GU(2,𝔽3)—91no—
63C3×#104—91no—
64C3×#104—91no—
65C3×#101—91no—
66#837—31no—
67#837—31no—
68#836—31no—
69C22×C3×D3—61no—
70C32×Q8—31no—
71C32×Q8—61no—
72C32×Q8#6711yescharacteristic
73C32×D4—31no—
74C32×D4—41no—
75C32×D4—61no—
76C32×D4—81no—
77C32×D4S411yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—32no—
80C2×C32×C4—61no—
81C4⋊(C3⋊D3)—41no—
82C2×C3×Dic3—61no—
83C3×D12—61no—
84C3×D12—61no—
85C3×SL(2,𝔽3)—21no—
86C3×SL(2,𝔽3)C3×D411yescharacteristic
87C3⋊(C3⋊C8)—41no—
88C3×C3⋊C8—61no—
89C3×C3⋊C8—61no—
90#454—91noSylow
91C2×H(3,𝔽3)—41no—
92C2×H(3,𝔽3)—161no—
93(C3⋊D3)⋊C3—161no—
94C3×#41—31no—
95C3×#41—31no—
96C3×#41—31no—
97C3×#41—61no—
98C3×#41—61no—
99C3×#41—62no—
100C3×#41—61no—
101C3×#41C2×C3×D311yescharacteristic
102C2×C3×Q8—21no—
103C2×C3×Q8—31no—
104C2×C3×Q8—31no—
105C2×C3×Q8—61no—
106C2×C3×Q8—61no—
107C2×C3×D4—91no—
108C2×D12—31no—
109#41⋊C3—31no—
110#41⋊C3—61no—
111C2×SL(2,𝔽3)—61no—
112C2×SL(2,𝔽3)—62no—
113GL(2,𝔽3)—181no—
114C3×SD16—181no—
115C3×SD16—181no—
116C3×D8—181no—
117C3×D8—181no—
118C3×M16—91no—
119C3×M16—91no—
120C3×C22⋊C4—182no—
121C3×C42—91no—
122#215—61no—
123#215—61no—
124#213—61no—
125#213—61no—
126#213—361no—
127(C2×D3)⋊C4—62no—
128C4×Dic3—31no—
129#208—31no—
130#208—31no—
131C22×C32—21no—
132C22×C32—31no—
133C22×C32—81no—
134C22×C32—82no—
135C2×C3⋊D3—81no—
136C2×C3×D3—122no—
137C2×C3×D3—121no—
138C32×C4—31no—
139C32×C4—41no—
140C32×C4—61no—
141C32×C4—81no—
142C32×C4C2×S411yescharacteristic
143C3×Dic3—121no—
1442−1+4(C3⋊D3)⋊C311yescharacteristic, complemented
145AGL(1,ℤ/8)—91no—
146AGL(1,ℤ/8)—91no—
147#124—91no—
148#104—91no—
149#104—91no—
150#101—91no—
151H(3,𝔽3)—41noSylow
152C23×C3—181no—
153C22×D3—61no—
154C3×Q8—31no—
155C3×Q8—31no—
156C3×Q8—31no—
157C3×Q8—61no—
158C3×Q8—61no—
159C3×Q8—61no—
160C3×Q8—62no—
161C3×Q8C3×#6711yescharacteristic
162C3×D4—31no—
163C3×D4—31no—
164C3×D4—31no—
165C3×D4—61no—
166C3×D4—61no—
167C3×D4—61no—
168C3×D4—62no—
169C3×D4—121no—
170C3×D4—181no—
171C3×D4—181no—
172C3×D4—241no—
173C3×D4C3×S411yescharacteristic
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—61no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—181no—
184C2×Dic3—61no—
185D12—61no—
186D12—61no—
187D12—361no—
188SL(2,𝔽3)—31no—
189SL(2,𝔽3)—61no—
190C3×C8—181no—
191C3×C8—181no—
192C3⋊C8—61no—
193C3⋊C8—61no—
194C3⋊C8—361no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—61no—
198C2×C32—81no—
199C2×C32—161no—
200C2×C32—162no—
201C2×C32GL(2,ℤ/4)11yescharacteristic
202C3⋊D3—161no—
203C3×D3—241no—
204#41—31no—
205#41—61no—
206#41C2×(C3⋊D3)⋊C311yescharacteristic
207C2×Q8—21no—
208C2×Q8—31no—
209C2×D4—91no—
210SD16—181no—
211SD16—181no—
212D8—181no—
213D8—181no—
214M16—91no—
215M16—91no—
216C22⋊C4—182no—
217C42—91no—
218C22×C3—21no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3—241no—
224C22×C3—242no—
225C22×C3—361no—
226C22×C3—362no—
227C2×D3—122no—
228C2×D3—121no—
229C2×D3—721no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4—62no—
235C3×C4—61no—
236C3×C4—61no—
237C3×C4—121no—
238C3×C4—241no—
239C3×C4—361no—
240C3×C4C2×C3×S411yescharacteristic
241Dic3—121no—
242C32—41no—
243C32—81no—
244C32#543111yescharacteristic, complemented
245C23—181no—
246Q8—31no—
247Q8—61no—
248Q8#638⋊C311yescharacteristic
249D4—31no—
250D4—61no—
251D4—181no—
252D4—181no—
253D4(C3⋊S4)⋊C311yescharacteristic
254C2×C4—31no—
255C2×C4—32no—
256C2×C4—61no—
257C2×C4—181no—
258C8—181no—
259C8—181no—
260C2×C3—31no—
261C2×C3—41no—
262C2×C3—61no—
263C2×C3—61no—
264C2×C3—121no—
265C2×C3—121no—
266C2×C3—121no—
267C2×C3—241no—
268C2×C3—481no—
269C2×C3—482no—
270C2×C3—721no—
271C2×C3C3×GL(2,ℤ/4)11yescharacteristic
272D3—241no—
273D3—1441no—
274C22—21no—
275C22—31no—
276C22—362no—
277C22—361no—
278C4—31no—
279C4—61no—
280C4—361no—
281C4C2×(C3⋊S4)⋊C311yescharacteristic
282C3—31no—
283C3—121no—
284C3—241no—
285C3C3×#543111yescharacteristic
286C2—41no—
287C2—61no—
288C2—721no—
289C2#64269⋊C311yescharacteristic
subgroupG/H
centerC2#64269⋊C3
commutator, cosocleC3×#41⋊C3C22×C3
FrattiniC3×C4C2×C3×S4
FittingC32×2−1+4D3
socleC2×C3C3×GL(2,ℤ/4)
hypercenterD4(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2semidirect
2. #64267⋊C3C2semidirect
3. (C3⋊GU(2,𝔽3))⋊C3C2semidirect
4. #10593665C3semidirect
10. C32×2−1+4D3semidirect
12. C3×2−1+4⋊C3C2×C3semidirect
13. #64267C2×C3semidirect
17. C3⋊GU(2,𝔽3)C2×C3semidirect
144. 2−1+4(C3⋊D3)⋊C3semidirect
244. C32#5431semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#64267⋊C32
(C3⋊GU(2,𝔽3))⋊C32
#105936653
C3×#52453
#3753⋊C34