Finite Group Databasebeta

← C3×#10589682 #13152686

Subgroups — C3×#10589682 #13152686

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
2C3×#63509C211yesmaximal, characteristic, complemented
3C3×#63508C211yesmaximal, characteristic, complemented
4C3×#63460C212yesmaximal, complemented
5C3×#63457C211yesmaximal, characteristic
6C3×#63456C212yesmaximal, complemented
7C3×#63452C211yesmaximal, characteristic, complemented
8C3×C4×D36C212yesmaximal, complemented
9C3×C4×D36C211yesmaximal, characteristic, complemented
10C3×#63434C212yesmaximal, complemented
11C3×C42×D9C211yesmaximal, characteristic, complemented
12C3×#5580—31nomaximal
13#10589682C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2212yes—
23C3×C42×C9C2211yescharacteristic
24C3×C42×C9C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4⋊Dic9C2212yes—
32C3×Dic9⋊C4C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2211yescharacteristic
37C32×C4×Q8D311yescharacteristic
38C3×#947—31no—
39C3×#946—31no—
40C3×#898—32no—
41C3×#895—31no—
42C3×#894—32no—
43C3×#890—31no—
44C3×C4×D12—32no—
45C3×C4×D12—31no—
46C3×#872—32no—
47C3×C42×D3—31no—
48C4×Q8×C9—21no—
49C4×Q8×C9C2×C311yescharacteristic, complemented
50#63509C2×C311yescharacteristic, complemented
51#63508C2×C311yescharacteristic, complemented
52#63460C2×C312yescomplemented
53#63457C2×C311yescharacteristic
54#63456C2×C312yescomplemented
55#63452C2×C311yescharacteristic, complemented
56C4×D36C2×C311yescharacteristic, complemented
57C4×D36C2×C312yescomplemented
58#63434C2×C312yescomplemented
59C42×D9C2×C311yescharacteristic, complemented
60C22×C3×D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C3×Q8×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C3×D36—24no—
73C3×D36—22no—
74C3×C4×D9—22no—
75C3×C4×D9—22no—
76C3×C4×D9—22no—
77C3×C4×D9—24no—
78C3×#529—91no—
79#5580—31no—
80C2×C32×Q8C2×D311yescharacteristic
81C2×C3×D12—32no—
82C2×C3×D12—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D312yes—
89C32×C42C2×D311yescharacteristic
90C32×C42C2×D312yes—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—31no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C2×Q8×C9—21no—
104C2×Q8×C9C22×C311yescharacteristic
105C2×D36C22×C311yescharacteristic
106C2×D36C22×C312yes—
107C2×C4×D9C22×C311yescharacteristic
108C2×C4×D9C22×C311yescharacteristic
109C2×C4×D9C22×C311yescharacteristic
110C2×C4×D9C22×C312yes—
111C9×C4⋊C4—21no—
112C9×C4⋊C4—22no—
113C9×C4⋊C4C22×C311yescharacteristic
114C9×C4⋊C4C22×C312yes—
115C42×C9—21no—
116C42×C9—22no—
117C42×C9C22×C311yescharacteristic
118C42×C9C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C312yes—
121(C2×D9)⋊C4C22×C312yes—
122(C2×D9)⋊C4C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124C4⋊Dic9C22×C311yescharacteristic
125C4⋊Dic9C22×C312yes—
126Dic9⋊C4C22×C312yes—
127Dic9⋊C4C22×C312yes—
128C4×Dic9C22×C311yescharacteristic
129C4×Dic9C22×C311yescharacteristic
130C4×Dic9C22×C311yescharacteristic
131C22×C3×C9C2411yescharacteristic
132C2×C3×D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—22no—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9#4112yes—
138C3×C4×C9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—22no—
141C3×C4×C9#4112yes—
142C3×C4×C9#4112yes—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×Dic9—21no—
147C3×#126—92no—
148C3×#125—91no—
149C3×#124—91no—
150C3×#123—92no—
151C3×C22⋊Q8—91no—
152C3×#121—91no—
153C3×C4×Q8—21no—
154C3×C4×Q8D911yescharacteristic, complemented
155C3×C4×Q8C3×D311yescharacteristic
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×#117—92no—
159C2×C3×C42—91no—
160#947—31no—
161#946—31no—
162#898—32no—
163#895—31no—
164#894—32no—
165#890—31no—
166C4×D12—32no—
167C4×D12—31no—
168#872—32no—
169C42×D3—31no—
170C22×C3×D3—31no—
171C22×C3×D3—32no—
172C32×Q8—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C3×D12—64no—
183C3×D12—62no—
184C3×C4×D3—62no—
185C3×C4×D3—62no—
186C3×C4×D3—62no—
187C3×C4×D3—64no—
188C22×D9C23×C312yes—
189C22×D9C23×C311yescharacteristic
190Q8×C9—24no—
191Q8×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—22no—
197C2×C4×C9C23×C311yescharacteristic
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C312yes—
200C2×C4×C9C23×C311yescharacteristic
201C2×C4×C9C23×C311yescharacteristic
202C2×Dic9C23×C311yescharacteristic
203C2×Dic9C23×C311yescharacteristic
204C2×Dic9C23×C311yescharacteristic
205C2×Dic9C23×C312yes—
206D36—24no—
207D36—22no—
208C4×D9—22no—
209C4×D9—22no—
210C4×D9—22no—
211C4×D9—24no—
212#529—91noSylow
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216C3×D9—22no—
217C3×D9—42no—
218C2×C3×Q8—21no—
219C2×C3×Q8C2×D911yescharacteristic
220C2×C3×Q8C2×C3×D311yescharacteristic
221C2×C3×D4—92no—
222C2×C3×D4—91no—
223C22×C3×C4—91no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—92no—
227C2×D12—32no—
228C2×D12—31no—
229C2×C4×D3—31no—
230C2×C4×D3—31no—
231C2×C4×D3—31no—
232C2×C4×D3—32no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4C2×D911yescharacteristic
240C3×C4⋊C4C2×D912yes—
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C3×C4⋊C4C2×C3×D312yes—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C42—21no—
249C3×C42—22no—
250C3×C42—91no—
251C3×C42—91no—
252C3×C42—91no—
253C3×C42C2×D911yescharacteristic
254C3×C42C2×D912yes—
255C3×C42C2×C3×D311yescharacteristic
256C3×C42C2×C3×D312yes—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—32no—
262C4⋊Dic3—32no—
263C4⋊Dic3—31no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—31no—
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—32no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—22no—
279C32×C4#23512yes—
280C32×C4#23912yes—
281C3×Dic3—61no—
282C3×Dic3—61no—
283C3×Dic3—62no—
284C3×Dic3—61no—
285C22×C9—21no—
286C22×C9C24×C311yescharacteristic
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C2×D9C3×#4112yes—
293C4×C9—24no—
294C4×C9—21no—
295C4×C9—22no—
296C4×C9—22no—
297C4×C9—22no—
298C4×C9—22no—
299C4×C9—41no—
300C4×C9—42no—
301C4×C9C3×#4112yes—
302C4×C9C3×#4112yes—
303Dic9—21no—
304Dic9—21no—
305Dic9—22no—
306Dic9—21no—
307#126—92no—
308#125—91no—
309#124—91no—
310#123—92no—
311C22⋊Q8—91no—
312#121—91no—
313C4×Q8C3×D911yescharacteristic, complemented
314C4×D4—91no—
315C4×D4—92no—
316#117—92no—
317C2×C42—91no—
318C3×C9#52911yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C23×C3—92no—
321C22×D3—31no—
322C22×D3—32no—
323C3×Q8—24no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×D4—184no—
327C3×D4—182no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—182no—
338C2×C3×C4—182no—
339C2×C3×C4—182no—
340C2×C3×C4—184no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D912yes—
343C2×C3×C4C22×D912yes—
344C2×C3×C4C22×D911yescharacteristic
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C22×C3×D311yescharacteristic
347C2×C3×C4C22×C3×D312yes—
348C2×C3×C4C22×C3×D312yes—
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×C3×C4C22×C3×D311yescharacteristic
351C2×Dic3—31no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—32no—
355D12—64no—
356D12—62no—
357C4×D3—62no—
358C4×D3—62no—
359C4×D3—62no—
360C4×D3—64no—
361C2×C32C2×#23511yescharacteristic
362C2×C32C2×#23911yescharacteristic
363C2×C32D3×#4111yescharacteristic
364C3×D3—62no—
365C3×D3—122no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9C2×C3×#4111yescharacteristic
370C2×C9C2×C3×#4111yescharacteristic
371C2×C9C2×C3×#4111yescharacteristic
372D9—22no—
373D9—42no—
374C2×Q8C2×C3×D911yescharacteristic
375C2×D4—92no—
376C2×D4—91no—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—92no—
381C4⋊C4—92no—
382C4⋊C4—92no—
383C4⋊C4—91no—
384C4⋊C4—92no—
385C4⋊C4C2×C3×D911yescharacteristic
386C4⋊C4C2×C3×D912yes—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—92no—
391C22⋊C4—92no—
392C42—91no—
393C42—91no—
394C42—91no—
395C42C2×C3×D911yescharacteristic
396C42C2×C3×D912yes—
397C22×C3—21no—
398C22×C3—92no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3C23×D911yescharacteristic
405C22×C3C23×C3×D311yescharacteristic
406C2×D3—32no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C3×C4—24no—
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—21no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—41no—
422C3×C4—42no—
423C3×C4—181no—
424C3×C4—181no—
425C3×C4—182no—
426C3×C4—181no—
427C3×C4#369712yes—
428C3×C4#370112yes—
429C3×C4C3×#23512yes—
430C3×C4C3×#23912yes—
431Dic3—61no—
432Dic3—61no—
433Dic3—62no—
434Dic3—61no—
435C32#558011yescharacteristic
436C9—21no—
437C9C3×#52911yescharacteristic, complemented
438C23—91no—
439C23—92no—
440Q8—22no—
441D4—184no—
442D4—182no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—92no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—184no—
451C2×C4C22×C3×D911yescharacteristic
452C2×C4C22×C3×D912yes—
453C2×C4C22×C3×D912yes—
454C2×C4C22×C3×D911yescharacteristic
455C2×C4C22×C3×D911yescharacteristic
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—182no—
460C2×C3—362no—
461C2×C3C2×#369711yescharacteristic
462C2×C3C2×#370111yescharacteristic
463C2×C3#41×D911yescharacteristic
464C2×C3C2×C3×#23511yescharacteristic
465C2×C3C2×C3×#23911yescharacteristic
466C2×C3C3×D3×#4111yescharacteristic
467D3—62no—
468D3—122no—
469C22—92no—
470C22—181no—
471C22—181no—
472C22—182no—
473C22—182no—
474C22—182no—
475C22C23×C3×D911yescharacteristic
476C4—22no—
477C4—21no—
478C4—22no—
479C4—181no—
480C4—181no—
481C4—182no—
482C4—181no—
483C4C3×#369712yes—
484C4C3×#370112yes—
485C3—21no—
486C3#1058968211yescharacteristic, direct product factor
487C3C3×#558011yescharacteristic
488C2—182no—
489C2—362no—
490C2C2×C3×#369711yescharacteristic
491C2C2×C3×#370111yescharacteristic
492C2C3×#41×D911yescharacteristic
subgroupG/H
centerC2×C3×C4C22×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socleC22×C32C23×D3
prime radicalC3×#63457C2
hypercenterC3×C4×Q8D9
hypocenterC9C3×#529
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C4×Q8×C9C2semidirect
2. C3×#63509C2semidirect
3. C3×#63508C2semidirect
4. C3×#63460C2semidirect
6. C3×#63456C2semidirect
7. C3×#63452C2semidirect
8. C3×C4×D36C2semidirect
9. C3×C4×D36C2semidirect
10. C3×#63434C2semidirect
11. C3×C42×D9C2semidirect
13. #10589682C3direct (N × H)
49. C4×Q8×C9C2×C3semidirect
50. #63509C2×C3semidirect
51. #63508C2×C3semidirect
52. #63460C2×C3semidirect
54. #63456C2×C3semidirect
55. #63452C2×C3semidirect
56. C4×D36C2×C3semidirect
57. C4×D36C2×C3semidirect
58. #63434C2×C3semidirect
59. C42×D9C2×C3semidirect
154. C3×C4×Q8D9semidirect
313. C4×Q8C3×D9semidirect
318. C3×C9#529semidirect
437. C9C3×#529semidirect
486. C3#10589682direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×Q8×C92
C3×#635092
C3×#635082
C3×#634602
C3×#634572
C3×#634562
C3×#634522
C3×C4×D362
C3×#634342
C3×C42×D92
C3×#55803
#105896823