Finite Group Databasebeta

← #13137554

Subgroups — #13137554

Order 1728.

294 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 89 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×#63452C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic
3#11720526C211yesmaximal, characteristic
4C9×#807C211yesmaximal, characteristic
5#11720279C211yesmaximal, characteristic
6C3⋊C8×Dic9C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic
8#10590413—31nomaximal
9#10588558—31nomaximal
10C3×C9×C4⋊C4C2211yescharacteristic
11C4⋊(C3⋊Dic9)C2211yescharacteristic
12C2×C3⋊(C9⋊C8)C2211yescharacteristic
13C9×C4⋊Dic3C2211yescharacteristic
14C3×C4⋊Dic9—21no—
15C3×C4⋊Dic9C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×#890—31no—
22#63869—31no—
23#63635—31no—
24C3×#807D311yescharacteristic
25#63575—31no—
26#63574—31no—
27Dic3×C3⋊C8—31no—
28#63452—31no—
29#63452D311yescharacteristic
30C9×#107—31no—
31#63378—31no—
32C8×Dic9—31no—
33#63367—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⋊(C9⋊C8)Q812yes—
44C9×C3⋊C8Q812yes—
45#5065—91no—
46#4878—91no—
47C32×C4⋊C4C2×D311yescharacteristic
48C4⋊(C3⋊Dic3)—31no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×C4⋊Dic3—31no—
51C3×C4⋊Dic3—61no—
52C3×C4⋊Dic3C2×D311yescharacteristic
53C3×Dic3⋊C4—61no—
54C3×C4×Dic3—31no—
55C2×C3×C3⋊C8C2×D311yescharacteristic
56#3724—61no—
57Dic3⋊Dic3—61no—
58C2×C8×C9—31no—
59C9×C4⋊C4—21no—
60C9×C4⋊C4—31no—
61C9×C4⋊C4C2×D311yescharacteristic
62C4⋊Dic9—21no—
63C4⋊Dic9—31no—
64C4⋊Dic9—61no—
65C4⋊Dic9—61no—
66C4⋊Dic9C2×D311yescharacteristic
67Dic9⋊C4—21no—
68Dic9⋊C4—61no—
69C4×Dic9C2×D311yescharacteristic
70C2×C9⋊C8—31no—
71C2×C9⋊C8—61no—
72C22×C3×C9C2×D411yescharacteristic
73C3×C4×C9—41no—
74C3×C4×C9C2×Q811yescharacteristic
75C3×C4×C9C2×Q811yescharacteristic
76C3⋊Dic9—41no—
77C9×Dic3—41no—
78C3×Dic9—21no—
79C3×Dic9—21no—
80C3×Dic9—41no—
81C3×#125—91no—
82#890—31no—
83#890—91no—
84#870—91no—
85C3×#107—31no—
86#818—91no—
87C8×Dic3—91no—
88#808—91no—
89#808—91no—
90#808—91no—
91#808—91no—
92#808—181no—
93#807—91no—
94#807D911yescharacteristic
95C4×C3⋊C8—91no—
96C2×C32×C4—21no—
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—61no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—31no—
101C2×C3×Dic3—31no—
102C2×C3×Dic3—61no—
103C3⋊(C3⋊C8)—32no—
104C3×C3⋊C8Dic612yes—
105C2×C4×C9—21no—
106C2×C4×C9—21no—
107C2×C4×C9—41no—
108C2×C4×C9—61no—
109C2×C4×C9C22×D311yescharacteristic
110C2×Dic9—21no—
111C2×Dic9—61no—
112C2×Dic9—121no—
113C2×Dic9#6711yescharacteristic
114C2×Dic9#6711yescharacteristic
115C8×C9—32no—
116C9⋊C8—32no—
117C9⋊C8—62no—
118#499—271noSylow
119C2×C3×C9C4⋊Q811yescharacteristic
120C2×C3×C9#13511yescharacteristic
121C2×C3×C9#13511yescharacteristic
122C2×C3×C8—31no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—91no—
126C3×C4⋊C4—181no—
127C3×C4⋊C4—181no—
128C3×C4⋊C4C2×D911yescharacteristic
129C3×C4⋊C4D3211yescharacteristic
130C3×C42—91no—
131C4⋊Dic3—31no—
132C4⋊Dic3—61no—
133C4⋊Dic3—91no—
134C4⋊Dic3—91no—
135C4⋊Dic3—181no—
136C4⋊Dic3—181no—
137C4⋊Dic3C2×D911yescharacteristic
138Dic3⋊C4—61no—
139Dic3⋊C4—181no—
140Dic3⋊C4—181no—
141Dic3⋊C4—181no—
142C4×Dic3—31no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—181no—
146C2×C3⋊C8C2×D911yescharacteristic
147C22×C32D3×D411yescharacteristic
148C32×C4—41no—
149C32×C4C2×Dic611yescharacteristic
150C32×C4D3×Q811yescharacteristic
151C3⋊Dic3—121no—
152C3×Dic3—41no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—121no—
156C22×C9—21no—
157C22×C9C2×#6711yescharacteristic
158C4×C9—21no—
159C4×C9—21no—
160C4×C9—41no—
161C4×C9—81no—
162C4×C9—121no—
163C4×C9D3×Q811yescharacteristic
164C4×C9D3×Q811yescharacteristic
165Dic9—21no—
166Dic9—21no—
167Dic9—41no—
168Dic9—121no—
169Dic9—241no—
170#125—91no—
171#125—271no—
172#107—31no—
173#107—271no—
174#106—271no—
175#106—271no—
176C4×C8—271no—
177C3×C9#49911yescharacteristic, complemented, Sylow
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—61no—
183C2×C3×C4—91no—
184C2×C3×C4—91no—
185C2×C3×C4—181no—
186C2×C3×C4C22×D911yescharacteristic
187C2×C3×C4C2×D3211yescharacteristic
188C2×Dic3—21no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—61no—
192C2×Dic3—181no—
193C2×Dic3—181no—
194C2×Dic3—361no—
195C3×C8—32no—
196C3⋊C8—92no—
197C3⋊C8—92no—
198C3⋊C8—182no—
199C3⋊C8Dic1812yes—
200C2×C32#88811yescharacteristic
201C2×C32#91211yescharacteristic
202C2×C32#91911yescharacteristic
203C2×C9—21no—
204C2×C9—21no—
205C2×C9—21no—
206C2×C9#94411yescharacteristic
207C2×C9#95011yescharacteristic
208C2×C9#95011yescharacteristic
209C2×C8—31no—
210C2×C8—271no—
211C4⋊C4—31no—
212C4⋊C4—91no—
213C4⋊C4—181no—
214C4⋊C4—181no—
215C4⋊C4—271no—
216C4⋊C4—541no—
217C4⋊C4—541no—
218C4⋊C4D3×D911yescharacteristic
219C42—91no—
220C22×C3—21no—
221C22×C3D4×D911yescharacteristic
222C22×C3D3×#6711yescharacteristic
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—81no—
228C3×C4—121no—
229C3×C4—181no—
230C3×C4—181no—
231C3×C4—361no—
232C3×C4C2×Dic1811yescharacteristic
233C3×C4Q8×D911yescharacteristic
234C3×C4D3×Dic611yescharacteristic
235C3×C4#379711yescharacteristic
236Dic3—41no—
237Dic3—61no—
238Dic3—61no—
239Dic3—121no—
240Dic3—361no—
241Dic3—361no—
242Dic3—721no—
243C32#487811yescharacteristic
244C9—21no—
245C9#506511yescharacteristic, complemented
246C2×C4—21no—
247C2×C4—61no—
248C2×C4—91no—
249C2×C4—91no—
250C2×C4—181no—
251C2×C4—541no—
252C2×C4C2×D3×D911yescharacteristic
253C8—32no—
254C8—272no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3#6345011yescharacteristic
259C2×C3#6347411yescharacteristic
260C2×C3#6348111yescharacteristic
261C2×C3#6383911yescharacteristic
262C2×C3#6393311yescharacteristic
263C2×C3#6394911yescharacteristic
264C22D9×#6711yescharacteristic
265C4—41no—
266C4—121no—
267C4—181no—
268C4—181no—
269C4—361no—
270C4—1081no—
271C4(C3⋊D9)⋊Q811yescharacteristic
272C4D3×Dic1811yescharacteristic
273C3—21no—
274C3#1058855811yescharacteristic, complemented
275C3#1059041311yescharacteristic
276C2#1172090411yescharacteristic
277C2#1172117711yescharacteristic
278C2#1172122311yescharacteristic
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#499
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
177. C3×C9#499semidirect
245. C9#5065semidirect
274. C3#10588558semidirect

Maximal subgroups

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

subgroupindex
C3×#634522
#117209352
#117205262
C9×#8072
#117202792
C3⋊C8×Dic92
#117202652
#105904133
#105885583