Finite Group Databasebeta

← #10589786⋊C3 #13153142

Subgroups — #10589786⋊C3 #13153142

Order 1728.

698 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×#124C211yesmaximal, characteristic, complemented
2#63508⋊C3C212yesmaximal, complemented
3#63501⋊C3C212yesmaximal, complemented
4#63448⋊C3C212yesmaximal, complemented
5#63446⋊C3C212yesmaximal, complemented
6#63445⋊C3C212yesmaximal, complemented
7#63440⋊C3C212yesmaximal, complemented
8#63437⋊C3C211yesmaximal, characteristic, complemented
9#63432⋊C3C211yesmaximal, characteristic, complemented
10C3×#5684—31nomaximal
11#10589786C311yesmaximal, characteristic, complemented
12C2×Q8×3−1+2C2211yescharacteristic
13C2×D4×3−1+2C2211yescharacteristic
14C2×#611⋊C3C2212yes—
15C2×#611⋊C3C2212yes—
16C22×3−1+2⋊C4C2212yes—
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)C2212yes—
19C2×3−1+2⋊Q8C2211yescharacteristic
20C22⋊C4×3−1+2C2212yes—
21C22⋊C4×3−1+2C2212yes—
22C42×3−1+2C2211yescharacteristic
23#3676⋊C3C2212yes—
24(C2×AGL(1,ℤ/9))⋊C4C2212yes—
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27C4⋊(3−1+2⋊C4)C2211yescharacteristic
28C4⋊(3−1+2⋊C4)C2211yescharacteristic
29C4⋊(3−1+2⋊C4)C2212yes—
30(3−1+2⋊C4)⋊C4C2212yes—
31(3−1+2⋊C4)⋊C4C2212yes—
32(3−1+2⋊C4)⋊C4C2212yes—
33C32×#124D311yescharacteristic
34C3×#946—32no—
35C3×#939—32no—
36C3×#886—32no—
37C3×#884—32no—
38C3×#883—32no—
39C3×#878—32no—
40C3×#875—31no—
41C3×#870—31no—
42C9×#124—21no—
43C9×#124C2×C311yescharacteristic, complemented
44#63508C2×C312yescomplemented
45#63501C2×C312yescomplemented
46#63448C2×C312yescomplemented
47#63446C2×C312yescomplemented
48#63445C2×C312yescomplemented
49#63440C2×C312yescomplemented
50#63437C2×C311yescharacteristic, complemented
51#63432C2×C311yescharacteristic, complemented
52C23×3−1+2C2312yes—
53C22×AGL(1,ℤ/9)C2312yes—
54Q8×3−1+2—41no—
55D4×3−1+2—41no—
56C2×C4×3−1+2C2311yescharacteristic
57C2×C4×3−1+2C2312yes—
58C2×C4×3−1+2C2311yescharacteristic
59C2×C4×3−1+2C2311yescharacteristic
60#611⋊C3—42no—
61#611⋊C3—42no—
62C2×3−1+2⋊C4—42no—
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—41no—
69C3×#562—91no—
70#5684—31no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C2×C3×#67—32no—
74C2×C3×#67—32no—
75C22×C3×Dic3—32no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—32no—
78C2×C3×Dic6—31no—
79C32×C22⋊C4C2×D312yes—
80C32×C22⋊C4C2×D312yes—
81C32×C42C2×D311yescharacteristic
82C3×#217—32no—
83C3×(C2×D3)⋊C4—32no—
84C3×(C2×D3)⋊C4—32no—
85C3×(C2×D3)⋊C4—32no—
86C3×C4⋊Dic3—32no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C2×Q8×C9—21no—
93C2×Q8×C9C22×C311yescharacteristic
94C2×D4×C9—21no—
95C2×D4×C9C22×C311yescharacteristic
96C2×#611C22×C312yes—
97C2×#611C22×C312yes—
98C22×Dic9C22×C312yes—
99C2×D36C22×C311yescharacteristic
100C2×C4×D9C22×C312yes—
101C2×Dic18C22×C311yescharacteristic
102C9×C22⋊C4—22no—
103C9×C22⋊C4—22no—
104C9×C22⋊C4C22×C312yes—
105C9×C22⋊C4C22×C312yes—
106C42×C9—21no—
107C42×C9C22×C311yescharacteristic
108#3676C22×C312yes—
109(C2×D9)⋊C4C22×C312yes—
110(C2×D9)⋊C4C22×C312yes—
111(C2×D9)⋊C4C22×C312yes—
112C4⋊Dic9C22×C312yes—
113C4⋊Dic9C22×C311yescharacteristic
114C4⋊Dic9C22×C311yescharacteristic
115Dic9⋊C4C22×C312yes—
116Dic9⋊C4C22×C312yes—
117Dic9⋊C4C22×C312yes—
118C22×3−1+2—22no—
119C22×3−1+2—22no—
120C22×3−1+2—22no—
121C22×3−1+2C2411yescharacteristic
122C2×AGL(1,ℤ/9)—22no—
123C2×AGL(1,ℤ/9)—22no—
124C2×AGL(1,ℤ/9)—22no—
125C4×3−1+2—21no—
126C4×3−1+2—21no—
127C4×3−1+2—22no—
128C4×3−1+2—21no—
1293−1+2⋊C4—22no—
1303−1+2⋊C4—22no—
1313−1+2⋊C4—22no—
132C3×#125—91no—
133C3×#124—31no—
134C3×#124—91no—
135C3×#124C3×D311yescharacteristic
136C3×#123—92no—
137C3×#123—92no—
138C3×C22⋊Q8—92no—
139C3×C22⋊Q8—92no—
140C3×#121—92no—
141C3×#121—92no—
142#946—32no—
143#939—32no—
144#886—32no—
145#884—32no—
146#883—32no—
147#878—32no—
148#875—31no—
149#870—31no—
150C23×C32C22×D312yes—
151C22×C3×D3—32no—
152C32×Q8—41no—
153C32×D4—41no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D311yescharacteristic
158C3×#67—122no—
159C3×#67—122no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—122no—
164C3×D12—121no—
165C3×C4×D3—122no—
166C3×Dic6—121no—
167C23×C9—22no—
168C23×C9C23×C312yes—
169C22×D9C23×C312yes—
170Q8×C9—41no—
171Q8×C9—42no—
172D4×C9—41no—
173D4×C9—42no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—22no—
178C2×C4×C9C23×C311yescharacteristic
179C2×C4×C9C23×C311yescharacteristic
180C2×C4×C9C23×C312yes—
181C2×C4×C9C23×C311yescharacteristic
182#611—42no—
183#611—42no—
184C2×Dic9—42no—
185C2×Dic9C23×C312yes—
186C2×Dic9C23×C312yes—
187C2×Dic9C23×C312yes—
188D36—41no—
189C4×D9—42no—
190Dic18—41no—
191#562—91noSylow
192C2×3−1+2—42no—
193C2×3−1+2H(5,𝔽2)11yescharacteristic
194C2×3−1+2H(5,𝔽2)11yescharacteristic
195C2×3−1+22−1+411yescharacteristic
196AGL(1,ℤ/9)—42no—
197C2×C3×Q8—31no—
198C2×C3×Q8—91no—
199C2×C3×Q8C2×C3×D311yescharacteristic
200C2×C3×D4—31no—
201C2×C3×D4—91no—
202C2×C3×D4—92no—
203C2×C3×D4—92no—
204C2×C3×D4C2×C3×D311yescharacteristic
205C22×C3×C4—92no—
206C22×C3×C4—92no—
207C2×#67—32no—
208C2×#67—32no—
209C22×Dic3—32no—
210C2×D12—31no—
211C2×C4×D3—32no—
212C2×Dic6—31no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C22⋊C4—32no—
220C3×C22⋊C4—32no—
221C3×C22⋊C4—92no—
222C3×C22⋊C4—92no—
223C3×C22⋊C4—92no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4C2×C3×D312yes—
226C3×C22⋊C4C2×C3×D312yes—
227C3×C42—31no—
228C3×C42C2×C3×D311yescharacteristic
229#217—32no—
230(C2×D3)⋊C4—32no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—32no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—32no—
236Dic3⋊C4—32no—
237Dic3⋊C4—32no—
238Dic3⋊C4—32no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32C23×D311yescharacteristic
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C32×C4—22no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C22×C9—22no—
254C22×C9—22no—
255C22×C9—22no—
256C22×C9—21no—
257C22×C9—42no—
258C22×C9—42no—
259C22×C9—42no—
260C22×C9C24×C311yescharacteristic
261C2×D9—22no—
262C2×D9—22no—
263C2×D9—22no—
264C4×C9—22no—
265C4×C9—21no—
266C4×C9—22no—
267C4×C9—21no—
268C4×C9—21no—
269C4×C9—41no—
270C4×C9—41no—
271C4×C9—42no—
272Dic9—22no—
273Dic9—22no—
274Dic9—22no—
275#125—91no—
276#124—91no—
277#124AGL(1,ℤ/9)11yescharacteristic, complemented
278#123—92no—
279#123—92no—
280C22⋊Q8—92no—
281C22⋊Q8—92no—
282#121—92no—
283#121—92no—
2843−1+2#56211yescharacteristic, complemented, Sylow
285C23×C3—32no—
286C23×C3—92no—
287C23×C3C22×C3×D312yes—
288C22×D3—32no—
289C3×Q8—41no—
290C3×Q8—121no—
291C3×Q8—361no—
292C3×D4—41no—
293C3×D4—121no—
294C3×D4—361no—
295C3×D4—362no—
296C3×D4—362no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—32no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—362no—
305C2×C3×C4—362no—
306C2×C3×C4C22×C3×D311yescharacteristic
307C2×C3×C4C22×C3×D311yescharacteristic
308C2×C3×C4C22×C3×D312yes—
309C2×C3×C4C22×C3×D311yescharacteristic
310#67—122no—
311#67—122no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—122no—
316D12—121no—
317C4×D3—122no—
318Dic6—121no—
319C2×C32—42no—
320C2×C32#100411yescharacteristic
321C2×C32#100911yescharacteristic
322C2×C32#101011yescharacteristic
323C3×D3—122no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—21no—
327C2×C9—42no—
328C2×C9—82no—
329C2×C9C3×H(5,𝔽2)11yescharacteristic
330C2×C9C3×H(5,𝔽2)11yescharacteristic
331C2×C9C3×2−1+411yescharacteristic
332D9—42no—
333C2×Q8—91no—
334C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
335C2×D4—92no—
336C2×D4—92no—
337C2×D4—91no—
338C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
339C22×C4—92no—
340C22×C4—92no—
341C4⋊C4—92no—
342C4⋊C4—92no—
343C4⋊C4—92no—
344C4⋊C4—92no—
345C4⋊C4—91no—
346C4⋊C4—91no—
347C22⋊C4—92no—
348C22⋊C4—92no—
349C22⋊C4—92no—
350C22⋊C4—92no—
351C22⋊C4C2×AGL(1,ℤ/9)12yes—
352C22⋊C4C2×AGL(1,ℤ/9)12yes—
353C42C2×AGL(1,ℤ/9)11yescharacteristic
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—31no—
358C22×C3—62no—
359C22×C3—62no—
360C22×C3—62no—
361C22×C3—182no—
362C22×C3—182no—
363C22×C3—182no—
364C22×C3C23×C3×D311yescharacteristic
365C2×D3—62no—
366C2×D3—62no—
367C2×D3—62no—
368C3×C4—22no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—62no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—182no—
377C3×C4—182no—
378C3×C4—182no—
379Dic3—62no—
380Dic3—62no—
381Dic3—62no—
382C32#568411yescharacteristic
383C9—21no—
384C9C3×#56211yescharacteristic, complemented
385C23—92no—
386C23C22×AGL(1,ℤ/9)12yes—
387Q8—41no—
388Q8—361no—
389D4—41no—
390D4—361no—
391D4—362no—
392D4—362no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—92no—
396C2×C4—362no—
397C2×C4—362no—
398C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
399C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
400C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
401C2×C4C22×AGL(1,ℤ/9)12yes—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—42no—
406C2×C3—122no—
407C2×C3—362no—
408C2×C3C3×#100411yescharacteristic
409C2×C3C3×#100911yescharacteristic
410C2×C3C3×#101011yescharacteristic
411D3—122no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—182no—
416C22—182no—
417C22—182no—
418C22C23×AGL(1,ℤ/9)11yescharacteristic
419C4—22no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—182no—
424C4—182no—
425C4—182no—
426C3—31no—
427C3C3×#568411yescharacteristic
428C2—42no—
429C2—362no—
430C2#63710⋊C311yescharacteristic
431C2#1172264111yescharacteristic
432C2#1172264211yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#124C2
hypercenter#124AGL(1,ℤ/9)
hypocenterC9C3×#562
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#124C2semidirect
2. #63508⋊C3C2semidirect
3. #63501⋊C3C2semidirect
4. #63448⋊C3C2semidirect
5. #63446⋊C3C2semidirect
6. #63445⋊C3C2semidirect
7. #63440⋊C3C2semidirect
8. #63437⋊C3C2semidirect
9. #63432⋊C3C2semidirect
11. #10589786C3semidirect
43. C9×#124C2×C3semidirect
44. #63508C2×C3semidirect
45. #63501C2×C3semidirect
46. #63448C2×C3semidirect
47. #63446C2×C3semidirect
48. #63445C2×C3semidirect
49. #63440C2×C3semidirect
50. #63437C2×C3semidirect
51. #63432C2×C3semidirect
277. #124AGL(1,ℤ/9)semidirect
284. 3−1+2#562semidirect
384. C9C3×#562semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1242
#63508⋊C32
#63501⋊C32
#63448⋊C32
#63446⋊C32
#63445⋊C32
#63440⋊C32
#63437⋊C32
#63432⋊C32
C3×#56843
#105897863