Finite Group Databasebeta

← #13137552

Subgroups — #13137552

Order 1728.

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

normal subgroups only

This group has 306 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×#63450C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic
3#11720525C211yesmaximal, characteristic
4C9×#808C211yesmaximal, characteristic
5#11720279C211yesmaximal, characteristic
6#11720274C211yesmaximal, characteristic
7#11720265C211yesmaximal, characteristic
8#10590421—31nomaximal
9#10588536—31nomaximal
10C2×C3×Dic18—21no—
11C3×C9×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×C4⋊Dic9C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×#888—31no—
22#63869—31no—
23#63634—31no—
24C3×#808D311yescharacteristic
25#63575—31no—
26#63574—31no—
27#63554—31no—
28#63452—31no—
29#63450D311yescharacteristic
30C9×#106—31no—
31#63378—31no—
32#63375—31no—
33#63367—31no—
34#63366—31no—
35#63366—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—42no—
44C3⋊(C9⋊C8)Q812yescomplemented
45C9×C3⋊C8Q812yescomplemented
46#5081—91no—
47#4856—91no—
48C2×C3×Dic6—61no—
49C32×C4⋊C4C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—31no—
51C2×C3⋊(C3⋊C8)—31no—
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3C2×D311yescharacteristic
54C3×Dic3⋊C4—61no—
55C3×C4×Dic3—31no—
56C2×C3×C3⋊C8C2×D311yescharacteristic
57#3724—61no—
58Dic3⋊Dic3—61no—
59C2×Dic18—21no—
60C2×C8×C9—31no—
61C9×C4⋊C4—21no—
62C9×C4⋊C4—31no—
63C9×C4⋊C4C2×D311yescharacteristic
64C4⋊Dic9—31no—
65C4⋊Dic9—61no—
66C4⋊Dic9—61no—
67C4⋊Dic9C2×D311yescharacteristic
68Dic9⋊C4—21no—
69Dic9⋊C4—61no—
70C4×Dic9C2×D311yescharacteristic
71C2×C9⋊C8—31no—
72C2×C9⋊C8—61no—
73C22×C3×C9C2×D411yescharacteristic
74C3×C4×C9—41no—
75C3×C4×C9C2×Q811yescharacteristic
76C3×C4×C9C2×Q811yescharacteristic
77C3⋊Dic9—41no—
78C9×Dic3—41no—
79C3×Dic9—21no—
80C3×Dic9—21no—
81C3×Dic9—41no—
82C3×C4⋊Q8—91no—
83#890—91no—
84#888—31no—
85#870—91no—
86C3×#106—31no—
87#818—91no—
88#815—91no—
89#808—91no—
90#808—91no—
91#808D911yescharacteristic
92#807—91no—
93#807—91no—
94#807—91no—
95#807—181no—
96#803—91no—
97C2×C32×C4—21no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3—61no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—31no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—61no—
104C3×Dic6—122no—
105C3⋊(C3⋊C8)—32no—
106C3×C3⋊C8Dic612yes—
107C2×C4×C9—21no—
108C2×C4×C9—21no—
109C2×C4×C9—41no—
110C2×C4×C9—61no—
111C2×C4×C9C22×D311yescharacteristic
112C2×Dic9—21no—
113C2×Dic9—61no—
114C2×Dic9—121no—
115C2×Dic9#6711yescharacteristic
116C2×Dic9#6711yescharacteristic
117Dic18—42no—
118C8×C9—32no—
119C9⋊C8—32no—
120C9⋊C8—62no—
121#501—271noSylow
122C2×C3×C9C4⋊Q811yescharacteristic
123C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
124C2×C3×C9#13711yescharacteristic
125C2×C3×Q8—181no—
126C2×Dic6—61no—
127C2×C3×C8—31no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—181no—
132C3×C4⋊C4C2×D911yescharacteristic
133C3×C4⋊C4D3211yescharacteristic
134C3×C42—91no—
135C4⋊Dic3—31no—
136C4⋊Dic3—91no—
137C4⋊Dic3—91no—
138C4⋊Dic3—181no—
139C4⋊Dic3—181no—
140C4⋊Dic3C2×D911yescharacteristic
141Dic3⋊C4—61no—
142Dic3⋊C4—181no—
143Dic3⋊C4—181no—
144Dic3⋊C4—181no—
145C4×Dic3—31no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—181no—
149C2×C3⋊C8C2×D911yescharacteristic
150C22×C32D3×D411yescharacteristic
151C32×C4—41no—
152C32×C4C2×Dic611yescharacteristic
153C32×C4D3×Q811yescharacteristic
154C3⋊Dic3—121no—
155C3×Dic3—41no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—121no—
159C22×C9—21no—
160C22×C9C2×#6711yescharacteristic
161C4×C9—21no—
162C4×C9—21no—
163C4×C9—41no—
164C4×C9—81no—
165C4×C9—121no—
166C4×C9D3×Q811yescharacteristic
167C4×C9D3×Q811yescharacteristic
168Dic9—21no—
169Dic9—21no—
170Dic9—41no—
171Dic9—121no—
172Dic9—241no—
173C4⋊Q8—91no—
174#125—271no—
175#107—271no—
176#107—271no—
177#106—31no—
178#106—271no—
179#97—271no—
180C3×C9#50111yescharacteristic, complemented, Sylow
181C3×Q8—362no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—61no—
187C2×C3×C4—91no—
188C2×C3×C4—91no—
189C2×C3×C4—181no—
190C2×C3×C4C22×D911yescharacteristic
191C2×C3×C4C2×D3211yescharacteristic
192C2×Dic3—21no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—61no—
196C2×Dic3—181no—
197C2×Dic3—181no—
198C2×Dic3—361no—
199Dic6—122no—
200C3×C8—32no—
201C3⋊C8—92no—
202C3⋊C8—92no—
203C3⋊C8—182no—
204C3⋊C8Dic1812yescomplemented
205C2×C32#88811yescharacteristic
206C2×C32#91411yescharacteristic
207C2×C32#91511yescharacteristic
208C2×C9—21no—
209C2×C9—21no—
210C2×C9—21no—
211C2×C9#93211yescharacteristic
212C2×C9#94211yescharacteristic
213C2×C9#94411yescharacteristic
214C2×Q8—181no—
215C2×C8—31no—
216C2×C8—271no—
217C4⋊C4—31no—
218C4⋊C4—91no—
219C4⋊C4—181no—
220C4⋊C4—271no—
221C4⋊C4—541no—
222C4⋊C4—541no—
223C4⋊C4D3×D911yescharacteristic
224C42—91no—
225C22×C3—21no—
226C22×C3D4×D911yescharacteristic
227C22×C3D3×#6711yescharacteristic
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—81no—
233C3×C4—121no—
234C3×C4—181no—
235C3×C4—181no—
236C3×C4—361no—
237C3×C4C2×Dic1811yescharacteristic
238C3×C4Q8×D911yescharacteristic
239C3×C4D3×Dic611yescharacteristic
240C3×C4#379711yescharacteristic
241Dic3—41no—
242Dic3—61no—
243Dic3—61no—
244Dic3—121no—
245Dic3—361no—
246Dic3—361no—
247Dic3—721no—
248C32#485611yescharacteristic
249C9—21no—
250C9#508111yescharacteristic, complemented
251Q8—362no—
252C2×C4—21no—
253C2×C4—61no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—181no—
257C2×C4—541no—
258C2×C4C2×D3×D911yescharacteristic
259C8—32no—
260C8—272no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3#6345011yescharacteristic
265C2×C3#6347611yescharacteristic
266C2×C3#6347711yescharacteristic
267C2×C3#6383911yescharacteristic
268C2×C3#6393411yescharacteristic
269C2×C3#6394611yescharacteristic
270C22D9×#6711yescharacteristic
271C4—41no—
272C4—121no—
273C4—181no—
274C4—181no—
275C4—361no—
276C4—1081no—
277C4(C3⋊D9)⋊Q811yescharacteristic
278C4D3×Dic1811yescharacteristic
279C3—21no—
280C3#1058853611yescharacteristic, complemented
281C3#1059042111yescharacteristic
282C2#1172090411yescharacteristic
283C2#1172118411yescharacteristic
284C2#1172121411yescharacteristic
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#501
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
44. C3⋊(C9⋊C8)Q8semidirect
45. C9×C3⋊C8Q8semidirect
180. C3×C9#501semidirect
204. C3⋊C8Dic18semidirect
250. C9#5081semidirect
280. C3#10588536semidirect

Maximal subgroups

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

subgroupindex
C3×#634502
#117209352
#117205252
C9×#8082
#117202792
#117202742
#117202652
#105904213
#105885363