Finite Group Databasebeta

← #10588672⋊C3 #13142663

Subgroups — #10588672⋊C3 #13142663

Order 1728.

556 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 136 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×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
43−1+2⋊#95C212yesmaximal, complemented
53−1+2⋊#95C211yesmaximal, characteristic, complemented
6C3×#4992—31nomaximal
7#10588672C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)C2211yescharacteristic
9C22×C4×3−1+2C2212yes—
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4C2212yes—
12C22×3−1+2⋊C4C2211yescharacteristic
13C4⋊C4×3−1+2—22no—
14(C2×AGL(1,ℤ/9))⋊C4—24no—
15(C2×AGL(1,ℤ/9))⋊C4—22no—
16C2×C32×C4⋊C4D311yescharacteristic
17C2×C3×(C2×D3)⋊C4—32no—
18C2×C3×(C2×D3)⋊C4—31no—
19C3×C3⋊#95—32no—
20C3×C3⋊#95—31no—
21C2×C9×C4⋊C4—21no—
22C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
23C2×(C2×D9)⋊C4C2×C312yes—
24C2×(C2×D9)⋊C4C2×C311yescharacteristic
25C9⋊#95C2×C312yescomplemented
26C9⋊#95C2×C311yescharacteristic, complemented
27C23×3−1+2C2311yescharacteristic
28C22×AGL(1,ℤ/9)—21no—
29C22×AGL(1,ℤ/9)—21no—
30C22×AGL(1,ℤ/9)—22no—
31C22×AGL(1,ℤ/9)—22no—
32C22×AGL(1,ℤ/9)D412yes—
33C2×C4×3−1+2—22no—
34C2×C4×3−1+2—22no—
35C2×C4×3−1+2—22no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2D412yes—
38C2×3−1+2⋊C4—22no—
39C2×3−1+2⋊C4—21no—
40C2×3−1+2⋊C4—22no—
41C2×3−1+2⋊C4—22no—
42C2×3−1+2⋊C4—22no—
43C3×#397—91no—
44#4992—31no—
45C23×C3×D3—31no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C2×D311yescharacteristic
48C22×C3×Dic3—32no—
49C22×C3×Dic3—31no—
50C23×D9C22×C311yescharacteristic
51C32×C4⋊C4—22no—
52C3×(C2×D3)⋊C4—64no—
53C3×(C2×D3)⋊C4—62no—
54C22×C4×C9—21no—
55C22×C4×C9—22no—
56C22×C4×C9C22×C312yes—
57C22×C4×C9C22×C311yescharacteristic
58C22×Dic9C22×C312yes—
59C22×Dic9C22×C311yescharacteristic
60C9×C4⋊C4—24no—
61C9×C4⋊C4—22no—
62(C2×D9)⋊C4—24no—
63(C2×D9)⋊C4—22no—
64C22×3−1+2C2×D411yescharacteristic
65C22×3−1+2C2×D411yescharacteristic
66C22×3−1+2#4111yescharacteristic
67C22×3−1+2#4112yes—
68C22×3−1+2#4112yes—
69C2×AGL(1,ℤ/9)—22no—
70C2×AGL(1,ℤ/9)—24no—
71C2×AGL(1,ℤ/9)—41no—
72C2×AGL(1,ℤ/9)—41no—
73C2×AGL(1,ℤ/9)—42no—
74C4×3−1+2—22no—
75C4×3−1+2—42no—
763−1+2⋊C4—42no—
773−1+2⋊C4—41no—
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4C3×D311yescharacteristic
80C2×C3×C22⋊C4—91no—
81C2×C3×C22⋊C4—92no—
82C2×(C2×D3)⋊C4—32no—
83C2×(C2×D3)⋊C4—31no—
84C3×#95—91no—
85C3×#95—92no—
86C3⋊#95—32no—
87C3⋊#95—31no—
88C23×C32C22×D311yescharacteristic
89C22×C3×D3—32no—
90C22×C3×D3—62no—
91C22×C3×D3—61no—
92C22×C3×D3—62no—
93C22×C3×D3—61no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4#6712yes—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—61no—
104C23×C9—21no—
105C23×C9C23×C311yescharacteristic
106C22×D9—21no—
107C22×D9—22no—
108C22×D9—21no—
109C22×D9—22no—
110C22×D9C3×D412yes—
111C2×C4×C9—24no—
112C2×C4×C9—22no—
113C2×C4×C9—24no—
114C2×C4×C9—22no—
115C2×C4×C9—22no—
116C2×C4×C9—22no—
117C2×C4×C9—22no—
118C2×C4×C9—42no—
119C2×C4×C9—42no—
120C2×C4×C9C3×D412yes—
121C2×Dic9—22no—
122C2×Dic9—21no—
123C2×Dic9—22no—
124C2×Dic9—22no—
125C2×Dic9—22no—
126#397—91noSylow
127C2×3−1+2#12111yescharacteristic
128C2×3−1+2#12312yes—
129C2×3−1+2#12311yescharacteristic
130C2×3−1+2#12411yescharacteristic
131C2×3−1+2#12612yes—
132AGL(1,ℤ/9)—42no—
133C24×C3—91no—
134C23×D3—31no—
135C22×C3×C4—31no—
136C22×C3×C4—32no—
137C22×C3×C4—92no—
138C22×C3×C4—91no—
139C22×C3×C4C2×C3×D312yes—
140C22×C3×C4C2×C3×D311yescharacteristic
141C22×Dic3—32no—
142C22×Dic3—31no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—62no—
145C3×C22⋊C4—182no—
146C3×C22⋊C4—184no—
147(C2×D3)⋊C4—64no—
148(C2×D3)⋊C4—62no—
149C22×C32#23512yes—
150C22×C32D3×D411yescharacteristic
151C22×C32#23711yescharacteristic
152C22×C32#23912yes—
153C22×C32C2×#6711yescharacteristic
154C2×C3×D3—64no—
155C2×C3×D3—62no—
156C2×C3×D3—122no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C32×C4—22no—
160C32×C4—42no—
161C3×Dic3—122no—
162C3×Dic3—121no—
163C22×C9—22no—
164C22×C9—21no—
165C22×C9—21no—
166C22×C9—22no—
167C22×C9—21no—
168C22×C9C2×C3×D411yescharacteristic
169C22×C9C2×C3×D411yescharacteristic
170C22×C9C3×#4112yes—
171C22×C9C3×#4111yescharacteristic
172C22×C9C3×#4112yes—
173C2×D9—22no—
174C2×D9—24no—
175C2×D9—41no—
176C2×D9—41no—
177C2×D9—42no—
178C4×C9—24no—
179C4×C9—22no—
180C4×C9—44no—
181C4×C9—42no—
182Dic9—42no—
183Dic9—41no—
184C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
185C2×C22⋊C4—92no—
186C2×C22⋊C4—91no—
187#95—92no—
188#95—91no—
1893−1+2#39711yescharacteristic, complemented, Sylow
190C23×C3—31no—
191C23×C3—92no—
192C23×C3—181no—
193C23×C3—182no—
194C23×C3—181no—
195C23×C3—182no—
196C23×C3C22×C3×D311yescharacteristic
197C22×D3—32no—
198C22×D3—62no—
199C22×D3—61no—
200C22×D3—62no—
201C22×D3—61no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—32no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—182no—
212C2×C3×C4—181no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4—182no—
216C2×C3×C4C3×#6712yes—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—61no—
222C2×C32#89412yes—
223C2×C32#89812yes—
224C2×C32#92911yescharacteristic
225C2×C32#93811yescharacteristic
226C2×C32#94711yescharacteristic
227C3×D3—122no—
228C2×C9—22no—
229C2×C9—22no—
230C2×C9—21no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9C3×#12111yescharacteristic
234C2×C9C3×#12312yes—
235C2×C9C3×#12311yescharacteristic
236C2×C9C3×#12411yescharacteristic
237C2×C9C3×#12612yes—
238D9—42no—
239C24—91no—
240C22×C4—92no—
241C22×C4—91no—
242C22×C4C2×AGL(1,ℤ/9)12yes—
243C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
244C4⋊C4—22no—
245C22⋊C4—184no—
246C22⋊C4—182no—
247C22×C3—32no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—31no—
252C22×C3—184no—
253C22×C3—182no—
254C22×C3—362no—
255C22×C3—361no—
256C22×C3—361no—
257C22×C3C3×#23512yes—
258C22×C3C3×D3×D411yescharacteristic
259C22×C3C3×#23711yescharacteristic
260C22×C3C3×#23912yes—
261C22×C3C2×C3×#6711yescharacteristic
262C2×D3—64no—
263C2×D3—62no—
264C2×D3—122no—
265C2×D3—121no—
266C2×D3—121no—
267C3×C4—22no—
268C3×C4—42no—
269C3×C4—62no—
270C3×C4—122no—
271C3×C4—362no—
272C3×C4—361no—
273Dic3—122no—
274Dic3—121no—
275C32#499211yescharacteristic
276C9—21no—
277C9C3×#39711yescharacteristic, complemented
278C23—92no—
279C23—182no—
280C23—181no—
281C23—182no—
282C23—181no—
283C23C22×AGL(1,ℤ/9)11yescharacteristic
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—182no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4—182no—
292C2×C4—181no—
293C2×C4#611⋊C312yes—
294C2×C3—32no—
295C2×C3—32no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—362no—
300C2×C3C3×#89412yes—
301C2×C3C3×#89812yes—
302C2×C3C3×#92911yescharacteristic
303C2×C3C3×#93811yescharacteristic
304C2×C3C3×#94711yescharacteristic
305D3—122no—
306C22—184no—
307C22—182no—
308C22—362no—
309C22—361no—
310C22—361no—
311C22#3697⋊C312yes—
312C22D4×AGL(1,ℤ/9)11yescharacteristic
313C22#3699⋊C311yescharacteristic
314C22#8872412yes—
315C22C2×#611⋊C311yescharacteristic
316C4—22no—
317C4—42no—
318C4—362no—
319C4—361no—
320C3—31no—
321C3C3×#499211yescharacteristic
322C2—362no—
323C2#63456⋊C312yes—
324C2#63460⋊C312yes—
325C2#63491⋊C311yescharacteristic
326C2#63500⋊C311yescharacteristic
327C2#63509⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC23×AGL(1,ℤ/9)C22
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#397
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
4. 3−1+2⋊#95C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588672C3semidirect
22. C2×C9×C4⋊C4C2×C3semidirect
25. C9⋊#95C2×C3semidirect
26. C9⋊#95C2×C3semidirect
184. C2×C4⋊C4AGL(1,ℤ/9)semidirect
189. 3−1+2#397semidirect
277. C9C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
3−1+2⋊#952
C3×#49923
#105886723