Finite Group Databasebeta

← C3×#10588659 #13142554

Subgroups — C3×#10588659 #13142554

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic
3C2×C3×Dic9⋊C4C212yesmaximal
4C3×C9⋊#95C212yesmaximal
5C3×C9⋊#95C211yesmaximal, characteristic
6C3×#4979—31nomaximal
7#10588659C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×C9C2211yescharacteristic
9C22×C3×C4×C9C2211yescharacteristic
10C22×C3×C4×C9C2211yescharacteristic
11C22×C3×Dic9C2212yes—
12C22×C3×Dic9C2212yes—
13C2×C3×Dic18—22no—
14C2×C3×Dic18—24no—
15C3×C9×C4⋊C4—22no—
16C3×Dic9⋊C4—24no—
17C22×C3×Dic6—31no—
18C2×C32×C4⋊C4D311yescharacteristic
19C2×C3×Dic3⋊C4—32no—
20C22×Dic18C2×C311yescharacteristic
21C3×C3⋊#95—32no—
22C3×C3⋊#95—31no—
23C2×C9×C4⋊C4—21no—
24C2×C9×C4⋊C4C2×C311yescharacteristic
25C2×Dic9⋊C4C2×C312yes—
26C9⋊#95C2×C312yes—
27C9⋊#95C2×C311yescharacteristic
28C23×C3×C9C2311yescharacteristic
29C2×C3×C4×C9—21no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9—21no—
32C2×C3×C4×C9—21no—
33C2×C3×C4×C9—22no—
34C2×C3×C4×C9D412yes—
35C2×C3×C4×C9Q812yescomplemented
36C2×C3×Dic9—22no—
37C2×C3×Dic9—22no—
38C2×C3×Dic9—22no—
39C2×C3×Dic9—22no—
40C2×C3×Dic9D414yes—
41C2×C3×Dic9Q814yescomplemented
42C3×Dic18—28no—
43C3×#395—91no—
44#4979—31no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C2×C3×Dic6—64no—
51C2×C3×Dic6—62no—
52C32×C4⋊C4—22no—
53C3×Dic3⋊C4—64no—
54C22×C4×C9—21no—
55C22×C4×C9—21no—
56C22×C4×C9—21no—
57C22×C4×C9C22×C311yescharacteristic
58C22×C4×C9C22×C311yescharacteristic
59C22×C4×C9C22×C311yescharacteristic
60C22×Dic9C22×C312yes—
61C22×Dic9C22×C312yes—
62C2×Dic18—22no—
63C2×Dic18—24no—
64C9×C4⋊C4—24no—
65C9×C4⋊C4—22no—
66Dic9⋊C4—24no—
67C22×C3×C9C2×D412yes—
68C22×C3×C9C2×D411yescharacteristic
69C22×C3×C9C2×Q811yescharacteristic
70C22×C3×C9C2×Q812yes—
71C22×C3×C9#4111yescharacteristic
72C3×C4×C9—22no—
73C3×C4×C9—22no—
74C3×C4×C9—41no—
75C3×Dic9—24no—
76C3×Dic9—24no—
77C22×C3×Q8—91no—
78C22×Dic6—31no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—92no—
81C2×C3×C4⋊C4D911yescharacteristic
82C2×C3×C4⋊C4C3×D311yescharacteristic
83C2×Dic3⋊C4—32no—
84C3×#95—91no—
85C3×#95—92no—
86C3⋊#95—31no—
87C3⋊#95—32no—
88C23×C32C22×D311yescharacteristic
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4Dic612yes—
95C2×C32×C4#6712yes—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C3×Dic6—68no—
103C23×C9—21no—
104C23×C9C23×C311yescharacteristic
105C2×C4×C9—22no—
106C2×C4×C9—22no—
107C2×C4×C9—21no—
108C2×C4×C9—22no—
109C2×C4×C9—21no—
110C2×C4×C9—22no—
111C2×C4×C9—22no—
112C2×C4×C9—21no—
113C2×C4×C9—22no—
114C2×C4×C9—22no—
115C2×C4×C9—42no—
116C2×C4×C9—42no—
117C2×C4×C9C3×D412yes—
118C2×C4×C9C3×Q812yescomplemented
119C2×Dic9—22no—
120C2×Dic9—22no—
121C2×Dic9—22no—
122C2×Dic9—22no—
123C2×Dic9C3×D414yes—
124C2×Dic9C3×Q814yescomplemented
125Dic18—28no—
126#395—91noSylow
127C2×C3×C9#12011yescharacteristic
128C2×C3×C9C22⋊Q812yes—
129C2×C3×C9C22⋊Q811yescharacteristic
130C2×C3×C9C4⋊Q812yes—
131C2×C3×C9C4⋊Q811yescharacteristic
132C2×C3×Q8—184no—
133C2×C3×Q8—182no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—92no—
138C22×C3×C4—92no—
139C22×C3×C4C2×D911yescharacteristic
140C22×C3×C4C2×D911yescharacteristic
141C22×C3×C4C2×D911yescharacteristic
142C22×C3×C4C2×C3×D311yescharacteristic
143C22×C3×C4C2×C3×D311yescharacteristic
144C22×C3×C4C2×C3×D311yescharacteristic
145C22×Dic3—32no—
146C22×Dic3—32no—
147C2×Dic6—64no—
148C2×Dic6—62no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—184no—
153Dic3⋊C4—64no—
154C22×C32C2×Dic611yescharacteristic
155C22×C32#23511yescharacteristic
156C22×C32D3×D412yes—
157C22×C32D3×Q812yes—
158C22×C32C2×#6711yescharacteristic
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—41no—
162C3×Dic3—64no—
163C3×Dic3—64no—
164C22×C9—21no—
165C22×C9—21no—
166C22×C9—22no—
167C22×C9—22no—
168C22×C9—21no—
169C22×C9C2×C3×D412yes—
170C22×C9C2×C3×D411yescharacteristic
171C22×C9C2×C3×Q811yescharacteristic
172C22×C9C2×C3×Q812yes—
173C22×C9C3×#4111yescharacteristic
174C4×C9—24no—
175C4×C9—22no—
176C4×C9—22no—
177C4×C9—42no—
178C4×C9—42no—
179C4×C9—41no—
180Dic9—24no—
181Dic9—24no—
182C22×Q8—91no—
183C2×C4⋊C4—92no—
184C2×C4⋊C4C3×D911yescharacteristic
185#95—91no—
186#95—92no—
187C3×C9#39511yescharacteristic, complemented, Sylow
188C23×C3—21no—
189C23×C3C22×D911yescharacteristic
190C23×C3C22×C3×D311yescharacteristic
191C3×Q8—188no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—94no—
210C2×C3×C4—94no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4Dic1812yescomplemented
216C2×C3×C4#61112yes—
217C2×C3×C4C3×Dic612yes—
218C2×C3×C4C3×#6712yes—
219C2×Dic3—34no—
220C2×Dic3—34no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225Dic6—68no—
226C2×C32#88812yes—
227C2×C32#89612yes—
228C2×C32#92411yescharacteristic
229C2×C32#93711yescharacteristic
230C2×C32#94411yescharacteristic
231C2×C9—22no—
232C2×C9—22no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9C3×#12011yescharacteristic
237C2×C9C3×C22⋊Q812yes—
238C2×C9C3×C22⋊Q811yescharacteristic
239C2×C9C3×C4⋊Q812yes—
240C2×C9C3×C4⋊Q811yescharacteristic
241C2×Q8—184no—
242C2×Q8—182no—
243C22×C4—92no—
244C22×C4—92no—
245C22×C4C2×C3×D911yescharacteristic
246C22×C4C2×C3×D911yescharacteristic
247C22×C4C2×C3×D911yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—184no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—21no—
255C22×C3C2×Dic1811yescharacteristic
256C22×C3#369711yescharacteristic
257C22×C3D4×D912yes—
258C22×C3Q8×D912yes—
259C22×C3C2×#61111yescharacteristic
260C22×C3C2×C3×Dic611yescharacteristic
261C22×C3C3×#23511yescharacteristic
262C22×C3C3×D3×D412yes—
263C22×C3C3×D3×Q812yes—
264C22×C3C2×C3×#6711yescharacteristic
265C3×C4—24no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—42no—
271C3×C4—42no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—184no—
275C3×C4—184no—
276Dic3—64no—
277Dic3—64no—
278C32#497911yescharacteristic
279C9—21no—
280C9C3×#39511yescharacteristic, complemented
281C23C22×C3×D911yescharacteristic
282Q8—188no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—94no—
289C2×C4—94no—
290C2×C4—182no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4C3×Dic1812yescomplemented
295C2×C4C3×#61112yes—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—22no—
300C2×C3—22no—
301C2×C3#6345012yes—
302C2×C3#6345812yes—
303C2×C3#6348611yescharacteristic
304C2×C3#6349911yescharacteristic
305C2×C3#6350611yescharacteristic
306C2×C3C3×#88812yes—
307C2×C3C3×#89612yes—
308C2×C3C3×#92411yescharacteristic
309C2×C3C3×#93711yescharacteristic
310C2×C3C3×#94411yescharacteristic
311C22C2×C3×Dic1811yescharacteristic
312C22C3×#369711yescharacteristic
313C22C3×D4×D912yes—
314C22C3×Q8×D912yes—
315C22C2×C3×#61111yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—184no—
320C4—184no—
321C3—21no—
322C3#1058865911yescharacteristic, direct product factor
323C3C3×#497911yescharacteristic
324C2C3×#6345012yes—
325C2C3×#6345812yes—
326C2C3×#6348611yescharacteristic
327C2C3×#6349911yescharacteristic
328C2C3×#6350611yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socle, prime radicalC23×C32C22×D3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#395
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10588659C3direct (N × H)
35. C2×C3×C4×C9Q8semidirect
41. C2×C3×Dic9Q8semidirect
118. C2×C4×C9C3×Q8semidirect
124. C2×Dic9C3×Q8semidirect
187. C3×C9#395semidirect
215. C2×C3×C4Dic18semidirect
280. C9C3×#395semidirect
294. C2×C4C3×Dic18semidirect
322. C3#10588659direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×C3×C9×C4⋊C42
C2×C3×Dic9⋊C42
C3×C9⋊#952
C3×#49793
#105886593