Finite Group Databasebeta

← #13137553

Subgroups — #13137553

Order 1728.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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×#63449C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic
3#11720526C211yesmaximal, characteristic
4C9×#807C211yesmaximal, characteristic
5#11720276C211yesmaximal, characteristic
6Dic9⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720268C211yesmaximal, characteristic
8#10590544—31nomaximal
9#10588557—31nomaximal
10C2×C3×Dic18C2211yescharacteristic
11C3×C9×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×Dic9⋊C4—21no—
16C3×C4×Dic9—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×#887—31no—
22#63869—31no—
23#63635—31no—
24C3×#807D311yescharacteristic
25Dic3⋊(C3⋊C8)—31no—
26#63566—31no—
27#63565—31no—
28#63452—31no—
29#63449D311yescharacteristic
30C9×#107—31no—
31#63376—31no—
32Dic9⋊C8—31no—
33#63368—31no—
34#63367—31no—
35#63367—61no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C9×Dic3—21no—
40C2×C3×Dic9—21no—
41C2×C3×Dic9D411yescharacteristic
42C2×C3×Dic9D411yescharacteristic
43C3×Dic18—21no—
44C3×Dic18Q812yes—
45C3⋊(C9⋊C8)—21no—
46C9×C3⋊C8—21no—
47#5022—91no—
48#4877—91no—
49C2×C3×Dic6—31no—
50C32×C4⋊C4C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)—31no—
52C2×C3⋊(C3⋊C8)—31no—
53C3×C4⋊Dic3C2×D311yescharacteristic
54C3×Dic3⋊C4—61no—
55C3×C4×Dic3—31no—
56C3×C4×Dic3—61no—
57C2×C3×C3⋊C8C2×D311yescharacteristic
58#3724—61no—
59Dic3⋊Dic3—61no—
60C2×Dic18C2×D311yescharacteristic
61C2×C8×C9—31no—
62C9×C4⋊C4—21no—
63C9×C4⋊C4—31no—
64C9×C4⋊C4C2×D311yescharacteristic
65C4⋊Dic9—31no—
66C4⋊Dic9—61no—
67C4⋊Dic9—61no—
68Dic9⋊C4—21no—
69Dic9⋊C4—61no—
70C4×Dic9—21no—
71C4×Dic9C2×D311yescharacteristic
72C2×C9⋊C8—31no—
73C2×C9⋊C8—61no—
74C22×C3×C9C2×D411yescharacteristic
75C3×C4×C9—41no—
76C3×C4×C9C2×Q811yescharacteristic
77C3×C4×C9#4111yescharacteristic
78C3⋊Dic9—41no—
79C9×Dic3—41no—
80C3×Dic9—22no—
81C3×Dic9—21no—
82C3×Dic9—21no—
83C3×C4×Q8—91no—
84#890—91no—
85#887—31no—
86#870—91no—
87C3×#107—31no—
88#835—91no—
89#835—91no—
90#816—91no—
91Dic3⋊C8—91no—
92#810—91no—
93#808—91no—
94#808—91no—
95#808—181no—
96#807D911yescharacteristic
97C4⋊(C3⋊C8)—91no—
98C2×C32×C4—21no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—61no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—61no—
105C3×Dic6—32no—
106C3×Dic6—61no—
107C3⋊(C3⋊C8)—61no—
108C3×C3⋊C8—21no—
109C2×C4×C9—21no—
110C2×C4×C9—21no—
111C2×C4×C9—41no—
112C2×C4×C9—61no—
113C2×C4×C9C22×D311yescharacteristic
114C2×Dic9—21no—
115C2×Dic9—61no—
116C2×Dic9—121no—
117C2×Dic9#6711yescharacteristic
118C2×Dic9#6711yescharacteristic
119Dic18—21no—
120Dic18Dic612yes—
121C8×C9—61no—
122C9⋊C8—62no—
123C9⋊C8—61no—
124#479—271noSylow
125C2×C3×C9C22⋊Q811yescharacteristic
126C2×C3×C9#13511yescharacteristic
127C2×C3×C9#13711yescharacteristic
128C2×C3×Q8—91no—
129C2×Dic6—31no—
130C2×C3×C8—31no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—181no—
134C3×C4⋊C4C2×D911yescharacteristic
135C3×C4⋊C4D3211yescharacteristic
136C3×C42—91no—
137C3×C42—181no—
138C4⋊Dic3—91no—
139C4⋊Dic3—91no—
140C4⋊Dic3—181no—
141C4⋊Dic3—181no—
142C4⋊Dic3C2×D911yescharacteristic
143Dic3⋊C4—61no—
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—181no—
147C4×Dic3—31no—
148C4×Dic3—61no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—181no—
152C2×C3⋊C8C2×D911yescharacteristic
153C22×C32D3×D411yescharacteristic
154C32×C4—41no—
155C32×C4#23511yescharacteristic
156C32×C4D3×Q811yescharacteristic
157C3⋊Dic3—121no—
158C3×Dic3—41no—
159C3×Dic3—62no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C22×C9—21no—
163C22×C9C2×#6711yescharacteristic
164C4×C9—21no—
165C4×C9—21no—
166C4×C9—41no—
167C4×C9—81no—
168C4×C9—121no—
169C4×C9C2×Dic611yescharacteristic
170C4×C9#23511yescharacteristic
171Dic9—22no—
172Dic9—21no—
173Dic9—21no—
174Dic9—121no—
175Dic9—241no—
176#125—271no—
177C4×Q8—91no—
178#107—31no—
179#106—271no—
180C4⋊C8—271no—
181#103—271no—
182#103—271no—
183C3×C9#47911yescharacteristic, complemented, Sylow
184C3×Q8—92no—
185C3×Q8—181no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—61no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—181no—
194C2×C3×C4C22×D911yescharacteristic
195C2×C3×C4C2×D3211yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—31no—
198C2×Dic3—31no—
199C2×Dic3—61no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—361no—
203Dic6—32no—
204Dic6—61no—
205C3×C8—61no—
206C3⋊C8—21no—
207C3⋊C8—182no—
208C3⋊C8—181no—
209C3⋊C8—181no—
210C2×C32#89611yescharacteristic
211C2×C32#91211yescharacteristic
212C2×C32#91811yescharacteristic
213C2×C9—21no—
214C2×C9—21no—
215C2×C9—21no—
216C2×C9#92411yescharacteristic
217C2×C9#94211yescharacteristic
218C2×C9#95011yescharacteristic
219C2×Q8—91no—
220C2×C8—31no—
221C2×C8—271no—
222C4⋊C4—31no—
223C4⋊C4—181no—
224C4⋊C4—271no—
225C4⋊C4—541no—
226C4⋊C4—541no—
227C4⋊C4D3×D911yescharacteristic
228C42—91no—
229C42—181no—
230C22×C3—21no—
231C22×C3D4×D911yescharacteristic
232C22×C3D3×#6711yescharacteristic
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—81no—
238C3×C4—121no—
239C3×C4—182no—
240C3×C4—181no—
241C3×C4—181no—
242C3×C4#369711yescharacteristic
243C3×C4Q8×D911yescharacteristic
244C3×C4D3×Dic611yescharacteristic
245C3×C4#379811yescharacteristic
246Dic3—41no—
247Dic3—62no—
248Dic3—61no—
249Dic3—61no—
250Dic3—361no—
251Dic3—361no—
252Dic3—721no—
253C32#487711yescharacteristic
254C9—21no—
255C9#502211yescharacteristic, complemented
256Q8—92no—
257Q8—181no—
258C2×C4—21no—
259C2×C4—61no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—181no—
263C2×C4—541no—
264C2×C4C2×D3×D911yescharacteristic
265C8—61no—
266C8—541no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3#6345811yescharacteristic
271C2×C3#6347411yescharacteristic
272C2×C3#6348011yescharacteristic
273C2×C3#6385111yescharacteristic
274C2×C3#6393311yescharacteristic
275C2×C3#6394811yescharacteristic
276C22D9×#6711yescharacteristic
277C4—41no—
278C4—121no—
279C4—181no—
280C4—181no—
281C4—182no—
282C4—1081no—
283C4D9×Dic611yescharacteristic
284C4#8864011yescharacteristic
285C3—21no—
286C3#1058855711yescharacteristic, complemented
287C3#1059054411yescharacteristic
288C2#1172093911yescharacteristic
289C2#1172117711yescharacteristic
290C2#1172122211yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#479
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
183. C3×C9#479semidirect
255. C9#5022semidirect
286. C3#10588557semidirect

Maximal subgroups

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

subgroupindex
C3×#634492
#117209352
#117205262
C9×#8072
#117202762
Dic9⋊(C3⋊C8)2
#117202682
#105905443
#105885573