Finite Group Databasebeta

← #10589616⋊C3 #13152465

Subgroups — #10589616⋊C3 #13152465

Order 1728.

760 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2#63492⋊C3C211yesmaximal, characteristic, complemented
3#63492⋊C3C212yesmaximal, complemented
4C4×#611⋊C3C211yesmaximal, characteristic, complemented
5C4×#611⋊C3C212yesmaximal, complemented
6#63460⋊C3C212yesmaximal, complemented
7#63457⋊C3C211yesmaximal, characteristic, complemented
8#63456⋊C3C212yesmaximal, complemented
9#63455⋊C3C211yesmaximal, characteristic, complemented
10#63454⋊C3C211yesmaximal, characteristic, complemented
11#63452⋊C3C211yesmaximal, characteristic, complemented
12C3×#5514—31nomaximal
13#10589616C311yesmaximal, characteristic, complemented
14C22×C4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2212yes—
16C2×#611⋊C3C2211yescharacteristic
17C2×#611⋊C3C2212yes—
18C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
19C2×C4⋊AGL(1,ℤ/9)C2212yes—
20C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
21C2×C4×AGL(1,ℤ/9)C2212yes—
22C4⋊C4×3−1+2C2211yescharacteristic
23C4⋊C4×3−1+2C2211yescharacteristic
24C4⋊C4×3−1+2C2212yes—
25#3676⋊C3C2211yescharacteristic
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
29(C2×AGL(1,ℤ/9))⋊C4C2212yes—
30(C2×AGL(1,ℤ/9))⋊C4C2212yes—
31C4⋊(3−1+2⋊C4)C2211yescharacteristic
32C4⋊(3−1+2⋊C4)C2212yes—
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊C4C2211yescharacteristic
35C4×3−1+2⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2212yes—
37C2×C32×C4⋊C4D311yescharacteristic
38C3×#930—31no—
39C3×#930—32no—
40C3×C4×#67—31no—
41C3×C4×#67—32no—
42C3×#898—32no—
43C3×#895—31no—
44C3×#894—32no—
45C3×#893—31no—
46C3×#892—31no—
47C3×#890—31no—
48C2×C9×C4⋊C4—21no—
49C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
50#63492C2×C311yescharacteristic, complemented
51#63492C2×C312yescomplemented
52C4×#611C2×C311yescharacteristic, complemented
53C4×#611C2×C312yescomplemented
54#63460C2×C312yescomplemented
55#63457C2×C311yescharacteristic, complemented
56#63456C2×C312yescomplemented
57#63455C2×C311yescharacteristic, complemented
58#63454C2×C311yescharacteristic, complemented
59#63452C2×C311yescharacteristic, complemented
60C23×3−1+2C2311yescharacteristic
61C22×AGL(1,ℤ/9)C2311yescharacteristic
62C22×AGL(1,ℤ/9)C2312yes—
63C2×C4×3−1+2—22no—
64C2×C4×3−1+2—42no—
65C2×C4×3−1+2C2312yes—
66C2×C4×3−1+2C2312yes—
67C2×C4×3−1+2C2311yescharacteristic
68C2×C4×3−1+2C2311yescharacteristic
69#611⋊C3—22no—
70#611⋊C3—24no—
71C2×3−1+2⋊C4C2311yescharacteristic
72C2×3−1+2⋊C4C2311yescharacteristic
73C2×3−1+2⋊C4C2311yescharacteristic
74C2×3−1+2⋊C4C2312yes—
75C4⋊AGL(1,ℤ/9)—22no—
76C4⋊AGL(1,ℤ/9)—42no—
77C4×AGL(1,ℤ/9)—22no—
78C4×AGL(1,ℤ/9)—42no—
79C3×#553—91no—
80#5514—31no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C2×D312yes—
83C2×C3×#67—31no—
84C2×C3×#67—32no—
85C2×C3×D12—31no—
86C2×C3×D12—32no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D312yes—
92C3×#217—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—32no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—32no—
104C22×C4×C9—21no—
105C22×C4×C9—22no—
106C22×C4×C9C22×C311yescharacteristic
107C22×C4×C9C22×C312yes—
108C2×#611C22×C311yescharacteristic
109C2×#611C22×C312yes—
110C2×D36C22×C311yescharacteristic
111C2×D36C22×C312yes—
112C2×C4×D9C22×C311yescharacteristic
113C2×C4×D9C22×C312yes—
114C9×C4⋊C4—21no—
115C9×C4⋊C4—21no—
116C9×C4⋊C4—22no—
117C9×C4⋊C4C22×C311yescharacteristic
118C9×C4⋊C4C22×C311yescharacteristic
119C9×C4⋊C4C22×C312yes—
120#3676C22×C311yescharacteristic
121(C2×D9)⋊C4C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C312yes—
126C4⋊Dic9C22×C311yescharacteristic
127C4⋊Dic9C22×C312yes—
128Dic9⋊C4C22×C312yes—
129Dic9⋊C4C22×C311yescharacteristic
130C4×Dic9C22×C311yescharacteristic
131C4×Dic9C22×C312yes—
132C22×3−1+2—21no—
133C22×3−1+2—21no—
134C22×3−1+2#4112yes—
135C22×3−1+2C2411yescharacteristic
136C2×AGL(1,ℤ/9)—21no—
137C2×AGL(1,ℤ/9)—22no—
138C2×AGL(1,ℤ/9)—21no—
139C2×AGL(1,ℤ/9)—22no—
140C2×AGL(1,ℤ/9)—22no—
141C2×AGL(1,ℤ/9)—21no—
142C4×3−1+2—21no—
143C4×3−1+2—22no—
144C4×3−1+2—22no—
145C4×3−1+2#4112yes—
1463−1+2⋊C4—21no—
1473−1+2⋊C4—21no—
1483−1+2⋊C4—22no—
1493−1+2⋊C4—21no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—91no—
154C3×#121—92no—
155C3×#121—91no—
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×C4×D4—91no—
159C3×#117—91no—
160C2×C3×C4⋊C4—31no—
161C2×C3×C4⋊C4C3×D311yescharacteristic
162#930—31no—
163#930—32no—
164C4×#67—31no—
165C4×#67—32no—
166#898—32no—
167#895—31no—
168#894—32no—
169#893—31no—
170#892—31no—
171#890—31no—
172C23×C32C22×D311yescharacteristic
173C22×C3×D3—31no—
174C22×C3×D3—32no—
175C2×C32×C4—22no—
176C2×C32×C4—42no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C3×#67—62no—
182C3×#67—64no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C3×D12—62no—
188C3×D12—122no—
189C3×C4×D3—62no—
190C3×C4×D3—122no—
191C23×C9—21no—
192C23×C9C23×C311yescharacteristic
193C22×D9C23×C311yescharacteristic
194C22×D9C23×C312yes—
195C2×C4×C9—24no—
196C2×C4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—22no—
200C2×C4×C9—22no—
201C2×C4×C9—42no—
202C2×C4×C9—44no—
203C2×C4×C9C23×C312yes—
204C2×C4×C9C23×C312yes—
205C2×C4×C9C23×C311yescharacteristic
206C2×C4×C9C23×C311yescharacteristic
207#611—22no—
208#611—24no—
209C2×Dic9C23×C311yescharacteristic
210C2×Dic9C23×C311yescharacteristic
211C2×Dic9C23×C311yescharacteristic
212C2×Dic9C23×C312yes—
213D36—22no—
214D36—42no—
215C4×D9—22no—
216C4×D9—42no—
217#553—91noSylow
218C2×3−1+2—22no—
219C2×3−1+2C2×#4111yescharacteristic
220C2×3−1+2C2×#4111yescharacteristic
221C2×3−1+2H(5,𝔽2)11yescharacteristic
222AGL(1,ℤ/9)—41no—
223AGL(1,ℤ/9)—42no—
224C2×C3×D4—91no—
225C2×C3×D4—92no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C22×C3×C4—31no—
229C22×C3×C4—32no—
230C22×C3×C4—91no—
231C22×C3×C4—92no—
232C22×C3×C4C2×C3×D311yescharacteristic
233C22×C3×C4C2×C3×D312yes—
234C2×#67—31no—
235C2×#67—32no—
236C2×D12—31no—
237C2×D12—32no—
238C2×C4×D3—31no—
239C2×C4×D3—32no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4C2×C3×D311yescharacteristic
248C3×C4⋊C4C2×C3×D311yescharacteristic
249C3×C4⋊C4C2×C3×D312yes—
250C3×C22⋊C4—92no—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—91no—
256C3×C42—92no—
257C3×C42—91no—
258#217—31no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—31no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264C4⋊Dic3—31no—
265C4⋊Dic3—32no—
266Dic3⋊C4—32no—
267Dic3⋊C4—31no—
268C4×Dic3—31no—
269C4×Dic3—32no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32#23912yes—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—61no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—62no—
280C32×C4—21no—
281C32×C4—22no—
282C32×C4—22no—
283C32×C4#23512yes—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—61no—
287C3×Dic3—62no—
288C22×C9—22no—
289C22×C9—22no—
290C22×C9—21no—
291C22×C9—21no—
292C22×C9—21no—
293C22×C9—22no—
294C22×C9C3×#4112yes—
295C22×C9C24×C311yescharacteristic
296C2×D9—21no—
297C2×D9—22no—
298C2×D9—21no—
299C2×D9—22no—
300C2×D9—22no—
301C2×D9—21no—
302C4×C9—22no—
303C4×C9—21no—
304C4×C9—22no—
305C4×C9—22no—
306C4×C9—22no—
307C4×C9—42no—
308C4×C9—42no—
309C4×C9C3×#4112yes—
310Dic9—21no—
311Dic9—21no—
312Dic9—22no—
313Dic9—21no—
314#126—92no—
315#125—91no—
316#123—92no—
317#121—92no—
318#121—91no—
319#121—91no—
320C4×D4—92no—
321C4×D4—91no—
322C4×D4—91no—
323#117—91no—
324C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
3253−1+2#55311yescharacteristic, complemented, Sylow
326C23×C3—31no—
327C23×C3—91no—
328C23×C3—92no—
329C23×C3C22×C3×D311yescharacteristic
330C22×D3—31no—
331C22×D3—32no—
332C3×D4—182no—
333C3×D4—184no—
334C3×D4—182no—
335C3×D4—362no—
336C2×C3×C4—22no—
337C2×C3×C4—31no—
338C2×C3×C4—32no—
339C2×C3×C4—31no—
340C2×C3×C4—32no—
341C2×C3×C4—42no—
342C2×C3×C4—62no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—91no—
346C2×C3×C4—91no—
347C2×C3×C4—122no—
348C2×C3×C4—182no—
349C2×C3×C4—362no—
350C2×C3×C4C22×C3×D312yes—
351C2×C3×C4C22×C3×D312yes—
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×C3×C4C22×C3×D311yescharacteristic
354#67—62no—
355#67—64no—
356C2×Dic3—31no—
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—32no—
360D12—62no—
361D12—122no—
362C4×D3—62no—
363C4×D3—122no—
364C2×C32—22no—
365C2×C32C2×#23511yescharacteristic
366C2×C32#100411yescharacteristic
367C2×C32C2×#23911yescharacteristic
368C3×D3—121no—
369C3×D3—122no—
370C2×C9—24no—
371C2×C9—22no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9C2×C3×#4111yescharacteristic
376C2×C9C2×C3×#4111yescharacteristic
377C2×C9C3×H(5,𝔽2)11yescharacteristic
378D9—41no—
379D9—42no—
380C2×D4—92no—
381C2×D4—91no—
382C2×D4—91no—
383C2×D4—92no—
384C22×C4—91no—
385C22×C4—92no—
386C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
387C22×C4C2×AGL(1,ℤ/9)12yes—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4—91no—
391C4⋊C4—92no—
392C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
393C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
394C4⋊C4C2×AGL(1,ℤ/9)12yes—
395C22⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—92no—
398C22⋊C4—92no—
399C22⋊C4—91no—
400C22⋊C4—92no—
401C42—92no—
402C42—91no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—31no—
406C22×C3—32no—
407C22×C3—61no—
408C22×C3—61no—
409C22×C3—181no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—181no—
413C22×C3—182no—
414C22×C3—181no—
415C22×C3C3×#23912yes—
416C22×C3C23×C3×D311yescharacteristic
417C2×D3—61no—
418C2×D3—62no—
419C2×D3—62no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—62no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—32no—
427C3×C4—61no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4—181no—
431C3×C4—181no—
432C3×C4—182no—
433C3×C4—181no—
434C3×C4C3×#23512yes—
435Dic3—61no—
436Dic3—61no—
437Dic3—61no—
438Dic3—62no—
439C32#551411yescharacteristic
440C9—21no—
441C9C3×#55311yescharacteristic, complemented
442C23—91no—
443C23—92no—
444C23C22×AGL(1,ℤ/9)11yescharacteristic
445D4—184no—
446D4—182no—
447D4—182no—
448D4—362no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—91no—
452C2×C4—92no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—182no—
456C2×C4—362no—
457C2×C4C22×AGL(1,ℤ/9)12yes—
458C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
459C2×C4C22×AGL(1,ℤ/9)12yes—
460C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
461C2×C3—22no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—62no—
466C2×C3—361no—
467C2×C3—362no—
468C2×C3C2×C3×#23511yescharacteristic
469C2×C3C3×#100411yescharacteristic
470C2×C3C2×C3×#23911yescharacteristic
471D3—121no—
472D3—122no—
473C22—21no—
474C22—21no—
475C22—181no—
476C22—182no—
477C22—182no—
478C22—181no—
479C22—181no—
480C22—182no—
481C22#8872412yes—
482C22C23×AGL(1,ℤ/9)11yescharacteristic
483C4—21no—
484C4—22no—
485C4—22no—
486C4—181no—
487C4—181no—
488C4—182no—
489C4—181no—
490C4#3697⋊C312yes—
491C3—31no—
492C3C3×#551411yescharacteristic
493C2—22no—
494C2—361no—
495C2—362no—
496C2C2×#3697⋊C311yescharacteristic
497C2#63710⋊C311yescharacteristic
498C2C2×#8872411yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×3−1+2C2
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
2. #63492⋊C3C2semidirect
3. #63492⋊C3C2semidirect
4. C4×#611⋊C3C2semidirect
5. C4×#611⋊C3C2semidirect
6. #63460⋊C3C2semidirect
7. #63457⋊C3C2semidirect
8. #63456⋊C3C2semidirect
9. #63455⋊C3C2semidirect
10. #63454⋊C3C2semidirect
11. #63452⋊C3C2semidirect
13. #10589616C3semidirect
49. C2×C9×C4⋊C4C2×C3semidirect
50. #63492C2×C3semidirect
51. #63492C2×C3semidirect
52. C4×#611C2×C3semidirect
53. C4×#611C2×C3semidirect
54. #63460C2×C3semidirect
55. #63457C2×C3semidirect
56. #63456C2×C3semidirect
57. #63455C2×C3semidirect
58. #63454C2×C3semidirect
59. #63452C2×C3semidirect
324. C2×C4⋊C4AGL(1,ℤ/9)semidirect
325. 3−1+2#553semidirect
441. C9C3×#553semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
#63492⋊C32
C4×#611⋊C32
#63460⋊C32
#63457⋊C32
#63456⋊C32
#63455⋊C32
#63454⋊C32
#63452⋊C32
C3×#55143
#105896163