Finite Group Databasebeta

← #13138253

Subgroups — #13138253

Order 1728.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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#11720969C211yesmaximal, characteristic
2#11720913C211yesmaximal, characteristic
3#11720539C211yesmaximal, characteristic
4C9×#817C211yesmaximal, characteristic
5C3×Dic9⋊C8C211yesmaximal, characteristic
6#11720269C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10590802—31nomaximal
9#10588534—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×#6189—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4×Dic9C2211yescharacteristic
17C2×C3×C9⋊C8C2211yescharacteristic
18#88442—21no—
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#63918—31no—
22#63844—31no—
23#63643—31no—
24C3×#817D311yescharacteristic
25C3×Dic3⋊C8—31no—
26#63573—31no—
27#63567—31no—
28#63450—31no—
29#63449—31no—
30C9×#106—31no—
31#63376—31no—
32#63376—61no—
33Dic9⋊C8D311yescharacteristic
34#63368—31no—
35#63367—31no—
36C2×C3×C4×C9C2311yescharacteristic
37C2×C3⋊Dic9—21no—
38C2×C3⋊Dic9—21no—
39#6226—21no—
40#6226Q812yescomplemented
41C2×C9×Dic3—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45#6189—42no—
46C3×C8×C9—21no—
47C3×C9⋊C8—21no—
48#4854—91no—
49#4851—91no—
50C2×#634—31no—
51C2×#627—61no—
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54C3×C4⋊Dic3C2×D311yescharacteristic
55C3×C4⋊Dic3C2×D311yescharacteristic
56C3×C4×Dic3—31no—
57C2×C3×C3⋊C8—31no—
58#3724—61no—
59Dic3⋊Dic3—61no—
60Dic32—61no—
61C2×Dic18—31no—
62C2×Dic18—61no—
63C2×Dic18—61no—
64C2×C8×C9—21no—
65C2×C8×C9C2×D311yescharacteristic
66C9×C4⋊C4—31no—
67C9×C4⋊C4—31no—
68C4⋊Dic9—31no—
69C4⋊Dic9—61no—
70Dic9⋊C4—61no—
71Dic9⋊C4—61no—
72C4×Dic9—61no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8C2×D311yescharacteristic
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9C2×Q811yescharacteristic
77C3×C4×C9#4111yescharacteristic
78C3⋊Dic9—22no—
79C3⋊Dic9—41no—
80C9×Dic3—41no—
81C9×Dic3—41no—
82C3×Dic9—21no—
83C3×Dic9SD1612yes—
84#888—91no—
85#887—91no—
86#869—91no—
87C4×Dic6—91no—
88C3×#106—31no—
89C3×C4⋊C8—91no—
90#817—91no—
91#817D911yescharacteristic
92#816—91no—
93#816—91no—
94#816—91no—
95#816—181no—
96Dic3⋊C8—31no—
97#810—91no—
98#808—91no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—61no—
102#634—32no—
103#634—61no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108#627—122no—
109C32×C8—21no—
110C3×C3⋊C8—61no—
111C2×C4×C9—21no—
112C2×C4×C9—61no—
113C2×C4×C9—61no—
114C2×C4×C9C22×D311yescharacteristic
115C2×Dic9—61no—
116C2×Dic9—61no—
117C2×Dic9—121no—
118C2×Dic9—121no—
119C2×Dic9D1211yescharacteristic
120C2×Dic9D1211yescharacteristic
121Dic18—32no—
122Dic18—62no—
123Dic18—61no—
124Dic18—121no—
125Dic18—122no—
126C8×C9—21no—
127C8×C9—41no—
128C9⋊C8—21no—
129#475—271noSylow
130C2×C3×C9C22⋊Q811yescharacteristic
131C2×C3×C9C2×SD1611yescharacteristic
132C2×C3×C9#13711yescharacteristic
133C2×Dic6—91no—
134C2×Dic6—91no—
135C2×Dic6—181no—
136C2×Dic6—181no—
137C2×Dic6—181no—
138C2×C3×C8—21no—
139C2×C3×C8—91no—
140C2×C3×C8C2×D911yescharacteristic
141C2×C3×C8D3211yescharacteristic
142C3×C4⋊C4—31no—
143C3×C4⋊C4—31no—
144C3×C42—91no—
145C4⋊Dic3—91no—
146C4⋊Dic3—91no—
147C4⋊Dic3—181no—
148C4⋊Dic3—181no—
149C4⋊Dic3C2×D911yescharacteristic
150C4⋊Dic3C2×D911yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154C4×Dic3—31no—
155C4×Dic3—181no—
156C4×Dic3—181no—
157C2×C3⋊C8—31no—
158C22×C32D3×D411yescharacteristic
159C32×C4#23511yescharacteristic
160C32×C4D3×Q811yescharacteristic
161C3⋊Dic3—62no—
162C3⋊Dic3—121no—
163C3×Dic3—32no—
164C3×Dic3—41no—
165C3×Dic3—41no—
166C3×Dic3—61no—
167C22×C9—21no—
168C22×C9C2×D1211yescharacteristic
169C4×C9—21no—
170C4×C9—21no—
171C4×C9—121no—
172C4×C9—121no—
173C4×C9#23711yescharacteristic
174C4×C9D3×Q811yescharacteristic
175Dic9—21no—
176Dic9—62no—
177Dic9—122no—
178Dic9—121no—
179Dic9—241no—
180Dic9#20412yes—
181C4⋊Q8—271no—
182C4×Q8—271no—
183#106—31no—
184#106—271no—
185C4⋊C8—91no—
186#103—271no—
187#103—271no—
188C3×C9#47511yescharacteristic, complemented, Sylow
189C2×C3×C4—21no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4C22×D911yescharacteristic
195C2×C3×C4C2×D3211yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—181no—
201C2×Dic3—181no—
202C2×Dic3—181no—
203C2×Dic3—181no—
204C2×Dic3—361no—
205C2×Dic3—361no—
206Dic6—92no—
207Dic6—92no—
208Dic6—182no—
209Dic6—181no—
210Dic6—181no—
211Dic6—361no—
212Dic6—362no—
213Dic6—362no—
214C3×C8—21no—
215C3×C8—21no—
216C3×C8—41no—
217C3×C8—181no—
218C3⋊C8—61no—
219C2×C32#89611yescharacteristic
220C2×C32D3×SD1611yescharacteristic
221C2×C32#91511yescharacteristic
222C2×C9—21no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9#89711yescharacteristic
226C2×C9C2×#20411yescharacteristic
227C2×C9#90911yescharacteristic
228C2×Q8—271no—
229C2×Q8—541no—
230C2×C8—91no—
231C2×C8D3×D911yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—271no—
235C4⋊C4—541no—
236C4⋊C4—541no—
237C42—91no—
238C42—541no—
239C22×C3—21no—
240C22×C3D4×D911yescharacteristic
241C22×C3D3×D1211yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—92no—
245C3×C4—121no—
246C3×C4—121no—
247C3×C4—181no—
248C3×C4#369711yescharacteristic
249C3×C4Q8×D911yescharacteristic
250C3×C4#379711yescharacteristic
251C3×C4#380411yescharacteristic
252Dic3—32no—
253Dic3—41no—
254Dic3—41no—
255Dic3—61no—
256Dic3—182no—
257Dic3—182no—
258Dic3—362no—
259Dic3—361no—
260Dic3—361no—
261Dic3—721no—
262C32#485411yescharacteristic
263C9—21no—
264C9#485111yescharacteristic, complemented
265Q8—272no—
266Q8—541no—
267Q8—1082no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—541no—
273C2×C4—541no—
274C2×C4C2×D3×D911yescharacteristic
275C8—21no—
276C8—181no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3#6345811yescharacteristic
281C2×C3SD16×D911yescharacteristic
282C2×C3#6347711yescharacteristic
283C2×C3D3×#20411yescharacteristic
284C2×C3#6380011yescharacteristic
285C2×C3#6389511yescharacteristic
286C22D9×D1211yescharacteristic
287C4—92no—
288C4—121no—
289C4—121no—
290C4—181no—
291C4—542no—
292C4—1081no—
293C4(C3⋊D9)⋊Q811yescharacteristic
294C4#8865811yescharacteristic
295C3—21no—
296C3#1058853411yescharacteristic, complemented
297C3#1059080211yescharacteristic
298C2D9×#20411yescharacteristic
299C2#1172082711yescharacteristic
300C2#1172099411yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#475
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
40. #6226Q8semidirect
188. C3×C9#475semidirect
264. C9#4851semidirect
296. C3#10588534semidirect

Maximal subgroups

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

subgroupindex
#117209692
#117209132
#117205392
C9×#8172
C3×Dic9⋊C82
#117202692
#117202642
#105908023
#105885343