Finite Group Databasebeta

← C3×#11724243 #72260098

Subgroups — C3×#11724243 #72260098

Order 2592.

460 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C22×C32×#157C211yesmaximal, characteristic
2C2×C3×C3⋊AGL(1,𝔽9)C212yesmaximal, complemented
3#11724243C311yesmaximal, characteristic, direct product factor
4C3×C22⋊AGL(1,𝔽9)—31nomaximal
5C22×C32×C3⋊D3C411yescharacteristic
6C2×C32×#157—22no—
7C2×C32×#157C411yescharacteristic
8C2×C32×#157C2211yescharacteristic
9C3×C3⋊AGL(1,𝔽9)—22no—
10C22×C3×#157—21no—
11C22×C3×#157D311yescharacteristic
12C22×C3×#157C2×C311yescharacteristic
13C2×C3⋊AGL(1,𝔽9)C2×C312yescomplemented
14C2×C3×AGL(1,𝔽9)—32no—
15C22×C34C811yescharacteristic, complemented
16C2×C32×C3⋊D3—21no—
17C2×C32×C3⋊D3—21no—
18C2×C32×C3⋊D3C811yescharacteristic, complemented
19C2×C32×C3⋊D3C2×C411yescharacteristic
20C32×#157—21no—
21C32×#157D412yes—
22C22⋊AGL(1,𝔽9)—31no—
23C3×#830—91nomaximal
24C22×C3×C3⋊D3—21no—
25C22×C3×C3⋊D3Dic311yescharacteristic
26C22×C3×C3⋊D3C3×C411yescharacteristic
27C22×C32×D3—121no—
28C2×C3×#157—24no—
29C2×C3×#157—21no—
30C2×C3×#157—21no—
31C2×C3×#157—22no—
32C2×C3×#157—22no—
33C2×C3×#157Dic311yescharacteristic
34C2×C3×#157C3×C411yescharacteristic
35C2×C3×#157C2×D311yescharacteristic
36C2×C3×#157C22×C311yescharacteristic
37C3⋊AGL(1,𝔽9)—22no—
38C3×AGL(1,𝔽9)—62no—
39C2×C34—21no—
40C2×C34C2×C811yescharacteristic
41C32×C3⋊D3—21no—
42C32×C3⋊D3C22⋊C411yescharacteristic
43C32×C3⋊D3M1611yescharacteristic
44C22×#157C3×D311yescharacteristic
45C2×AGL(1,𝔽9)—32no—
46C22×C32×C4—91no—
47C2×C3×C3⋊C8—92no—
48C22×C33—21no—
49C22×C33—41no—
50C22×C33—82no—
51C22×C33—81no—
52C22×C33—81no—
53C22×C33C3⋊C811yescharacteristic, complemented
54C22×C33C3×C811yescharacteristic, complemented
55C2×C3×C3⋊D3—22no—
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3—21no—
58C2×C3×C3⋊D3—21no—
59C2×C3×C3⋊D3—21no—
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3—21no—
62C2×C3×C3⋊D3—21no—
63C2×C3×C3⋊D3C3⋊C811yescharacteristic, complemented
64C2×C3×C3⋊D3C3×C811yescharacteristic, complemented
65C2×C3×C3⋊D3C2×Dic311yescharacteristic
66C2×C3×C3⋊D3C2×C3×C411yescharacteristic
67C2×C32×D3—121no—
68C2×C32×D3—121no—
69C2×C32×D3—122no—
70C2×C32×D3—122no—
71C3×#157—22no—
72C3×#157—22no—
73C3×#157—21no—
74C3×#157—21no—
75C3×#157#6712yes—
76C3×#157C3×D412yes—
77C3×C22⋊C8—271no—
78#830—91no—
79C34C22⋊C811yescharacteristic, complemented, Sylow
80C23×C32—91no—
81C22×C3⋊D3C3×Dic311yescharacteristic
82C22×C3×D3—121no—
83C22×C3×D3—121no—
84C22×C3×D3—122no—
85C2×#157—22no—
86C2×#157C3×Dic311yescharacteristic
87C2×#157C2×C3×D311yescharacteristic
88AGL(1,𝔽9)—62no—
89C2×C32×C4—91no—
90C2×C32×C4—91no—
91C2×C32×C4—182no—
92C3×C3⋊C8—182no—
93C2×C33—22no—
94C2×C33—21no—
95C2×C33—21no—
96C2×C33—21no—
97C2×C33—42no—
98C2×C33—41no—
99C2×C33—84no—
100C2×C33—82no—
101C2×C33—82no—
102C2×C33—81no—
103C2×C33—81no—
104C2×C33—82no—
105C2×C33C2×C3⋊C811yescharacteristic
106C2×C33C2×C3×C811yescharacteristic
107C3×C3⋊D3—22no—
108C3×C3⋊D3—21no—
109C3×C3⋊D3—21no—
110C3×C3⋊D3—21no—
111C3×C3⋊D3—21no—
112C3×C3⋊D3#20811yescharacteristic
113C3×C3⋊D3#21711yescharacteristic
114C3×C3⋊D3C3×C22⋊C411yescharacteristic
115C3×C3⋊D3C3×M1611yescharacteristic
116C32×D3—121no—
117C32×D3—121no—
118C32×D3—122no—
119C22×C3×C4—91no—
120C22×C3×C4—91no—
121C22×C3×C4—181no—
122C2×C3×C8—272no—
123C2×C3⋊C8—92no—
124C22×C32—42no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—82no—
128C22×C32—82no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32—82no—
132C22×C32—81no—
133C22×C32—81no—
134C22×C32—84no—
135C22×C32—91no—
136C22×C32—91no—
137C22×C32—181no—
138C22×C32—181no—
139C22×C32C3×C3⋊C811yescharacteristic, complemented
140C22×C32AGL(1,𝔽9)11yescharacteristic, complemented
141C2×C3⋊D3—21no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3C3×C3⋊C811yescharacteristic, complemented
144C2×C3⋊D3C2×C3×Dic311yescharacteristic
145C2×C3×D3—121no—
146C2×C3×D3—121no—
147C2×C3×D3—121no—
148C2×C3×D3—121no—
149C2×C3×D3—122no—
150C2×C3×D3—122no—
151C2×C3×D3—122no—
152C2×C3×D3—122no—
153C2×C3×D3—122no—
154C2×C3×D3—122no—
155C2×C3×D3—124no—
156C2×C3×D3—124no—
157#157—21no—
158#157C3×#6712yes—
159C32×C4—92no—
160C32×C4—181no—
161C22⋊C8—271noSylow
162C33—21no—
163C33—41no—
164C33—82no—
165C33—81no—
166C33—81no—
167C33#83011yescharacteristic, complemented
168C33C3×C22⋊C811yescharacteristic, complemented
169C23×C3—91no—
170C23×C3—91no—
171C23×C3—181no—
172C22×D3—121no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—182no—
180C2×C3×C4—182no—
181C2×C3×C4—184no—
182C3×C8—542no—
183C3⋊C8—182no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—44no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32—42no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—84no—
193C2×C32—82no—
194C2×C32—82no—
195C2×C32—82no—
196C2×C32—82no—
197C2×C32—82no—
198C2×C32—84no—
199C2×C32—82no—
200C2×C32—81no—
201C2×C32—81no—
202C2×C32—88no—
203C2×C32—84no—
204C2×C32—82no—
205C2×C32—81no—
206C2×C32—81no—
207C2×C32—84no—
208C2×C32—91no—
209C2×C32—91no—
210C2×C32—181no—
211C2×C32C2×C3×C3⋊C811yescharacteristic
212C2×C32C2×AGL(1,𝔽9)11yescharacteristic
213C3⋊D3—21no—
214C3⋊D3C3×#20811yescharacteristic
215C3⋊D3C3×#21711yescharacteristic
216C3×D3—121no—
217C3×D3—121no—
218C3×D3—121no—
219C3×D3—121no—
220C3×D3—122no—
221C3×D3—122no—
222C3×D3—124no—
223C3×D3—122no—
224C3×D3—122no—
225C22×C4—91no—
226C2×C8—272no—
227C22×C3—21no—
228C22×C3—41no—
229C22×C3—82no—
230C22×C3—81no—
231C22×C3—81no—
232C22×C3—91no—
233C22×C3—91no—
234C22×C3—91no—
235C22×C3—91no—
236C22×C3—181no—
237C22×C3—181no—
238C22×C3—181no—
239C22×C3—181no—
240C22×C3—181no—
241C22×C3—181no—
242C22×C3—182no—
243C22×C3—182no—
244C22×C3C3×AGL(1,𝔽9)11yescharacteristic, complemented
245C22×C3C3⋊AGL(1,𝔽9)11yescharacteristic, complemented
246C2×D3—121no—
247C2×D3—121no—
248C2×D3—122no—
249C2×D3—122no—
250C3×C4—92no—
251C3×C4—92no—
252C3×C4—182no—
253C3×C4—181no—
254C3×C4—181no—
255C3×C4—182no—
256C32—42no—
257C32—41no—
258C32—41no—
259C32—82no—
260C32—81no—
261C32—81no—
262C32—82no—
263C32—84no—
264C32—82no—
265C32—81no—
266C32—81no—
267C32C3×#83011yescharacteristic, complemented
268C32C22⋊AGL(1,𝔽9)11yescharacteristic, complemented
269C23—91no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—182no—
273C8—542no—
274C2×C3—22no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—42no—
279C2×C3—41no—
280C2×C3—84no—
281C2×C3—82no—
282C2×C3—82no—
283C2×C3—82no—
284C2×C3—81no—
285C2×C3—81no—
286C2×C3—91no—
287C2×C3—91no—
288C2×C3—91no—
289C2×C3—91no—
290C2×C3—181no—
291C2×C3—181no—
292C2×C3—182no—
293C2×C3—181no—
294C2×C3—181no—
295C2×C3C2×C3×AGL(1,𝔽9)11yescharacteristic
296C2×C3C2×C3⋊AGL(1,𝔽9)11yescharacteristic
297D3—121no—
298D3—121no—
299D3—122no—
300C22—91no—
301C22—91no—
302C22—181no—
303C22—181no—
304C22C3×C3⋊AGL(1,𝔽9)11yescharacteristic, complemented
305C4—92no—
306C4—181no—
307C3—21no—
308C3—41no—
309C3—82no—
310C3—81no—
311C3—81no—
312C3C3×C22⋊AGL(1,𝔽9)11yescharacteristic, complemented
313C3#1172424311yescharacteristic, direct product factor
314C2—21no—
315C2—91no—
316C2—91no—
317C2—181no—
318C2C2×C3×C3⋊AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊AGL(1,𝔽9)
commutatorC2×C33C2×C3×C8
FrattiniC2C2×C3×C3⋊AGL(1,𝔽9)
FittingC22×C34C8
socleC2×C34C2×C8
prime radicalC22×C32×C3⋊D3C4
cosocleC2×C3×#157C22×C3
hypercenterC22×C3C3⋊AGL(1,𝔽9)
hypocenterC33C3×C22⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊AGL(1,𝔽9)C2semidirect
3. #11724243C3direct (N × H)
13. C2×C3⋊AGL(1,𝔽9)C2×C3semidirect
15. C22×C34C8semidirect
18. C2×C32×C3⋊D3C8semidirect
53. C22×C33C3⋊C8semidirect
54. C22×C33C3×C8semidirect
63. C2×C3×C3⋊D3C3⋊C8semidirect
64. C2×C3×C3⋊D3C3×C8semidirect
79. C34C22⋊C8semidirect
139. C22×C32C3×C3⋊C8semidirect
140. C22×C32AGL(1,𝔽9)semidirect
143. C2×C3⋊D3C3×C3⋊C8semidirect
167. C33#830semidirect
168. C33C3×C22⋊C8semidirect
244. C22×C3C3×AGL(1,𝔽9)semidirect
245. C22×C3C3⋊AGL(1,𝔽9)semidirect
267. C32C3×#830semidirect
268. C32C22⋊AGL(1,𝔽9)semidirect
304. C22C3×C3⋊AGL(1,𝔽9)semidirect
312. C3C3×C22⋊AGL(1,𝔽9)semidirect
313. C3#11724243direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×#1572
C2×C3×C3⋊AGL(1,𝔽9)2
#117242433
C3×C22⋊AGL(1,𝔽9)3
C3×#8309