Finite Group Databasebeta

← #13138268

Subgroups — #13138268

Order 1728.

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

normal subgroups only

This group has 326 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
1#11720971C211yesmaximal, characteristic
2#11720910C211yesmaximal, characteristic
3#11720537C211yesmaximal, characteristic
4C9×#818C211yesmaximal, characteristic
5C3×#63376C211yesmaximal, characteristic
6#11720268C211yesmaximal, characteristic
7#11720263C211yesmaximal, characteristic
8#10590817—31nomaximal
9#10588555—31nomaximal
10C2×C3×Dic18C2211yescharacteristic
11C2×#6189—21no—
12C2×C3×C8×C9C2211yescharacteristic
13C4×C3⋊Dic9C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18#88442—21no—
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#63919—31no—
22#63843—31no—
23#63641—31no—
24C3×#818D311yescharacteristic
25C3×#816—31no—
26#63576—31no—
27#63565—31no—
28#63450—31no—
29#63449—31no—
30C9×#107—31no—
31#63376D311yescharacteristic
32Dic9⋊C8—31no—
33Dic9⋊C8—61no—
34#63368—31no—
35#63366—31no—
36C2×C3×C4×C9C2311yescharacteristic
37C2×C3⋊Dic9D411yescharacteristic
38C2×C3⋊Dic9D411yescharacteristic
39C2×C9×Dic3—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9—21no—
43C3×Dic18—21no—
44C3×Dic18Q812yescomplemented
45#6189—42no—
46C3×C8×C9—21no—
47C3⋊(C9⋊C8)—21no—
48#4875—91no—
49#4793—91no—
50C2×C3×Dic6—31no—
51C2×#627—61no—
52C2×C32×C8C2×D311yescharacteristic
53C4×C3⋊Dic3—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C3×C4⋊Dic3C2×D311yescharacteristic
58#3724—61no—
59Dic3⋊Dic3—61no—
60Dic32—61no—
61C2×Dic18—61no—
62C2×Dic18C2×D311yescharacteristic
63C2×C8×C9—21no—
64C2×C8×C9C2×D311yescharacteristic
65C9×C4⋊C4—31no—
66C9×C4⋊C4—31no—
67C4⋊Dic9C2×D311yescharacteristic
68Dic9⋊C4—61no—
69Dic9⋊C4—61no—
70C4×Dic9—31no—
71C4×Dic9—61no—
72C4×Dic9—61no—
73C2×C9⋊C8—31no—
74C2×C9⋊C8—61no—
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9C2×Q811yescharacteristic
77C3×C4×C9#4111yescharacteristic
78C3⋊Dic9—21no—
79C3⋊Dic9Q1612yes—
80C9×Dic3—41no—
81C9×Dic3—41no—
82C3×Dic9—22no—
83C3×Dic9—41no—
84Q8×Dic3—91no—
85#888—91no—
86#888—91no—
87#887—91no—
88C3×#107—31no—
89C3×#103—91no—
90#835—91no—
91#818D911yescharacteristic
92#816—31no—
93Dic3⋊C8—91no—
94Dic3⋊C8—91no—
95Dic3⋊C8—181no—
96#810—91no—
97#807—91no—
98#807—91no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—31no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×Dic6—32no—
107C3×Dic6—61no—
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—21no—
116C2×Dic9—21no—
117C2×Dic9—31no—
118C2×Dic9—31no—
119C2×Dic9—61no—
120C2×Dic9—61no—
121Dic18—21no—
122Dic18—122no—
123Dic18Dic612yescomplemented
124C8×C9—21no—
125C8×C9—41no—
126C9⋊C8—61no—
127C9⋊C8—121no—
128#477—271noSylow
129C2×C3×C9C22⋊Q811yescharacteristic
130C2×C3×C9C2×Q1611yescharacteristic
131C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
132C2×C3×Q8—91no—
133C2×Dic6—31no—
134C2×Dic6—181no—
135C2×Dic6—181no—
136C2×C3×C8—21no—
137C2×C3×C8C2×D911yescharacteristic
138C2×C3×C8D3211yescharacteristic
139C3×C4⋊C4—31no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—91no—
142C4⋊Dic3—31no—
143C4⋊Dic3—181no—
144C4⋊Dic3C2×D911yescharacteristic
145C4⋊Dic3C2×D911yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149C4×Dic3—91no—
150C4×Dic3—91no—
151C4×Dic3—181no—
152C4×Dic3—181no—
153C4×Dic3—181no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—181no—
157C22×C32D3×D411yescharacteristic
158C32×C4#23511yescharacteristic
159C32×C4D3×Q811yescharacteristic
160C3⋊Dic3—32no—
161C3⋊Dic3—61no—
162C3×Dic3—41no—
163C3×Dic3—41no—
164C3×Dic3—62no—
165C3×Dic3—121no—
166C22×C9—21no—
167C22×C9D3×D411yescharacteristic
168C4×C9—21no—
169C4×C9—21no—
170C4×C9—121no—
171C4×C9—121no—
172C4×C9C2×Dic611yescharacteristic
173C4×C9#23711yescharacteristic
174Dic9—22no—
175Dic9—32no—
176Dic9—41no—
177Dic9—62no—
178Dic9—61no—
179Dic9—121no—
180C4⋊Q8—271no—
181C4×Q8—271no—
182#107—31no—
183#107—271no—
184C4⋊C8—271no—
185#103—91no—
186#103—271no—
187C3×C9#47711yescharacteristic, complemented, Sylow
188C3×Q8—92no—
189C3×Q8—181no—
190C2×C3×C4—21no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C22×D911yescharacteristic
196C2×C3×C4C2×D3211yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—91no—
202C2×Dic3—91no—
203C2×Dic3—91no—
204C2×Dic3—91no—
205C2×Dic3—181no—
206C2×Dic3—181no—
207Dic6—32no—
208Dic6—61no—
209Dic6—362no—
210Dic6—362no—
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—41no—
214C3⋊C8—181no—
215C3⋊C8—181no—
216C3⋊C8—361no—
217C2×C32#89611yescharacteristic
218C2×C32#91111yescharacteristic
219C2×C32D3×Q1611yescharacteristic
220C2×C9—21no—
221C2×C9—21no—
222C2×C9—21no—
223C2×C9#87811yescharacteristic
224C2×C9#91411yescharacteristic
225C2×C9D3×Q1611yescharacteristic
226C2×Q8—91no—
227C2×Q8—541no—
228C2×C8—271no—
229C2×C8D3×D911yescharacteristic
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—91no—
233C4⋊C4—541no—
234C4⋊C4—541no—
235C42—271no—
236C42—541no—
237C22×C3—21no—
238C22×C3D4×D911yescharacteristic
239C22×C3#380211yescharacteristic
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—121no—
243C3×C4—121no—
244C3×C4—182no—
245C3×C4—361no—
246C3×C4#369711yescharacteristic
247C3×C4Q8×D911yescharacteristic
248C3×C4D3×Dic611yescharacteristic
249C3×C4#380411yescharacteristic
250Dic3—41no—
251Dic3—41no—
252Dic3—62no—
253Dic3—92no—
254Dic3—92no—
255Dic3—121no—
256Dic3—182no—
257Dic3—181no—
258Dic3—181no—
259Dic3—361no—
260C32#487511yescharacteristic
261C9—21no—
262C9#479311yescharacteristic, complemented
263Q8—92no—
264Q8—181no—
265Q8—1082no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—271no—
271C2×C4—271no—
272C2×C4C2×D3×D911yescharacteristic
273C8—21no—
274C8—541no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3#6345811yescharacteristic
279C2×C3#6347311yescharacteristic
280C2×C3Q16×D911yescharacteristic
281C2×C3#6379811yescharacteristic
282C2×C3#6380211yescharacteristic
283C2×C3#6389911yescharacteristic
284C22#8864611yescharacteristic
285C4—121no—
286C4—121no—
287C4—182no—
288C4—272no—
289C4—361no—
290C4—541no—
291C4D9×Dic611yescharacteristic
292C4#8865811yescharacteristic
293C3—21no—
294C3#1058855511yescharacteristic, complemented
295C3#1059081711yescharacteristic
296C2#1172081911yescharacteristic
297C2#1172082911yescharacteristic
298C2#1172099911yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#477
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
44. C3×Dic18Q8semidirect
123. Dic18Dic6semidirect
187. C3×C9#477semidirect
262. C9#4793semidirect
294. C3#10588555semidirect

Maximal subgroups

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

subgroupindex
#117209712
#117209102
#117205372
C9×#8182
C3×#633762
#117202682
#117202632
#105908173
#105885553