Finite Group Databasebeta

← #10589750⋊C3 #13153056

Subgroups — #10589750⋊C3 #13153056

Order 1728.

849 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
1C2×#3699⋊C3C211yesmaximal, characteristic
2C22×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
33−1+2×#123C211yesmaximal, characteristic, complemented
4#63497⋊C3C211yesmaximal, characteristic, complemented
5#63491⋊C3C211yesmaximal, characteristic, complemented
6#63458⋊C3C212yesmaximal, complemented
7#63450⋊C3C212yesmaximal, complemented
8#63448⋊C3C211yesmaximal, characteristic, complemented
9#63447⋊C3C212yesmaximal, complemented
10#63440⋊C3C212yesmaximal, complemented
11#63439⋊C3C211yesmaximal, characteristic, complemented
12C3×#5648—31nomaximal
13#10589750C311yesmaximal, characteristic, complemented
14C2×D4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3C2211yescharacteristic
17C2×#611⋊C3C2211yescharacteristic
18C22×3−1+2⋊C4C2211yescharacteristic
19C22×3−1+2⋊C4C2211yescharacteristic
20C22×3−1+2⋊C4C2211yescharacteristic
21#3699⋊C3—24no—
22C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
23C2×3−1+2⋊Q8—24no—
24C2×3−1+2⋊Q8C2211yescharacteristic
25C2×3−1+2⋊Q8C2212yes—
26C2×3−1+2⋊Q8C2212yes—
27C4⋊C4×3−1+2C2212yes—
28C22⋊C4×3−1+2C2211yescharacteristic
29C22⋊C4×3−1+2C2212yes—
30#3676⋊C3C2211yescharacteristic
31#3676⋊C3C2212yes—
32(C2×AGL(1,ℤ/9))⋊C4C2212yes—
33(C2×AGL(1,ℤ/9))⋊C4C2212yes—
34C4⋊(3−1+2⋊C4)C2212yes—
35(3−1+2⋊C4)⋊C4C2212yes—
36(3−1+2⋊C4)⋊C4C2212yes—
37(3−1+2⋊C4)⋊C4C2212yes—
38C4×3−1+2⋊C4C2212yes—
39C2×C3×#237—31no—
40C22×C3×Dic6—31no—
41C32×#123D311yescharacteristic
42C3×#935—31no—
43C3×#929—31no—
44C3×#896—32no—
45C3×#888—32no—
46C3×#886—31no—
47C3×#885—32no—
48C3×#878—32no—
49C3×#877—31no—
50C2×#3699C2×C311yescharacteristic
51C22×Dic18C2×C311yescharacteristic, complemented
52C9×#123—21no—
53C9×#123C2×C311yescharacteristic, complemented
54#63497C2×C311yescharacteristic, complemented
55#63491C2×C311yescharacteristic, complemented
56#63458C2×C312yescomplemented
57#63450C2×C312yescomplemented
58#63448C2×C311yescharacteristic, complemented
59#63447C2×C312yescomplemented
60#63440C2×C312yescomplemented
61#63439C2×C311yescharacteristic, complemented
62C23×3−1+2C2311yescharacteristic
63C23×3−1+2C2311yescharacteristic
64C22×AGL(1,ℤ/9)C2311yescharacteristic
65D4×3−1+2—22no—
66C2×C4×3−1+2—22no—
67C2×C4×3−1+2C2312yes—
68C2×C4×3−1+2C2311yescharacteristic
69C2×C4×3−1+2C2312yes—
70#611⋊C3—22no—
71#611⋊C3—22no—
72C2×3−1+2⋊C4—22no—
73C2×3−1+2⋊C4—22no—
74C2×3−1+2⋊C4D414yes—
75C2×3−1+2⋊C4C2312yes—
76C2×3−1+2⋊C4C2311yescharacteristic
77C2×3−1+2⋊C4C2312yes—
78C2×3−1+2⋊C4C2311yescharacteristic
79C2×3−1+2⋊C4C2311yescharacteristic
80C4×AGL(1,ℤ/9)—22no—
813−1+2⋊Q8—24no—
823−1+2⋊Q8—24no—
833−1+2⋊Q8—22no—
84C3×#536—91no—
85#5648—31no—
86C2×C32×D4C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×C3×#67—31no—
89C2×C3×#67—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C3×#237—64no—
94C2×C3×C4×D3—31no—
95C2×C3×Dic6—32no—
96C2×C3×Dic6—32no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6—64no—
99C32×C4⋊C4C2×D312yes—
100C32×C22⋊C4C2×D311yescharacteristic
101C32×C22⋊C4C2×D312yes—
102C3×#217—31no—
103C3×#217—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—32no—
111C2×D4×C9—21no—
112C2×D4×C9C22×C311yescharacteristic
113C22×C4×C9—21no—
114C22×C4×C9C22×C311yescharacteristic
115C2×#611C22×C311yescharacteristic
116C2×#611C22×C311yescharacteristic
117C22×Dic9C22×C311yescharacteristic
118C22×Dic9C22×C311yescharacteristic
119C22×Dic9C22×C311yescharacteristic
120#3699—24no—
121C2×C4×D9C22×C311yescharacteristic
122C2×Dic18—24no—
123C2×Dic18C22×C312yes—
124C2×Dic18C22×C312yes—
125C2×Dic18C22×C311yescharacteristic
126C9×C4⋊C4—22no—
127C9×C4⋊C4C22×C312yes—
128C9×C22⋊C4—21no—
129C9×C22⋊C4—22no—
130C9×C22⋊C4C22×C311yescharacteristic
131C9×C22⋊C4C22×C312yes—
132#3676C22×C311yescharacteristic
133#3676C22×C312yes—
134(C2×D9)⋊C4C22×C312yes—
135(C2×D9)⋊C4C22×C312yes—
136C4⋊Dic9C22×C312yes—
137Dic9⋊C4C22×C312yes—
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140C4×Dic9C22×C312yes—
141C22×3−1+2—22no—
142C22×3−1+2—21no—
143C22×3−1+2—22no—
144C22×3−1+2C2×D412yes—
145C22×3−1+2C2411yescharacteristic
146C2×AGL(1,ℤ/9)—21no—
147C2×AGL(1,ℤ/9)—22no—
148C4×3−1+2—22no—
149C4×3−1+2—21no—
150C4×3−1+2—22no—
1513−1+2⋊C4—22no—
1523−1+2⋊C4—21no—
1533−1+2⋊C4—22no—
1543−1+2⋊C4C2×D412yes—
1553−1+2⋊C4C2×D412yes—
156C2×C3×#41—91no—
157C22×C3×Q8—91no—
158C2×#237—31no—
159C22×Dic6—31no—
160C3×C4⋊Q8—92no—
161C3×#124—92no—
162C3×#123—31no—
163C3×#123—91no—
164C3×#123—91no—
165C3×#123—91no—
166C3×#123C3×D311yescharacteristic
167C3×C22⋊Q8—92no—
168C3×C22⋊Q8—92no—
169C3×#117—91no—
170#935—31no—
171#929—31no—
172#896—32no—
173#888—32no—
174#886—31no—
175#885—32no—
176#878—32no—
177#877—31no—
178C23×C32C22×D311yescharacteristic
179C23×C32C22×D311yescharacteristic
180C22×C3×D3—31no—
181C32×D4—22no—
182C2×C32×C4—22no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C3×#67—62no—
187C3×#67—62no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—62no—
196C3×C4×D3—62no—
197C3×Dic6—64no—
198C3×Dic6—64no—
199C3×Dic6—62no—
200C23×C9—21no—
201C23×C9—21no—
202C23×C9C23×C311yescharacteristic
203C23×C9C23×C311yescharacteristic
204C22×D9C23×C311yescharacteristic
205D4×C9—22no—
206D4×C9—42no—
207C2×C4×C9—22no—
208C2×C4×C9—22no—
209C2×C4×C9—22no—
210C2×C4×C9—21no—
211C2×C4×C9—42no—
212C2×C4×C9C23×C312yes—
213C2×C4×C9C23×C311yescharacteristic
214C2×C4×C9C23×C312yes—
215#611—22no—
216#611—22no—
217C2×Dic9—22no—
218C2×Dic9—22no—
219C2×Dic9C3×D414yes—
220C2×Dic9C23×C312yes—
221C2×Dic9C23×C311yescharacteristic
222C2×Dic9C23×C312yes—
223C2×Dic9C23×C311yescharacteristic
224C2×Dic9C23×C311yescharacteristic
225C4×D9—22no—
226Dic18—24no—
227Dic18—24no—
228Dic18—22no—
229#536—91noSylow
230C2×3−1+2—22no—
231C2×3−1+2—41no—
232C2×3−1+2C22×D411yescharacteristic
233C2×3−1+22−1+412yes—
234AGL(1,ℤ/9)—41no—
235C3×#41—184no—
236C2×C3×Q8—92no—
237C2×C3×Q8—92no—
238C2×C3×Q8—91no—
239C2×C3×Q8—184no—
240C2×C3×D4—31no—
241C2×C3×D4—91no—
242C2×C3×D4—91no—
243C2×C3×D4C2×C3×D311yescharacteristic
244C22×C3×C4—31no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C22×C3×C4—91no—
248C22×C3×C4—91no—
249C22×C3×C4C2×C3×D311yescharacteristic
250C2×#67—31no—
251C2×#67—31no—
252C22×Dic3—31no—
253C22×Dic3—31no—
254C22×Dic3—31no—
255#237—64no—
256C2×C4×D3—31no—
257C2×Dic6—32no—
258C2×Dic6—32no—
259C2×Dic6—31no—
260C2×Dic6—64no—
261C3×C4⋊C4—32no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4C2×C3×D312yes—
267C3×C22⋊C4—31no—
268C3×C22⋊C4—32no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—91no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4C2×C3×D311yescharacteristic
274C3×C22⋊C4C2×C3×D312yes—
275C3×C42—92no—
276#217—31no—
277#217—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280C4⋊Dic3—32no—
281Dic3⋊C4—32no—
282Dic3⋊C4—32no—
283Dic3⋊C4—32no—
284C4×Dic3—32no—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—22no—
288C22×C32D3×D412yes—
289C22×C32C23×D311yescharacteristic
290C2×C3×D3—61no—
291C2×C3×D3—62no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—22no—
295C3×Dic3—32no—
296C3×Dic3—32no—
297C3×Dic3—62no—
298C3×Dic3—61no—
299C3×Dic3—62no—
300C22×C9—22no—
301C22×C9—21no—
302C22×C9—22no—
303C22×C9—21no—
304C22×C9—22no—
305C22×C9—42no—
306C22×C9—41no—
307C22×C9—42no—
308C22×C9C2×C3×D412yes—
309C22×C9C24×C311yescharacteristic
310C2×D9—21no—
311C2×D9—22no—
312C4×C9—22no—
313C4×C9—21no—
314C4×C9—22no—
315C4×C9—42no—
316C4×C9—42no—
317C4×C9—41no—
318Dic9—22no—
319Dic9—21no—
320Dic9—22no—
321Dic9C2×C3×D412yes—
322Dic9C2×C3×D412yes—
323C2×#41—91no—
324C22×Q8—91no—
325C4⋊Q8—92no—
326#124—92no—
327#123—91no—
328#123—91no—
329#123—91no—
330#123AGL(1,ℤ/9)11yescharacteristic, complemented
331C22⋊Q8—92no—
332C22⋊Q8—92no—
333#117—91no—
3343−1+2#53611yescharacteristic, complemented, Sylow
335C23×C3—31no—
336C23×C3—31no—
337C23×C3—91no—
338C23×C3C22×C3×D311yescharacteristic
339C23×C3C22×C3×D311yescharacteristic
340C22×D3—31no—
341C3×Q8—184no—
342C3×Q8—184no—
343C3×Q8—182no—
344C3×D4—22no—
345C3×D4—62no—
346C3×D4—182no—
347C3×D4—182no—
348C2×C3×C4—22no—
349C2×C3×C4—32no—
350C2×C3×C4—32no—
351C2×C3×C4—31no—
352C2×C3×C4—62no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—94no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—91no—
359C2×C3×C4—182no—
360C2×C3×C4—182no—
361C2×C3×C4—182no—
362C2×C3×C4C22×C3×D312yes—
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D312yes—
365#67—62no—
366#67—62no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—34no—
370C2×Dic3—32no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—62no—
374C2×Dic3—62no—
375C4×D3—62no—
376Dic6—64no—
377Dic6—64no—
378Dic6—62no—
379C2×C32—22no—
380C2×C32—41no—
381C2×C32C2×D3×D411yescharacteristic
382C2×C32#101012yes—
383C3×D3—121no—
384C2×C9—22no—
385C2×C9—21no—
386C2×C9—22no—
387C2×C9—42no—
388C2×C9—41no—
389C2×C9—81no—
390C2×C9C22×C3×D411yescharacteristic
391C2×C9C3×2−1+412yes—
392D9—41no—
393#41—184no—
394C2×Q8—92no—
395C2×Q8—92no—
396C2×Q8—91no—
397C2×Q8—184no—
398C2×D4—91no—
399C2×D4—91no—
400C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
401C22×C4—91no—
402C22×C4—91no—
403C22×C4—91no—
404C22×C4—91no—
405C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
406C4⋊C4—92no—
407C4⋊C4—92no—
408C4⋊C4—92no—
409C4⋊C4—92no—
410C4⋊C4C2×AGL(1,ℤ/9)12yes—
411C22⋊C4—91no—
412C22⋊C4—92no—
413C22⋊C4—92no—
414C22⋊C4—92no—
415C22⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
416C22⋊C4C2×AGL(1,ℤ/9)12yes—
417C42—92no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—31no—
422C22×C3—32no—
423C22×C3—62no—
424C22×C3—61no—
425C22×C3—62no—
426C22×C3—181no—
427C22×C3—182no—
428C22×C3C3×D3×D412yes—
429C22×C3C23×C3×D311yescharacteristic
430C2×D3—61no—
431C2×D3—62no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—22no—
435C3×C4—62no—
436C3×C4—61no—
437C3×C4—62no—
438C3×C4—92no—
439C3×C4—92no—
440C3×C4—181no—
441C3×C4—182no—
442C3×C4—182no—
443Dic3—32no—
444Dic3—32no—
445Dic3—62no—
446Dic3—61no—
447Dic3—62no—
448C32#564811yescharacteristic
449C9—21no—
450C9C3×#53611yescharacteristic, complemented
451C23—91no—
452C23C22×AGL(1,ℤ/9)11yescharacteristic
453C23C22×AGL(1,ℤ/9)11yescharacteristic
454Q8—184no—
455Q8—184no—
456Q8—182no—
457D4—22no—
458D4—182no—
459D4—182no—
460C2×C4—22no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—94no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—91no—
467C2×C4—182no—
468C2×C4—182no—
469C2×C4—182no—
470C2×C4C22×AGL(1,ℤ/9)12yes—
471C2×C4C22×AGL(1,ℤ/9)12yes—
472C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
473C2×C3—22no—
474C2×C3—31no—
475C2×C3—32no—
476C2×C3—41no—
477C2×C3—62no—
478C2×C3—121no—
479C2×C3—361no—
480C2×C3C2×C3×D3×D411yescharacteristic
481C2×C3C3×#101012yes—
482D3—121no—
483C22—22no—
484C22—21no—
485C22—22no—
486C22—181no—
487C22—182no—
488C22D4×AGL(1,ℤ/9)12yes—
489C22C23×AGL(1,ℤ/9)11yescharacteristic
490C4—22no—
491C4—21no—
492C4—22no—
493C4—92no—
494C4—92no—
495C4—181no—
496C4—182no—
497C4—182no—
498C3—31no—
499C3C3×#564811yescharacteristic
500C2—22no—
501C2—41no—
502C2—361no—
503C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
504C2#1172264212yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#123C2
prime radicalC2×#3699⋊C3C2
hypercenter#123AGL(1,ℤ/9)
hypocenterC9C3×#536
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×3−1+2⋊Q8C2semidirect
3. 3−1+2×#123C2semidirect
4. #63497⋊C3C2semidirect
5. #63491⋊C3C2semidirect
6. #63458⋊C3C2semidirect
7. #63450⋊C3C2semidirect
8. #63448⋊C3C2semidirect
9. #63447⋊C3C2semidirect
10. #63440⋊C3C2semidirect
11. #63439⋊C3C2semidirect
13. #10589750C3semidirect
51. C22×Dic18C2×C3semidirect
53. C9×#123C2×C3semidirect
54. #63497C2×C3semidirect
55. #63491C2×C3semidirect
56. #63458C2×C3semidirect
57. #63450C2×C3semidirect
58. #63448C2×C3semidirect
59. #63447C2×C3semidirect
60. #63440C2×C3semidirect
61. #63439C2×C3semidirect
330. #123AGL(1,ℤ/9)semidirect
334. 3−1+2#536semidirect
450. C9C3×#536semidirect

Maximal subgroups

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

subgroupindex
C2×#3699⋊C32
C22×3−1+2⋊Q82
3−1+2×#1232
#63497⋊C32
#63491⋊C32
#63458⋊C32
#63450⋊C32
#63448⋊C32
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