Finite Group Databasebeta

← #10589766⋊C3 #13153061

Subgroups — #10589766⋊C3 #13153061

Order 1728.

747 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×#123C211yesmaximal, characteristic, complemented
2#63502⋊C3C211yesmaximal, characteristic
3#63497⋊C3C211yesmaximal, characteristic, complemented
4C4×#611⋊C3C212yesmaximal, complemented
5#63456⋊C3C212yesmaximal, complemented
6#63449⋊C3C212yesmaximal, complemented
7#63447⋊C3C212yesmaximal, complemented
8#63447⋊C3C212yesmaximal, complemented
9(#611⋊C3)⋊C4C212yesmaximal, complemented
10C3×#5664—31nomaximal
11#10589766C311yesmaximal, characteristic, complemented
12C2×D4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C2×#611⋊C3C2212yes—
15C2×#611⋊C3C2212yes—
16C22×3−1+2⋊C4C2211yescharacteristic
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)C2212yes—
19C2×3−1+2⋊Q8C2212yes—
20C4⋊C4×3−1+2C2212yes—
21C22⋊C4×3−1+2C2212yes—
22C22⋊C4×3−1+2C2211yescharacteristic
23#3676⋊C3C2212yes—
24#3676⋊C3C2211yescharacteristic
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(3−1+2⋊C4)⋊C4C2212yes—
29(3−1+2⋊C4)⋊C4C2212yes—
30C4×3−1+2⋊C4C2211yescharacteristic
31C4×3−1+2⋊C4C2212yes—
32C4×3−1+2⋊C4C2212yes—
33C32×#123D311yescharacteristic
34C3×#940—31no—
35C3×#935—31no—
36C3×C4×#67—32no—
37C3×#894—32no—
38C3×#887—32no—
39C3×#885—32no—
40C3×#885—32no—
41C3×#67⋊C4—32no—
42C9×#123—21no—
43C9×#123C2×C311yescharacteristic, complemented
44#63502C2×C311yescharacteristic
45#63497C2×C311yescharacteristic, complemented
46C4×#611C2×C312yescomplemented
47#63456C2×C312yescomplemented
48#63449C2×C312yescomplemented
49#63447C2×C312yescomplemented
50#63447C2×C312yescomplemented
51#611⋊C4C2×C312yescomplemented
52C23×3−1+2C2311yescharacteristic
53C23×3−1+2C2311yescharacteristic
54C22×AGL(1,ℤ/9)C2312yes—
55D4×3−1+2—41no—
56C2×C4×3−1+2—41no—
57C2×C4×3−1+2C2312yes—
58C2×C4×3−1+2C2312yes—
59C2×C4×3−1+2C2311yescharacteristic
60#611⋊C3—24no—
61#611⋊C3—24no—
62C2×3−1+2⋊C4—41no—
63C2×3−1+2⋊C4C2312yes—
64C2×3−1+2⋊C4C2312yes—
65C2×3−1+2⋊C4C2312yes—
66C4⋊AGL(1,ℤ/9)—41no—
67C4×AGL(1,ℤ/9)—42no—
683−1+2⋊Q8—24no—
69C3×#559—91no—
70#5664—31no—
71C2×C32×D4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C2×C3×#67—32no—
74C2×C3×#67—32no—
75C22×C3×Dic3—31no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—32no—
78C2×C3×Dic6—32no—
79C32×C4⋊C4C2×D312yes—
80C32×C22⋊C4C2×D312yes—
81C32×C22⋊C4C2×D311yescharacteristic
82C3×#217—31no—
83C3×#217—32no—
84C3×(C2×D3)⋊C4—32no—
85C3×(C2×D3)⋊C4—32no—
86C3×(C2×D3)⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—31no—
92C2×D4×C9—21no—
93C2×D4×C9C22×C311yescharacteristic
94C22×C4×C9—21no—
95C22×C4×C9C22×C311yescharacteristic
96C2×#611C22×C312yes—
97C2×#611C22×C312yes—
98C22×Dic9C22×C311yescharacteristic
99C2×D36C22×C311yescharacteristic
100C2×C4×D9C22×C312yes—
101C2×Dic18C22×C312yes—
102C9×C4⋊C4—22no—
103C9×C4⋊C4C22×C312yes—
104C9×C22⋊C4—21no—
105C9×C22⋊C4—22no—
106C9×C22⋊C4C22×C312yes—
107C9×C22⋊C4C22×C311yescharacteristic
108#3676C22×C312yes—
109#3676C22×C311yescharacteristic
110(C2×D9)⋊C4C22×C312yes—
111(C2×D9)⋊C4C22×C312yes—
112(C2×D9)⋊C4C22×C312yes—
113Dic9⋊C4C22×C312yes—
114Dic9⋊C4C22×C312yes—
115C4×Dic9C22×C312yes—
116C4×Dic9C22×C311yescharacteristic
117C4×Dic9C22×C312yes—
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)—22no—
126C4×3−1+2—22no—
127C4×3−1+2—22no—
128C4×3−1+2—21no—
1293−1+2⋊C4—22no—
1303−1+2⋊C4—22no—
1313−1+2⋊C4#4114yes—
132C3×#127—91no—
133C3×#124—92no—
134C3×#124—92no—
135C3×#123—31no—
136C3×#123—92no—
137C3×#123—91no—
138C3×#123C3×D311yescharacteristic
139C3×C4×Q8—92no—
140C3×C4×D4—92no—
141C3×C4×D4—92no—
142#940—31no—
143#935—31no—
144C4×#67—32no—
145#894—32no—
146#887—32no—
147#885—32no—
148#885—32no—
149#67⋊C4—32no—
150C23×C32C22×D311yescharacteristic
151C23×C32C22×D311yescharacteristic
152C22×C3×D3—32no—
153C32×D4—41no—
154C2×C32×C4—41no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C3×#67—64no—
159C3×#67—64no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—121no—
164C3×D12—121no—
165C3×C4×D3—122no—
166C3×Dic6—64no—
167C23×C9—21no—
168C23×C9—21no—
169C23×C9C23×C311yescharacteristic
170C23×C9C23×C311yescharacteristic
171C22×D9C23×C312yes—
172D4×C9—41no—
173D4×C9—42no—
174C2×C4×C9—22no—
175C2×C4×C9—22no—
176C2×C4×C9—21no—
177C2×C4×C9—41no—
178C2×C4×C9—42no—
179C2×C4×C9C23×C311yescharacteristic
180C2×C4×C9C23×C312yes—
181C2×C4×C9C23×C312yes—
182#611—24no—
183#611—24no—
184C2×Dic9—41no—
185C2×Dic9C23×C312yes—
186C2×Dic9C23×C312yes—
187C2×Dic9C23×C312yes—
188D36—41no—
189C4×D9—42no—
190Dic18—24no—
191#559—91noSylow
192C2×3−1+2—41no—
193C2×3−1+2—41no—
194C2×3−1+2C2×#4112yes—
195C2×3−1+2H(5,𝔽2)11yescharacteristic
196AGL(1,ℤ/9)—42no—
197C2×C3×Q8—92no—
198C2×C3×D4—31no—
199C2×C3×D4—91no—
200C2×C3×D4—92no—
201C2×C3×D4—92no—
202C2×C3×D4C2×C3×D311yescharacteristic
203C22×C3×C4—31no—
204C22×C3×C4—92no—
205C22×C3×C4—91no—
206C22×C3×C4C2×C3×D311yescharacteristic
207C2×#67—32no—
208C2×#67—32no—
209C22×Dic3—31no—
210C2×D12—31no—
211C2×C4×D3—32no—
212C2×Dic6—32no—
213C3×C4⋊C4—32no—
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—91no—
224C3×C22⋊C4C2×C3×D312yes—
225C3×C22⋊C4C2×C3×D311yescharacteristic
226C3×C42—92no—
227C3×C42—92no—
228C3×C42—91no—
229#217—31no—
230#217—32no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—32no—
233(C2×D3)⋊C4—32no—
234Dic3⋊C4—32no—
235Dic3⋊C4—32no—
236C4×Dic3—32no—
237C4×Dic3—32no—
238C4×Dic3—31no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32C23×D311yescharacteristic
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C32×C4—22no—
248C32×C4—21no—
249C32×C4—22no—
250C3×Dic3—34no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C22×C9—22no—
254C22×C9—22no—
255C22×C9—22no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—42no—
260C22×C9—41no—
261C22×C9—42no—
262C22×C9C24×C311yescharacteristic
263C2×D9—22no—
264C2×D9—22no—
265C2×D9—22no—
266C4×C9—22no—
267C4×C9—22no—
268C4×C9—21no—
269C4×C9—42no—
270C4×C9—42no—
271C4×C9—41no—
272Dic9—22no—
273Dic9—22no—
274Dic9C3×#4114yes—
275#127—91no—
276#124—92no—
277#124—92no—
278#123—91no—
279#123—92no—
280#123AGL(1,ℤ/9)11yescharacteristic, complemented
281C4×Q8—92no—
282C4×D4—92no—
283C4×D4—92no—
2843−1+2#55911yescharacteristic, complemented, Sylow
285C23×C3—31no—
286C23×C3—31no—
287C23×C3—92no—
288C23×C3C22×C3×D311yescharacteristic
289C23×C3C22×C3×D311yescharacteristic
290C22×D3—32no—
291C3×Q8—184no—
292C3×D4—41no—
293C3×D4—121no—
294C3×D4—184no—
295C3×D4—184no—
296C3×D4—361no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—31no—
300C2×C3×C4—41no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—121no—
305C2×C3×C4—362no—
306C2×C3×C4—361no—
307C2×C3×C4C22×C3×D311yescharacteristic
308C2×C3×C4C22×C3×D312yes—
309C2×C3×C4C22×C3×D312yes—
310#67—64no—
311#67—64no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—121no—
316D12—121no—
317C4×D3—122no—
318Dic6—64no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32#100411yescharacteristic
322C2×C32D3×#4112yes—
323C3×D3—122no—
324C2×C9—21no—
325C2×C9—22no—
326C2×C9—42no—
327C2×C9—41no—
328C2×C9—41no—
329C2×C9—81no—
330C2×C9C2×C3×#4112yes—
331C2×C9C3×H(5,𝔽2)11yescharacteristic
332D9—42no—
333C2×Q8—92no—
334C2×D4—92no—
335C2×D4—92no—
336C2×D4—91no—
337C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
338C22×C4—91no—
339C22×C4—92no—
340C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
341C4⋊C4—92no—
342C4⋊C4—92no—
343C4⋊C4C2×AGL(1,ℤ/9)12yes—
344C22⋊C4—92no—
345C22⋊C4—91no—
346C22⋊C4—92no—
347C22⋊C4—92no—
348C22⋊C4—92no—
349C22⋊C4C2×AGL(1,ℤ/9)12yes—
350C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
351C42—92no—
352C42—92no—
353C42—91no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—31no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—62no—
362C22×C3—62no—
363C22×C3—182no—
364C22×C3—182no—
365C22×C3—182no—
366C22×C3C23×C3×D311yescharacteristic
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—62no—
370C3×C4—22no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—62no—
374C3×C4—61no—
375C3×C4—62no—
376C3×C4—94no—
377C3×C4—182no—
378C3×C4—182no—
379Dic3—34no—
380Dic3—62no—
381Dic3—62no—
382C32#566411yescharacteristic
383C9—21no—
384C9C3×#55911yescharacteristic, complemented
385C23—92no—
386C23C22×AGL(1,ℤ/9)11yescharacteristic
387C23C22×AGL(1,ℤ/9)11yescharacteristic
388Q8—184no—
389D4—41no—
390D4—184no—
391D4—184no—
392D4—361no—
393C2×C4—41no—
394C2×C4—92no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4—362no—
398C2×C4—361no—
399C2×C4C22×AGL(1,ℤ/9)12yes—
400C2×C4C22×AGL(1,ℤ/9)12yes—
401C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
402C2×C3—31no—
403C2×C3—32no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—121no—
407C2×C3—121no—
408C2×C3—362no—
409C2×C3C3×#100411yescharacteristic
410C2×C3C3×D3×#4112yes—
411D3—122no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—182no—
417C22—182no—
418C22—182no—
419C22C23×AGL(1,ℤ/9)11yescharacteristic
420C4—22no—
421C4—21no—
422C4—22no—
423C4—94no—
424C4—182no—
425C4—182no—
426C3—31no—
427C3C3×#566411yescharacteristic
428C2—41no—
429C2—41no—
430C2—362no—
431C2#63710⋊C311yescharacteristic
432C2#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#63502⋊C3C2
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#559
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. #63497⋊C3C2semidirect
4. C4×#611⋊C3C2semidirect
5. #63456⋊C3C2semidirect
6. #63449⋊C3C2semidirect
7. #63447⋊C3C2semidirect
8. #63447⋊C3C2semidirect
9. (#611⋊C3)⋊C4C2semidirect
11. #10589766C3semidirect
43. C9×#123C2×C3semidirect
45. #63497C2×C3semidirect
46. C4×#611C2×C3semidirect
47. #63456C2×C3semidirect
48. #63449C2×C3semidirect
49. #63447C2×C3semidirect
50. #63447C2×C3semidirect
51. #611⋊C4C2×C3semidirect
280. #123AGL(1,ℤ/9)semidirect
284. 3−1+2#559semidirect
384. C9C3×#559semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1232
#63502⋊C32
#63497⋊C32
C4×#611⋊C32
#63456⋊C32
#63449⋊C32
#63447⋊C32
(#611⋊C3)⋊C42
C3×#56643
#105897663