Finite Group Databasebeta

← C3×#10589683 #13152687

Subgroups — C3×#10589683 #13152687

Order 1728.

712 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
2C3×#63508C211yesmaximal, characteristic, complemented
3C3×#63508C211yesmaximal, characteristic, complemented
4C3×#63456C212yesmaximal
5C3×#63454C211yesmaximal, characteristic, complemented
6C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
7C3×#63451C212yesmaximal, complemented
8C3×#63438C212yesmaximal, complemented
9C3×C4×D36C211yesmaximal, characteristic
10C3×#63434C212yesmaximal, complemented
11C3×#63432C211yesmaximal, characteristic, complemented
12C3×#5581—31nomaximal
13#10589683C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C42×C9C2211yescharacteristic
22C3×C42×C9C2212yes—
23C3×(C2×D9)⋊C4C2211yescharacteristic
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32C3×Dic9⋊C4C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×Dic9⋊C4C2212yes—
35C3×Dic9⋊C4C2212yes—
36C3×C4×Dic9C2212yes—
37C32×C4×Q8D311yescharacteristic
38C3×#946—31no—
39C3×#946—31no—
40C3×#894—32no—
41C3×#892—31no—
42C3×D3×C4⋊C4—31no—
43C3×#889—32no—
44C3×#876—32no—
45C3×C4×D12—31no—
46C3×#872—32no—
47C3×#870—31no—
48C4×Q8×C9—21no—
49C4×Q8×C9C2×C311yescharacteristic, complemented
50#63508C2×C311yescharacteristic, complemented
51#63508C2×C311yescharacteristic, complemented
52#63456C2×C312yes—
53#63454C2×C311yescharacteristic, complemented
54C4⋊C4×D9C2×C311yescharacteristic, complemented
55#63451C2×C312yescomplemented
56#63438C2×C312yescomplemented
57C4×D36C2×C311yescharacteristic
58#63434C2×C312yescomplemented
59#63432C2×C311yescharacteristic, complemented
60C22×C3×D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C3×Q8×C9—22no—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C3×D36—22no—
73C3×C4×D9—24no—
74C3×C4×D9—22no—
75C3×C4×D9—22no—
76C3×#552—91no—
77#5581—31no—
78C2×C32×Q8C2×D311yescharacteristic
79C2×C3×D12—31no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—31no—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C42C2×D311yescharacteristic
86C32×C42C2×D312yes—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—31no—
91C3×(C2×D3)⋊C4—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×C4×Dic3—32no—
101C2×Q8×C9—21no—
102C2×Q8×C9C22×C311yescharacteristic
103C2×D36C22×C311yescharacteristic
104C2×C4×D9C22×C311yescharacteristic
105C2×C4×D9C22×C311yescharacteristic
106C2×C4×D9C22×C312yes—
107C9×C4⋊C4—22no—
108C9×C4⋊C4—21no—
109C9×C4⋊C4C22×C312yes—
110C9×C4⋊C4C22×C311yescharacteristic
111C42×C9—21no—
112C42×C9—22no—
113C42×C9C22×C311yescharacteristic
114C42×C9C22×C312yes—
115(C2×D9)⋊C4C22×C312yes—
116(C2×D9)⋊C4C22×C312yes—
117(C2×D9)⋊C4C22×C311yescharacteristic
118(C2×D9)⋊C4C22×C311yescharacteristic
119(C2×D9)⋊C4C22×C312yes—
120C4⋊Dic9C22×C311yescharacteristic
121C4⋊Dic9C22×C311yescharacteristic
122C4⋊Dic9C22×C311yescharacteristic
123Dic9⋊C4C22×C312yes—
124Dic9⋊C4C22×C312yes—
125Dic9⋊C4C22×C312yes—
126Dic9⋊C4C22×C312yes—
127Dic9⋊C4C22×C312yes—
128C4×Dic9C22×C312yes—
129C22×C3×C9C2411yescharacteristic
130C2×C3×D9—21no—
131C2×C3×D9—21no—
132C2×C3×D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9#4112yes—
136C3×C4×C9—22no—
137C3×C4×C9—22no—
138C3×C4×C9—21no—
139C3×C4×C9—21no—
140C3×C4×C9#4112yes—
141C3×Dic9—21no—
142C3×Dic9—21no—
143C3×Dic9—22no—
144C3×Dic9—22no—
145C3×#126—92no—
146C3×#125—92no—
147C3×#125—91no—
148C3×#123—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—21no—
152C3×C4×Q8D911yescharacteristic, complemented
153C3×C4×Q8C3×D311yescharacteristic
154C3×C4×D4—91no—
155C3×#117—91no—
156C3×#117—92no—
157C2×C3×C4⋊C4—91no—
158#946—31no—
159#946—31no—
160#894—32no—
161#892—31no—
162D3×C4⋊C4—31no—
163#889—32no—
164#876—32no—
165C4×D12—31no—
166#872—32no—
167#870—31no—
168C22×C3×D3—31no—
169C22×C3×D3—31no—
170C32×Q8—22no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3×Dic3—31no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—32no—
180C3×D12—62no—
181C3×C4×D3—64no—
182C3×C4×D3—62no—
183C3×C4×D3—62no—
184C22×D9C23×C311yescharacteristic
185C22×D9C23×C311yescharacteristic
186Q8×C9—24no—
187Q8×C9—22no—
188C2×C4×C9—21no—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9C23×C312yes—
194C2×C4×C9C23×C311yescharacteristic
195C2×C4×C9C23×C311yescharacteristic
196C2×C4×C9C23×C312yes—
197C2×C4×C9C23×C311yescharacteristic
198C2×Dic9C23×C311yescharacteristic
199C2×Dic9C23×C312yes—
200C2×Dic9C23×C312yes—
201C2×Dic9C23×C311yescharacteristic
202D36—22no—
203C4×D9—24no—
204C4×D9—22no—
205C4×D9—22no—
206#552—91noSylow
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C3×D9—22no—
211C3×D9—41no—
212C2×C3×Q8—21no—
213C2×C3×Q8C2×D911yescharacteristic
214C2×C3×Q8C2×C3×D311yescharacteristic
215C2×C3×D4—91no—
216C22×C3×C4—92no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C2×D12—31no—
220C2×C4×D3—32no—
221C2×C4×D3—31no—
222C2×C4×D3—31no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4C2×D911yescharacteristic
234C3×C4⋊C4C2×D912yes—
235C3×C4⋊C4C2×C3×D312yes—
236C3×C4⋊C4C2×C3×D311yescharacteristic
237C3×C22⋊C4—91no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C42—22no—
243C3×C42—21no—
244C3×C42—92no—
245C3×C42C2×D912yes—
246C3×C42C2×D911yescharacteristic
247C3×C42C2×C3×D311yescharacteristic
248C3×C42C2×C3×D312yes—
249(C2×D3)⋊C4—31no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—31no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261Dic3⋊C4—32no—
262C4×Dic3—32no—
263C22×C32C23×D311yescharacteristic
264C2×C3×D3—32no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4#23512yes—
275C3×Dic3—61no—
276C3×Dic3—61no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C22×C9—21no—
280C22×C9C24×C311yescharacteristic
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—21no—
284C2×D9—21no—
285C2×D9—21no—
286C2×D9C3×#4112yes—
287C4×C9—22no—
288C4×C9—24no—
289C4×C9—22no—
290C4×C9—21no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—42no—
295C4×C9—41no—
296C4×C9C3×#4112yes—
297Dic9—21no—
298Dic9—21no—
299Dic9—22no—
300Dic9—22no—
301#126—92no—
302#125—92no—
303#125—91no—
304#123—92no—
305C22⋊Q8—91no—
306C22⋊Q8—91no—
307C4×Q8C3×D911yescharacteristic, complemented
308C4×D4—91no—
309#117—91no—
310#117—92no—
311C2×C4⋊C4—91no—
312C3×C9#55211yescharacteristic, complemented, Sylow
313C23×C3—91no—
314C23×C3—91no—
315C22×D3—31no—
316C22×D3—31no—
317C3×Q8—24no—
318C3×Q8—22no—
319C3×Q8—22no—
320C3×D4—182no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—91no—
329C2×C3×C4—92no—
330C2×C3×C4—184no—
331C2×C3×C4—182no—
332C2×C3×C4—182no—
333C2×C3×C4C22×D912yes—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D912yes—
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C22×C3×D312yes—
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×C3×C4C22×C3×D311yescharacteristic
341C2×C3×C4C22×C3×D312yes—
342C2×C3×C4C22×C3×D311yescharacteristic
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—31no—
346C2×Dic3—32no—
347D12—62no—
348C4×D3—64no—
349C4×D3—62no—
350C4×D3—62no—
351C2×C32C2×#23511yescharacteristic
352C2×C32#100711yescharacteristic
353C2×C32D3×#4111yescharacteristic
354C3×D3—62no—
355C3×D3—121no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9C2×C3×#4111yescharacteristic
360C2×C9C2×C3×#4111yescharacteristic
361C2×C9C3×2−1+411yescharacteristic
362D9—22no—
363D9—41no—
364C2×Q8C2×C3×D911yescharacteristic
365C2×D4—91no—
366C22×C4—92no—
367C22×C4—91no—
368C22×C4—91no—
369C4⋊C4—91no—
370C4⋊C4—92no—
371C4⋊C4—92no—
372C4⋊C4—91no—
373C4⋊C4—92no—
374C4⋊C4—91no—
375C4⋊C4—92no—
376C4⋊C4—92no—
377C4⋊C4C2×C3×D911yescharacteristic
378C4⋊C4C2×C3×D912yes—
379C22⋊C4—91no—
380C22⋊C4—92no—
381C22⋊C4—92no—
382C22⋊C4—91no—
383C22⋊C4—92no—
384C42—92no—
385C42C2×C3×D912yes—
386C42C2×C3×D911yescharacteristic
387C22×C3—21no—
388C22×C3—92no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×D911yescharacteristic
395C22×C3C23×C3×D311yescharacteristic
396C2×D3—32no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
402C3×C4—24no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—22no—
407C3×C4—21no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—22no—
413C3×C4—42no—
414C3×C4—41no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4—182no—
418C3×C4—182no—
419C3×C4#369712yes—
420C3×C4C3×#23512yes—
421Dic3—61no—
422Dic3—61no—
423Dic3—62no—
424Dic3—62no—
425C32#558111yescharacteristic
426C9—21no—
427C9C3×#55211yescharacteristic, complemented
428C23—91no—
429C23—91no—
430Q8—22no—
431D4—182no—
432C2×C4—91no—
433C2×C4—92no—
434C2×C4—91no—
435C2×C4—92no—
436C2×C4—184no—
437C2×C4—182no—
438C2×C4—182no—
439C2×C4C22×C3×D912yes—
440C2×C4C22×C3×D911yescharacteristic
441C2×C4C22×C3×D912yes—
442C2×C4C22×C3×D911yescharacteristic
443C2×C4C22×C3×D911yescharacteristic
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—182no—
448C2×C3—361no—
449C2×C3C2×#369711yescharacteristic
450C2×C3#6371311yescharacteristic
451C2×C3#41×D911yescharacteristic
452C2×C3C2×C3×#23511yescharacteristic
453C2×C3C3×#100711yescharacteristic
454C2×C3C3×D3×#4111yescharacteristic
455D3—62no—
456D3—121no—
457C22—92no—
458C22—181no—
459C22—181no—
460C22—181no—
461C22—181no—
462C22—181no—
463C22C23×C3×D911yescharacteristic
464C4—22no—
465C4—21no—
466C4—22no—
467C4—21no—
468C4—181no—
469C4—181no—
470C4—182no—
471C4—182no—
472C4C3×#369712yes—
473C3—21no—
474C3#1058968311yescharacteristic, direct product factor
475C3C3×#558111yescharacteristic
476C2—182no—
477C2—361no—
478C2C2×C3×#369711yescharacteristic
479C2C3×#6371311yescharacteristic
480C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC3×C4×Q8D9
hypocenterC9C3×#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C4×Q8×C9C2semidirect
2. C3×#63508C2semidirect
3. C3×#63508C2semidirect
5. C3×#63454C2semidirect
6. C3×C4⋊C4×D9C2semidirect
7. C3×#63451C2semidirect
8. C3×#63438C2semidirect
10. C3×#63434C2semidirect
11. C3×#63432C2semidirect
13. #10589683C3direct (N × H)
49. C4×Q8×C9C2×C3semidirect
50. #63508C2×C3semidirect
51. #63508C2×C3semidirect
53. #63454C2×C3semidirect
54. C4⋊C4×D9C2×C3semidirect
55. #63451C2×C3semidirect
56. #63438C2×C3semidirect
58. #63434C2×C3semidirect
59. #63432C2×C3semidirect
152. C3×C4×Q8D9semidirect
307. C4×Q8C3×D9semidirect
312. C3×C9#552semidirect
427. C9C3×#552semidirect
474. C3#10589683direct (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×#635082
C3×#634562
C3×#634542
C3×C4⋊C4×D92
C3×#634512
C3×#634382
C3×C4×D362
C3×#634342
C3×#634322
C3×#55813
#105896833