Finite Group Databasebeta

← #10589713⋊C3 #13152882

Subgroups — #10589713⋊C3 #13152882

Order 1728.

923 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3699⋊C3C212yesmaximal, complemented
23−1+2×#121C211yesmaximal, characteristic, complemented
3#63500⋊C3C211yesmaximal, characteristic, complemented
4#63497⋊C3C212yesmaximal, complemented
5D4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
6C4×#611⋊C3C211yesmaximal, characteristic, complemented
7C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
8C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
9#63450⋊C3C211yesmaximal, characteristic, complemented
10#63445⋊C3C212yesmaximal, complemented
11#63440⋊C3C212yesmaximal, complemented
12C3×#5611—31nomaximal
13#10589713C311yesmaximal, characteristic, complemented
14C2×D4×3−1+2C2212yes—
15C2×D4×3−1+2C2211yescharacteristic
16C22×C4×3−1+2C2211yescharacteristic
17C2×#611⋊C3C2212yes—
18C2×#611⋊C3C2211yescharacteristic
19C22×3−1+2⋊C4C2212yes—
20C22×3−1+2⋊C4C2212yes—
21#3699⋊C3—28no—
22C2×C4×AGL(1,ℤ/9)C2212yes—
23C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
24C2×3−1+2⋊Q8C2212yes—
25C4⋊C4×3−1+2C2211yescharacteristic
26C22⋊C4×3−1+2C2212yes—
27#3676⋊C3C2211yescharacteristic
28#3676⋊C3C2212yes—
29#3676⋊C3C2212yes—
30(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
31C4⋊(3−1+2⋊C4)C2211yescharacteristic
32C4⋊(3−1+2⋊C4)C2211yescharacteristic
33C4⋊(3−1+2⋊C4)C2212yes—
34(3−1+2⋊C4)⋊C4C2212yes—
35(3−1+2⋊C4)⋊C4C2212yes—
36(3−1+2⋊C4)⋊C4C2211yescharacteristic
37C4×3−1+2⋊C4C2211yescharacteristic
38C2×C3×#237—32no—
39C32×#121D311yescharacteristic
40C3×#938—31no—
41C3×#935—32no—
42C3×D4×Dic3—31no—
43C3×C4×#67—31no—
44C2×C3×C4⋊Dic3—31no—
45C3×D3×C4⋊C4—31no—
46C3×#888—31no—
47C3×#883—32no—
48C3×#878—32no—
49C2×#3699C2×C312yescomplemented
50C9×#121—21no—
51C9×#121C2×C311yescharacteristic, complemented
52#63500C2×C311yescharacteristic, complemented
53#63497C2×C312yescomplemented
54D4×Dic9C2×C311yescharacteristic, complemented
55C4×#611C2×C311yescharacteristic, complemented
56C2×C4⋊Dic9C2×C311yescharacteristic, complemented
57C4⋊C4×D9C2×C311yescharacteristic, complemented
58#63450C2×C311yescharacteristic, complemented
59#63445C2×C312yescomplemented
60#63440C2×C312yescomplemented
61C23×3−1+2C2312yes—
62C23×3−1+2C2311yescharacteristic
63C22×AGL(1,ℤ/9)C2311yescharacteristic
64D4×3−1+2—24no—
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+2C2311yescharacteristic
70#611⋊C3—24no—
71#611⋊C3D414yescomplemented
72C2×3−1+2⋊C4—24no—
73C2×3−1+2⋊C4—24no—
74C2×3−1+2⋊C4C2312yes—
75C2×3−1+2⋊C4C2312yes—
76C2×3−1+2⋊C4C2311yescharacteristic
77C2×3−1+2⋊C4C2311yescharacteristic
78C2×3−1+2⋊C4C2311yescharacteristic
79C4×AGL(1,ℤ/9)—24no—
80C4×AGL(1,ℤ/9)—22no—
813−1+2⋊Q8—24no—
82C3×#547—91no—
83#5611—31no—
84C2×C32×D4C2×D311yescharacteristic
85C2×C32×D4C2×D312yes—
86C22×C32×C4C2×D311yescharacteristic
87C2×C3×#67—31no—
88C2×C3×#67—32no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—32no—
91C3×#237—68no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3—31no—
94C2×C3×Dic6—32no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C22⋊C4C2×D312yes—
97C3×#217—32no—
98C3×#217—32no—
99C3×#217—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×C4⋊Dic3—32no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×Dic3⋊C4—31no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×C4×Dic3—31no—
108C2×D4×C9—21no—
109C2×D4×C9—22no—
110C2×D4×C9C22×C311yescharacteristic
111C2×D4×C9C22×C312yes—
112C22×C4×C9—21no—
113C22×C4×C9C22×C311yescharacteristic
114C2×#611C22×C312yes—
115C2×#611C22×C311yescharacteristic
116C22×Dic9C22×C312yes—
117C22×Dic9C22×C312yes—
118#3699—28no—
119C2×C4×D9C22×C311yescharacteristic
120C2×C4×D9C22×C312yes—
121C2×Dic18C22×C312yes—
122C9×C4⋊C4—21no—
123C9×C4⋊C4C22×C311yescharacteristic
124C9×C22⋊C4—22no—
125C9×C22⋊C4C22×C312yes—
126#3676C22×C311yescharacteristic
127#3676C22×C312yes—
128#3676C22×C312yes—
129(C2×D9)⋊C4C22×C311yescharacteristic
130C4⋊Dic9C22×C311yescharacteristic
131C4⋊Dic9C22×C311yescharacteristic
132C4⋊Dic9C22×C312yes—
133Dic9⋊C4C22×C311yescharacteristic
134Dic9⋊C4C22×C312yes—
135Dic9⋊C4C22×C312yes—
136C4×Dic9C22×C311yescharacteristic
137C22×3−1+2—21no—
138C22×3−1+2—21no—
139C22×3−1+2—22no—
140C22×3−1+2—22no—
141C22×3−1+2—22no—
142C22×3−1+2C2×D412yes—
143C22×3−1+2C2411yescharacteristic
144C2×AGL(1,ℤ/9)—21no—
145C2×AGL(1,ℤ/9)—21no—
146C2×AGL(1,ℤ/9)C2×D412yes—
147C4×3−1+2—22no—
148C4×3−1+2—21no—
149C4×3−1+2#4112yescomplemented
1503−1+2⋊C4—22no—
1513−1+2⋊C4—22no—
1523−1+2⋊C4—21no—
1533−1+2⋊C4—21no—
1543−1+2⋊C4C2×D412yes—
155C2×C3×#41—92no—
156C2×#237—32no—
157C3×C4⋊Q8—91no—
158C3×#123—92no—
159C3×#123—92no—
160C3×C22⋊Q8—92no—
161C3×#121—31no—
162C3×#121—91no—
163C3×#121C3×D311yescharacteristic
164C3×C4×D4—91no—
165C3×C4×D4—91no—
166C2×C3×C4⋊C4—91no—
167C2×C3×C4⋊C4—91no—
168#938—31no—
169#935—32no—
170D4×Dic3—31no—
171C4×#67—31no—
172C2×C4⋊Dic3—31no—
173D3×C4⋊C4—31no—
174#888—31no—
175#883—32no—
176#878—32no—
177C23×C32C22×D312yes—
178C23×C32C22×D311yescharacteristic
179C22×C3×D3—31no—
180C32×D4—24no—
181C32×D4—22no—
182C2×C32×C4—22no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C3×#67—34no—
187C3×#67—64no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C3×C4×D3—64no—
196C3×C4×D3—62no—
197C3×Dic6—64no—
198C23×C9—21no—
199C23×C9—22no—
200C23×C9C23×C312yes—
201C23×C9C23×C311yescharacteristic
202C22×D9C23×C311yescharacteristic
203D4×C9—24no—
204D4×C9—22no—
205D4×C9—42no—
206D4×C9—44no—
207C2×C4×C9—22no—
208C2×C4×C9—21no—
209C2×C4×C9—21no—
210C2×C4×C9—22no—
211C2×C4×C9—42no—
212C2×C4×C9C23×C312yes—
213C2×C4×C9C23×C311yescharacteristic
214C2×C4×C9C23×C311yescharacteristic
215#611—24no—
216#611C3×D414yescomplemented
217C2×Dic9—24no—
218C2×Dic9—24no—
219C2×Dic9C23×C311yescharacteristic
220C2×Dic9C23×C311yescharacteristic
221C2×Dic9C23×C312yes—
222C2×Dic9C23×C312yes—
223C2×Dic9C23×C311yescharacteristic
224C4×D9—24no—
225C4×D9—22no—
226Dic18—24no—
227#547—91noSylow
228C2×3−1+2—22no—
229C2×3−1+2—42no—
230C2×3−1+2C22×D411yescharacteristic
231C2×3−1+2C2×#4111yescharacteristic
232C2×3−1+22−1+411yescharacteristic
233AGL(1,ℤ/9)—22no—
234C3×#41—188no—
235C2×C3×Q8—92no—
236C2×C3×D4—31no—
237C2×C3×D4—32no—
238C2×C3×D4—91no—
239C2×C3×D4—92no—
240C2×C3×D4C2×C3×D311yescharacteristic
241C2×C3×D4C2×C3×D312yes—
242C22×C3×C4—31no—
243C22×C3×C4—91no—
244C22×C3×C4—92no—
245C22×C3×C4—92no—
246C22×C3×C4—92no—
247C22×C3×C4C2×C3×D311yescharacteristic
248C2×#67—31no—
249C2×#67—32no—
250C22×Dic3—32no—
251C22×Dic3—32no—
252#237—68no—
253C2×C4×D3—32no—
254C2×C4×D3—31no—
255C2×Dic6—32no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4C2×C3×D311yescharacteristic
264C3×C22⋊C4—32no—
265C3×C22⋊C4—91no—
266C3×C22⋊C4—91no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4C2×C3×D312yes—
270C3×C42—91no—
271#217—32no—
272#217—32no—
273#217—31no—
274(C2×D3)⋊C4—31no—
275C4⋊Dic3—32no—
276C4⋊Dic3—31no—
277C4⋊Dic3—31no—
278Dic3⋊C4—31no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281C4×Dic3—31no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—22no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32D3×D412yes—
288C22×C32C23×D311yescharacteristic
289C2×C3×D3—32no—
290C2×C3×D3—61no—
291C2×C3×D3—61no—
292C32×C4—22no—
293C32×C4—21no—
294C32×C4#23712yes—
295C3×Dic3—32no—
296C3×Dic3—62no—
297C3×Dic3—62no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C22×C9—21no—
301C22×C9—21no—
302C22×C9—22no—
303C22×C9—22no—
304C22×C9—22no—
305C22×C9—21no—
306C22×C9—22no—
307C22×C9—41no—
308C22×C9—41no—
309C22×C9—42no—
310C22×C9—42no—
311C22×C9—42no—
312C22×C9C2×C3×D412yes—
313C22×C9C24×C311yescharacteristic
314C2×D9—21no—
315C2×D9—21no—
316C2×D9C2×C3×D412yes—
317C4×C9—22no—
318C4×C9—21no—
319C4×C9—22no—
320C4×C9—41no—
321C4×C9—42no—
322C4×C9C3×#4112yescomplemented
323Dic9—22no—
324Dic9—22no—
325Dic9—21no—
326Dic9—21no—
327Dic9C2×C3×D412yes—
328C2×#41—92no—
329C4⋊Q8—91no—
330#123—92no—
331#123—92no—
332C22⋊Q8—92no—
333#121—91no—
334#121AGL(1,ℤ/9)11yescharacteristic, complemented
335C4×D4—91no—
336C4×D4—91no—
337C2×C4⋊C4—91no—
338C2×C4⋊C4—91no—
3393−1+2#54711yescharacteristic, complemented, Sylow
340C23×C3—31no—
341C23×C3—32no—
342C23×C3—91no—
343C23×C3C22×C3×D312yes—
344C23×C3C22×C3×D311yescharacteristic
345C22×D3—31no—
346C3×Q8—184no—
347C3×D4—24no—
348C3×D4—22no—
349C3×D4—62no—
350C3×D4—64no—
351C3×D4—94no—
352C3×D4—184no—
353C2×C3×C4—22no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—62no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—182no—
364C2×C3×C4—184no—
365C2×C3×C4—184no—
366C2×C3×C4—184no—
367C2×C3×C4C22×C3×D312yes—
368C2×C3×C4C22×C3×D311yescharacteristic
369C2×C3×C4C22×C3×D311yescharacteristic
370#67—34no—
371#67—64no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—64no—
378C2×Dic3—64no—
379C4×D3—64no—
380C4×D3—62no—
381Dic6—64no—
382C2×C32—22no—
383C2×C32—42no—
384C2×C32C2×D3×D411yescharacteristic
385C2×C32C2×#23711yescharacteristic
386C2×C32#101011yescharacteristic
387C3×D3—62no—
388C2×C9—22no—
389C2×C9—21no—
390C2×C9—21no—
391C2×C9—21no—
392C2×C9—42no—
393C2×C9—42no—
394C2×C9—82no—
395C2×C9C22×C3×D411yescharacteristic
396C2×C9C2×C3×#4111yescharacteristic
397C2×C9C3×2−1+411yescharacteristic
398D9—22no—
399#41—188no—
400C2×Q8—92no—
401C2×D4—91no—
402C2×D4—92no—
403C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
404C2×D4C2×AGL(1,ℤ/9)12yes—
405C22×C4—92no—
406C22×C4—92no—
407C22×C4—91no—
408C22×C4—92no—
409C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
410C4⋊C4—91no—
411C4⋊C4—91no—
412C4⋊C4—92no—
413C4⋊C4—92no—
414C4⋊C4—91no—
415C4⋊C4—92no—
416C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
417C22⋊C4—91no—
418C22⋊C4—92no—
419C22⋊C4—92no—
420C22⋊C4—91no—
421C22⋊C4C2×AGL(1,ℤ/9)12yes—
422C42—91no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—31no—
429C22×C3—32no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—92no—
436C22×C3—181no—
437C22×C3—181no—
438C22×C3C3×D3×D412yes—
439C22×C3C23×C3×D311yescharacteristic
440C2×D3—32no—
441C2×D3—61no—
442C2×D3—61no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—32no—
446C3×C4—61no—
447C3×C4—62no—
448C3×C4—92no—
449C3×C4—182no—
450C3×C4—182no—
451C3×C4—181no—
452C3×C4—181no—
453C3×C4C3×#23712yes—
454Dic3—32no—
455Dic3—62no—
456Dic3—62no—
457Dic3—61no—
458Dic3—61no—
459C32#561111yescharacteristic
460C9—21no—
461C9C3×#54711yescharacteristic, complemented
462C23—91no—
463C23C22×AGL(1,ℤ/9)12yes—
464C23C22×AGL(1,ℤ/9)11yescharacteristic
465Q8—184no—
466D4—24no—
467D4—22no—
468D4—94no—
469D4—184no—
470C2×C4—22no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—91no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—184no—
477C2×C4—184no—
478C2×C4—184no—
479C2×C4—182no—
480C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
481C2×C4C22×AGL(1,ℤ/9)12yes—
482C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
483C2×C3—22no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—42no—
488C2×C3—62no—
489C2×C3—122no—
490C2×C3—182no—
491C2×C3C2×C3×D3×D411yescharacteristic
492C2×C3C2×C3×#23711yescharacteristic
493C2×C3C3×#101011yescharacteristic
494D3—62no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—92no—
501C22—181no—
502C22—181no—
503C22D4×AGL(1,ℤ/9)12yes—
504C22C23×AGL(1,ℤ/9)11yescharacteristic
505C4—22no—
506C4—21no—
507C4—92no—
508C4—182no—
509C4—182no—
510C4—181no—
511C4—181no—
512C4#3699⋊C312yescomplemented
513C3—31no—
514C3C3×#561111yescharacteristic
515C2—22no—
516C2—42no—
517C2—182no—
518C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
519C2C2×#3699⋊C311yescharacteristic
520C2#1172264211yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#121C2
hypercenter#121AGL(1,ℤ/9)
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#3699⋊C3C2semidirect
2. 3−1+2×#121C2semidirect
3. #63500⋊C3C2semidirect
4. #63497⋊C3C2semidirect
5. D4×3−1+2⋊C4C2semidirect
6. C4×#611⋊C3C2semidirect
7. C2×C4⋊(3−1+2⋊C4)C2semidirect
8. C4⋊C4×AGL(1,ℤ/9)C2semidirect
9. #63450⋊C3C2semidirect
10. #63445⋊C3C2semidirect
11. #63440⋊C3C2semidirect
13. #10589713C3semidirect
49. C2×#3699C2×C3semidirect
51. C9×#121C2×C3semidirect
52. #63500C2×C3semidirect
53. #63497C2×C3semidirect
54. D4×Dic9C2×C3semidirect
55. C4×#611C2×C3semidirect
56. C2×C4⋊Dic9C2×C3semidirect
57. C4⋊C4×D9C2×C3semidirect
58. #63450C2×C3semidirect
59. #63445C2×C3semidirect
60. #63440C2×C3semidirect
71. #611⋊C3D4semidirect
149. C4×3−1+2#41semidirect
216. #611C3×D4semidirect
322. C4×C9C3×#41semidirect
334. #121AGL(1,ℤ/9)semidirect
339. 3−1+2#547semidirect
461. C9C3×#547semidirect
512. C4#3699⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3699⋊C32
3−1+2×#1212
#63500⋊C32
#63497⋊C32
D4×3−1+2⋊C42
C4×#611⋊C32
C2×C4⋊(3−1+2⋊C4)2
C4⋊C4×AGL(1,ℤ/9)2
#63450⋊C32
#63445⋊C32
#63440⋊C32
C3×#56113
#105897133