Finite Group Databasebeta

← #10593664⋊C3 #13146936

Subgroups — #10593664⋊C3 #13146936

Order 1728.

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

normal subgroups only

This group has 304 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
2#64265⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊GU(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
4#10593664C311yesmaximal, characteristic, complemented
5C3×#5246—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)—21no—
8(C3⋊#226)⋊C3—21no—
9#3754⋊C3—41nomaximal
10C32×2−1+4D311yescharacteristic
11C3×2−1+4⋊C3—21no—
12C3×2−1+4⋊C3C2×C311yescharacteristic
13#64265C2×C311yescharacteristic, complemented
14C3×#951—31no—
15C3×#951—31no—
16C3×#944—31no—
17C3⋊GU(2,𝔽3)C2×C311yescharacteristic, complemented
18C3×#837—31no—
19C3×#837—31no—
20C3×#836—31no—
21D4×H(3,𝔽3)—41no—
22#634⋊C3—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5430—91no—
26C3×#456—91no—
27#5246—31no—
28C32×#41—31no—
29C32×#41—61no—
30C32×#41C2×D311yescharacteristic
31C2×C32×Q8—21no—
32C2×C32×Q8—31no—
33C2×C3×Dic6—31no—
34C3×#41⋊C3—21no—
35C3×#41⋊C3C22×C311yescharacteristic
36C2×C3×SL(2,𝔽3)—22no—
37C2×C3×SL(2,𝔽3)—21no—
38C3⋊#226—21no—
39#3754—41no—
40C3×#216—61no—
41C3×#216—61no—
42C3×#214—61no—
43C3×#214—61no—
44C3×Dic3⋊C4—62no—
45C3×C4×Dic3—31no—
46C3×#208—31no—
47C3×#208—31no—
48C22×H(3,𝔽3)—81no—
49C4×H(3,𝔽3)—41no—
50(C3⋊Dic3)⋊C3—81no—
51C3×2−1+4—31no—
52C3×2−1+4C3×D311yescharacteristic
532−1+4⋊C3—31no—
542−1+4⋊C3—61no—
55#984—91no—
56C3×#137—91no—
57C3×#137—91no—
58C3×C4⋊Q8—91no—
59#951—31no—
60#951—31no—
61#944—31no—
62GU(2,𝔽3)—91no—
63C3×#104—91no—
64C3×#104—91no—
65C3×#101—91no—
66#837—31no—
67#837—31no—
68#836—31no—
69C32×Q8—31no—
70C32×Q8—61no—
71C32×Q8#6711yescharacteristic
72C32×D4—31no—
73C32×D4—41no—
74C32×D4—61no—
75C32×D4—81no—
76C32×D4S411yescharacteristic
77C2×C32×C4—31no—
78C2×C32×C4—32no—
79C2×C32×C4—61no—
80#634—41no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C3×Dic6—61no—
84C3×Dic6—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#456—91noSylow
91C2×H(3,𝔽3)—41no—
92C2×H(3,𝔽3)—161no—
93C3×#41—31no—
94C3×#41—31no—
95C3×#41—31no—
96C3×#41—61no—
97C3×#41—61no—
98C3×#41—62no—
99C3×#41—61no—
100C3×#41C2×C3×D311yescharacteristic
101C2×C3×Q8—21no—
102C2×C3×Q8—31no—
103C2×C3×Q8—31no—
104C2×C3×Q8—61no—
105C2×C3×Q8—61no—
106C2×C3×Q8—91no—
107C2×Dic6—31no—
108#41⋊C3—31no—
109#41⋊C3—61no—
110C2×SL(2,𝔽3)—61no—
111C2×SL(2,𝔽3)—62no—
112#226—181no—
113C3×Q16—181no—
114C3×Q16—181no—
115C3×SD16—181no—
116C3×SD16—181no—
117C3×M16—91no—
118C3×M16—91no—
119C3×C4⋊C4—182no—
120C3×C42—91no—
121#216—61no—
122#216—61no—
123#214—61no—
124#214—61no—
125#214—361no—
126Dic3⋊C4—62no—
127C4×Dic3—31no—
128#208—31no—
129#208—31no—
130C22×C32—21no—
131C22×C32—31no—
132C22×C32—81no—
133C22×C32—82no—
134C32×C4—31no—
135C32×C4—41no—
136C32×C4—61no—
137C32×C4—81no—
138C32×C4C2×S411yescharacteristic
139C3⋊Dic3—81no—
140C3×Dic3—62no—
141C3×Dic3—121no—
1422−1+4(C3⋊D3)⋊C311yescharacteristic
143#137—91no—
144#137—91no—
145C4⋊Q8—91no—
146#104—91no—
147#104—91no—
148#101—91no—
149H(3,𝔽3)—41noSylow
150C3×Q8—31no—
151C3×Q8—31no—
152C3×Q8—31no—
153C3×Q8—61no—
154C3×Q8—61no—
155C3×Q8—61no—
156C3×Q8—62no—
157C3×Q8—181no—
158C3×Q8—181no—
159C3×Q8C3×#6711yescharacteristic
160C3×D4—31no—
161C3×D4—31no—
162C3×D4—31no—
163C3×D4—61no—
164C3×D4—61no—
165C3×D4—61no—
166C3×D4—62no—
167C3×D4—121no—
168C3×D4—241no—
169C3×D4C3×S411yescharacteristic
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—61no—
175C2×C3×C4—62no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183Dic6—61no—
184Dic6—61no—
185Dic6—361no—
186SL(2,𝔽3)—31no—
187SL(2,𝔽3)—61no—
188C3×C8—181no—
189C3×C8—181no—
190C3⋊C8—61no—
191C3⋊C8—61no—
192C3⋊C8—361no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—61no—
196C2×C32—81no—
197C2×C32—161no—
198C2×C32—162no—
199C2×C32GL(2,ℤ/4)11yescharacteristic
200#41—31no—
201#41—61no—
202#41C2×(C3⋊D3)⋊C311yescharacteristic
203C2×Q8—21no—
204C2×Q8—31no—
205C2×Q8—91no—
206Q16—181no—
207Q16—181no—
208SD16—181no—
209SD16—181no—
210M16—91no—
211M16—91no—
212C4⋊C4—182no—
213C42—91no—
214C22×C3—21no—
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—241no—
220C22×C3—242no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—61no—
225C3×C4—62no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—121no—
229C3×C4—182no—
230C3×C4—241no—
231C3×C4—361no—
232C3×C4C2×C3×S411yescharacteristic
233Dic3—62no—
234Dic3—121no—
235Dic3—721no—
236C32—41no—
237C32—81no—
238C32#543011yescharacteristic, complemented
239Q8—31no—
240Q8—61no—
241Q8—181no—
242Q8—181no—
243Q8#638⋊C311yescharacteristic
244D4—31no—
245D4—61no—
246D4(C3⋊S4)⋊C311yescharacteristic
247C2×C4—31no—
248C2×C4—32no—
249C2×C4—61no—
250C2×C4—181no—
251C2×C4—181no—
252C8—181no—
253C8—181no—
254C2×C3—31no—
255C2×C3—41no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3—121no—
259C2×C3—121no—
260C2×C3—121no—
261C2×C3—241no—
262C2×C3—481no—
263C2×C3—482no—
264C2×C3C3×GL(2,ℤ/4)11yescharacteristic
265C22—21no—
266C22—31no—
267C4—31no—
268C4—61no—
269C4—182no—
270C4—361no—
271C4C2×(C3⋊S4)⋊C311yescharacteristic
272C3—31no—
273C3—121no—
274C3—241no—
275C3C3×#543011yescharacteristic
276C2—41no—
277C2—61no—
278C2#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)
prime radical2−1+4⋊H(3,𝔽3)C2
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
2. #64265⋊C3C2semidirect
3. (C3⋊GU(2,𝔽3))⋊C3C2semidirect
4. #10593664C3semidirect
13. #64265C2×C3semidirect
17. C3⋊GU(2,𝔽3)C2×C3semidirect
238. C32#5430semidirect

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
#64265⋊C32
(C3⋊GU(2,𝔽3))⋊C32
#105936643
C3×#52463
#3754⋊C34