Finite Group Databasebeta

← #10589761⋊C3 #13153069

Subgroups — #10589761⋊C3 #13153069

Order 1728.

783 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×#123C211yesmaximal, characteristic, complemented
2#63499⋊C3C211yesmaximal, characteristic
3#63498⋊C3C211yesmaximal, characteristic, complemented
4C4×#611⋊C3C212yesmaximal, complemented
5#63458⋊C3C212yesmaximal, complemented
6#63454⋊C3C212yesmaximal, complemented
7#63445⋊C3C212yesmaximal, complemented
8C22⋊C4×AGL(1,ℤ/9)C212yesmaximal, complemented
9#63441⋊C3C212yesmaximal, complemented
10C3×#5659—31nomaximal
11#10589761C311yesmaximal, characteristic, complemented
12C23×AGL(1,ℤ/9)C2211yescharacteristic
13C2×D4×3−1+2C2211yescharacteristic
14C22×C4×3−1+2C2211yescharacteristic, complemented
15C2×#611⋊C3C2212yes—
16C2×C4×AGL(1,ℤ/9)C2212yes—
17C2×C4×AGL(1,ℤ/9)C2212yes—
18C2×3−1+2⋊Q8C2211yescharacteristic, complemented
19C4⋊C4×3−1+2C2212yescomplemented
20C22⋊C4×3−1+2C2212yes—
21C22⋊C4×3−1+2C2211yescharacteristic, complemented
22#3676⋊C3C2212yes—
23#3676⋊C3C2211yescharacteristic
24#3676⋊C3C2212yescomplemented
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28C4⋊(3−1+2⋊C4)C2212yescomplemented
29(3−1+2⋊C4)⋊C4C2212yescomplemented
30(3−1+2⋊C4)⋊C4C2212yescomplemented
31C4×3−1+2⋊C4C2211yescharacteristic, complemented
32C4×3−1+2⋊C4C2212yescomplemented
33C32×#123D311yescharacteristic
34C3×#937—31no—
35C3×#936—31no—
36C3×C4×#67—32no—
37C3×#896—32no—
38C3×#892—32no—
39C3×#883—32no—
40C3×D3×C22⋊C4—32no—
41C3×#879—32no—
42C9×#123—21no—
43C9×#123C2×C311yescharacteristic, complemented
44#63499C2×C311yescharacteristic
45#63498C2×C311yescharacteristic, complemented
46C4×#611C2×C312yescomplemented
47#63458C2×C312yescomplemented
48#63454C2×C312yescomplemented
49#63445C2×C312yescomplemented
50C22⋊C4×D9C2×C312yescomplemented
51#63441C2×C312yescomplemented
52C23×3−1+2C2311yescharacteristic
53C23×3−1+2C2311yescharacteristic
54C22×AGL(1,ℤ/9)—24no—
55C22×AGL(1,ℤ/9)—41no—
56C22×AGL(1,ℤ/9)C2312yes—
57D4×3−1+2—41no—
58C2×C4×3−1+2—41no—
59C2×C4×3−1+2C2311yescharacteristic
60C2×C4×3−1+2C2312yes—
61C2×C4×3−1+2C2312yes—
62#611⋊C3—24no—
63C2×3−1+2⋊C4C2312yes—
64C2×3−1+2⋊C4C2312yes—
65C2×3−1+2⋊C4C2312yes—
66C4×AGL(1,ℤ/9)—24no—
67C4×AGL(1,ℤ/9)—42no—
683−1+2⋊Q8—41no—
69C3×#551—91no—
70#5659—31no—
71C23×C3×D3—31no—
72C2×C32×D4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C2×C3×#67—32no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—32no—
77C2×C3×Dic6—31no—
78C23×D9C22×C311yescharacteristic
79C32×C4⋊C4C2×D312yes—
80C32×C22⋊C4C2×D312yes—
81C32×C22⋊C4C2×D311yescharacteristic
82C3×#217—31no—
83C3×#217—32no—
84C3×#217—32no—
85C3×(C2×D3)⋊C4—32no—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×C4⋊Dic3—32no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—31no—
93C2×D4×C9—21no—
94C2×D4×C9C22×C311yescharacteristic
95C22×C4×C9—21no—
96C22×C4×C9C22×C311yescharacteristic, complemented
97C2×#611C22×C312yes—
98C2×C4×D9C22×C312yes—
99C2×C4×D9C22×C312yes—
100C2×Dic18C22×C311yescharacteristic, complemented
101C9×C4⋊C4—22no—
102C9×C4⋊C4C22×C312yescomplemented
103C9×C22⋊C4—22no—
104C9×C22⋊C4—21no—
105C9×C22⋊C4C22×C312yes—
106C9×C22⋊C4C22×C311yescharacteristic, complemented
107#3676C22×C312yes—
108#3676C22×C311yescharacteristic
109#3676C22×C312yescomplemented
110(C2×D9)⋊C4C22×C312yes—
111(C2×D9)⋊C4C22×C312yes—
112(C2×D9)⋊C4C22×C312yes—
113C4⋊Dic9C22×C312yescomplemented
114Dic9⋊C4C22×C312yescomplemented
115Dic9⋊C4C22×C312yescomplemented
116C4×Dic9C22×C312yescomplemented
117C4×Dic9C22×C311yescharacteristic, complemented
118C22×3−1+2—22no—
119C22×3−1+2—22no—
120C22×3−1+2—21no—
121C22×3−1+2—21no—
122C22×3−1+2C2411yescharacteristic
123C2×AGL(1,ℤ/9)—22no—
124C2×AGL(1,ℤ/9)—22no—
125C2×AGL(1,ℤ/9)—44no—
126C2×AGL(1,ℤ/9)#4114yes—
127C4×3−1+2—22no—
128C4×3−1+2—21no—
129C4×3−1+2—22no—
1303−1+2⋊C4—22no—
1313−1+2⋊C4—22no—
1323−1+2⋊C4—22no—
133C3×#126—92no—
134C3×#124—91no—
135C3×#123—31no—
136C3×#123—92no—
137C3×#123C3×D311yescharacteristic
138C3×C22⋊Q8—92no—
139C3×#120—91no—
140C3×C4×D4—92no—
141C3×#117—92no—
142C2×C3×C22⋊C4—92no—
143#937—31no—
144#936—31no—
145C4×#67—32no—
146#896—32no—
147#892—32no—
148#883—32no—
149D3×C22⋊C4—32no—
150#879—32no—
151C23×C32C22×D311yescharacteristic
152C23×C32C22×D311yescharacteristic
153C22×C3×D3—32no—
154C22×C3×D3—64no—
155C22×C3×D3—121no—
156C32×D4—41no—
157C2×C32×C4—41no—
158C2×C32×C4C22×D312yes—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D312yes—
161C3×#67—64no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C3×C4×D3—64no—
166C3×C4×D3—122no—
167C3×Dic6—121no—
168C23×C9—21no—
169C23×C9—21no—
170C23×C9C23×C311yescharacteristic
171C23×C9C23×C311yescharacteristic
172C22×D9—24no—
173C22×D9—41no—
174C22×D9C23×C312yes—
175D4×C9—41no—
176D4×C9—42no—
177C2×C4×C9—22no—
178C2×C4×C9—22no—
179C2×C4×C9—21no—
180C2×C4×C9—41no—
181C2×C4×C9—42no—
182C2×C4×C9C23×C312yes—
183C2×C4×C9C23×C311yescharacteristic
184C2×C4×C9C23×C312yes—
185#611—24no—
186C2×Dic9C23×C312yes—
187C2×Dic9C23×C312yes—
188C2×Dic9C23×C312yes—
189C4×D9—24no—
190C4×D9—42no—
191Dic18—41no—
192#551—91noSylow
193C2×3−1+2—41no—
194C2×3−1+2—41no—
195C2×3−1+2C2×#4112yes—
196C2×3−1+2H(5,𝔽2)11yescharacteristic
197AGL(1,ℤ/9)—24no—
198C24×C3—91no—
199C23×D3—31no—
200C2×C3×Q8—91no—
201C2×C3×D4—31no—
202C2×C3×D4—92no—
203C2×C3×D4C2×C3×D311yescharacteristic
204C22×C3×C4—31no—
205C22×C3×C4—92no—
206C22×C3×C4—92no—
207C22×C3×C4C2×C3×D311yescharacteristic
208C2×#67—32no—
209C2×C4×D3—32no—
210C2×C4×D3—32no—
211C2×Dic6—31no—
212C3×C4⋊C4—32no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4C2×C3×D312yes—
217C3×C22⋊C4—32no—
218C3×C22⋊C4—31no—
219C3×C22⋊C4—92no—
220C3×C22⋊C4—92no—
221C3×C22⋊C4—92no—
222C3×C22⋊C4—92no—
223C3×C22⋊C4—92no—
224C3×C22⋊C4—91no—
225C3×C22⋊C4C2×C3×D312yes—
226C3×C22⋊C4C2×C3×D311yescharacteristic
227C3×C42—91no—
228C3×C42—92no—
229#217—31no—
230#217—32no—
231#217—32no—
232(C2×D3)⋊C4—32no—
233(C2×D3)⋊C4—32no—
234(C2×D3)⋊C4—32no—
235C4⋊Dic3—32no—
236Dic3⋊C4—32no—
237Dic3⋊C4—32no—
238C4×Dic3—32no—
239C4×Dic3—31no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32C23×D311yescharacteristic
245C2×C3×D3—34no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—124no—
249C32×C4—21no—
250C32×C4—22no—
251C32×C4—22no—
252C3×Dic3—62no—
253C3×Dic3—62no—
254C3×Dic3—62no—
255C22×C9—22no—
256C22×C9—22no—
257C22×C9—22no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—42no—
262C22×C9—41no—
263C22×C9—42no—
264C22×C9C24×C311yescharacteristic
265C2×D9—22no—
266C2×D9—22no—
267C2×D9—44no—
268C2×D9C3×#4114yes—
269C4×C9—22no—
270C4×C9—21no—
271C4×C9—22no—
272C4×C9—42no—
273C4×C9—42no—
274C4×C9—41no—
275Dic9—22no—
276Dic9—22no—
277Dic9—22no—
278#126—92no—
279#124—91no—
280#123—92no—
281#123AGL(1,ℤ/9)11yescharacteristic, complemented
282C22⋊Q8—92no—
283#120—91no—
284C4×D4—92no—
285#117—92no—
286C2×C22⋊C4—92no—
2873−1+2#55111yescharacteristic, complemented, Sylow
288C23×C3—31no—
289C23×C3—31no—
290C23×C3—92no—
291C23×C3—184no—
292C23×C3—361no—
293C23×C3C22×C3×D311yescharacteristic
294C23×C3C22×C3×D311yescharacteristic
295C22×D3—32no—
296C22×D3—64no—
297C22×D3—121no—
298C3×Q8—361no—
299C3×D4—41no—
300C3×D4—121no—
301C3×D4—184no—
302C2×C3×C4—32no—
303C2×C3×C4—31no—
304C2×C3×C4—32no—
305C2×C3×C4—41no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—92no—
309C2×C3×C4—121no—
310C2×C3×C4—184no—
311C2×C3×C4—362no—
312C2×C3×C4C22×C3×D312yes—
313C2×C3×C4C22×C3×D311yescharacteristic
314C2×C3×C4C22×C3×D312yes—
315#67—64no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C4×D3—64no—
320C4×D3—122no—
321Dic6—121no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32#100411yescharacteristic
325C2×C32D3×#4112yes—
326C3×D3—64no—
327C2×C9—21no—
328C2×C9—22no—
329C2×C9—42no—
330C2×C9—41no—
331C2×C9—41no—
332C2×C9—81no—
333C2×C9C2×C3×#4112yes—
334C2×C9C3×H(5,𝔽2)11yescharacteristic
335D9—24no—
336C24—91no—
337C2×Q8—91no—
338C2×D4—92no—
339C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
340C22×C4—92no—
341C22×C4—92no—
342C22×C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
343C4⋊C4—92no—
344C4⋊C4—92no—
345C4⋊C4—92no—
346C4⋊C4C2×AGL(1,ℤ/9)12yescomplemented
347C22⋊C4—92no—
348C22⋊C4—92no—
349C22⋊C4—91no—
350C22⋊C4—92no—
351C22⋊C4—92no—
352C22⋊C4—92no—
353C22⋊C4C2×AGL(1,ℤ/9)12yes—
354C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
355C42—91no—
356C42—92no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—31no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—62no—
365C22×C3—62no—
366C22×C3—94no—
367C22×C3—182no—
368C22×C3—182no—
369C22×C3—364no—
370C22×C3C23×C3×D311yescharacteristic
371C2×D3—34no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—124no—
375C3×C4—21no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—62no—
379C3×C4—62no—
380C3×C4—61no—
381C3×C4—182no—
382C3×C4—182no—
383C3×C4—182no—
384Dic3—62no—
385Dic3—62no—
386Dic3—62no—
387C32#565911yescharacteristic
388C9—21no—
389C9C3×#55111yescharacteristic, complemented
390C23—92no—
391C23—184no—
392C23—361no—
393C23C22×AGL(1,ℤ/9)11yescharacteristic
394C23C22×AGL(1,ℤ/9)11yescharacteristic
395Q8—361no—
396D4—41no—
397D4—184no—
398C2×C4—41no—
399C2×C4—92no—
400C2×C4—92no—
401C2×C4—92no—
402C2×C4—184no—
403C2×C4—362no—
404C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
405C2×C4C22×AGL(1,ℤ/9)12yes—
406C2×C4C22×AGL(1,ℤ/9)12yes—
407C2×C3—31no—
408C2×C3—32no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—121no—
412C2×C3—121no—
413C2×C3—184no—
414C2×C3C3×#100411yescharacteristic
415C2×C3C3×D3×#4112yes—
416D3—64no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—22no—
421C22—94no—
422C22—182no—
423C22—182no—
424C22—364no—
425C22C23×AGL(1,ℤ/9)11yescharacteristic
426C4—22no—
427C4—21no—
428C4—22no—
429C4—182no—
430C4—182no—
431C4—182no—
432C3—31no—
433C3C3×#565911yescharacteristic
434C2—41no—
435C2—41no—
436C2—184no—
437C2#63710⋊C311yescharacteristic
438C2#41×AGL(1,ℤ/9)12yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#123C2
prime radical#63499⋊C3C2
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#551
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#123C2semidirect
3. #63498⋊C3C2semidirect
4. C4×#611⋊C3C2semidirect
5. #63458⋊C3C2semidirect
6. #63454⋊C3C2semidirect
7. #63445⋊C3C2semidirect
8. C22⋊C4×AGL(1,ℤ/9)C2semidirect
9. #63441⋊C3C2semidirect
11. #10589761C3semidirect
14. C22×C4×3−1+2C22semidirect
18. C2×3−1+2⋊Q8C22semidirect
19. C4⋊C4×3−1+2C22semidirect
21. C22⋊C4×3−1+2C22semidirect
24. #3676⋊C3C22semidirect
28. C4⋊(3−1+2⋊C4)C22semidirect
29. (3−1+2⋊C4)⋊C4C22semidirect
30. (3−1+2⋊C4)⋊C4C22semidirect
31. C4×3−1+2⋊C4C22semidirect
32. C4×3−1+2⋊C4C22semidirect
43. C9×#123C2×C3semidirect
45. #63498C2×C3semidirect
46. C4×#611C2×C3semidirect
47. #63458C2×C3semidirect
48. #63454C2×C3semidirect
49. #63445C2×C3semidirect
50. C22⋊C4×D9C2×C3semidirect
51. #63441C2×C3semidirect
96. C22×C4×C9C22×C3semidirect
100. C2×Dic18C22×C3semidirect
102. C9×C4⋊C4C22×C3semidirect
106. C9×C22⋊C4C22×C3semidirect
109. #3676C22×C3semidirect
113. C4⋊Dic9C22×C3semidirect
114. Dic9⋊C4C22×C3semidirect
115. Dic9⋊C4C22×C3semidirect
116. C4×Dic9C22×C3semidirect
117. C4×Dic9C22×C3semidirect
281. #123AGL(1,ℤ/9)semidirect
287. 3−1+2#551semidirect
342. C22×C4C2×AGL(1,ℤ/9)semidirect
346. C4⋊C4C2×AGL(1,ℤ/9)semidirect
354. C22⋊C4C2×AGL(1,ℤ/9)semidirect
389. C9C3×#551semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1232
#63499⋊C32
#63498⋊C32
C4×#611⋊C32
#63458⋊C32
#63454⋊C32
#63445⋊C32
C22⋊C4×AGL(1,ℤ/9)2
#63441⋊C32
C3×#56593
#105897613