Finite Group Databasebeta

← #13147447

Subgroups — #13147447

Order 1728.

652 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 161 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#11721709C211yesmaximal, characteristic
2C9×#870C211yesmaximal, characteristic, complemented
3C3×#63431C211yesmaximal, characteristic, complemented
4#11720936C212yesmaximal
5#11720921C214yesmaximal
6#11720916C214yesmaximal, complemented
7#11720910C212yesmaximal, complemented
8#10593867—31nomaximal
9#10589800—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×Dic18C2212yes—
12C3×C42×C9C2211yescharacteristic
13#88536C2212yes—
14#88534C2212yes—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2212yes—
17C9×Dic3⋊C4C2214yes—
18C3×C4⋊Dic9C2214yes—
19#88445C2214yes—
20Dic3⋊Dic9C2214yes—
21Dic9⋊Dic3C2214yes—
22Dic3×Dic9C2214yes—
23#64081—31no—
24C3×#870D311yescharacteristic
25C3×#869—31no—
26#63871—32no—
27#63848—34no—
28#63846—34no—
29#63843—32no—
30C9×#125—31no—
31#63460—34no—
32#63459—32no—
33#63452—34no—
34#63449—32no—
35#63434—31no—
36#63434—61no—
37#63431D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C2×C3×C4×C9C2312yes—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9C2312yes—
42C2×C3⋊Dic9C2311yescharacteristic
43C4×C3⋊D9—22no—
44C2×C9×Dic3C2314yes—
45C2×C3×Dic9C2314yes—
46C3×Dic18—24no—
47#5731—91no—
48#5698—91no—
49C2×C4×C3⋊D3—31no—
50C2×C3×Dic6—32no—
51C32×C42C2×D311yescharacteristic
52#3752—32no—
53#3750—32no—
54C4×C3⋊Dic3—31no—
55C3×C4⋊Dic3—34no—
56C3×C4⋊Dic3C2×D312yes—
57C3×Dic3⋊C4C2×D314yes—
58Dic3⋊Dic3—34no—
59Dic3⋊Dic3—34no—
60#3722—34no—
61Dic32—34no—
62C2×C4×D9—31no—
63C2×C4×D9—61no—
64C2×Dic18C2×D312yes—
65C9×C4⋊C4—34no—
66C9×C4⋊C4—32no—
67C42×C9—21no—
68C42×C9C2×D311yescharacteristic
69(C2×D9)⋊C4—32no—
70(C2×D9)⋊C4—34no—
71(C2×D9)⋊C4—62no—
72C4⋊Dic9—34no—
73C4⋊Dic9C2×D314yes—
74Dic9⋊C4—32no—
75Dic9⋊C4—34no—
76Dic9⋊C4—62no—
77C4×Dic9—31no—
78C4×Dic9—34no—
79C4×Dic9—61no—
80C22×C3×C9C2411yescharacteristic
81C2×C3⋊D9—21no—
82C2×C3⋊D9—22no—
83C3×C4×C9—22no—
84C3×C4×C9#4112yes—
85C3⋊Dic9—21no—
86C3⋊Dic9—22no—
87C9×Dic3—24no—
88C3×Dic9—24no—
89C3×C4⋊Q8—91no—
90C3×#125—31no—
91#946—92no—
92Q8×Dic3—92no—
93#898—94no—
94#898—94no—
95#897—92no—
96#890—94no—
97#889—94no—
98#887—92no—
99#872—91no—
100#872—91no—
101#872—181no—
102#870D911yescharacteristic, complemented
103#869—31no—
104C22×C3⋊D3—31no—
105C2×C32×C4C22×D312yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—32no—
108C2×C3⋊Dic3—31no—
109C4×C3⋊D3—62no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3C22×D314yes—
112C3×Dic6—64no—
113C22×D9—31no—
114C22×D9—61no—
115C2×C4×C9—21no—
116C2×C4×C9—22no—
117C2×C4×C9—34no—
118C2×C4×C9C22×D312yes—
119C2×C4×C9C22×D311yescharacteristic
120C2×Dic9—32no—
121C2×Dic9—31no—
122C2×Dic9—61no—
123C2×Dic9—62no—
124C2×Dic9C22×D314yes—
125C4×D9—64no—
126C4×D9—62no—
127Dic18—24no—
128#541—271noSylow
129C2×C3×C9C2×#4111yescharacteristic
130C2×C3×C92−1+412yes—
131C3⋊D9—41no—
132C2×C3×Q8—92no—
133C2×C4×D3—91no—
134C2×C4×D3—91no—
135C2×C4×D3—181no—
136C2×Dic6—32no—
137C3×C4⋊C4—32no—
138C3×C4⋊C4—34no—
139C3×C4⋊C4—94no—
140C3×C42—21no—
141C3×C42C2×D911yescharacteristic
142C3×C42D3211yescharacteristic
143(C2×D3)⋊C4—94no—
144(C2×D3)⋊C4—92no—
145(C2×D3)⋊C4—92no—
146(C2×D3)⋊C4—94no—
147(C2×D3)⋊C4—182no—
148C4⋊Dic3—34no—
149C4⋊Dic3—94no—
150C4⋊Dic3—94no—
151C4⋊Dic3C2×D912yes—
152Dic3⋊C4—94no—
153Dic3⋊C4—92no—
154Dic3⋊C4—92no—
155Dic3⋊C4—94no—
156Dic3⋊C4—182no—
157Dic3⋊C4C2×D914yes—
158C4×Dic3—94no—
159C4×Dic3—91no—
160C4×Dic3—91no—
161C4×Dic3—94no—
162C4×Dic3—181no—
163C22×C32C23×D311yescharacteristic
164C2×C3⋊D3—61no—
165C2×C3⋊D3—62no—
166C32×C4—22no—
167C32×C4#23912yes—
168C3⋊Dic3—61no—
169C3⋊Dic3—62no—
170C3×Dic3—24no—
171C3×Dic3—64no—
172C22×C9—21no—
173C22×C9C23×D311yescharacteristic
174C2×D9—62no—
175C2×D9—61no—
176C2×D9—62no—
177C2×D9—122no—
178C4×C9—22no—
179C4×C9—22no—
180C4×C9—42no—
181C4×C9—64no—
182C4×C9#23712yes—
183Dic9—24no—
184Dic9—62no—
185Dic9—61no—
186Dic9—62no—
187Dic9—122no—
188C4⋊Q8—91no—
189#126—274no—
190#125—31no—
191#125—274no—
192C22⋊Q8—272no—
193C4×Q8—272no—
194#117—271no—
195C3×C9#54111yescharacteristic, complemented, Sylow
196C22×D3—91no—
197C22×D3—91no—
198C22×D3—181no—
199C3×Q8—184no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—34no—
203C2×C3×C4—94no—
204C2×C3×C4C22×D912yes—
205C2×C3×C4C22×D911yescharacteristic
206C2×C3×C4C2×D3212yes—
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—34no—
209C2×Dic3—92no—
210C2×Dic3—91no—
211C2×Dic3—92no—
212C2×Dic3—91no—
213C2×Dic3—181no—
214C2×Dic3—182no—
215C2×Dic3C22×D914yes—
216C4×D3—184no—
217C4×D3—182no—
218C4×D3—182no—
219Dic6—64no—
220C2×C32C2×#23911yescharacteristic
221C2×C32#101012yes—
222C3⋊D3—121no—
223C2×C9—21no—
224C2×C9—22no—
225C2×C9C2×#23711yescharacteristic
226C2×C9#100712yes—
227D9—122no—
228D9—121no—
229C2×Q8—92no—
230C22×C4—271no—
231C4⋊C4—32no—
232C4⋊C4—34no—
233C4⋊C4—94no—
234C4⋊C4—274no—
235C4⋊C4—274no—
236C4⋊C4—272no—
237C22⋊C4—274no—
238C22⋊C4—272no—
239C42—274no—
240C42—271no—
241C42D3×D911yescharacteristic
242C22×C3—21no—
243C22×C3C23×D911yescharacteristic
244C22×C3C22×D3211yescharacteristic
245C2×D3—182no—
246C2×D3—181no—
247C2×D3—182no—
248C2×D3—181no—
249C2×D3—182no—
250C2×D3—362no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—42no—
255C3×C4—64no—
256C3×C4—184no—
257C3×C4#370112yes—
258C3×C4#379612yes—
259Dic3—24no—
260Dic3—64no—
261Dic3—182no—
262Dic3—181no—
263Dic3—182no—
264Dic3—181no—
265Dic3—182no—
266Dic3—362no—
267C32#569811yescharacteristic
268C9—21no—
269C9#573111yescharacteristic, complemented
270C23—271no—
271Q8—184no—
272C2×C4—34no—
273C2×C4—94no—
274C2×C4—272no—
275C2×C4—271no—
276C2×C4—542no—
277C2×C4C2×D3×D912yes—
278C2×C4C2×D3×D911yescharacteristic
279C2×C3—21no—
280C2×C3—22no—
281C2×C3C2×#370111yescharacteristic
282C2×C3#6371612yes—
283C2×C3C2×#379611yescharacteristic
284C2×C3#6429712yes—
285D3—362no—
286D3—361no—
287D3—361no—
288C22—541no—
289C22—542no—
290C22C22×D3×D911yescharacteristic
291C4—22no—
292C4—64no—
293C4—184no—
294C4—541no—
295C4—542no—
296C4#8863412yes—
297C3—21no—
298C3#1058980011yescharacteristic, complemented
299C3#1059386711yescharacteristic
300C2—1081no—
301C2C2×#8863411yescharacteristic
302C2#1172246412yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC42D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C9×#870C2semidirect
3. C3×#63431C2semidirect
6. #11720916C2semidirect
7. #11720910C2semidirect
37. #63431D3semidirect
102. #870D9semidirect
195. C3×C9#541semidirect
269. C9#5731semidirect
298. C3#10589800semidirect

Maximal subgroups

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

subgroupindex
#117217092
C9×#8702
C3×#634312
#117209362
#117209212
#117209162
#117209102
#105938673
#105898003