Finite Group Databasebeta

← #13144219

Subgroups — #13144219

Order 1728.

410 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 95 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#11721805C211yesmaximal, characteristic
2Q16×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×C9×Q32C211yesmaximal, characteristic, complemented
4#11720568C211yesmaximal, characteristic, complemented
5#11720567C211yesmaximal, characteristic, complemented
6#11720508C211yesmaximal, characteristic, complemented
7#11720506C211yesmaximal, characteristic, complemented
8#10592617—31nomaximal
9#10588598—33nomaximal
10#88746—21no—
11Q8×C3⋊D9—21no—
12C3×C9×Q16C2211yescharacteristic
13C3×C9×Q16C2211yescharacteristic
14#88548—21no—
15#88547—21no—
16#88525C2211yescharacteristic
17C8×C3⋊D9C2211yescharacteristic
18#88521C2211yescharacteristic
19C3×C9×C16C2211yescharacteristic
20C3⋊(C9⋊C16)C2211yescharacteristic
21#64128—31no—
22Q16×C3⋊D3—31no—
23C32×Q32D311yescharacteristic
24#63656—31no—
25#63655—31no—
26#63627—31no—
27#63625—31no—
28#63481—33no—
29Q16×D9—33no—
30C9×Q32D313yescomplemented
31#63388—33no—
32#63387—33no—
33#63359—33no—
34#63357—33no—
35C3×Q8×C9—21no—
36C3×Q8×C9—21no—
37C4⋊(C3⋊D9)—21no—
38C4⋊(C3⋊D9)—41no—
39C4×C3⋊D9—41no—
40C4×C3⋊D9—41no—
41C4×C3⋊D9D411yescharacteristic
42#6226—21no—
43#6226—41no—
44C3×C8×C9C2311yescharacteristic
45C3⋊(C9⋊C8)D411yescharacteristic
46#4918—93no—
47#4918—91no—
48#3832—61no—
49Q8×C3⋊D3—61no—
50C32×Q16C2×D311yescharacteristic
51C32×Q16C2×D311yescharacteristic
52#3756—61no—
53#3755—61no—
54#3746—31no—
55#3745—31no—
56C8×C3⋊D3—31no—
57#3701—63no—
58Q8×D9—63no—
59C32×C16C2×D311yescharacteristic
60C3⋊(C3⋊C16)—31no—
61C9×Q16C2×D313yes—
62C9×Q16C2×D313yes—
63#3675—63no—
64#3674—63no—
65D72—33no—
66C8×D9—33no—
67Dic36—33no—
68C9×C16C2×D313yes—
69C9⋊C16—33no—
70C2×C3⋊D9—41no—
71C2×C3⋊D9D811yescharacteristic
72C3×C4×C9—41no—
73C3×C4×C9—41no—
74C3×C4×C9C2×D411yescharacteristic
75C3⋊Dic9—41no—
76C3⋊Dic9D811yescharacteristic
77#919—93no—
78#919—91no—
79D3×Q16—93no—
80D3×Q16—91no—
81C3×Q32D913yescomplemented
82C3×Q32C3⋊D311yescharacteristic
83#829—93no—
84#829—91no—
85#828—93no—
86#828—91no—
87#800—93no—
88#800—91no—
89#798—93no—
90#798—91no—
91C32×Q8—21no—
92C32×Q8—21no—
93C4⋊(C3⋊D3)—61no—
94C4⋊(C3⋊D3)—121no—
95C4×C3⋊D3—31no—
96C4×C3⋊D3—121no—
97C4×C3⋊D3—121no—
98#634—61no—
99#634—121no—
100C32×C8C22×D311yescharacteristic
101C3⋊(C3⋊C8)—31no—
102Q8×C9—23no—
103Q8×C9—23no—
104D36—63no—
105D36—123no—
106C4×D9—33no—
107C4×D9—123no—
108C4×D9—123no—
109Dic18—63no—
110Dic18—123no—
111C8×C9C22×D313yes—
112C9⋊C8—33no—
113#510—271noSylow
114C2×C3×C9C2×D811yescharacteristic
115C3⋊D9—21no—
116C3⋊D9—81no—
117#239—181no—
118#239—183no—
119D3×Q8—181no—
120D3×Q8—183no—
121C3×Q16C2×D913yes—
122C3×Q16C2×D913yes—
123C3×Q16C2×C3⋊D311yescharacteristic
124C3×Q16C2×C3⋊D311yescharacteristic
125#216—181no—
126#216—183no—
127#215—181no—
128#215—183no—
129Dic12—93no—
130Dic12—91no—
131D24—93no—
132D24—91no—
133D3×C8—93no—
134D3×C8—91no—
135C3×C16C2×D913yes—
136C3×C16C2×C3⋊D311yescharacteristic
137C3⋊C16—93no—
138C3⋊C16—91no—
139C2×C3⋊D3—31no—
140C2×C3⋊D3—121no—
141C32×C4—41no—
142C32×C4—41no—
143C32×C4D3×D411yescharacteristic
144C3⋊Dic3—31no—
145C3⋊Dic3—121no—
146C2×D9—33no—
147C2×D9—123no—
148C4×C9—43no—
149C4×C9—43no—
150C4×C9D3×D413yes—
151Dic9—33no—
152Dic9—123no—
153#135—271no—
154C2×Q16—271no—
155Q32—271no—
156Q32C3⋊D911yescharacteristic, complemented
157SD32—271no—
158SD32—271no—
159M32—271no—
160C3×C9#51011yescharacteristic, complemented, Sylow
161C3×Q8—23no—
162C3×Q8—23no—
163C3×Q8—21no—
164C3×Q8—21no—
165D12—183no—
166D12—181no—
167D12—361no—
168D12—363no—
169C4×D3—93no—
170C4×D3—91no—
171C4×D3—361no—
172C4×D3—363no—
173C4×D3—361no—
174C4×D3—363no—
175Dic6—183no—
176Dic6—181no—
177Dic6—361no—
178Dic6—363no—
179C3×C8C22×D913yes—
180C3×C8C22×C3⋊D311yescharacteristic
181C3⋊C8—93no—
182C3⋊C8—91no—
183C2×C32D3×D811yescharacteristic
184C3⋊D3—61no—
185C3⋊D3—241no—
186C2×C9D3×D813yes—
187D9—63no—
188D9—243no—
189#41—541no—
190C2×Q8—541no—
191Q16—271no—
192Q16—541no—
193Q16C2×C3⋊D911yescharacteristic
194Q16C2×C3⋊D911yescharacteristic
195SD16—541no—
196D8—271no—
197C2×C8—271no—
198C16—271no—
199C16C2×C3⋊D911yescharacteristic
200C2×D3—91no—
201C2×D3—93no—
202C2×D3—361no—
203C2×D3—363no—
204C3×C4—41no—
205C3×C4—41no—
206C3×C4—43no—
207C3×C4—43no—
208C3×C4D4×D913yes—
209C3×C4D4×C3⋊D311yescharacteristic
210Dic3—91no—
211Dic3—93no—
212Dic3—361no—
213Dic3—363no—
214C32#491811yescharacteristic
215C9#491813yescomplemented
216Q8—21no—
217Q8—21no—
218Q8—541no—
219Q8—1081no—
220D4—541no—
221D4—1081no—
222C2×C4—271no—
223C2×C4—1081no—
224C2×C4—1081no—
225C8—271no—
226C8C22×C3⋊D911yescharacteristic
227C2×C3D8×D913yes—
228C2×C3D8×C3⋊D311yescharacteristic
229D3—181no—
230D3—183no—
231D3—721no—
232D3—723no—
233C22—271no—
234C22—1081no—
235C4—41no—
236C4—41no—
237C4—271no—
238C4—1081no—
239C4D4×C3⋊D911yescharacteristic
240C3#1058859813yescomplemented
241C3#1059261711yescharacteristic
242C2—541no—
243C2—2161no—
244C2D8×C3⋊D911yescharacteristic
subgroupG/H
centerC2D8×C3⋊D9
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C22×C3⋊D3
FittingC3×C9×Q32C2
socleC2×C32D3×D8
prime radical#11721805C2
hypercenterQ32C3⋊D9
hypocenterC3×C9#510
LCSG ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32

Complements

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

NH ≅ G/Ntype
2. Q16×C3⋊D9C2semidirect
3. C3×C9×Q32C2semidirect
4. #11720568C2semidirect
5. #11720567C2semidirect
6. #11720508C2semidirect
7. #11720506C2semidirect
30. C9×Q32D3semidirect
81. C3×Q32D9semidirect
156. Q32C3⋊D9semidirect
160. C3×C9#510semidirect
215. C9#4918semidirect
240. C3#10588598semidirect

Maximal subgroups

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

subgroupindex
#117218052
Q16×C3⋊D92
C3×C9×Q322
#117205682
#117205672
#117205082
#117205062
#105926173
#105885983