Finite Group Databasebeta

← #13139270

Subgroups — #13139270

Order 1728.

270 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C3×#63481C211yesmaximal, characteristic
2D9×Dic12C211yesmaximal, characteristic, complemented
3#11720568C211yesmaximal, characteristic, complemented
4C9×#829C211yesmaximal, characteristic, complemented
5#11720242C211yesmaximal, characteristic, complemented
6#11720233C211yesmaximal, characteristic, complemented
7#11720230C211yesmaximal, characteristic, complemented
8#10591042—31nomaximal
9#10588598—31nomaximal
10C3×#3701—21no—
11D9×Dic6—21no—
12C3×C9×Q16C2211yescharacteristic
13#88521C2211yescharacteristic
14C3×#3675—21no—
15C9×Dic12C2211yescharacteristic
16C3×D72C2211yescharacteristic
17C3×C8×D9C2211yescharacteristic
18#88428—21no—
19C3⋊(C9⋊C16)C2211yescharacteristic
20C9×C3⋊C16C2211yescharacteristic
21C3×#919—31no—
22D3×Dic12—31no—
23#63656—31no—
24C3×#829D311yescharacteristic
25#63548—31no—
26#63547—31no—
27#63543—31no—
28#63481D311yescharacteristic
29Q16×D9—31no—
30C9×Q32—31no—
31#63388—31no—
32#63388—61no—
33#63387—31no—
34#63359—31no—
35#63357—31no—
36C3×Q8×C9—21no—
37#6226—21no—
38C3×D36—21no—
39C3×D36—41no—
40C3×C4×D9—41no—
41C3×C4×D9D411yescharacteristic
42C9×Dic6—21no—
43Dic3×D9—41no—
44#6189—41no—
45C3×C8×C9C2311yescharacteristic
46C3×C9⋊C8D411yescharacteristic
47#5195—91no—
48#4918—91no—
49C3×#239—61no—
50D3×Dic6—61no—
51C32×Q16C2×D311yescharacteristic
52#3746—31no—
53C3×#215—61no—
54C3×Dic12C2×D311yescharacteristic
55C3×D24—31no—
56C3×D3×C8—31no—
57#3719—61no—
58#3701—21no—
59Q8×D9—61no—
60C3⋊(C3⋊C16)—31no—
61C3×C3⋊C16C2×D311yescharacteristic
62C9×Q16—21no—
63C9×Q16—31no—
64C9×Q16C2×D311yescharacteristic
65#3675—21no—
66#3674—61no—
67D72C2×D311yescharacteristic
68C8×D9C2×D311yescharacteristic
69Dic36—31no—
70Dic36—61no—
71C9×C16—31no—
72C9⋊C16—31no—
73C9⋊C16—61no—
74C2×C3×D9—41no—
75C2×C3×D9D811yescharacteristic
76C3×C4×C9—41no—
77C3×C4×C9C2×D411yescharacteristic
78C3⋊Dic9—41no—
79C9×Dic3—41no—
80C3×Dic9D811yescharacteristic
81C3×#135—91no—
82#919—31no—
83D3×Q16—91no—
84C2×Dic12—91no—
85C3×Q32—31no—
86#829—91no—
87#829—91no—
88#829—181no—
89#829D911yescharacteristic, complemented
90#828—91no—
91#827—91no—
92#827—91no—
93#812—91no—
94#800—91no—
95#798—91no—
96C32×Q8—21no—
97#634—61no—
98C3×D12—61no—
99C3×D12—121no—
100C3×C4×D3—31no—
101C3×C4×D3—121no—
102C3×Dic6—21no—
103#627—121no—
104D3×Dic3—121no—
105C32×C8C22×D311yescharacteristic
106C3×C3⋊C8—31no—
107Q8×C9—21no—
108Q8×C9—41no—
109Q8×C9—61no—
110D36—21no—
111D36—41no—
112C4×D9—41no—
113C4×D9—121no—
114C4×D9#6711yescharacteristic
115Dic18—61no—
116Dic18—121no—
117Dic18—121no—
118C8×C9—21no—
119C8×C9C22×D311yescharacteristic
120C9⋊C8#6711yescharacteristic
121#510—271noSylow
122C2×C3×C9C2×D811yescharacteristic
123C3×D9—21no—
124C3×D9—81no—
125C3×#41—181no—
126#239—61no—
127D3×Q8—181no—
128C2×Dic6—181no—
129C3×Q16—21no—
130C3×Q16—31no—
131C3×Q16C2×D911yescharacteristic
132C3×Q16D3211yescharacteristic
133C3×SD16—181no—
134C3×D8—91no—
135C2×C3×C8—91no—
136#216—181no—
137#215—61no—
138Dic12—91no—
139Dic12—91no—
140Dic12—181no—
141Dic12—181no—
142Dic12C2×D911yescharacteristic
143D24—31no—
144D3×C8—31no—
145C3×C16—31no—
146C3⋊C16—91no—
147C3⋊C16—91no—
148C3⋊C16—181no—
149C3⋊C16C2×D911yescharacteristic
150C2×C3×D3—31no—
151C2×C3×D3—121no—
152C32×C4—41no—
153C32×C4D3×D411yescharacteristic
154C3⋊Dic3—121no—
155C3×Dic3—31no—
156C3×Dic3—41no—
157C2×D9—41no—
158C2×D9#21311yescharacteristic
159C4×C9—21no—
160C4×C9—41no—
161C4×C9—81no—
162C4×C9—121no—
163C4×C9C2×#6711yescharacteristic
164Dic9—121no—
165Dic9—241no—
166Dic9#21311yescharacteristic
167#135—91no—
168C2×Q16—271no—
169Q32—31no—
170Q32—271no—
171SD32—271no—
172SD32—271no—
173M32—271no—
174C3×C9#51011yescharacteristic, complemented, Sylow
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—41no—
178C3×Q8—61no—
179C3×D4—181no—
180C3×D4—361no—
181C2×C3×C4—91no—
182C2×C3×C4—361no—
183C2×Dic3—361no—
184D12—61no—
185D12—121no—
186C4×D3—31no—
187C4×D3—121no—
188C4×D3—361no—
189Dic6—21no—
190Dic6—181no—
191Dic6—181no—
192Dic6—361no—
193Dic6—361no—
194Dic6—361no—
195C3×C8—21no—
196C3×C8—91no—
197C3×C8C22×D911yescharacteristic
198C3×C8C2×D3211yescharacteristic
199C3⋊C8—31no—
200C2×C32D3×D811yescharacteristic
201C3×D3—61no—
202C3×D3—241no—
203C2×C9—21no—
204C2×C9C2×#21311yescharacteristic
205D9—21no—
206D9—81no—
207#41—181no—
208C2×Q8—541no—
209Q16—31no—
210Q16—271no—
211Q16—541no—
212Q16D3×D911yescharacteristic
213SD16—181no—
214D8—91no—
215C2×C8—91no—
216C16—31no—
217C16—271no—
218C22×C3—91no—
219C22×C3—361no—
220C2×D3—31no—
221C2×D3—121no—
222C3×C4—21no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—81no—
226C3×C4—91no—
227C3×C4—121no—
228C3×C4D4×D911yescharacteristic
229C3×C4D3×#6711yescharacteristic
230Dic3—31no—
231Dic3—41no—
232Dic3—361no—
233Dic3—361no—
234Dic3—721no—
235C32#491811yescharacteristic
236C9—21no—
237C9#519511yescharacteristic, complemented
238Q8—21no—
239Q8—61no—
240Q8—541no—
241Q8—1081no—
242D4—181no—
243D4—361no—
244C2×C4—91no—
245C2×C4—361no—
246C2×C4—1081no—
247C8—91no—
248C8C2×D3×D911yescharacteristic
249C2×C3—21no—
250C2×C3—181no—
251C2×C3—721no—
252C2×C3D8×D911yescharacteristic
253C2×C3D3×#21311yescharacteristic
254D3—61no—
255D3—241no—
256C22—91no—
257C22—361no—
258C4—41no—
259C4—91no—
260C4—121no—
261C4—1081no—
262C4D9×#6711yescharacteristic
263C3—21no—
264C3#1058859811yescharacteristic, complemented
265C3#1059104211yescharacteristic
266C2—181no—
267C2—721no—
268C2D9×#21311yescharacteristic
subgroupG/H
centerC2D9×#213
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×Q16C22
socleC2×C32D3×D8
prime radicalC3×#63481C2
hypercenterQ16D3×D9
hypocenterC3×C9#510
LCSG ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. D9×Dic12C2semidirect
3. #11720568C2semidirect
4. C9×#829C2semidirect
5. #11720242C2semidirect
6. #11720233C2semidirect
7. #11720230C2semidirect
89. #829D9semidirect
174. C3×C9#510semidirect
237. C9#5195semidirect
264. C3#10588598semidirect

Maximal subgroups

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

subgroupindex
C3×#634812
D9×Dic122
#117205682
C9×#8292
#117202422
#117202332
#117202302
#105910423
#105885983