Finite Group Databasebeta

← #13139232

Subgroups — #13139232

Order 1728.

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

normal subgroups only

This group has 272 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×#63474C211yesmaximal, characteristic
2D9×Dic12C211yesmaximal, characteristic, complemented
3#11720564C211yesmaximal, characteristic, complemented
4C9×#827C211yesmaximal, characteristic, complemented
5#11720244C211yesmaximal, characteristic, complemented
6#11720235C211yesmaximal, characteristic, complemented
7#11720230C211yesmaximal, characteristic, complemented
8#10591028—31nomaximal
9#10588595—31nomaximal
10C3×#3699—21no—
11D9×Dic6—21no—
12C3×C9×D8C2211yescharacteristic
13#88521C2211yescharacteristic
14C3×#3672—21no—
15C9×Dic12C2211yescharacteristic
16C3×C8×D9C2211yescharacteristic
17C3×Dic36C2211yescharacteristic
18#88428—21no—
19C3⋊(C9⋊C16)C2211yescharacteristic
20C9×C3⋊C16C2211yescharacteristic
21C3×#912—31no—
22D3×Dic12—31no—
23#63654—31no—
24C3×#827D311yescharacteristic
25#63551—31no—
26#63550—31no—
27#63543—31no—
28Q16×D9—31no—
29#63474D311yescharacteristic
30C9×SD32—31no—
31#63388—31no—
32#63386—31no—
33#63386—61no—
34Dic72—31no—
35#63357—31no—
36C3×D4×C9—21no—
37#6226—21no—
38C3×#611—41no—
39C2×C3×Dic9—41no—
40C3×C4×D9D411yescharacteristic
41C9×Dic6—21no—
42C3×Dic18—21no—
43Dic3×D9—41no—
44#6189—41no—
45C3×C8×C9C2311yescharacteristic
46C3×C9⋊C8D411yescharacteristic
47#5195—91no—
48#4915—91no—
49C3×#237—61no—
50D3×Dic6—61no—
51C32×D8C2×D311yescharacteristic
52#3746—31no—
53C3×#214—61no—
54C3×Dic12—31no—
55C3×Dic12C2×D311yescharacteristic
56C3×D3×C8—31no—
57#3719—61no—
58Q8×D9—61no—
59#3699—21no—
60C3⋊(C3⋊C16)—31no—
61C3×C3⋊C16C2×D311yescharacteristic
62C9×Q16—31no—
63C9×D8—21no—
64C9×D8C2×D311yescharacteristic
65#3674—61no—
66#3672—21no—
67C8×D9C2×D311yescharacteristic
68Dic36—31no—
69Dic36—61no—
70Dic36C2×D311yescharacteristic
71C9×C16—31no—
72C9⋊C16—31no—
73C9⋊C16—61no—
74C22×C3×C9—41no—
75C2×C3×D9D811yescharacteristic
76C3×C4×C9C2×D411yescharacteristic
77C3⋊Dic9—41no—
78C9×Dic3—41no—
79C3×Dic9—41no—
80C3×Dic9D811yescharacteristic
81C3×#135—91no—
82D3×Q16—91no—
83#912—31no—
84C2×Dic12—91no—
85C3×SD32—31no—
86#829—91no—
87#829—91no—
88#829—91no—
89#827—91no—
90#827—91no—
91#827—181no—
92#827D911yescharacteristic, complemented
93#812—91no—
94Dic24—91no—
95#798—91no—
96C32×D4—21no—
97#634—61no—
98C3×#67—121no—
99C2×C3×Dic3—121no—
100C3×C4×D3—31no—
101C3×Dic6—21no—
102C3×Dic6—61no—
103#627—121no—
104D3×Dic3—121no—
105C32×C8C22×D311yescharacteristic
106C3×C3⋊C8—31no—
107Q8×C9—61no—
108D4×C9—21no—
109D4×C9—41no—
110#611—41no—
111C2×Dic9—41no—
112C4×D9—121no—
113C4×D9#6711yescharacteristic
114Dic18—21no—
115Dic18—61no—
116Dic18—121no—
117Dic18—121no—
118C8×C9—21no—
119C8×C9C22×D311yescharacteristic
120C9⋊C8#6711yescharacteristic
121#510—271noSylow
122C2×C3×C9—81no—
123C2×C3×C9C2×D811yescharacteristic
124C3×D9—21no—
125C3×#41—181no—
126D3×Q8—181no—
127#237—61no—
128C2×Dic6—181no—
129C3×Q16—31no—
130C3×Q16—91no—
131C3×SD16—181no—
132C3×D8—21no—
133C3×D8C2×D911yescharacteristic
134C3×D8D3211yescharacteristic
135C2×C3×C8—91no—
136#216—181no—
137#214—61no—
138Dic12—31no—
139Dic12—91no—
140Dic12—91no—
141Dic12—181no—
142Dic12—181no—
143Dic12C2×D911yescharacteristic
144D3×C8—31no—
145C3×C16—31no—
146C3⋊C16—91no—
147C3⋊C16—91no—
148C3⋊C16—181no—
149C3⋊C16C2×D911yescharacteristic
150C22×C32—41no—
151C2×C3×D3—31no—
152C32×C4D3×D411yescharacteristic
153C3⋊Dic3—121no—
154C3×Dic3—31no—
155C3×Dic3—41no—
156C3×Dic3—121no—
157C22×C9—41no—
158C22×C9—81no—
159C2×D9#21311yescharacteristic
160C4×C9—21no—
161C4×C9—121no—
162C4×C9C2×#6711yescharacteristic
163Dic9—41no—
164Dic9—121no—
165Dic9—241no—
166Dic9#21311yescharacteristic
167#135—91no—
168C2×Q16—271no—
169Q32—271no—
170Q32—271no—
171SD32—31no—
172SD32—271no—
173M32—271no—
174C3×C9#51011yescharacteristic, complemented, Sylow
175C3×Q8—61no—
176C3×Q8—181no—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—41no—
180C3×D4—361no—
181C2×C3×C4—91no—
182C2×C3×C4—361no—
183#67—121no—
184C2×Dic3—121no—
185C2×Dic3—361no—
186C4×D3—31no—
187C4×D3—361no—
188Dic6—21no—
189Dic6—61no—
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×C32—81no—
201C2×C32D3×D811yescharacteristic
202C3×D3—61no—
203C2×C9—21no—
204C2×C9—81no—
205C2×C9—161no—
206C2×C9C2×#21311yescharacteristic
207D9—21no—
208#41—181no—
209C2×Q8—541no—
210Q16—31no—
211Q16—91no—
212Q16—271no—
213Q16—541no—
214SD16—181no—
215D8D3×D911yescharacteristic
216C2×C8—91no—
217C16—31no—
218C16—271no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—81no—
222C22×C3—91no—
223C2×D3—31no—
224C3×C4—21no—
225C3×C4—91no—
226C3×C4—121no—
227C3×C4—361no—
228C3×C4D4×D911yescharacteristic
229C3×C4D3×#6711yescharacteristic
230Dic3—31no—
231Dic3—41no—
232Dic3—121no—
233Dic3—361no—
234Dic3—361no—
235Dic3—721no—
236C32#491511yescharacteristic
237C9—21no—
238C9#519511yescharacteristic, complemented
239Q8—61no—
240Q8—181no—
241Q8—541no—
242Q8—1081no—
243D4—21no—
244D4—361no—
245C2×C4—91no—
246C2×C4—361no—
247C2×C4—1081no—
248C8—91no—
249C8C2×D3×D911yescharacteristic
250C2×C3—21no—
251C2×C3—81no—
252C2×C3—81no—
253C2×C3—161no—
254C2×C3—181no—
255C2×C3D8×D911yescharacteristic
256C2×C3D3×#21311yescharacteristic
257D3—61no—
258C22—41no—
259C22—91no—
260C4—91no—
261C4—121no—
262C4—361no—
263C4—1081no—
264C4D9×#6711yescharacteristic
265C3—21no—
266C3#1058859511yescharacteristic, complemented
267C3#1059102811yescharacteristic
268C2—81no—
269C2—181no—
270C2D9×#21311yescharacteristic
subgroupG/H
centerC2D9×#213
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×D8C22
socleC2×C32D3×D8
prime radicalC3×#63474C2
hypercenterD8D3×D9
hypocenterC3×C9#510
LCSG ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
2. D9×Dic12C2semidirect
3. #11720564C2semidirect
4. C9×#827C2semidirect
5. #11720244C2semidirect
6. #11720235C2semidirect
7. #11720230C2semidirect
92. #827D9semidirect
174. C3×C9#510semidirect
238. C9#5195semidirect
266. C3#10588595semidirect

Maximal subgroups

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

subgroupindex
C3×#634742
D9×Dic122
#117205642
C9×#8272
#117202442
#117202352
#117202302
#105910283
#105885953