Finite Group Databasebeta

← C3×#10589684 #13152688

Subgroups — C3×#10589684 #13152688

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 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×#63458C212yesmaximal, complemented
5C3×#63457C211yesmaximal, characteristic
6C3×#63456C212yesmaximal, complemented
7C3×#63450C211yesmaximal, characteristic, complemented
8C3×#63438C212yesmaximal, complemented
9C3×#63437C212yesmaximal, complemented
10C3×C4×D36C211yesmaximal, characteristic, complemented
11C3×#63434C211yesmaximal, characteristic, complemented
12C3×#5582—31nomaximal
13#10589684C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C3×Dic18C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C4⋊C4C2212yes—
22C3×C42×C9C2211yescharacteristic
23C3×C42×C9C2212yes—
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C3×(C2×D9)⋊C4C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C3×C4⋊Dic9C2211yescharacteristic
32C3×Dic9⋊C4C2211yescharacteristic
33C3×Dic9⋊C4C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×Dic9⋊C4C2212yes—
36C3×C4×Dic9C2211yescharacteristic
37C32×C4×Q8D311yescharacteristic
38C3×#947—31no—
39C3×#946—31no—
40C3×#896—32no—
41C3×#895—31no—
42C3×#894—32no—
43C3×#888—31no—
44C3×#876—32no—
45C3×#875—32no—
46C3×C4×D12—31no—
47C3×#872—31no—
48C4×Q8×C9—21no—
49C4×Q8×C9C2×C311yescharacteristic, complemented
50#63509C2×C311yescharacteristic, complemented
51#63508C2×C311yescharacteristic, complemented
52#63458C2×C312yescomplemented
53#63457C2×C311yescharacteristic
54#63456C2×C312yescomplemented
55#63450C2×C311yescharacteristic, complemented
56#63438C2×C312yescomplemented
57#63437C2×C312yescomplemented
58C4×D36C2×C311yescharacteristic, complemented
59#63434C2×C311yescharacteristic, complemented
60C22×C3×D9C2311yescharacteristic
61C22×C3×D9C2312yes—
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×Dic9C2311yescharacteristic
72C3×D36—22no—
73C3×D36—42no—
74C3×C4×D9—22no—
75C3×C4×D9—42no—
76C3×Dic18—42no—
77C3×#542—91no—
78#5582—31no—
79C2×C32×Q8C2×D311yescharacteristic
80C2×C3×D12—31no—
81C2×C3×D12—32no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D312yes—
87C32×C42C2×D311yescharacteristic
88C32×C42C2×D312yes—
89C3×(C2×D3)⋊C4—31no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—31no—
95C3×(C2×D3)⋊C4—32no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—31no—
101C3×C4×Dic3—31no—
102C2×Q8×C9—21no—
103C2×Q8×C9C22×C311yescharacteristic
104C2×D36C22×C311yescharacteristic
105C2×D36C22×C312yes—
106C2×C4×D9C22×C311yescharacteristic
107C2×C4×D9C22×C312yes—
108C2×Dic18C22×C312yes—
109C9×C4⋊C4—21no—
110C9×C4⋊C4—22no—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C4⋊C4C22×C312yes—
113C42×C9—21no—
114C42×C9—22no—
115C42×C9C22×C311yescharacteristic
116C42×C9C22×C312yes—
117(C2×D9)⋊C4C22×C312yes—
118(C2×D9)⋊C4C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C311yescharacteristic
121(C2×D9)⋊C4C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124C4⋊Dic9C22×C311yescharacteristic
125Dic9⋊C4C22×C312yes—
126Dic9⋊C4C22×C311yescharacteristic
127Dic9⋊C4C22×C311yescharacteristic
128Dic9⋊C4C22×C312yes—
129C4×Dic9C22×C311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3×D9—21no—
132C2×C3×D9—22no—
133C2×C3×D9—22no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C2×C3×D9—22no—
137C3×C4×C9—22no—
138C3×C4×C9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9—22no—
145C3×Dic9—21no—
146C3×C4⋊Q8—91no—
147C3×#126—92no—
148C3×#124—91no—
149C3×#124—92no—
150C3×#123—92no—
151C3×C22⋊Q8—91no—
152C3×C22⋊Q8—92no—
153C3×#121—91no—
154C3×C4×Q8—21no—
155C3×C4×Q8D911yescharacteristic, complemented
156C3×C4×Q8C3×D311yescharacteristic
157C3×C4×D4—91no—
158C3×#117—91no—
159#947—31no—
160#946—31no—
161#896—32no—
162#895—31no—
163#894—32no—
164#888—31no—
165#876—32no—
166#875—32no—
167C4×D12—31no—
168#872—31no—
169C22×C3×D3—31no—
170C22×C3×D3—32no—
171C32×Q8—22no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C3×D12—62no—
182C3×D12—122no—
183C3×C4×D3—62no—
184C3×C4×D3—122no—
185C3×Dic6—122no—
186C22×D9C23×C311yescharacteristic
187C22×D9C23×C312yes—
188Q8×C9—24no—
189Q8×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9C23×C311yescharacteristic
196C2×C4×C9C23×C312yes—
197C2×C4×C9C23×C312yes—
198C2×C4×C9C23×C311yescharacteristic
199C2×C4×C9C23×C311yescharacteristic
200C2×Dic9C23×C311yescharacteristic
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C311yescharacteristic
203C2×Dic9C23×C312yes—
204D36—22no—
205D36—42no—
206C4×D9—22no—
207C4×D9—42no—
208Dic18—42no—
209#542—91noSylow
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C2×C3×C92−1+411yescharacteristic
213C3×D9—41no—
214C3×D9—42no—
215C2×C3×Q8—21no—
216C2×C3×Q8—92no—
217C2×C3×Q8C2×D911yescharacteristic
218C2×C3×Q8C2×C3×D311yescharacteristic
219C2×C3×D4—91no—
220C2×C3×D4—92no—
221C22×C3×C4—91no—
222C22×C3×C4—92no—
223C2×D12—31no—
224C2×D12—32no—
225C2×C4×D3—31no—
226C2×C4×D3—32no—
227C2×Dic6—32no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4C2×D911yescharacteristic
236C3×C4⋊C4C2×D912yes—
237C3×C4⋊C4C2×C3×D311yescharacteristic
238C3×C4⋊C4C2×C3×D312yes—
239C3×C22⋊C4—91no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C42—21no—
247C3×C42—22no—
248C3×C42—91no—
249C3×C42C2×D911yescharacteristic
250C3×C42C2×D912yes—
251C3×C42C2×C3×D311yescharacteristic
252C3×C42C2×C3×D312yes—
253(C2×D3)⋊C4—31no—
254(C2×D3)⋊C4—31no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260C4⋊Dic3—31no—
261Dic3⋊C4—31no—
262Dic3⋊C4—31no—
263Dic3⋊C4—32no—
264Dic3⋊C4—32no—
265C4×Dic3—31no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—61no—
268C2×C3×D3—62no—
269C2×C3×D3—61no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—61no—
273C32×C4—22no—
274C32×C4—21no—
275C32×C4—22no—
276C32×C4—21no—
277C32×C4#23512yes—
278C3×Dic3—61no—
279C3×Dic3—61no—
280C3×Dic3—61no—
281C3×Dic3—62no—
282C22×C9—21no—
283C22×C9C24×C311yescharacteristic
284C2×D9—21no—
285C2×D9—22no—
286C2×D9—22no—
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—22no—
290C4×C9—22no—
291C4×C9—24no—
292C4×C9—22no—
293C4×C9—22no—
294C4×C9—21no—
295C4×C9—21no—
296C4×C9—22no—
297C4×C9—42no—
298C4×C9—41no—
299C4×C9C3×#4112yes—
300Dic9—21no—
301Dic9—21no—
302Dic9—22no—
303Dic9—21no—
304C4⋊Q8—91no—
305#126—92no—
306#124—91no—
307#124—92no—
308#123—92no—
309C22⋊Q8—91no—
310C22⋊Q8—92no—
311#121—91no—
312C4×Q8C3×D911yescharacteristic, complemented
313C4×D4—91no—
314#117—91no—
315C3×C9#54211yescharacteristic, complemented, Sylow
316C23×C3—91no—
317C23×C3—92no—
318C22×D3—31no—
319C22×D3—32no—
320C3×Q8—24no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—362no—
324C3×D4—182no—
325C3×D4—362no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—182no—
336C2×C3×C4—362no—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C22×D912yes—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C22×D912yes—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×C3×D311yescharacteristic
343C2×C3×C4C22×C3×D312yes—
344C2×C3×C4C22×C3×D312yes—
345C2×C3×C4C22×C3×D311yescharacteristic
346C2×C3×C4C22×C3×D311yescharacteristic
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—32no—
351D12—62no—
352D12—122no—
353C4×D3—62no—
354C4×D3—122no—
355Dic6—122no—
356C2×C32C2×#23511yescharacteristic
357C2×C32#100711yescharacteristic
358C2×C32#100911yescharacteristic
359C3×D3—121no—
360C3×D3—122no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9C2×C3×#4111yescharacteristic
365C2×C9C3×H(5,𝔽2)11yescharacteristic
366C2×C9C3×2−1+411yescharacteristic
367D9—41no—
368D9—42no—
369C2×Q8—92no—
370C2×Q8C2×C3×D911yescharacteristic
371C2×D4—91no—
372C2×D4—92no—
373C22×C4—91no—
374C22×C4—92no—
375C4⋊C4—91no—
376C4⋊C4—91no—
377C4⋊C4—91no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4C2×C3×D911yescharacteristic
381C4⋊C4C2×C3×D912yes—
382C22⋊C4—91no—
383C22⋊C4—91no—
384C22⋊C4—92no—
385C22⋊C4—92no—
386C22⋊C4—92no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C42—91no—
390C42C2×C3×D911yescharacteristic
391C42C2×C3×D912yes—
392C22×C3—21no—
393C22×C3—181no—
394C22×C3—182no—
395C22×C3—181no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—181no—
399C22×C3C23×D911yescharacteristic
400C22×C3C23×C3×D311yescharacteristic
401C2×D3—61no—
402C2×D3—62no—
403C2×D3—61no—
404C2×D3—62no—
405C2×D3—62no—
406C2×D3—61no—
407C3×C4—24no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—42no—
420C3×C4—181no—
421C3×C4—181no—
422C3×C4—182no—
423C3×C4—181no—
424C3×C4#369712yes—
425C3×C4C3×#23512yes—
426Dic3—61no—
427Dic3—61no—
428Dic3—61no—
429Dic3—62no—
430C32#558211yescharacteristic
431C9—21no—
432C9C3×#54211yescharacteristic, complemented
433C23—91no—
434C23—92no—
435Q8—22no—
436Q8—362no—
437D4—182no—
438D4—362no—
439C2×C4—91no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—182no—
444C2×C4—362no—
445C2×C4C22×C3×D911yescharacteristic
446C2×C4C22×C3×D912yes—
447C2×C4C22×C3×D911yescharacteristic
448C2×C4C22×C3×D912yes—
449C2×C4C22×C3×D911yescharacteristic
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—361no—
454C2×C3—362no—
455C2×C3C2×#369711yescharacteristic
456C2×C3#6371311yescharacteristic
457C2×C3#6371511yescharacteristic
458C2×C3C2×C3×#23511yescharacteristic
459C2×C3C3×#100711yescharacteristic
460C2×C3C3×#100911yescharacteristic
461D3—121no—
462D3—122no—
463C22—181no—
464C22—182no—
465C22—181no—
466C22—182no—
467C22—182no—
468C22—181no—
469C22C23×C3×D911yescharacteristic
470C4—22no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—181no—
475C4—181no—
476C4—182no—
477C4—181no—
478C4C3×#369712yes—
479C3—21no—
480C3#1058968411yescharacteristic, direct product factor
481C3C3×#558211yescharacteristic
482C2—361no—
483C2—362no—
484C2C2×C3×#369711yescharacteristic
485C2C3×#6371311yescharacteristic
486C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socleC22×C32C23×D3
prime radicalC3×#63457C2
hypercenterC3×C4×Q8D9
hypocenterC9C3×#542
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×#63509C2semidirect
3. C3×#63508C2semidirect
4. C3×#63458C2semidirect
6. C3×#63456C2semidirect
7. C3×#63450C2semidirect
8. C3×#63438C2semidirect
9. C3×#63437C2semidirect
10. C3×C4×D36C2semidirect
11. C3×#63434C2semidirect
13. #10589684C3direct (N × H)
49. C4×Q8×C9C2×C3semidirect
50. #63509C2×C3semidirect
51. #63508C2×C3semidirect
52. #63458C2×C3semidirect
54. #63456C2×C3semidirect
55. #63450C2×C3semidirect
56. #63438C2×C3semidirect
57. #63437C2×C3semidirect
58. C4×D36C2×C3semidirect
59. #63434C2×C3semidirect
155. C3×C4×Q8D9semidirect
312. C4×Q8C3×D9semidirect
315. C3×C9#542semidirect
432. C9C3×#542semidirect
480. C3#10589684direct (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×#634582
C3×#634572
C3×#634562
C3×#634502
C3×#634382
C3×#634372
C3×C4×D362
C3×#634342
C3×#55823
#105896843