Finite Group Databasebeta

← #13139271

Subgroups — #13139271

Order 1728.

306 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q16×C3⋊D9C211yesmaximal, characteristic
2#11720829C211yesmaximal, characteristic
3C9×#829C211yesmaximal, characteristic, complemented
4C3×#63388C211yesmaximal, characteristic, complemented
5#11720245C211yesmaximal, characteristic, complemented
6#11720235C211yesmaximal, characteristic, complemented
7(C3⋊D9)⋊C16C211yesmaximal, characteristic
8#10591043—31nomaximal
9D9×Q32—31nomaximal
10Q8×C3⋊D9—21no—
11(C3⋊D9)⋊Q8—21no—
12C3×C9×Q16C2211yescharacteristic
13#88547—21no—
14C8×C3⋊D9C2211yescharacteristic
15#88521C2211yescharacteristic
16C9×Dic12C2211yescharacteristic
17C3×Dic36C2211yescharacteristic
18#88427—21no—
19C9×C3⋊C16C2211yescharacteristic
20C3×C9⋊C16C2211yescharacteristic
21Q16×C3⋊D3—31no—
22#63802—31no—
23C3×#829—31no—
24C3×#829D311yescharacteristic
25#63551—31no—
26#63551—31no—
27#63542—31no—
28Q16×D9—31no—
29Q16×D9—31no—
30Q16×D9—61no—
31C9×Q32—31no—
32#63388—31no—
33#63388D311yescharacteristic, complemented
34Dic72—31no—
35C16×D9—31no—
36C3×Q8×C9—21no—
37C4×C3⋊D9—41no—
38C4×C3⋊D9D411yescharacteristic
39#6226—21no—
40#6226—41no—
41C9×Dic6—21no—
42C3×Dic18—21no—
43(C3⋊D9)⋊C4—41no—
44#6189—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47D3×Q32—91no—
48D3×Q32—91no—
49Q8×C3⋊D3—61no—
50#3797—61no—
51C32×Q16C2×D311yescharacteristic
52#3756—61no—
53#3746—31no—
54C8×C3⋊D3—31no—
55C3×Dic12—31no—
56C3×Dic12C2×D311yescharacteristic
57#3718—61no—
58Q8×D9—61no—
59Q8×D9—61no—
60Q8×D9—62no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—21no—
64C9×Q16—31no—
65C9×Q16C2×D311yescharacteristic
66#3674—61no—
67#3674—61no—
68#3674—62no—
69C8×D9—31no—
70C8×D9—61no—
71Dic36—31no—
72Dic36—61no—
73Dic36C2×D311yescharacteristic
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic
76C2×C3⋊D9D811yescharacteristic
77C3×C4×C9—41no—
78C3×C4×C9C2×D411yescharacteristic
79C3⋊Dic9—41no—
80C3⋊Dic9D811yescharacteristic
81C9×Dic3—41no—
82C3×Dic9—41no—
83D3×Q16—91no—
84D3×Q16—91no—
85D3×Q16—91no—
86D3×Q16—91no—
87D3×Q16—181no—
88C3×Q32—31no—
89C3×Q32—91no—
90#829—31no—
91#829—91no—
92#829—91no—
93#829D911yescharacteristic, complemented
94Dic24—91no—
95Dic24—91no—
96D3×C16—91no—
97D3×C16—91no—
98C32×Q8—21no—
99C4×C3⋊D3—31no—
100C4×C3⋊D3—121no—
101#634—61no—
102#634—121no—
103C3×Dic6—21no—
104C3×Dic6—61no—
105#627—121no—
106#624—121no—
107C32×C8C22×D311yescharacteristic
108C3⋊(C3⋊C8)—31no—
109Q8×C9—22no—
110Q8×C9—21no—
111Q8×C9—61no—
112C4×D9—31no—
113C4×D9—61no—
114C4×D9—121no—
115C4×D9—121no—
116C4×D9—122no—
117Dic18—21no—
118Dic18—62no—
119Dic18—61no—
120Dic18—121no—
121Dic18—121no—
122Dic18—122no—
123C8×C9—21no—
124C8×C9C22×D311yescharacteristic
125C9⋊C8—31no—
126C9⋊C8—61no—
127C2×Q32—271noSylow
128C2×C3×C9C2×D811yescharacteristic
129C3⋊D9Q3212yescomplemented
130D3×Q8—181no—
131D3×Q8—181no—
132D3×Q8—181no—
133D3×Q8—181no—
134D3×Q8—182no—
135C3×Q16—21no—
136C3×Q16—31no—
137C3×Q16—91no—
138C3×Q16C2×D911yescharacteristic
139C3×Q16D3211yescharacteristic
140#216—181no—
141#216—181no—
142#216—181no—
143#216—181no—
144#216—182no—
145Dic12—31no—
146Dic12—91no—
147Dic12—91no—
148Dic12—181no—
149Dic12C2×D911yescharacteristic
150D3×C8—91no—
151D3×C8—91no—
152D3×C8—181no—
153C3×C16—31no—
154C3×C16—91no—
155C3⋊C16—31no—
156C3⋊C16C2×D911yescharacteristic
157C2×C3⋊D3—31no—
158C32×C4—41no—
159C32×C4D3×D411yescharacteristic
160C3⋊Dic3—31no—
161C3⋊Dic3—121no—
162C3×Dic3—41no—
163C3×Dic3—121no—
164C2×D9—31no—
165C2×D9—61no—
166C4×C9—21no—
167C4×C9—42no—
168C4×C9—41no—
169C4×C9—121no—
170C4×C9D3×D411yescharacteristic
171Dic9—31no—
172Dic9—41no—
173Dic9—61no—
174Dic9—122no—
175Dic9—121no—
176C2×Q16—271no—
177C2×Q16—271no—
178Q32—31no—
179Q32—91no—
180Q32—271no—
181Q32—271no—
182C2×C16—271no—
183C3×C9C2×Q3211yescharacteristic, complemented, Sylow
184C3×Q8—22no—
185C3×Q8—21no—
186C3×Q8—21no—
187C3×Q8—61no—
188C3×Q8—181no—
189C4×D3—91no—
190C4×D3—91no—
191C4×D3—181no—
192C4×D3—361no—
193C4×D3—361no—
194C4×D3—361no—
195C4×D3—361no—
196C4×D3—362no—
197Dic6—21no—
198Dic6—61no—
199Dic6—182no—
200Dic6—181no—
201Dic6—181no—
202Dic6—361no—
203Dic6—361no—
204Dic6—361no—
205Dic6—361no—
206Dic6—362no—
207C3×C8—21no—
208C3×C8C22×D911yescharacteristic
209C3×C8C2×D3211yescharacteristic
210C3⋊C8—91no—
211C3⋊C8—91no—
212C3⋊C8—181no—
213C2×C32D3×D811yescharacteristic
214C3⋊D3—32no—
215C2×C9—21no—
216C2×C9D3×D811yescharacteristic
217D9—32no—
218D9—62no—
219C2×Q8—541no—
220C2×Q8—541no—
221Q16—31no—
222Q16—91no—
223Q16—271no—
224Q16—541no—
225Q16—541no—
226Q16D3×D911yescharacteristic
227C2×C8—271no—
228C16—31no—
229C16—91no—
230C2×D3—91no—
231C2×D3—91no—
232C2×D3—181no—
233C3×C4—21no—
234C3×C4—42no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—121no—
238C3×C4—361no—
239C3×C4D4×D911yescharacteristic
240C3×C4#381111yescharacteristic
241Dic3—41no—
242Dic3—91no—
243Dic3—91no—
244Dic3—121no—
245Dic3—181no—
246Dic3—362no—
247Dic3—361no—
248Dic3—361no—
249C32D3×Q3211yescharacteristic
250C9—21no—
251C9D3×Q3211yescharacteristic, complemented
252Q8—21no—
253Q8—61no—
254Q8—181no—
255Q8—541no—
256Q8—1081no—
257Q8—1081no—
258C2×C4—271no—
259C2×C4—1081no—
260C2×C4—1081no—
261C8—271no—
262C8C2×D3×D911yescharacteristic
263C2×C3—21no—
264C2×C3D8×D911yescharacteristic
265C2×C3#6392611yescharacteristic
266D3—92no—
267D3—92no—
268D3—182no—
269C22—271no—
270C4—41no—
271C4—121no—
272C4—271no—
273C4—361no—
274C4—1081no—
275C4#8866811yescharacteristic
276C3—21no—
277C3D9×Q3211yescharacteristic, complemented
278C3#1059104311yescharacteristic
279C2—272no—
280C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×Q16C22
socleC2×C32D3×D8
prime radicalC2×C3⋊D9D8
hypercenterQ16D3×D9
hypocenterC3×C9C2×Q32
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
3. C9×#829C2semidirect
4. C3×#63388C2semidirect
5. #11720245C2semidirect
6. #11720235C2semidirect
33. #63388D3semidirect
93. #829D9semidirect
129. C3⋊D9Q32semidirect
183. C3×C9C2×Q32semidirect
251. C9D3×Q32semidirect
277. C3D9×Q32semidirect

Maximal subgroups

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

subgroupindex
Q16×C3⋊D92
#117208292
C9×#8292
C3×#633882
#117202452
#117202352
(C3⋊D9)⋊C162
#105910433
D9×Q323