Finite Group Databasebeta

← #13140687

Subgroups — #13140687

Order 1728.

390 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#88547C211yesmaximal, characteristic
2C3×Q8×Dic9C211yesmaximal, characteristic
3#11720913C211yesmaximal, characteristic
4C9×#835C211yesmaximal, characteristic
5#11720277C211yesmaximal, characteristic
6C3⋊C8×Dic9C211yesmaximal, characteristic
7#11720263C211yesmaximal, characteristic
8#10591537—31nomaximal
9#10588470—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12#88547C414yescomplemented
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×C4⋊Dic9—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4×Dic9—21no—
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20#88442—21no—
21Dic3×Dic9—21no—
22C2×#3756—31no—
23C3×Q8×Dic3—31no—
24#63844—31no—
25C3×#835D311yescharacteristic
26#63576—31no—
27#63567—31no—
28Dic3×C3⋊C8—31no—
29Q8×Dic9D311yescharacteristic
30C2×#3674—31no—
31C2×#3674—61no—
32#63449—31no—
33C9×#103—31no—
34#63376—31no—
35C8×Dic9—31no—
36#63366—31no—
37C3×Q8×C9—21no—
38C3×Q8×C9C2×C412yes—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42#6226—21no—
43#6226C2×C412yes—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C2×C3×Dic9D411yescharacteristic
48C3⋊(C9⋊C8)C2×C412yes—
49C9×C3⋊C8—21no—
50C4×#216—91no—
51#4790—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×#634—31no—
54#3756—34no—
55C2×C3⋊(C3⋊C8)—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3—61no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C3×C4×Dic3—31no—
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62#3724—61no—
63Dic32—61no—
64C2×Q8×C9—21no—
65C2×Q8×C9C2×D311yescharacteristic
66C2×Dic18—31no—
67C2×Dic18—61no—
68C2×C8×C9—31no—
69C9×C4⋊C4—31no—
70#3674—34no—
71#3674—64no—
72C4⋊Dic9—21no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C4×Dic9—21no—
76C4×Dic9—61no—
77C4×Dic9C2×D311yescharacteristic
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80C22×C3×C9C2×D411yescharacteristic
81C3×C4×C9—22no—
82C3×C4×C9C22×C411yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—22no—
85C9×Dic3—41no—
86C3×Dic9—21no—
87C3×Dic9—41no—
88C3×Dic9Q1612yescomplemented
89C3×C4×Q8—91no—
90Q8×Dic3—31no—
91C2×#216—91no—
92C2×#216—91no—
93C2×#216—181no—
94#887—91no—
95C4×Dic6—91no—
96C3×#103—31no—
97#835D911yescharacteristic
98#816—91no—
99C8×Dic3—91no—
100#810—91no—
101#807—91no—
102#807—91no—
103C4×C3⋊C8—91no—
104C32×Q8—21no—
105C32×Q8C4×D312yes—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—61no—
109#634—32no—
110#634—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—61no—
115C3⋊(C3⋊C8)—32no—
116C3×C3⋊C8—21no—
117Q8×C9—21no—
118Q8×C9—22no—
119Q8×C9—41no—
120Q8×C9C4×D312yes—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—41no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9—61no—
128C2×Dic9—121no—
129C2×Dic9#6711yescharacteristic
130C2×Dic9#6711yescharacteristic
131Dic18—32no—
132Dic18—62no—
133Dic18—61no—
134Dic18—121no—
135C8×C9—61no—
136C9⋊C8—32no—
137C9⋊C8—62no—
138C4×Q16—271noSylow
139C2×C3×C9C4×D411yescharacteristic
140C2×C3×C9C2×Q1611yescharacteristic
141C2×C3×C9#13511yescharacteristic
142C2×C3×Q8—21no—
143C2×C3×Q8C2×D911yescharacteristic
144C2×C3×Q8D3211yescharacteristic
145C2×Dic6—91no—
146C2×Dic6—91no—
147C2×Dic6—181no—
148C2×C3×C8—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C42—91no—
153C3×C42—181no—
154#216—94no—
155#216—94no—
156#216—184no—
157C4⋊Dic3—31no—
158C4⋊Dic3—61no—
159C4⋊Dic3C2×D911yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162C4×Dic3—31no—
163C4×Dic3—61no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—181no—
169C2×C3⋊C8C2×D911yescharacteristic
170C22×C32D3×D411yescharacteristic
171C32×C4—22no—
172C32×C4C2×C4×D311yescharacteristic
173C32×C4#23711yescharacteristic
174C3⋊Dic3—62no—
175C3×Dic3—32no—
176C3×Dic3—41no—
177C3×Dic3—61no—
178C3×Dic3—121no—
179C22×C9—21no—
180C22×C9C2×#6711yescharacteristic
181C4×C9—22no—
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—42no—
185C4×C9—121no—
186C4×C9C2×C4×D311yescharacteristic
187C4×C9#23511yescharacteristic
188Dic9—21no—
189Dic9—41no—
190Dic9—62no—
191Dic9—122no—
192Dic9#21612yescomplemented
193C2×Q16—271no—
194C4×Q8—91no—
195C4×Q8—271no—
196#107—271no—
197#103—31no—
198#103—271no—
199C4×C8—271no—
200C3×C9C4×Q1611yescharacteristic, complemented, Sylow
201C3×Q8—21no—
202C3×Q8—21no—
203C3×Q8—22no—
204C3×Q8—41no—
205C3×Q8C4×D912yes—
206C3×Q8#62412yes—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—61no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—181no—
215C2×C3×C4C22×D911yescharacteristic
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—31no—
219C2×Dic3—31no—
220C2×Dic3—61no—
221C2×Dic3—181no—
222C2×Dic3—181no—
223C2×Dic3—361no—
224Dic6—92no—
225Dic6—92no—
226Dic6—182no—
227Dic6—181no—
228Dic6—181no—
229Dic6—361no—
230C3×C8—61no—
231C3⋊C8—21no—
232C3⋊C8—92no—
233C3⋊C8—92no—
234C3⋊C8—182no—
235C2×C32#67⋊C411yescharacteristic
236C2×C32#91611yescharacteristic
237C2×C32D3×Q1611yescharacteristic
238C2×C9—21no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9C4×#6711yescharacteristic
242C2×C9C2×#21611yescharacteristic
243C2×C9#95011yescharacteristic
244C2×Q8—271no—
245C2×Q8D3×D911yescharacteristic
246Q16—274no—
247C2×C8—31no—
248C2×C8—271no—
249C4⋊C4—31no—
250C4⋊C4—91no—
251C4⋊C4—181no—
252C4⋊C4—541no—
253C42—91no—
254C42—181no—
255C42—541no—
256C22×C3—21no—
257C22×C3D4×D911yescharacteristic
258C22×C3D3×#6711yescharacteristic
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—42no—
264C3×C4—92no—
265C3×C4—121no—
266C3×C4—181no—
267C3×C4—361no—
268C3×C4C2×C4×D911yescharacteristic
269C3×C4#369911yescharacteristic
270C3×C4#380411yescharacteristic
271C3×C4C2×#62411yescharacteristic
272Dic3—32no—
273Dic3—41no—
274Dic3—61no—
275Dic3—121no—
276Dic3—182no—
277Dic3—182no—
278Dic3—362no—
279C32#479011yescharacteristic
280C9—21no—
281C9C4×#21611yescharacteristic, complemented
282Q8—21no—
283Q8—272no—
284Q8—541no—
285Q8(C3⋊D9)⋊C412yes—
286C2×C4—21no—
287C2×C4—61no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4—541no—
292C2×C4C2×D3×D911yescharacteristic
293C8—61no—
294C8—272no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3#611⋊C411yescharacteristic
299C2×C3#6347811yescharacteristic
300C2×C3Q16×D911yescharacteristic
301C2×C3D3×#21611yescharacteristic
302C2×C3#6394711yescharacteristic
303C2×C3#6395211yescharacteristic
304C22D9×#6711yescharacteristic
305C4—22no—
306C4—92no—
307C4—121no—
308C4—181no—
309C4—361no—
310C4—542no—
311C4C2×(C3⋊D9)⋊C411yescharacteristic
312C4#8866211yescharacteristic
313C3—21no—
314C3#1058847011yescharacteristic, complemented
315C3#1059153711yescharacteristic
316C2#1172121511yescharacteristic
317C2D9×#21611yescharacteristic
318C2#1172124111yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9C4×Q16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
12. #88547C4semidirect
88. C3×Dic9Q16semidirect
192. Dic9#216semidirect
200. C3×C9C4×Q16semidirect
281. C9C4×#216semidirect
314. C3#10588470semidirect

Maximal subgroups

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

subgroupindex
C2×#885472
C3×Q8×Dic92
#117209132
C9×#8352
#117202772
C3⋊C8×Dic92
#117202632
#105915373
#105884703