Finite Group Databasebeta

← #13138863

Subgroups — #13138863

Order 1728.

404 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 83 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#64113⋊C3C211yesmaximal, characteristic
2#11721055C212yesmaximal, complemented
3#11720548C211yesmaximal, characteristic, complemented
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#11720305C212yesmaximal, complemented
6#10590918—31nomaximal
7#10590916—31nomaximal
8#3828⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#3748⋊C3C2212yes—
13#3745⋊C3C411yescharacteristic, complemented
14#3744⋊C3C412yescomplemented
15#3746⋊C3C411yescharacteristic, complemented
16#88430—22no—
17((C3⋊D3)⋊C3)⋊C8—22no—
18#64113D311yescharacteristic
19C3×#904—31no—
20#63817—32no—
21#63814—32no—
22#63646D311yescharacteristic, complemented
23C3×#819—31no—
24C3×#815—31no—
25C3×#815—31no—
26C3×#815—61no—
27#63570—32no—
28#63569—32no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—21no—
31#638⋊C3—22no—
32C4⋊((C3⋊D3)⋊C3)C2×C412yes—
33C4×(C3⋊D3)⋊C3—22no—
34#634⋊C3C2×C412yes—
35C8×H(3,𝔽3)C2×C411yescharacteristic
36C8×H(3,𝔽3)C2×C411yescharacteristic
37#6180D412yes—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5197—91no—
40#4894—91no—
41#4717—91no—
42#3828C2×D312yes—
43C3×#235—32no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47#3748C2×D312yes—
48#3746Dic311yescharacteristic, complemented
49#3745Dic311yescharacteristic, complemented
50#3744Dic312yescomplemented
51C3×C4×Dic3—31no—
52C3×C4×Dic3—31no—
53C3×C4×Dic3—61no—
54C3×#208—32no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—61no—
58C3×Dic12—31no—
59C3×D24—31no—
60C3×#204—32no—
61#3712—62no—
62#3711—62no—
63#3710—62no—
64D3×C3⋊C8—62no—
65C22×H(3,𝔽3)#4111yescharacteristic
66C2×(C3⋊D3)⋊C3—22no—
67C4×H(3,𝔽3)C22×C411yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69#1084—41no—
70(C3⋊Dic3)⋊C3—22no—
71C3×#135—91no—
72#948—92no—
73#907—92no—
74#904—31no—
75#904—91no—
76#901—92no—
77C3×#108—91no—
78C3×#97—91no—
79#837—92no—
80#825—92no—
81#819—31no—
82#819—91no—
83#815—31no—
84#815—31no—
85#815—61no—
86#805—92no—
87C2×C32×C4—21no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C22×D311yescharacteristic
90#638—22no—
91C4⋊(C3⋊D3)C2×Dic312yes—
92C4×C3⋊D3—22no—
93#634C2×Dic312yes—
94C3×#67—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C3×D12—32no—
99C3×C4×D3—62no—
100C3×Dic6—32no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8C4×D311yescharacteristic
104C32×C8C4×D311yescharacteristic
105C32×C8C2×Dic311yescharacteristic
106C32×C8C2×Dic311yescharacteristic
107C3⋊(C3⋊C8)—22no—
108C3×C3⋊C8—32no—
109C3×C3⋊C8—32no—
110C3×C3⋊C8—62no—
111C3×C3⋊C8—62no—
112#444—271noSylow
113C2×H(3,𝔽3)—21no—
114C2×H(3,𝔽3)C4×D411yescharacteristic
115(C3⋊D3)⋊C3—42no—
116C3×#41—92no—
117#235—32no—
118#235—92no—
119C3×Q16—91no—
120C3×SD16—92no—
121C3×D8—91no—
122C3×M16—92no—
123C2×C3×C8—31no—
124C2×C3×C8—31no—
125C2×C3×C8—61no—
126C2×C3×C8—91no—
127C2×C3×C8D3211yescharacteristic
128C3×C42—91no—
129C4×Dic3—31no—
130C4×Dic3—31no—
131C4×Dic3—61no—
132#208—32no—
133#208—92no—
134#208—182no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—61no—
138C2×C3⋊C8—182no—
139Dic12—31no—
140Dic12—91no—
141D24—31no—
142D24—91no—
143#204—32no—
144#204—92no—
145ΓL(1,𝔽25)—182no—
146ΓL(1,𝔽25)—182no—
147D3×C8—182no—
148D3×C8—182no—
149C22×C32—21no—
150C22×C32#23711yescharacteristic
151C22×C32#23911yescharacteristic
152C2×C3⋊D3—22no—
153C2×C3×D3—62no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4C2×C4×D311yescharacteristic
157C32×C4D3×D411yescharacteristic
158C32×C4D3×D411yescharacteristic
159C32×C4C22×Dic311yescharacteristic
160C3⋊Dic3—22no—
161C3×Dic3—62no—
162C3×Dic3—122no—
163C3×Dic3—121no—
164C3×Dic3—121no—
165#135—91no—
166#131—272no—
167#108—91no—
168#104—272no—
169#97—91no—
170H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
171C3×Q8—92no—
172C3×D4—92no—
173C3×D4—182no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—61no—
177C2×C3×C4—182no—
178C2×C3×C4—181no—
179C2×C3×C4C2×D3211yescharacteristic
180#67—62no—
181#67—182no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—62no—
185D12—32no—
186D12—92no—
187C4×D3—62no—
188C4×D3—182no—
189Dic6—32no—
190Dic6—92no—
191C3×C8—31no—
192C3×C8—31no—
193C3×C8—31no—
194C3×C8—31no—
195C3×C8—61no—
196C3×C8—61no—
197C3×C8—92no—
198C3×C8—182no—
199C3×C8D3×Dic311yescharacteristic
200C3×C8D3×Dic311yescharacteristic
201C3⋊C8—32no—
202C3⋊C8—32no—
203C3⋊C8—62no—
204C3⋊C8—62no—
205C3⋊C8—182no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32#89311yescharacteristic
211C2×C32D4×Dic311yescharacteristic
212C3⋊D3—42no—
213C3×D3—122no—
214#41—92no—
215Q16—91no—
216SD16—92no—
217D8—91no—
218M16—92no—
219M16—542no—
220C2×C8—91no—
221C2×C8—542no—
222C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C42—91no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—61no—
227C22×C3—182no—
228C22×C3#379611yescharacteristic
229C2×D3—62no—
230C2×D3—182no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—61no—
236C3×C4—61no—
237C3×C4—182no—
238C3×C4—361no—
239C3×C4#380211yescharacteristic
240C3×C4C2×D3×Dic311yescharacteristic
241Dic3—62no—
242Dic3—121no—
243Dic3—121no—
244Dic3—122no—
245Dic3—182no—
246C32—21no—
247C32#489411yescharacteristic, complemented
248C32#519711yescharacteristic, complemented
249Q8—92no—
250D4—92no—
251D4—182no—
252C2×C4—182no—
253C2×C4—181no—
254C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C8—92no—
256C8—182no—
257C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
258C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
259C2×C3—21no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—61no—
263C2×C3—61no—
264C2×C3—62no—
265C2×C3—61no—
266C2×C3—362no—
267C2×C3#6389811yescharacteristic
268D3—122no—
269D3—362no—
270C22—182no—
271C22#8864911yescharacteristic
272C4—182no—
273C4—361no—
274C4#8865511yescharacteristic
275C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
276C3—31no—
277C3—31no—
278C3—61no—
279C3#1059092011yescharacteristic
280C2—21no—
281C2—362no—
282C2#1172113611yescharacteristic
subgroupG/H
centerC4#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63898
prime radical#64113⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#444
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721055C2semidirect
3. #11720548C2semidirect
4. #63646⋊C3C2semidirect
5. #11720305C2semidirect
13. #3745⋊C3C4semidirect
14. #3744⋊C3C4semidirect
15. #3746⋊C3C4semidirect
22. #63646D3semidirect
48. #3746Dic3semidirect
49. #3745Dic3semidirect
50. #3744Dic3semidirect
170. H(3,𝔽3)#444semidirect
247. C32#4894semidirect
248. C32#5197semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
#117210552
#117205482
#63646⋊C32
#117203052
#105909183
#105909163