Finite Group Databasebeta

← #13138862

Subgroups — #13138862

Order 1728.

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

normal subgroups only

This group has 415 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
3C8×#1084C211yesmaximal, characteristic, complemented
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#11720305C212yesmaximal, complemented
6#10590917—31nomaximal
7#10590915—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×C8×Dic3—31no—
25C3×C8×Dic3—31no—
26C3×C8×Dic3—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#6180D411yescharacteristic
38#6180D411yescharacteristic
39(C3⋊(C3⋊C8))⋊C3—22no—
40#5196—91no—
41#4895—91no—
42#4700—91no—
43#3828C2×D312yes—
44C3×#235—32no—
45C2×C32×C8—21no—
46C2×C32×C8C2×D311yescharacteristic
47C2×C32×C8C2×D311yescharacteristic
48#3748C2×D312yes—
49#3746Dic311yescharacteristic, complemented
50#3745Dic311yescharacteristic, complemented
51#3744Dic312yescomplemented
52C3×C4×Dic3—31no—
53C3×C4×Dic3—31no—
54C3×C4×Dic3—61no—
55C3×#208—32no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—61no—
59C3×Dic12—31no—
60C3×D24—31no—
61C3×#204—32no—
62#3712—62no—
63#3711—62no—
64#3710—62no—
65D3×C3⋊C8—62no—
66C22×H(3,𝔽3)#4111yescharacteristic
67C2×(C3⋊D3)⋊C3—22no—
68C4×H(3,𝔽3)C22×C411yescharacteristic
69C4×H(3,𝔽3)C2×D411yescharacteristic
70#1084—22no—
71(C3⋊Dic3)⋊C3—22no—
72C3×#135—91no—
73#948—92no—
74#907—92no—
75#904—31no—
76#904—91no—
77#901—92no—
78C3×#108—91no—
79C3×C4×C8—91no—
80#837—92no—
81#825—92no—
82#819—31no—
83#819—91no—
84C8×Dic3—31no—
85C8×Dic3—31no—
86C8×Dic3—61no—
87#805—92no—
88C2×C32×C4—21no—
89C2×C32×C4C22×D311yescharacteristic
90C2×C32×C4C22×D311yescharacteristic
91#638—22no—
92C4⋊(C3⋊D3)C2×Dic312yes—
93C4×C3⋊D3—22no—
94#634C2×Dic312yes—
95C3×#67—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C3×D12—32no—
100C3×C4×D3—62no—
101C3×Dic6—32no—
102C32×C8—21no—
103C32×C8—21no—
104C32×C8C4×D311yescharacteristic
105C32×C8C4×D311yescharacteristic
106C32×C8C2×Dic311yescharacteristic
107C32×C8C2×Dic311yescharacteristic
108C3⋊(C3⋊C8)—22no—
109C3×C3⋊C8—31no—
110C3×C3⋊C8—31no—
111C3×C3⋊C8—31no—
112C3×C3⋊C8—31no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115C3×C3⋊C8—62no—
116#443—271noSylow
117C2×H(3,𝔽3)—21no—
118C2×H(3,𝔽3)C4×D411yescharacteristic
119(C3⋊D3)⋊C3—42no—
120C3×#41—92no—
121#235—32no—
122#235—92no—
123C3×Q16—91no—
124C3×SD16—92no—
125C3×D8—91no—
126C3×M16—92no—
127C2×C3×C8—31no—
128C2×C3×C8—31no—
129C2×C3×C8—61no—
130C2×C3×C8—91no—
131C2×C3×C8D3211yescharacteristic
132C3×C42—91no—
133C4×Dic3—31no—
134C4×Dic3—31no—
135C4×Dic3—61no—
136#208—32no—
137#208—92no—
138#208—182no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—61no—
142C2×C3⋊C8—182no—
143Dic12—31no—
144Dic12—91no—
145D24—31no—
146D24—91no—
147#204—32no—
148#204—92no—
149ΓL(1,𝔽25)—182no—
150ΓL(1,𝔽25)—182no—
151D3×C8—182no—
152D3×C8—182no—
153C22×C32—21no—
154C22×C32#23711yescharacteristic
155C22×C32#23911yescharacteristic
156C2×C3⋊D3—22no—
157C2×C3×D3—62no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4C2×C4×D311yescharacteristic
161C32×C4D3×D411yescharacteristic
162C32×C4D3×D411yescharacteristic
163C32×C4C22×Dic311yescharacteristic
164C3⋊Dic3—22no—
165C3×Dic3—64no—
166C3×Dic3—62no—
167C3×Dic3—62no—
168C3×Dic3—62no—
169#135—91no—
170#131—272no—
171#108—91no—
172#104—272no—
173C4×C8—91no—
174H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
175C3×Q8—92no—
176C3×D4—92no—
177C3×D4—182no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—61no—
181C2×C3×C4—182no—
182C2×C3×C4—181no—
183C2×C3×C4C2×D3211yescharacteristic
184#67—62no—
185#67—182no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—62no—
189D12—32no—
190D12—92no—
191C4×D3—62no—
192C4×D3—182no—
193Dic6—32no—
194Dic6—92no—
195C3×C8—31no—
196C3×C8—31no—
197C3×C8—31no—
198C3×C8—31no—
199C3×C8—61no—
200C3×C8—61no—
201C3×C8—91no—
202C3×C8—91no—
203C3×C8—182no—
204C3×C8D3×Dic311yescharacteristic
205C3×C8D3×Dic311yescharacteristic
206C3⋊C8—31no—
207C3⋊C8—31no—
208C3⋊C8—31no—
209C3⋊C8—31no—
210C3⋊C8—61no—
211C3⋊C8—61no—
212C3⋊C8—62no—
213C3⋊C8—182no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32#89311yescharacteristic
219C2×C32D4×Dic311yescharacteristic
220C3⋊D3—42no—
221C3×D3—122no—
222#41—92no—
223Q16—91no—
224SD16—92no—
225D8—91no—
226M16—92no—
227M16—542no—
228C2×C8—91no—
229C2×C8—542no—
230C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
231C42—91no—
232C22×C3—31no—
233C22×C3—31no—
234C22×C3—61no—
235C22×C3—182no—
236C22×C3#379611yescharacteristic
237C2×D3—62no—
238C2×D3—182no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—61no—
244C3×C4—61no—
245C3×C4—182no—
246C3×C4—182no—
247C3×C4#380211yescharacteristic
248C3×C4C2×D3×Dic311yescharacteristic
249Dic3—62no—
250Dic3—62no—
251Dic3—64no—
252Dic3—62no—
253Dic3—182no—
254C32—21no—
255C32#489511yescharacteristic, complemented
256C32#519611yescharacteristic, complemented
257Q8—92no—
258D4—92no—
259D4—182no—
260C2×C4—181no—
261C2×C4—182no—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C8—91no—
264C8—91no—
265C8—182no—
266C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
267C8((C3⋊D3)⋊C3)⋊C411yescharacteristic
268C2×C3—21no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—61no—
272C2×C3—61no—
273C2×C3—62no—
274C2×C3—61no—
275C2×C3—362no—
276C2×C3#6389811yescharacteristic
277D3—122no—
278D3—362no—
279C22—182no—
280C22#8864911yescharacteristic
281C4—182no—
282C4—182no—
283C4#8865511yescharacteristic
284C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
285C3—31no—
286C3—31no—
287C3—61no—
288C3#1059091911yescharacteristic
289C2—21no—
290C2—362no—
291C2#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)#443
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. C8×#1084C2semidirect
4. #63646⋊C3C2semidirect
5. #11720305C2semidirect
13. #3745⋊C3C4semidirect
14. #3744⋊C3C4semidirect
15. #3746⋊C3C4semidirect
22. #63646D3semidirect
49. #3746Dic3semidirect
50. #3745Dic3semidirect
51. #3744Dic3semidirect
174. H(3,𝔽3)#443semidirect
255. C32#4895semidirect
256. C32#5196semidirect

Maximal subgroups

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

subgroupindex
#64113⋊C32
#117210552
C8×#10842
#63646⋊C32
#117203052
#105909173
#105909153