Finite Group Databasebeta

← #10588354⋊C3 #13141393

Subgroups — #10588354⋊C3 #13141393

Order 1728.

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

normal subgroups only

This group has 548 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×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
2C2×(C2×AGL(1,ℤ/9))⋊C4C212yesmaximal
3C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
43−1+2×#95C211yesmaximal, characteristic, complemented
53−1+2⋊#95C212yesmaximal, complemented
6C3×#4674—31nomaximal
7#10588354C311yesmaximal, 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—
13(C2×AGL(1,ℤ/9))⋊C4—24no—
14(C2×AGL(1,ℤ/9))⋊C4—22no—
15C4⋊(3−1+2⋊C4)—22no—
16C2×C3×(C2×D3)⋊C4—32no—
17C2×C3×(C2×D3)⋊C4—31no—
18C2×C3×C4⋊Dic3—31no—
19C32×#95D311yescharacteristic
20C3×C3⋊#95—32no—
21C2×(C2×D9)⋊C4C2×C312yes—
22C2×(C2×D9)⋊C4C2×C311yescharacteristic
23C2×C4⋊Dic9C2×C311yescharacteristic, complemented
24C9×#95—21no—
25C9×#95C2×C311yescharacteristic, complemented
26C9⋊#95C2×C312yescomplemented
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—22no—
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4—22no—
42C2×3−1+2⋊C4—22no—
43C3×#397—91no—
44#4674—31no—
45C23×C3×D3—31no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C2×D311yescharacteristic
48C22×C3×Dic3—32no—
49C22×C3×Dic3—31no—
50C23×D9C22×C311yescharacteristic
51C3×(C2×D3)⋊C4—64no—
52C3×(C2×D3)⋊C4—62no—
53C3×C4⋊Dic3—62no—
54C22×C4×C9—22no—
55C22×C4×C9—21no—
56C22×C4×C9C22×C312yes—
57C22×C4×C9C22×C311yescharacteristic
58C22×Dic9C22×C311yescharacteristic
59C22×Dic9C22×C312yes—
60(C2×D9)⋊C4—24no—
61(C2×D9)⋊C4—22no—
62C4⋊Dic9—22no—
63C22×3−1+2C2×D411yescharacteristic
64C22×3−1+2C2×D411yescharacteristic
65C22×3−1+2#4111yescharacteristic
66C22×3−1+2#4112yes—
67C22×3−1+2#4112yes—
68C2×AGL(1,ℤ/9)—22no—
69C2×AGL(1,ℤ/9)—24no—
70C2×AGL(1,ℤ/9)—41no—
71C2×AGL(1,ℤ/9)—41no—
72C2×AGL(1,ℤ/9)—42no—
73C4×3−1+2—22no—
74C4×3−1+2—42no—
753−1+2⋊C4—42no—
763−1+2⋊C4—41no—
77C2×C3×C4⋊C4—91no—
78C2×C3×C22⋊C4—91no—
79C2×C3×C22⋊C4—92no—
80C2×(C2×D3)⋊C4—32no—
81C2×(C2×D3)⋊C4—31no—
82C2×C4⋊Dic3—31no—
83C3×#95—31no—
84C3×#95—92no—
85C3×#95C3×D311yescharacteristic
86C3⋊#95—32no—
87C23×C32C22×D311yescharacteristic
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C22×C3×D3—61no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4D1212yes—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C23×C9—21no—
104C23×C9C23×C311yescharacteristic
105C22×D9—21no—
106C22×D9—22no—
107C22×D9—22no—
108C22×D9—21no—
109C22×D9C3×D412yes—
110C2×C4×C9—22no—
111C2×C4×C9—24no—
112C2×C4×C9—22no—
113C2×C4×C9—22no—
114C2×C4×C9—22no—
115C2×C4×C9—22no—
116C2×C4×C9—42no—
117C2×C4×C9—42no—
118C2×C4×C9—42no—
119C2×C4×C9C3×D412yes—
120C2×Dic9—22no—
121C2×Dic9—21no—
122C2×Dic9—22no—
123C2×Dic9—22no—
124C2×Dic9—22no—
125#397—91noSylow
126C2×3−1+2#12111yescharacteristic
127C2×3−1+2#12312yes—
128C2×3−1+2#12311yescharacteristic
129C2×3−1+2#12411yescharacteristic
130C2×3−1+2#12612yes—
131AGL(1,ℤ/9)—42no—
132C24×C3—91no—
133C23×D3—31no—
134C22×C3×C4—31no—
135C22×C3×C4—32no—
136C22×C3×C4—91no—
137C22×C3×C4—92no—
138C22×C3×C4C2×C3×D312yes—
139C22×C3×C4C2×C3×D311yescharacteristic
140C22×Dic3—31no—
141C22×Dic3—32no—
142C3×C4⋊C4—182no—
143C3×C22⋊C4—182no—
144C3×C22⋊C4—184no—
145(C2×D3)⋊C4—64no—
146(C2×D3)⋊C4—62no—
147C4⋊Dic3—62no—
148C22×C32C2×D1211yescharacteristic
149C22×C32#23512yes—
150C22×C32D3×D411yescharacteristic
151C22×C32#23712yes—
152C22×C32#23911yescharacteristic
153C2×C3×D3—64no—
154C2×C3×D3—62no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—122no—
158C32×C4—22no—
159C32×C4—42no—
160C3×Dic3—121no—
161C3×Dic3—122no—
162C22×C9—21no—
163C22×C9—22no—
164C22×C9—21no—
165C22×C9—22no—
166C22×C9—21no—
167C22×C9C2×C3×D411yescharacteristic
168C22×C9C2×C3×D411yescharacteristic
169C22×C9C3×#4112yes—
170C22×C9C3×#4112yes—
171C22×C9C3×#4111yescharacteristic
172C2×D9—22no—
173C2×D9—24no—
174C2×D9—41no—
175C2×D9—41no—
176C2×D9—42no—
177C4×C9—22no—
178C4×C9—42no—
179C4×C9—44no—
180C4×C9—42no—
181Dic9—42no—
182Dic9—41no—
183C2×C4⋊C4—91no—
184C2×C22⋊C4—92no—
185C2×C22⋊C4—91no—
186#95—92no—
187#95AGL(1,ℤ/9)11yescharacteristic, complemented
1883−1+2#39711yescharacteristic, complemented, Sylow
189C23×C3—31no—
190C23×C3—92no—
191C23×C3—181no—
192C23×C3—182no—
193C23×C3—182no—
194C23×C3—181no—
195C23×C3C22×C3×D311yescharacteristic
196C22×D3—32no—
197C22×D3—61no—
198C22×D3—62no—
199C22×D3—62no—
200C22×D3—61no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—32no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—181no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4C3×D1212yes—
216C2×Dic3—62no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×C32#87511yescharacteristic
222C2×C32#88312yes—
223C2×C32#88611yescharacteristic
224C2×C32#89511yescharacteristic
225C2×C32#89812yes—
226C3×D3—122no—
227C2×C9—22no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—22no—
231C2×C9—21no—
232C2×C9C3×#12111yescharacteristic
233C2×C9C3×#12312yes—
234C2×C9C3×#12311yescharacteristic
235C2×C9C3×#12411yescharacteristic
236C2×C9C3×#12612yes—
237D9—42no—
238C24—91no—
239C22×C4—91no—
240C22×C4—92no—
241C22×C4C2×AGL(1,ℤ/9)12yes—
242C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
243C4⋊C4—182no—
244C22⋊C4—184no—
245C22⋊C4—182no—
246C22×C3—31no—
247C22×C3—32no—
248C22×C3—31no—
249C22×C3—32no—
250C22×C3—31no—
251C22×C3—182no—
252C22×C3—184no—
253C22×C3—362no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3C2×C3×D1211yescharacteristic
257C22×C3C3×#23512yes—
258C22×C3C3×D3×D411yescharacteristic
259C22×C3C3×#23712yes—
260C22×C3C3×#23911yescharacteristic
261C2×D3—64no—
262C2×D3—62no—
263C2×D3—121no—
264C2×D3—122no—
265C2×D3—121no—
266C3×C4—22no—
267C3×C4—42no—
268C3×C4—62no—
269C3×C4—122no—
270C3×C4—362no—
271C3×C4—361no—
272Dic3—122no—
273Dic3—121no—
274C32#467411yescharacteristic
275C9—21no—
276C9C3×#39711yescharacteristic, complemented
277C23—92no—
278C23—181no—
279C23—182no—
280C23—182no—
281C23—181no—
282C23C22×AGL(1,ℤ/9)11yescharacteristic
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—181no—
288C2×C4—182no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4—182no—
292C2×C4C4⋊AGL(1,ℤ/9)12yes—
293C2×C3—31no—
294C2×C3—32no—
295C2×C3—31no—
296C2×C3—32no—
297C2×C3—31no—
298C2×C3—362no—
299C2×C3C3×#87511yescharacteristic
300C2×C3C3×#88312yes—
301C2×C3C3×#88611yescharacteristic
302C2×C3C3×#89511yescharacteristic
303C2×C3C3×#89812yes—
304D3—122no—
305C22—182no—
306C22—184no—
307C22—362no—
308C22—361no—
309C22—361no—
310C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
311C22#3697⋊C312yes—
312C22D4×AGL(1,ℤ/9)11yescharacteristic
313C22#3699⋊C312yes—
314C22#8872411yescharacteristic
315C4—22no—
316C4—42no—
317C4—362no—
318C4—361no—
319C3—31no—
320C3C3×#467411yescharacteristic
321C2—362no—
322C2#63437⋊C311yescharacteristic
323C2#63445⋊C312yes—
324C2#63448⋊C311yescharacteristic
325C2#63457⋊C311yescharacteristic
326C2#63460⋊C312yes—
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
Fitting3−1+2×#95C2
prime radicalC23×AGL(1,ℤ/9)C22
hypercenter#95AGL(1,ℤ/9)
hypocenterC9C3×#397
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊(3−1+2⋊C4)C2semidirect
4. 3−1+2×#95C2semidirect
5. 3−1+2⋊#95C2semidirect
7. #10588354C3semidirect
23. C2×C4⋊Dic9C2×C3semidirect
25. C9×#95C2×C3semidirect
26. C9⋊#95C2×C3semidirect
187. #95AGL(1,ℤ/9)semidirect
188. 3−1+2#397semidirect
276. C9C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
3−1+2×#952
3−1+2⋊#952
C3×#46743
#105883543