Finite Group Databasebeta

← #10589664⋊C3 #13152663

Subgroups — #10589664⋊C3 #13152663

Order 1728.

767 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×3−1+2C211yesmaximal, characteristic, complemented
2#63502⋊C3C211yesmaximal, characteristic, complemented
3#63500⋊C3C211yesmaximal, characteristic, complemented
4#63492⋊C3C212yesmaximal, complemented
5#63491⋊C3C212yesmaximal, complemented
6#63457⋊C3C211yesmaximal, characteristic, complemented
7#63452⋊C3C211yesmaximal, characteristic, complemented
8#63446⋊C3C212yesmaximal, complemented
9#63445⋊C3C212yesmaximal, complemented
10C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
11#63434⋊C3C211yesmaximal, characteristic, complemented
12C3×#5562—31nomaximal
13#10589664C311yesmaximal, characteristic, complemented
14C2×D4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2212yes—
16C2×#611⋊C3C2212yes—
17C2×#611⋊C3C2212yes—
18C2×#611⋊C3C2212yes—
19C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
20C2×C4⋊AGL(1,ℤ/9)C2212yes—
21C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
22C2×C4×AGL(1,ℤ/9)C2212yes—
23C4⋊C4×3−1+2C2211yescharacteristic
24C22⋊C4×3−1+2C2212yes—
25C42×3−1+2C2211yescharacteristic
26#3676⋊C3C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2212yes—
29(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
30(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
31C4⋊(3−1+2⋊C4)C2212yes—
32C4⋊(3−1+2⋊C4)C2211yescharacteristic
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊C4C2211yescharacteristic
35(3−1+2⋊C4)⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2211yescharacteristic
37C32×C4×D4D311yescharacteristic
38C3×#940—31no—
39C3×#938—31no—
40C3×#930—32no—
41C3×#929—32no—
42C3×#895—31no—
43C3×#890—31no—
44C3×#884—32no—
45C3×#883—32no—
46C3×C4×D12—31no—
47C3×#872—31no—
48C4×D4×C9—21no—
49C4×D4×C9C2×C311yescharacteristic, complemented
50#63502C2×C311yescharacteristic, complemented
51#63500C2×C311yescharacteristic, complemented
52#63492C2×C312yescomplemented
53#63491C2×C312yescomplemented
54#63457C2×C311yescharacteristic, complemented
55#63452C2×C311yescharacteristic, complemented
56#63446C2×C312yescomplemented
57#63445C2×C312yescomplemented
58C4×D36C2×C311yescharacteristic, complemented
59#63434C2×C311yescharacteristic, complemented
60C23×3−1+2C2312yes—
61C22×AGL(1,ℤ/9)C2311yescharacteristic
62C22×AGL(1,ℤ/9)C2312yes—
63D4×3−1+2—22no—
64C2×C4×3−1+2—42no—
65C2×C4×3−1+2C2311yescharacteristic
66C2×C4×3−1+2C2312yes—
67C2×C4×3−1+2C2311yescharacteristic
68C2×C4×3−1+2C2311yescharacteristic
69#611⋊C3—42no—
70#611⋊C3—42no—
71#611⋊C3—42no—
72C2×3−1+2⋊C4C2312yes—
73C2×3−1+2⋊C4C2311yescharacteristic
74C2×3−1+2⋊C4C2311yescharacteristic
75C2×3−1+2⋊C4C2311yescharacteristic
76C4⋊AGL(1,ℤ/9)—22no—
77C4⋊AGL(1,ℤ/9)—42no—
78C4×AGL(1,ℤ/9)—22no—
79C4×AGL(1,ℤ/9)—42no—
80C3×#540—91no—
81#5562—31no—
82C2×C32×D4C2×D311yescharacteristic
83C22×C32×C4C2×D312yes—
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×D12—31no—
88C2×C3×D12—32no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—32no—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C22⋊C4C2×D312yes—
93C32×C42C2×D311yescharacteristic
94C3×#217—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—31no—
97C3×(C2×D3)⋊C4—32no—
98C3×(C2×D3)⋊C4—31no—
99C3×C4⋊Dic3—32no—
100C3×C4⋊Dic3—31no—
101C3×Dic3⋊C4—31no—
102C3×Dic3⋊C4—32no—
103C3×Dic3⋊C4—31no—
104C3×C4×Dic3—31no—
105C2×D4×C9—21no—
106C2×D4×C9C22×C311yescharacteristic
107C22×C4×C9—22no—
108C22×C4×C9C22×C312yes—
109C2×#611C22×C312yes—
110C2×#611C22×C312yes—
111C2×#611C22×C312yes—
112C2×D36C22×C311yescharacteristic
113C2×D36C22×C312yes—
114C2×C4×D9C22×C311yescharacteristic
115C2×C4×D9C22×C312yes—
116C9×C4⋊C4—21no—
117C9×C4⋊C4C22×C311yescharacteristic
118C9×C22⋊C4—22no—
119C9×C22⋊C4C22×C312yes—
120C42×C9—21no—
121C42×C9C22×C311yescharacteristic
122#3676C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C311yescharacteristic
126(C2×D9)⋊C4C22×C311yescharacteristic
127C4⋊Dic9C22×C312yes—
128C4⋊Dic9C22×C311yescharacteristic
129Dic9⋊C4C22×C311yescharacteristic
130Dic9⋊C4C22×C312yes—
131Dic9⋊C4C22×C311yescharacteristic
132C4×Dic9C22×C311yescharacteristic
133C22×3−1+2—22no—
134C22×3−1+2—22no—
135C22×3−1+2—22no—
136C22×3−1+2C2411yescharacteristic
137C2×AGL(1,ℤ/9)—21no—
138C2×AGL(1,ℤ/9)—22no—
139C2×AGL(1,ℤ/9)—22no—
140C2×AGL(1,ℤ/9)—21no—
141C2×AGL(1,ℤ/9)—22no—
142C2×AGL(1,ℤ/9)—21no—
143C4×3−1+2—21no—
144C4×3−1+2—21no—
145C4×3−1+2—22no—
146C4×3−1+2#4112yes—
1473−1+2⋊C4—21no—
1483−1+2⋊C4—22no—
1493−1+2⋊C4—21no—
1503−1+2⋊C4—21no—
151C3×#127—91no—
152C3×#125—91no—
153C3×#123—92no—
154C3×#123—92no—
155C3×#121—92no—
156C3×#121—91no—
157C3×#121—92no—
158C3×#121—91no—
159C3×C4×D4—31no—
160C3×C4×D4—91no—
161C3×C4×D4C3×D311yescharacteristic
162C3×#117—91no—
163#940—31no—
164#938—31no—
165#930—32no—
166#929—32no—
167#895—31no—
168#890—31no—
169#884—32no—
170#883—32no—
171C4×D12—31no—
172#872—31no—
173C23×C32C22×D312yes—
174C22×C3×D3—31no—
175C22×C3×D3—32no—
176C32×D4—22no—
177C2×C32×C4—42no—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C3×#67—122no—
183C3×#67—122no—
184C3×#67—122no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—32no—
189C3×D12—62no—
190C3×D12—122no—
191C3×C4×D3—62no—
192C3×C4×D3—122no—
193C23×C9—22no—
194C23×C9C23×C312yes—
195C22×D9C23×C311yescharacteristic
196C22×D9C23×C312yes—
197D4×C9—24no—
198D4×C9—22no—
199C2×C4×C9—21no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—22no—
203C2×C4×C9—42no—
204C2×C4×C9—44no—
205C2×C4×C9C23×C312yes—
206C2×C4×C9C23×C311yescharacteristic
207C2×C4×C9C23×C311yescharacteristic
208C2×C4×C9C23×C311yescharacteristic
209#611—42no—
210#611—42no—
211#611—42no—
212C2×Dic9C23×C311yescharacteristic
213C2×Dic9C23×C311yescharacteristic
214C2×Dic9C23×C311yescharacteristic
215C2×Dic9C23×C312yes—
216D36—22no—
217D36—42no—
218C4×D9—22no—
219C4×D9—42no—
220#540—91noSylow
221C2×3−1+2—42no—
222C2×3−1+2C2×#4111yescharacteristic
223C2×3−1+2H(5,𝔽2)11yescharacteristic
224C2×3−1+2H(5,𝔽2)11yescharacteristic
225AGL(1,ℤ/9)—41no—
226AGL(1,ℤ/9)—42no—
227C2×C3×D4—31no—
228C2×C3×D4—91no—
229C2×C3×D4—92no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4C2×C3×D311yescharacteristic
234C22×C3×C4—32no—
235C22×C3×C4—91no—
236C22×C3×C4—92no—
237C22×C3×C4C2×C3×D312yes—
238C2×#67—32no—
239C2×#67—32no—
240C2×#67—32no—
241C2×D12—31no—
242C2×D12—32no—
243C2×C4×D3—31no—
244C2×C4×D3—32no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4C2×C3×D311yescharacteristic
252C3×C22⋊C4—32no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4C2×C3×D312yes—
259C3×C42—31no—
260C3×C42—91no—
261C3×C42C2×C3×D311yescharacteristic
262#217—32no—
263(C2×D3)⋊C4—31no—
264(C2×D3)⋊C4—31no—
265(C2×D3)⋊C4—32no—
266(C2×D3)⋊C4—32no—
267C4⋊Dic3—31no—
268C4⋊Dic3—32no—
269Dic3⋊C4—31no—
270Dic3⋊C4—31no—
271Dic3⋊C4—32no—
272C4×Dic3—31no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32C23×D311yescharacteristic
277C2×C3×D3—61no—
278C2×C3×D3—62no—
279C2×C3×D3—61no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—61no—
283C32×C4—22no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4#23512yes—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C3×Dic3—62no—
290C3×Dic3—61no—
291C22×C9—24no—
292C22×C9—22no—
293C22×C9—22no—
294C22×C9—22no—
295C22×C9—21no—
296C22×C9—42no—
297C22×C9—42no—
298C22×C9C24×C311yescharacteristic
299C2×D9—21no—
300C2×D9—22no—
301C2×D9—22no—
302C2×D9—21no—
303C2×D9—22no—
304C2×D9—21no—
305C4×C9—22no—
306C4×C9—21no—
307C4×C9—22no—
308C4×C9—21no—
309C4×C9—22no—
310C4×C9—41no—
311C4×C9—42no—
312C4×C9C3×#4112yes—
313Dic9—21no—
314Dic9—22no—
315Dic9—21no—
316Dic9—21no—
317#127—91no—
318#125—91no—
319#123—92no—
320#123—92no—
321#121—91no—
322#121—92no—
323#121—91no—
324#121—92no—
325C4×D4—91no—
326C4×D4AGL(1,ℤ/9)11yescharacteristic, complemented
327#117—91no—
3283−1+2#54011yescharacteristic, complemented, Sylow
329C23×C3—32no—
330C23×C3—91no—
331C23×C3—92no—
332C23×C3C22×C3×D312yes—
333C22×D3—31no—
334C22×D3—32no—
335C3×D4—22no—
336C3×D4—62no—
337C3×D4—182no—
338C3×D4—362no—
339C3×D4—362no—
340C3×D4—362no—
341C3×D4—362no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—32no—
346C2×C3×C4—42no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—92no—
351C2×C3×C4—122no—
352C2×C3×C4—182no—
353C2×C3×C4—362no—
354C2×C3×C4C22×C3×D312yes—
355C2×C3×C4C22×C3×D311yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357C2×C3×C4C22×C3×D311yescharacteristic
358#67—122no—
359#67—122no—
360#67—122no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—31no—
364C2×Dic3—32no—
365D12—62no—
366D12—122no—
367C4×D3—62no—
368C4×D3—122no—
369C2×C32—42no—
370C2×C32C2×#23511yescharacteristic
371C2×C32#100411yescharacteristic
372C2×C32#100911yescharacteristic
373C3×D3—121no—
374C3×D3—122no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—44no—
379C2×C9—42no—
380C2×C9C2×C3×#4111yescharacteristic
381C2×C9C3×H(5,𝔽2)11yescharacteristic
382C2×C9C3×H(5,𝔽2)11yescharacteristic
383D9—41no—
384D9—42no—
385C2×D4—92no—
386C2×D4—92no—
387C2×D4—92no—
388C2×D4—91no—
389C2×D4—92no—
390C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
391C22×C4—91no—
392C22×C4—92no—
393C22×C4C2×AGL(1,ℤ/9)12yes—
394C4⋊C4—91no—
395C4⋊C4—91no—
396C4⋊C4—91no—
397C4⋊C4—92no—
398C4⋊C4—92no—
399C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
400C22⋊C4—92no—
401C22⋊C4—91no—
402C22⋊C4—91no—
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4C2×AGL(1,ℤ/9)12yes—
406C42—91no—
407C42C2×AGL(1,ℤ/9)11yescharacteristic
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—31no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—181no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—181no—
419C22×C3—182no—
420C22×C3—181no—
421C22×C3C23×C3×D311yescharacteristic
422C2×D3—61no—
423C2×D3—62no—
424C2×D3—61no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—61no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—32no—
432C3×C4—61no—
433C3×C4—62no—
434C3×C4—61no—
435C3×C4—181no—
436C3×C4—182no—
437C3×C4—181no—
438C3×C4—181no—
439C3×C4C3×#23512yes—
440Dic3—61no—
441Dic3—61no—
442Dic3—62no—
443Dic3—61no—
444C32#556211yescharacteristic
445C9—21no—
446C9C3×#54011yescharacteristic, complemented
447C23—91no—
448C23—92no—
449C23C22×AGL(1,ℤ/9)12yes—
450D4—22no—
451D4—182no—
452D4—362no—
453D4—362no—
454D4—362no—
455D4—362no—
456C2×C4—42no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—92no—
461C2×C4—182no—
462C2×C4—362no—
463C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
464C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
465C2×C4C22×AGL(1,ℤ/9)12yes—
466C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—42no—
471C2×C3—122no—
472C2×C3—361no—
473C2×C3—362no—
474C2×C3C2×C3×#23511yescharacteristic
475C2×C3C3×#100411yescharacteristic
476C2×C3C3×#100911yescharacteristic
477D3—121no—
478D3—122no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—181no—
483C22—182no—
484C22—181no—
485C22—182no—
486C22—182no—
487C22—181no—
488C22C23×AGL(1,ℤ/9)11yescharacteristic
489C4—21no—
490C4—22no—
491C4—21no—
492C4—181no—
493C4—181no—
494C4—182no—
495C4—181no—
496C4#3697⋊C312yes—
497C3—31no—
498C3C3×#556211yescharacteristic
499C2—42no—
500C2—361no—
501C2—362no—
502C2C2×#3697⋊C311yescharacteristic
503C2#63710⋊C311yescharacteristic
504C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×D4×3−1+2C2
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C3×#540
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×3−1+2C2semidirect
2. #63502⋊C3C2semidirect
3. #63500⋊C3C2semidirect
4. #63492⋊C3C2semidirect
5. #63491⋊C3C2semidirect
6. #63457⋊C3C2semidirect
7. #63452⋊C3C2semidirect
8. #63446⋊C3C2semidirect
9. #63445⋊C3C2semidirect
10. C4×C4⋊AGL(1,ℤ/9)C2semidirect
11. #63434⋊C3C2semidirect
13. #10589664C3semidirect
49. C4×D4×C9C2×C3semidirect
50. #63502C2×C3semidirect
51. #63500C2×C3semidirect
52. #63492C2×C3semidirect
53. #63491C2×C3semidirect
54. #63457C2×C3semidirect
55. #63452C2×C3semidirect
56. #63446C2×C3semidirect
57. #63445C2×C3semidirect
58. C4×D36C2×C3semidirect
59. #63434C2×C3semidirect
326. C4×D4AGL(1,ℤ/9)semidirect
328. 3−1+2#540semidirect
446. C9C3×#540semidirect

Maximal subgroups

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

subgroupindex
C4×D4×3−1+22
#63502⋊C32
#63500⋊C32
#63492⋊C32
#63491⋊C32
#63457⋊C32
#63452⋊C32
#63446⋊C32
#63445⋊C32
C4×C4⋊AGL(1,ℤ/9)2
#63434⋊C32
C3×#55623
#105896643