Finite Group Databasebeta

← #10588673⋊C3 #13142651

Subgroups — #10588673⋊C3 #13142651

Order 1728.

604 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal
4C2×(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
53−1+2⋊#95C211yesmaximal, characteristic, complemented
6C3×#4993—31nomaximal
7#10588673C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)C2211yescharacteristic
9C22×C4×3−1+2C2212yes—
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4C2211yescharacteristic
12C22×3−1+2⋊C4C2212yes—
13C4⋊C4×3−1+2—22no—
14(C2×AGL(1,ℤ/9))⋊C4—22no—
15(C2×AGL(1,ℤ/9))⋊C4—24no—
16(3−1+2⋊C4)⋊C4—24no—
17C2×C32×C4⋊C4D311yescharacteristic
18C2×C3×(C2×D3)⋊C4—32no—
19C2×C3×(C2×D3)⋊C4—31no—
20C2×C3×Dic3⋊C4—32no—
21C3×C3⋊#95—31no—
22C2×C9×C4⋊C4—21no—
23C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
24C2×(C2×D9)⋊C4C2×C311yescharacteristic
25C2×(C2×D9)⋊C4C2×C312yes—
26C2×Dic9⋊C4C2×C312yescomplemented
27C9⋊#95C2×C311yescharacteristic, complemented
28C23×3−1+2C2311yescharacteristic
29C22×AGL(1,ℤ/9)—22no—
30C22×AGL(1,ℤ/9)—21no—
31C22×AGL(1,ℤ/9)—22no—
32C22×AGL(1,ℤ/9)—21no—
33C22×AGL(1,ℤ/9)Q812yes—
34C2×C4×3−1+2—22no—
35C2×C4×3−1+2—22no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2—22no—
38C2×C4×3−1+2D412yes—
39C2×3−1+2⋊C4—22no—
40C2×3−1+2⋊C4—22no—
41C2×3−1+2⋊C4—22no—
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4D414yes—
44C3×#399—91no—
45#4993—31no—
46C23×C3×D3—31no—
47C22×C32×C4C2×D312yes—
48C22×C32×C4C2×D311yescharacteristic
49C22×C3×Dic3—32no—
50C22×C3×Dic3—31no—
51C23×D9C22×C311yescharacteristic
52C32×C4⋊C4—22no—
53C3×(C2×D3)⋊C4—64no—
54C3×(C2×D3)⋊C4—62no—
55C3×Dic3⋊C4—64no—
56C22×C4×C9—21no—
57C22×C4×C9—22no—
58C22×C4×C9C22×C312yes—
59C22×C4×C9C22×C311yescharacteristic
60C22×Dic9C22×C311yescharacteristic
61C22×Dic9C22×C312yes—
62C9×C4⋊C4—24no—
63C9×C4⋊C4—22no—
64(C2×D9)⋊C4—22no—
65(C2×D9)⋊C4—24no—
66Dic9⋊C4—24no—
67C22×3−1+2C2×D412yes—
68C22×3−1+2C2×D411yescharacteristic
69C22×3−1+2C2×Q811yescharacteristic
70C22×3−1+2#4112yes—
71C22×3−1+2#4111yescharacteristic
72C2×AGL(1,ℤ/9)—24no—
73C2×AGL(1,ℤ/9)—22no—
74C2×AGL(1,ℤ/9)—42no—
75C2×AGL(1,ℤ/9)—41no—
76C2×AGL(1,ℤ/9)—41no—
77C4×3−1+2—22no—
78C4×3−1+2—42no—
793−1+2⋊C4—24no—
803−1+2⋊C4—41no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C4⋊C4—92no—
83C2×C3×C4⋊C4C3×D311yescharacteristic
84C2×C3×C22⋊C4—91no—
85C2×C3×C22⋊C4—92no—
86C2×(C2×D3)⋊C4—32no—
87C2×(C2×D3)⋊C4—31no—
88C2×Dic3⋊C4—32no—
89C3×#95—91no—
90C3⋊#95—31no—
91C23×C32C22×D311yescharacteristic
92C22×C3×D3—32no—
93C22×C3×D3—62no—
94C22×C3×D3—62no—
95C22×C3×D3—61no—
96C22×C3×D3—61no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4#6712yes—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—61no—
107C23×C9—21no—
108C23×C9C23×C311yescharacteristic
109C22×D9—22no—
110C22×D9—21no—
111C22×D9—22no—
112C22×D9—21no—
113C22×D9C3×Q812yes—
114C2×C4×C9—24no—
115C2×C4×C9—22no—
116C2×C4×C9—24no—
117C2×C4×C9—22no—
118C2×C4×C9—22no—
119C2×C4×C9—22no—
120C2×C4×C9—22no—
121C2×C4×C9—42no—
122C2×C4×C9—42no—
123C2×C4×C9C3×D412yes—
124C2×Dic9—22no—
125C2×Dic9—22no—
126C2×Dic9—22no—
127C2×Dic9—21no—
128C2×Dic9C3×D414yes—
129#399—91noSylow
130C2×3−1+2C22⋊Q812yes—
131C2×3−1+2C22⋊Q811yescharacteristic
132C2×3−1+2#12312yes—
133C2×3−1+2#12311yescharacteristic
134C2×3−1+2#12711yescharacteristic
135AGL(1,ℤ/9)—42no—
136C24×C3—91no—
137C23×D3—31no—
138C22×C3×C4—31no—
139C22×C3×C4—32no—
140C22×C3×C4—92no—
141C22×C3×C4—91no—
142C22×C3×C4C2×C3×D312yes—
143C22×C3×C4C2×C3×D311yescharacteristic
144C22×Dic3—32no—
145C22×Dic3—31no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—62no—
148C3×C4⋊C4—184no—
149C3×C22⋊C4—182no—
150C3×C22⋊C4—184no—
151(C2×D3)⋊C4—64no—
152(C2×D3)⋊C4—62no—
153Dic3⋊C4—64no—
154C22×C32#23512yes—
155C22×C32D3×D412yes—
156C22×C32D3×Q811yescharacteristic
157C22×C32#23911yescharacteristic
158C22×C32C2×#6711yescharacteristic
159C2×C3×D3—64no—
160C2×C3×D3—62no—
161C2×C3×D3—122no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C32×C4—22no—
165C32×C4—42no—
166C3×Dic3—64no—
167C3×Dic3—121no—
168C22×C9—22no—
169C22×C9—22no—
170C22×C9—21no—
171C22×C9—21no—
172C22×C9—21no—
173C22×C9C2×C3×D412yes—
174C22×C9C2×C3×D411yescharacteristic
175C22×C9C2×C3×Q811yescharacteristic
176C22×C9C3×#4112yes—
177C22×C9C3×#4111yescharacteristic
178C2×D9—24no—
179C2×D9—22no—
180C2×D9—42no—
181C2×D9—41no—
182C2×D9—41no—
183C4×C9—24no—
184C4×C9—22no—
185C4×C9—44no—
186C4×C9—42no—
187Dic9—24no—
188Dic9—41no—
189C2×C4⋊C4—92no—
190C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
191C2×C22⋊C4—92no—
192C2×C22⋊C4—91no—
193#95—91no—
1943−1+2#39911yescharacteristic, complemented, Sylow
195C23×C3—31no—
196C23×C3—92no—
197C23×C3—182no—
198C23×C3—181no—
199C23×C3—182no—
200C23×C3—181no—
201C23×C3C22×C3×D311yescharacteristic
202C22×D3—32no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—61no—
206C22×D3—61no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—32no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—94no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—181no—
220C2×C3×C4—182no—
221C2×C3×C4C3×#6712yes—
222C2×Dic3—34no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—61no—
227C2×C32#89412yes—
228C2×C32#89612yes—
229C2×C32#92911yescharacteristic
230C2×C32#94011yescharacteristic
231C2×C32#94611yescharacteristic
232C3×D3—122no—
233C2×C9—22no—
234C2×C9—22no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9C3×C22⋊Q812yes—
239C2×C9C3×C22⋊Q811yescharacteristic
240C2×C9C3×#12312yes—
241C2×C9C3×#12311yescharacteristic
242C2×C9C3×#12711yescharacteristic
243D9—42no—
244C24—91no—
245C22×C4—92no—
246C22×C4—91no—
247C22×C4C2×AGL(1,ℤ/9)12yes—
248C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
249C4⋊C4—22no—
250C4⋊C4—184no—
251C22⋊C4—184no—
252C22⋊C4—182no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—184no—
259C22×C3—182no—
260C22×C3—362no—
261C22×C3—361no—
262C22×C3—361no—
263C22×C3C3×#23512yes—
264C22×C3C3×D3×D412yes—
265C22×C3C3×D3×Q811yescharacteristic
266C22×C3C3×#23911yescharacteristic
267C22×C3C2×C3×#6711yescharacteristic
268C2×D3—64no—
269C2×D3—62no—
270C2×D3—122no—
271C2×D3—121no—
272C2×D3—121no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—62no—
276C3×C4—122no—
277C3×C4—184no—
278C3×C4—361no—
279Dic3—64no—
280Dic3—121no—
281C32#499311yescharacteristic
282C9—21no—
283C9C3×#39911yescharacteristic, complemented
284C23—92no—
285C23—182no—
286C23—182no—
287C23—181no—
288C23—181no—
289C23C22×AGL(1,ℤ/9)11yescharacteristic
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—94no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4—182no—
298C2×C4—181no—
299C2×C4#611⋊C312yes—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—362no—
306C2×C3C3×#89412yes—
307C2×C3C3×#89612yes—
308C2×C3C3×#92911yescharacteristic
309C2×C3C3×#94011yescharacteristic
310C2×C3C3×#94611yescharacteristic
311D3—122no—
312C22—184no—
313C22—182no—
314C22—362no—
315C22—361no—
316C22—361no—
317C22#3697⋊C312yes—
318C22D4×AGL(1,ℤ/9)12yes—
319C22Q8×AGL(1,ℤ/9)11yescharacteristic
320C22#8872411yescharacteristic
321C22C2×#611⋊C311yescharacteristic
322C4—22no—
323C4—42no—
324C4—184no—
325C4—361no—
326C3—31no—
327C3C3×#499311yescharacteristic
328C2—362no—
329C2#63456⋊C312yes—
330C2#63458⋊C312yes—
331C2#63491⋊C311yescharacteristic
332C2#63502⋊C311yescharacteristic
333C2#63508⋊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×#399
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. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588673C3semidirect
23. C2×C9×C4⋊C4C2×C3semidirect
26. C2×Dic9⋊C4C2×C3semidirect
27. C9⋊#95C2×C3semidirect
190. C2×C4⋊C4AGL(1,ℤ/9)semidirect
194. 3−1+2#399semidirect
283. C9C3×#399semidirect

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
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49933
#105886733