Finite Group Databasebeta

← #13139264

Subgroups — #13139264

Order 1728.

332 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 57 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, complemented
2#11720818C211yesmaximal, characteristic, complemented
3C9×#828C211yesmaximal, characteristic, complemented
4C3×#63387C211yesmaximal, characteristic, complemented
5#11720239C211yesmaximal, characteristic, complemented
6#11720232C211yesmaximal, characteristic, complemented
7#11720231C211yesmaximal, characteristic, complemented
8#10591040—31nomaximal
9#10588594—31nomaximal
10#88746—21no—
11#88646—21no—
12C3×C9×Q16C2211yescharacteristic, complemented
13#88548—21no—
14#88525C2211yescharacteristic, complemented
15C8×C3⋊D9C2211yescharacteristic
16C9×D24C2211yescharacteristic
17C3×D72C2211yescharacteristic
18#88421—21no—
19C9×C3⋊C16C2211yescharacteristic, complemented
20C3×C9⋊C16C2211yescharacteristic, complemented
21#64128—31no—
22#63797—31no—
23C3×#828—31no—
24C3×#828D311yescharacteristic
25#63546—31no—
26#63546—31no—
27#63544—31no—
28#63481—31no—
29#63481—61no—
30D8×D9—31no—
31C9×SD32—31no—
32#63387D311yescharacteristic, complemented
33#63385—31no—
34D144—31no—
35#63357—31no—
36C2×D3×D9—41no—
37C3×Q8×C9—21no—
38C4⋊(C3⋊D9)—21no—
39C4⋊(C3⋊D9)—41no—
40C4×C3⋊D9—41no—
41C4×C3⋊D9D411yescharacteristic
42C9×D12—21no—
43C3×D36—21no—
44#6194—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47#4914—91no—
48#4914—91no—
49#3832—61no—
50#3802—61no—
51C32×Q16C2×D311yescharacteristic
52#3755—61no—
53#3745—31no—
54C8×C3⋊D3—31no—
55C3×D24—31no—
56C3×D24C2×D311yescharacteristic
57#3713—61no—
58#3701—61no—
59#3701—62no—
60D4×D9—61no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—21no—
64C9×Q16C2×D311yescharacteristic, complemented
65C9×D8—31no—
66#3675—61no—
67#3675—62no—
68#3673—61no—
69D72—31no—
70D72—61no—
71D72C2×D311yescharacteristic
72C8×D9—31no—
73C8×D9—61no—
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic, complemented
76C2×C3⋊D9—41no—
77C2×C3⋊D9D811yescharacteristic
78C2×D3×C9—41no—
79C2×C3×D9—41no—
80D3×D9—82no—
81C3×C4×C9—41no—
82C3×C4×C9C2×D411yescharacteristic
83C3⋊Dic9D811yescharacteristic
84#919—91no—
85#919—91no—
86#919—181no—
87D3×D8—91no—
88D3×D8—91no—
89C3×SD32—31no—
90C3×SD32—91no—
91#828—31no—
92#828D911yescharacteristic, complemented
93#826—91no—
94#826—91no—
95D48—91no—
96D48—91no—
97#798—91no—
98#798—91no—
99C2×D32—121no—
100C32×Q8—21no—
101C4⋊(C3⋊D3)—61no—
102C4⋊(C3⋊D3)—121no—
103C4×C3⋊D3—31no—
104C4×C3⋊D3—121no—
105C3×D12—21no—
106C3×D12—61no—
107#625—121no—
108C22×D9—121no—
109C32×C8C22×D311yescharacteristic
110C3⋊(C3⋊C8)—31no—
111Q8×C9—22no—
112Q8×C9—21no—
113D4×C9—61no—
114#611—121no—
115D36—21no—
116D36—62no—
117D36—61no—
118D36—121no—
119D36—122no—
120C4×D9—31no—
121C4×D9—61no—
122C4×D9—121no—
123C4×D9—122no—
124C8×C9—21no—
125C8×C9C22×D311yescharacteristic
126C9⋊C8—31no—
127C9⋊C8—61no—
128#509—271noSylow
129C2×C3×C9C2×D811yescharacteristic
130C3⋊D9—21no—
131C3⋊D9—81no—
132D3×C9—81no—
133C3×D9—81no—
134#239—181no—
135#239—181no—
136#239—182no—
137D3×D4—181no—
138D3×D4—181no—
139C3×Q16—21no—
140C3×Q16C2×D911yescharacteristic, complemented
141C3×Q16D3211yescharacteristic
142C3×D8—31no—
143C3×D8—91no—
144#215—181no—
145#215—181no—
146#215—182no—
147#213—181no—
148#213—181no—
149D24—31no—
150D24—91no—
151D24—91no—
152D24—181no—
153D24C2×D911yescharacteristic
154D3×C8—91no—
155D3×C8—91no—
156D3×C8—181no—
157C3×C16—31no—
158C3×C16—91no—
159C3⋊C16—31no—
160C3⋊C16C2×D911yescharacteristic, complemented
161C2×C3⋊D3—31no—
162C2×C3⋊D3—121no—
163C2×C3×D3—41no—
164C2×C3×D3—121no—
165D32—242no—
166C32×C4—41no—
167C32×C4D3×D411yescharacteristic
168C3⋊Dic3—31no—
169C22×C9—121no—
170C2×D9—31no—
171C2×D9—41no—
172C2×D9—61no—
173C2×D9—122no—
174C2×D9—121no—
175C2×D9—242no—
176C4×C9—21no—
177C4×C9—42no—
178C4×C9—41no—
179C4×C9D3×D411yescharacteristic
180Dic9—31no—
181Dic9—61no—
182#135—271no—
183C2×D8—271no—
184SD32—31no—
185SD32—91no—
186D16—271no—
187D16—271no—
188M32—271no—
189C3×C9#50911yescharacteristic, complemented, Sylow
190C22×D3—361no—
191C22×D3—361no—
192C3×Q8—22no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×D4—61no—
196C3×D4—181no—
197#67—361no—
198#67—361no—
199D12—21no—
200D12—61no—
201D12—182no—
202D12—181no—
203D12—181no—
204D12—361no—
205D12—361no—
206D12—362no—
207C4×D3—91no—
208C4×D3—91no—
209C4×D3—181no—
210C4×D3—361no—
211C4×D3—361no—
212C4×D3—362no—
213C3×C8—21no—
214C3×C8C22×D911yescharacteristic
215C3×C8C2×D3211yescharacteristic
216C3⋊C8—91no—
217C3⋊C8—91no—
218C3⋊C8—181no—
219C2×C32D3×D811yescharacteristic
220C3⋊D3—61no—
221C3⋊D3—241no—
222C3×D3—81no—
223C3×D3—241no—
224C2×C9—21no—
225C2×C9—241no—
226C2×C9D3×D811yescharacteristic
227D9—61no—
228D9—81no—
229D9—121no—
230D9—242no—
231D9—241no—
232#41—541no—
233C2×D4—541no—
234Q16D3×D911yescharacteristic, complemented
235SD16—541no—
236D8—31no—
237D8—91no—
238D8—271no—
239D8—541no—
240C2×C8—271no—
241C16—31no—
242C16—91no—
243C22×C3—121no—
244C22×C3—361no—
245C2×D3—41no—
246C2×D3—91no—
247C2×D3—91no—
248C2×D3—121no—
249C2×D3—181no—
250C2×D3—362no—
251C2×D3—361no—
252C2×D3—361no—
253C2×D3—722no—
254C2×D3—722no—
255C3×C4—21no—
256C3×C4—42no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4D4×D911yescharacteristic
260C3×C4#381111yescharacteristic
261Dic3—91no—
262Dic3—91no—
263Dic3—181no—
264C32#491411yescharacteristic
265C9—21no—
266C9#491411yescharacteristic, complemented
267C23—1081no—
268Q8—21no—
269D4—61no—
270D4—181no—
271D4—541no—
272D4—1081no—
273D4—1081no—
274C2×C4—271no—
275C2×C4—1081no—
276C8—271no—
277C8C2×D3×D911yescharacteristic
278C2×C3—21no—
279C2×C3—241no—
280C2×C3—721no—
281C2×C3D8×D911yescharacteristic
282C2×C3#6392611yescharacteristic
283D3—81no—
284D3—181no—
285D3—181no—
286D3—241no—
287D3—361no—
288D3—722no—
289D3—721no—
290D3—721no—
291C22—121no—
292C22—271no—
293C22—361no—
294C22—1081no—
295C22—2162no—
296C4—41no—
297C4—271no—
298C4#8866811yescharacteristic
299C3—21no—
300C3#1058859411yescharacteristic, complemented
301C3#1059104011yescharacteristic
302C2—241no—
303C2—541no—
304C2—721no—
305C2—2161no—
306C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×Q16C22
socleC2×C32D3×D8
hypercenterQ16D3×D9
hypocenterC3×C9#509
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
1. #11721805C2semidirect
2. #11720818C2semidirect
3. C9×#828C2semidirect
4. C3×#63387C2semidirect
5. #11720239C2semidirect
6. #11720232C2semidirect
7. #11720231C2semidirect
12. C3×C9×Q16C22semidirect
14. #88525C22semidirect
19. C9×C3⋊C16C22semidirect
20. C3×C9⋊C16C22semidirect
32. #63387D3semidirect
64. C9×Q16C2×D3semidirect
75. C9⋊C16C2×D3semidirect
92. #828D9semidirect
140. C3×Q16C2×D9semidirect
160. C3⋊C16C2×D9semidirect
189. C3×C9#509semidirect
234. Q16D3×D9semidirect
266. C9#4914semidirect
300. C3#10588594semidirect

Maximal subgroups

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

subgroupindex
#117218052
#117208182
C9×#8282
C3×#633872
#117202392
#117202322
#117202312
#105910403
#105885943