Finite Group Databasebeta

← #10592451⋊C3 #13141698

Subgroups — #10592451⋊C3 #13141698

Order 1728.

535 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#11722627C211yesmaximal, characteristic
2#64153⋊C3C211yesmaximal, characteristic, complemented
3#64118⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
5#63652⋊C3C212yesmaximal, complemented
6#10592451C311yesmaximal, characteristic, complemented
7C3×#4752—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9#88722—22no—
10Q8×(C3⋊D3)⋊C3—22no—
11#3828⋊C3—22no—
12#3828⋊C3C2212yes—
13C2×#634⋊C3C2211yescharacteristic
14M16×H(3,𝔽3)C2212yes—
15#3750⋊C3—22no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3744⋊C3—22no—
18#3746⋊C3—22no—
19#64364C2×C311yescharacteristic
20C3×#1007—31no—
21#64153C2×C311yescharacteristic, complemented
22#64118C2×C312yescomplemented
23C3×#944—31no—
24C3×#909—32no—
25C32×#101—21no—
26C32×#101D311yescharacteristic, complemented
27C32×#101C2×C311yescharacteristic, complemented
28#63652C2×C312yescomplemented
29C3×#825—32no—
30Q8×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)D412yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#638⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)D412yes—
38C4×(C3⋊D3)⋊C3—24no—
39C4×(C3⋊D3)⋊C3—22no—
40#634⋊C3—22no—
41#634⋊C3—22no—
42#634⋊C3D412yes—
43C8×H(3,𝔽3)—22no—
44C3×#456—91no—
45#4752—31no—
46#4752—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C22×C311yescharacteristic
50#3832—22no—
51Q8×C3⋊D3—22no—
52#3828—22no—
53#3828C22×C312yes—
54C2×#634C22×C311yescharacteristic
55C3×#239—62no—
56C3×D3×Q8—62no—
57C3×#235—32no—
58C3×#235—62no—
59C2×C3×Dic6—31no—
60C32×M16—22no—
61C32×M16C2×D312yes—
62C32×M16C22×C312yes—
63#3750—22no—
64C4×C3⋊Dic3C22×C311yescharacteristic
65#3746—22no—
66#3744—22no—
67C3×Dic3⋊C4—62no—
68C3×C4×Dic3—31no—
69C3×Dic12—62no—
70C3×#204—62no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×(C3⋊D3)⋊C3—22no—
73C4×H(3,𝔽3)—22no—
74C4×H(3,𝔽3)C2×D412yes—
75(C3⋊Dic3)⋊C3—22no—
76(C3⋊Dic3)⋊C3—22no—
77(C3⋊Dic3)⋊C3—41no—
78C3×2−1+4—91no—
79#1007—31no—
80#1007—91no—
81C3×#137—92no—
82C3×C4⋊Q8—91no—
83#944—31no—
84#944—91no—
85#909—32no—
86#909—92no—
87C3×#104—92no—
88C3×#101—31no—
89C3×#101—31no—
90C3×#101—61no—
91C3×#101C3×D311yescharacteristic
92#825—32no—
93#825—92no—
94C32×Q8—22no—
95C32×Q8—22no—
96C32×Q8—42no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4D1212yes—
100C2×C32×C4C3×D412yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C23×C311yescharacteristic
103#638—22no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C4⋊(C3⋊D3)—22no—
107C4⋊(C3⋊D3)C3×D412yes—
108C4×C3⋊D3—24no—
109C4×C3⋊D3—22no—
110#634—22no—
111#634—22no—
112#634C3×D412yes—
113C3×#67—62no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C3×D12—32no—
117C3×D12—62no—
118C3×C4×D3—64no—
119C3×C4×D3—62no—
120C3×Dic6—32no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C32×C8—22no—
124C32×C8—22no—
125C32×C8—42no—
126#456—91noSylow
127C2×H(3,𝔽3)—21no—
128C2×H(3,𝔽3)#12011yescharacteristic
129(C3⋊D3)⋊C3—42no—
130C3×#41—92no—
131C3×#41—182no—
132C3×#41—182no—
133C2×C3×Q8—31no—
134C2×C3×Q8—31no—
135C2×C3×Q8—61no—
136C2×C3×Q8—91no—
137C2×C3×Q8—182no—
138C2×C3×Q8C2×C3×D311yescharacteristic
139#239—62no—
140#239—182no—
141D3×Q8—62no—
142D3×Q8—182no—
143#235—32no—
144#235—62no—
145#235—92no—
146#235—182no—
147C2×Dic6—31no—
148C2×Dic6—91no—
149C3×Q16—182no—
150C3×SD16—182no—
151C3×M16—32no—
152C3×M16—32no—
153C3×M16—62no—
154C3×M16C2×C3×D312yes—
155C3×C4⋊C4—182no—
156C3×C42—91no—
157Dic3⋊C4—62no—
158Dic3⋊C4—182no—
159C4×Dic3—31no—
160C4×Dic3—91no—
161Dic12—62no—
162Dic12—182no—
163#204—62no—
164#204—182no—
165C22×C32—21no—
166C22×C32C2×D1211yescharacteristic
167C22×C32C2×C3×D411yescharacteristic
168C2×C3⋊D3—22no—
169C2×C3×D3—62no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—42no—
174C32×C4D3×D412yes—
175C32×C4C2×C3×D412yes—
176C3⋊Dic3—22no—
177C3⋊Dic3—22no—
178C3⋊Dic3—41no—
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—121no—
1822−1+4—91no—
183#137—92no—
184C4⋊Q8—91no—
185#104—92no—
186#101(C3⋊D3)⋊C311yescharacteristic, complemented
187H(3,𝔽3)#45611yescharacteristic, complemented, Sylow
188C3×Q8—22no—
189C3×Q8—62no—
190C3×Q8—62no—
191C3×Q8—92no—
192C3×Q8—122no—
193C3×Q8—182no—
194C3×Q8—182no—
195C3×D4—92no—
196C3×D4—182no—
197C3×D4—182no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—182no—
205C2×C3×C4—184no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4C3×D1212yes—
209C2×C3×C4C22×C3×D311yescharacteristic
210#67—62no—
211#67—182no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216D12—32no—
217D12—62no—
218D12—92no—
219D12—182no—
220C4×D3—64no—
221C4×D3—62no—
222C4×D3—182no—
223C4×D3—184no—
224Dic6—32no—
225Dic6—62no—
226Dic6—62no—
227Dic6—92no—
228Dic6—182no—
229Dic6—182no—
230C3×C8—22no—
231C3×C8—62no—
232C3×C8—62no—
233C3×C8—122no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—41no—
238C2×C32#88211yescharacteristic
239C2×C32C3×#12011yescharacteristic
240C3⋊D3—42no—
241C3×D3—122no—
242#41—92no—
243#41—182no—
244#41—182no—
245C2×Q8—91no—
246C2×Q8—182no—
247C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
248Q16—182no—
249SD16—182no—
250M16C2×(C3⋊D3)⋊C312yes—
251C4⋊C4—182no—
252C42—91no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—182no—
257C22×C3C2×C3×D1211yescharacteristic
258C2×D3—62no—
259C2×D3—182no—
260C3×C4—22no—
261C3×C4—32no—
262C3×C4—32no—
263C3×C4—62no—
264C3×C4—62no—
265C3×C4—62no—
266C3×C4—122no—
267C3×C4—182no—
268C3×C4—182no—
269C3×C4—361no—
270C3×C4C3×D3×D412yes—
271Dic3—62no—
272Dic3—62no—
273Dic3—121no—
274Dic3—182no—
275Dic3—182no—
276Dic3—361no—
277C32—21no—
278C32#475211yescharacteristic, complemented
279C32C3×#45611yescharacteristic, complemented
280Q8—22no—
281Q8—92no—
282Q8—182no—
283Q8—182no—
284D4—92no—
285D4—182no—
286D4—182no—
287C2×C4—184no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—181no—
291C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
292C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
293C8—22no—
294C2×C3—21no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—121no—
301C2×C3—362no—
302C2×C3C3×#88211yescharacteristic
303D3—122no—
304D3—362no—
305C22—182no—
306C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
307C4—22no—
308C4—182no—
309C4—182no—
310C4—361no—
311C4D4×(C3⋊D3)⋊C312yes—
312C3—31no—
313C3—31no—
314C3—61no—
315C3C3×#475211yescharacteristic
316C2—21no—
317C2—362no—
318C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#101C2
socleC2×C3C3×#882
prime radical#11722627C2
hypercenter#101(C3⋊D3)⋊C3
hypocenterC32C3×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. #64153⋊C3C2semidirect
3. #64118⋊C3C2semidirect
4. H(3,𝔽3)×#101C2semidirect
5. #63652⋊C3C2semidirect
6. #10592451C3semidirect
21. #64153C2×C3semidirect
22. #64118C2×C3semidirect
26. C32×#101D3semidirect
27. C32×#101C2×C3semidirect
28. #63652C2×C3semidirect
186. #101(C3⋊D3)⋊C3semidirect
187. H(3,𝔽3)#456semidirect
278. C32#4752semidirect
279. C32C3×#456semidirect

Maximal subgroups

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

subgroupindex
#117226272
#64153⋊C32
#64118⋊C32
H(3,𝔽3)×#1012
#63652⋊C32
#105924513
C3×#47523