Finite Group Databasebeta

← C3×#10589726 #13152895

Subgroups — C3×#10589726 #13152895

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63509C211yesmaximal, characteristic, complemented
3C3×Q8×Dic9C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic
5C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
6C3×#63460C212yesmaximal, complemented
7C3×#63455C211yesmaximal, characteristic, complemented
8C3×#63454C212yesmaximal, complemented
9C3×#63452C211yesmaximal, characteristic, complemented
10C3×#63448C212yesmaximal, complemented
11C3×#611⋊C4C212yesmaximal, complemented
12C3×#5624—31nomaximal
13#10589726C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2212yes—
17C22×C3×Dic9C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×C3×D36C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2212yes—
23C3×C9×C22⋊C4C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2212yes—
30C3×C4⋊Dic9C2212yes—
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×C22⋊Q8D311yescharacteristic
38C3×#947—31no—
39C3×Q8×Dic3—31no—
40C3×#930—31no—
41C2×C3×C4×Dic3—31no—
42C3×#898—32no—
43C3×#893—31no—
44C3×#892—32no—
45C3×#890—31no—
46C3×#886—32no—
47C3×#67⋊C4—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic, complemented
50#63509C2×C311yescharacteristic, complemented
51Q8×Dic9C2×C311yescharacteristic, complemented
52#63492C2×C311yescharacteristic
53C2×C4×Dic9C2×C311yescharacteristic, complemented
54#63460C2×C312yescomplemented
55#63455C2×C311yescharacteristic, complemented
56#63454C2×C312yescomplemented
57#63452C2×C311yescharacteristic, complemented
58#63448C2×C312yescomplemented
59#611⋊C4C2×C312yescomplemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C3×Q8×C9—22no—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C3×#611—24no—
69C2×C3×Dic9—22no—
70C2×C3×Dic9—22no—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C3×D36—22no—
77C3×C4×D9—24no—
78C3×#529—91no—
79#5624—31no—
80C2×C32×Q8C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C2×C3×#67—32no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C2×C3×D12—31no—
86C2×C3×C4×D3—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D312yes—
89C32×C22⋊C4C2×D312yes—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C2×Q8×C9—21no—
104C2×Q8×C9C22×C311yescharacteristic
105C22×C4×C9—21no—
106C22×C4×C9C22×C311yescharacteristic
107C2×#611C22×C312yes—
108C22×Dic9C22×C311yescharacteristic
109C22×Dic9C22×C311yescharacteristic
110C2×D36C22×C311yescharacteristic
111C2×C4×D9C22×C312yes—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—22no—
114C9×C4⋊C4C22×C311yescharacteristic
115C9×C4⋊C4C22×C312yes—
116C9×C22⋊C4—22no—
117C9×C22⋊C4C22×C312yes—
118(C2×D9)⋊C4C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C312yes—
121(C2×D9)⋊C4C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C312yes—
124C4⋊Dic9C22×C312yes—
125Dic9⋊C4C22×C312yes—
126C4×Dic9C22×C312yes—
127C4×Dic9C22×C311yescharacteristic
128C4×Dic9C22×C311yescharacteristic
129C4×Dic9C22×C311yescharacteristic
130C4×Dic9C22×C311yescharacteristic
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9#4112yes—
134C22×C3×C9C2411yescharacteristic
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C3×C4×C9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—22no—
141C3×C4×C9#4112yes—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×Dic9#4112yes—
147C3×#126—92no—
148C3×#125—91no—
149C3×#124—91no—
150C3×#123—92no—
151C3×C22⋊Q8—21no—
152C3×C22⋊Q8D911yescharacteristic, complemented
153C3×C22⋊Q8C3×D311yescharacteristic
154C3×#121—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×#117—92no—
159C2×C3×C42—91no—
160#947—31no—
161Q8×Dic3—31no—
162#930—31no—
163C2×C4×Dic3—31no—
164#898—32no—
165#893—31no—
166#892—32no—
167#890—31no—
168#886—32no—
169#67⋊C4—32no—
170C23×C32C22×D311yescharacteristic
171C22×C3×D3—32no—
172C32×Q8—22no—
173C2×C32×C4—22no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C3×#67—64no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—62no—
186C3×D12—62no—
187C3×C4×D3—64no—
188C23×C9—21no—
189C23×C9C23×C311yescharacteristic
190C22×D9C23×C312yes—
191Q8×C9—22no—
192Q8×C9—42no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—22no—
196C2×C4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—42no—
199C2×C4×C9C23×C311yescharacteristic
200C2×C4×C9C23×C312yes—
201C2×C4×C9C23×C312yes—
202C2×C4×C9C23×C311yescharacteristic
203#611—24no—
204C2×Dic9—22no—
205C2×Dic9—22no—
206C2×Dic9C23×C311yescharacteristic
207C2×Dic9C23×C311yescharacteristic
208C2×Dic9C23×C311yescharacteristic
209C2×Dic9C23×C312yes—
210C2×Dic9C23×C311yescharacteristic
211D36—22no—
212C4×D9—24no—
213#529—91noSylow
214C2×C3×C9—22no—
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C3×D9—42no—
219C2×C3×Q8—21no—
220C2×C3×Q8C2×D911yescharacteristic
221C2×C3×Q8C2×C3×D311yescharacteristic
222C2×C3×D4—92no—
223C2×C3×D4—91no—
224C22×C3×C4—21no—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—92no—
228C22×C3×C4C2×D911yescharacteristic
229C22×C3×C4C2×C3×D311yescharacteristic
230C2×#67—32no—
231C22×Dic3—31no—
232C22×Dic3—31no—
233C2×D12—31no—
234C2×C4×D3—32no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4C2×D912yes—
242C3×C4⋊C4C2×D911yescharacteristic
243C3×C4⋊C4C2×C3×D311yescharacteristic
244C3×C4⋊C4C2×C3×D312yes—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4C2×D912yes—
251C3×C22⋊C4C2×C3×D312yes—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—91no—
255C3×C42—91no—
256C3×C42—92no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261C4⋊Dic3—32no—
262C4⋊Dic3—32no—
263C4⋊Dic3—31no—
264Dic3⋊C4—32no—
265C4×Dic3—32no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—31no—
269C4×Dic3—31no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32#23912yes—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C32×C4—21no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4#23712yes—
281C3×Dic3—32no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C3×Dic3—62no—
285C3×Dic3—61no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—22no—
290C22×C9—41no—
291C22×C9—41no—
292C22×C9C3×#4112yes—
293C22×C9C24×C311yescharacteristic
294C2×D9—22no—
295C2×D9—22no—
296C2×D9—22no—
297C4×C9—22no—
298C4×C9—21no—
299C4×C9—22no—
300C4×C9—22no—
301C4×C9—42no—
302C4×C9—41no—
303C4×C9—42no—
304C4×C9C3×#4112yes—
305Dic9—21no—
306Dic9—21no—
307Dic9—21no—
308Dic9—22no—
309Dic9C3×#4112yes—
310#126—92no—
311#125—91no—
312#124—91no—
313#123—92no—
314C22⋊Q8C3×D911yescharacteristic, complemented
315#121—91no—
316C4×Q8—91no—
317C4×D4—91no—
318C4×D4—92no—
319#117—92no—
320C2×C42—91no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3—92no—
324C23×C3C22×D911yescharacteristic
325C23×C3C22×C3×D311yescharacteristic
326C22×D3—32no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—42no—
330C3×D4—184no—
331C3×D4—182no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—42no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4—92no—
344C2×C3×C4—184no—
345C2×C3×C4—182no—
346C2×C3×C4—182no—
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C22×D912yes—
349C2×C3×C4C22×D912yes—
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×C3×C4C22×C3×D312yes—
353C2×C3×C4C22×C3×D312yes—
354C2×C3×C4C22×C3×D311yescharacteristic
355#67—64no—
356C2×Dic3—31no—
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—32no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363D12—62no—
364C4×D3—64no—
365C2×C32—22no—
366C2×C32C2×#23711yescharacteristic
367C2×C32C2×#23911yescharacteristic
368C2×C32D3×#4111yescharacteristic
369C3×D3—122no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—42no—
375C2×C9C2×C3×#4111yescharacteristic
376C2×C9C2×C3×#4111yescharacteristic
377C2×C9C2×C3×#4111yescharacteristic
378D9—42no—
379C2×Q8C2×C3×D911yescharacteristic
380C2×D4—92no—
381C2×D4—91no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—92no—
385C22×C4C2×C3×D911yescharacteristic
386C4⋊C4—92no—
387C4⋊C4—92no—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4C2×C3×D912yes—
391C4⋊C4C2×C3×D911yescharacteristic
392C22⋊C4—92no—
393C22⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—92no—
396C22⋊C4C2×C3×D912yes—
397C42—91no—
398C42—91no—
399C42—91no—
400C42—91no—
401C42—92no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—182no—
413C22×C3#370112yes—
414C22×C3C23×D911yescharacteristic
415C22×C3C3×#23912yes—
416C22×C3C23×C3×D311yescharacteristic
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—62no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—41no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—92no—
431C3×C4—181no—
432C3×C4—181no—
433C3×C4—182no—
434C3×C4—181no—
435C3×C4#369912yes—
436C3×C4C3×#23712yes—
437Dic3—32no—
438Dic3—61no—
439Dic3—61no—
440Dic3—62no—
441Dic3—61no—
442C32#562411yescharacteristic
443C9—21no—
444C9C3×#52911yescharacteristic, complemented
445C23—92no—
446C23C22×C3×D911yescharacteristic
447Q8—22no—
448D4—184no—
449D4—182no—
450C2×C4—22no—
451C2×C4—91no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—92no—
456C2×C4—184no—
457C2×C4—182no—
458C2×C4—182no—
459C2×C4C22×C3×D911yescharacteristic
460C2×C4C22×C3×D912yes—
461C2×C4C22×C3×D912yes—
462C2×C4C22×C3×D911yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3—362no—
470C2×C3C2×#369911yescharacteristic
471C2×C3C2×#370111yescharacteristic
472C2×C3#41×D911yescharacteristic
473C2×C3C2×C3×#23711yescharacteristic
474C2×C3C2×C3×#23911yescharacteristic
475C2×C3C3×D3×#4111yescharacteristic
476D3—122no—
477C22—21no—
478C22—21no—
479C22—182no—
480C22—182no—
481C22—182no—
482C22C3×#370112yes—
483C22C23×C3×D911yescharacteristic
484C4—21no—
485C4—22no—
486C4—22no—
487C4—92no—
488C4—181no—
489C4—181no—
490C4—182no—
491C4—181no—
492C4C3×#369912yes—
493C3—21no—
494C3#1058972611yescharacteristic, direct product factor
495C3C3×#562411yescharacteristic
496C2—22no—
497C2—362no—
498C2C2×C3×#369911yescharacteristic
499C2C2×C3×#370111yescharacteristic
500C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
prime radicalC3×#63492C2
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#529
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C3×#63509C2semidirect
3. C3×Q8×Dic9C2semidirect
5. C2×C3×C4×Dic9C2semidirect
6. C3×#63460C2semidirect
7. C3×#63455C2semidirect
8. C3×#63454C2semidirect
9. C3×#63452C2semidirect
10. C3×#63448C2semidirect
11. C3×#611⋊C4C2semidirect
13. #10589726C3direct (N × H)
49. C9×C22⋊Q8C2×C3semidirect
50. #63509C2×C3semidirect
51. Q8×Dic9C2×C3semidirect
53. C2×C4×Dic9C2×C3semidirect
54. #63460C2×C3semidirect
55. #63455C2×C3semidirect
56. #63454C2×C3semidirect
57. #63452C2×C3semidirect
58. #63448C2×C3semidirect
59. #611⋊C4C2×C3semidirect
152. C3×C22⋊Q8D9semidirect
314. C22⋊Q8C3×D9semidirect
321. C3×C9#529semidirect
444. C9C3×#529semidirect
494. C3#10589726direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×#635092
C3×Q8×Dic92
C3×#634922
C2×C3×C4×Dic92
C3×#634602
C3×#634552
C3×#634542
C3×#634522
C3×#634482
C3×#611⋊C42
C3×#56243
#105897263