Finite Group Databasebeta

← C3×#80472 #11898415

Subgroups — C3×#80472 #11898415

Order 1152.

798 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5732C211yesmaximal, characteristic, complemented
2C3×#5585C211yesmaximal, characteristic, complemented
3C32×#462C211yesmaximal, characteristic, complemented
4C3×#5191C212yesmaximal, complemented
5C3×#5185C212yesmaximal, complemented
6C3×#5085C211yesmaximal, characteristic, complemented
7C3×C4×#215C211yesmaximal, characteristic, complemented
8C3×#4843C211yesmaximal, characteristic
9C3×#4830C211yesmaximal, characteristic, complemented
10C3×#4819C212yesmaximal, complemented
11C3×#4788C212yesmaximal, complemented
12C3×#3394—31nomaximal
13#80472C311yesmaximal, characteristic, direct product factor
14C2×C32×Q16C2212yes—
15C32×C4⋊Q8C2211yescharacteristic
16C32×C4×Q8C2211yescharacteristic
17C3×#947C2212yes—
18C3×#947C2212yes—
19C3×Q8×Dic3C2212yes—
20C2×C3×#215C2211yescharacteristic
21C2×C3×#215C2212yes—
22C2×C3×#215C2212yes—
23C2×C3×D24C2212yes—
24C3×#898—22no—
25C3×#897—21no—
26C3×#894—22no—
27C3×#893C2212yes—
28C3×#892—21no—
29C3×#875—21no—
30C3×#874C2211yescharacteristic
31C3×C4×D12—21no—
32C3×C4×D12C2211yescharacteristic
33C3×C4×Dic6C2211yescharacteristic
34C32×C4⋊C8C2211yescharacteristic
35C32×#103C2212yes—
36C3×#835C2211yescharacteristic
37C3×#821C2212yes—
38C3×Dic3⋊C8C2212yes—
39C3×C8×Dic3C2212yes—
40C3×#809C2211yescharacteristic
41C3×#808C2211yescharacteristic
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#559—31no—
44C3×#556—31no—
45#5732C2×C311yescharacteristic, complemented
46#5585C2×C311yescharacteristic, complemented
47C3×#495—32no—
48C3×#492—31no—
49C3×#464—32no—
50C3×#463—32no—
51C3×#462—21no—
52C3×#462D311yescharacteristic, complemented
53C3×#462C2×C311yescharacteristic, complemented
54C3×#459—31no—
55C3×C8×Q8—31no—
56C3×C4×SD16—31no—
57C3×C4×SD16—32no—
58#5191C2×C312yescomplemented
59#5185C2×C312yescomplemented
60#5085C2×C311yescharacteristic, complemented
61C4×#215C2×C311yescharacteristic, complemented
62#4843C2×C311yescharacteristic
63#4830C2×C311yescharacteristic, complemented
64#4819C2×C312yescomplemented
65#4788C2×C312yescomplemented
66C2×C32×Q8C2312yes—
67C2×C32×Q8C2311yescharacteristic
68C2×C3×D12—21no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C2×C3×D12C2311yescharacteristic
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3—22no—
74C2×C3×Dic6D411yescharacteristic
75C32×Q16—42no—
76C2×C32×C8C2312yes—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4C2311yescharacteristic
81C32×C42—21no—
82C32×C42C2311yescharacteristic
83C3×#215—22no—
84C3×#215—24no—
85C3×#215—42no—
86C3×(C2×D3)⋊C4—21no—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—22no—
89C3×(C2×D3)⋊C4—22no—
90C3×(C2×D3)⋊C4—22no—
91C3×(C2×D3)⋊C4—22no—
92C3×C4⋊Dic3—22no—
93C3×C4⋊Dic3D411yescharacteristic
94C3×C4⋊Dic3C2311yescharacteristic
95C3×Dic3⋊C4D412yes—
96C3×C4×Dic3—22no—
97C3×C4×Dic3C2312yes—
98C2×C3×C3⋊C8C2311yescharacteristic
99C2×C3×C3⋊C8C2311yescharacteristic
100C3×D24—42no—
101#3394—31noSylow
102C2×C3×Q16—22no—
103C2×C3×Q16C2×D312yes—
104C2×C3×Q16C22×C312yes—
105C2×C3×SD16—32no—
106C2×C3×SD16—31no—
107C2×C3×SD16—32no—
108C2×C3×D8—32no—
109C3×C4⋊Q8—21no—
110C3×C4⋊Q8C2×D311yescharacteristic
111C3×C4⋊Q8C22×C311yescharacteristic
112C3×#127—31no—
113C3×#126—62no—
114C3×#124—32no—
115C3×#124—32no—
116C3×#124—61no—
117C3×#123—62no—
118C3×C22⋊Q8—61no—
119C3×C4×Q8—21no—
120C3×C4×Q8—31no—
121C3×C4×Q8—32no—
122C3×C4×Q8C2×D311yescharacteristic
123C3×C4×Q8C22×C311yescharacteristic
124C3×C4×D4—32no—
125C3×C4×D4—31no—
126C3×C4×D4—61no—
127C3×#117—61no—
128#947C22×C312yes—
129#947C22×C312yes—
130Q8×Dic3C22×C312yes—
131C2×#215C22×C311yescharacteristic
132C2×#215C22×C312yes—
133C2×#215C22×C312yes—
134C2×D24C22×C312yes—
135#898—22no—
136#897—21no—
137#894—22no—
138#893C22×C312yes—
139#892—21no—
140#875—21no—
141#874C22×C311yescharacteristic
142C4×D12—21no—
143C4×D12C22×C311yescharacteristic
144C4×Dic6C22×C311yescharacteristic
145C3×#106—31no—
146C3×C4⋊C8—21no—
147C3×C4⋊C8—32no—
148C3×C4⋊C8C2×D311yescharacteristic
149C3×C4⋊C8C22×C311yescharacteristic
150C3×#103—22no—
151C3×#103—31no—
152C3×#103C2×D312yes—
153C3×#103C22×C312yes—
154C3×#102—32no—
155C3×#102—31no—
156C3×C4×C8—32no—
157C3×C4×C8—31no—
158#835C22×C311yescharacteristic
159#821C22×C312yes—
160Dic3⋊C8C22×C312yes—
161C8×Dic3C22×C312yes—
162#809C22×C311yescharacteristic
163#808C22×C311yescharacteristic
164C4×C3⋊C8C22×C311yescharacteristic
165C22×C3×D3—21no—
166C22×C3×D3—22no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—22no—
170C32×Q8—22no—
171C2×C32×C4—22no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C2×C3×Dic3—21no—
178C2×C3×Dic3C2×D412yes—
179C2×C3×Dic3C2×D412yes—
180C3×D12—22no—
181C3×D12—21no—
182C3×D12—21no—
183C3×D12—22no—
184C3×D12—42no—
185C3×D12—81no—
186C3×C4×D3—42no—
187C3×C4×D3—82no—
188C3×Dic6—22no—
189C32×C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8#4112yes—
192#559—31no—
193#556—31no—
194#495—32no—
195#492—31no—
196#464—32no—
197#463—32no—
198#462C3×D311yescharacteristic, complemented
199#459—31no—
200C8×Q8—31no—
201C4×SD16—31no—
202C4×SD16—32no—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—31no—
206C2×C3×Q8C22×D311yescharacteristic
207C2×C3×Q8C22×D312yes—
208C2×C3×Q8C23×C311yescharacteristic
209C2×C3×Q8C23×C312yes—
210C2×C3×D4—31no—
211C2×C3×D4—32no—
212C2×C3×D4—61no—
213C2×C3×D4—61no—
214C22×C3×C4—62no—
215C22×C3×C4—61no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C23×C311yescharacteristic
219C2×D12C23×C312yes—
220C2×C4×D3—21no—
221C2×C4×D3—22no—
222C2×Dic6C3×D411yescharacteristic
223C3×Q16—42no—
224C3×Q16—42no—
225C3×Q16—44no—
226C3×SD16—64no—
227C3×SD16—62no—
228C3×SD16—122no—
229C3×D8—122no—
230C2×C3×C8—22no—
231C2×C3×C8—31no—
232C2×C3×C8—31no—
233C2×C3×C8C22×D312yes—
234C2×C3×C8C23×C312yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4—62no—
249C3×C4⋊C4C22×D311yescharacteristic
250C3×C4⋊C4C23×C311yescharacteristic
251C3×C22⋊C4—62no—
252C3×C22⋊C4—62no—
253C3×C22⋊C4—62no—
254C3×C22⋊C4—62no—
255C3×C22⋊C4—62no—
256C3×C22⋊C4—61no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—32no—
261C3×C42—41no—
262C3×C42—62no—
263C3×C42C22×D311yescharacteristic
264C3×C42C23×C311yescharacteristic
265#215—22no—
266#215—24no—
267#215—42no—
268(C2×D3)⋊C4—21no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—22no—
274C4⋊Dic3—22no—
275C4⋊Dic3C3×D411yescharacteristic
276C4⋊Dic3C23×C311yescharacteristic
277Dic3⋊C4C3×D412yes—
278C4×Dic3—22no—
279C4×Dic3C23×C312yes—
280C2×C3⋊C8C23×C311yescharacteristic
281C2×C3⋊C8C23×C311yescharacteristic
282D24—42no—
283C22×C32C22×D411yescharacteristic
284C2×C3×D3—41no—
285C2×C3×D3—42no—
286C2×C3×D3—41no—
287C2×C3×D3—42no—
288C2×C3×D3—42no—
289C2×C3×D3—41no—
290C32×C4—21no—
291C32×C4—41no—
292C32×C4—42no—
293C32×C4—41no—
294C32×C4#13512yes—
295C32×C4C2×#4111yescharacteristic
296C32×C4H(5,𝔽2)11yescharacteristic
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—41no—
300C2×Q16C2×C3×D312yes—
301C2×SD16—32no—
302C2×SD16—31no—
303C2×SD16—32no—
304C2×D8—32no—
305C4⋊Q8C2×C3×D311yescharacteristic
306#127—31no—
307#126—62no—
308#124—32no—
309#124—32no—
310#124—61no—
311#123—62no—
312C22⋊Q8—61no—
313C4×Q8—31no—
314C4×Q8—32no—
315C4×Q8C2×C3×D311yescharacteristic
316C4×D4—32no—
317C4×D4—31no—
318C4×D4—61no—
319#117—61no—
320#106—31no—
321C4⋊C8—32no—
322C4⋊C8C2×C3×D311yescharacteristic
323#103—31no—
324#103C2×C3×D312yes—
325#102—32no—
326#102—31no—
327C4×C8—32no—
328C4×C8—31no—
329C23×C3—61no—
330C23×C3—62no—
331C22×D3—21no—
332C22×D3—22no—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—21no—
336C3×Q8—21no—
337C3×Q8—22no—
338C3×Q8—21no—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—21no—
342C3×Q8—41no—
343C3×Q8—42no—
344C3×Q8—42no—
345C3×Q8—62no—
346C3×D4—61no—
347C3×D4—62no—
348C3×D4—61no—
349C3×D4—62no—
350C3×D4—122no—
351C3×D4—241no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—32no—
362C2×C3×C4—32no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—42no—
366C2×C3×C4—61no—
367C2×C3×C4—122no—
368C2×C3×C4—242no—
369C2×C3×C4D3×D411yescharacteristic
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4C2×C3×D411yescharacteristic
372C2×C3×C4C2×C3×D411yescharacteristic
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C24×C311yescharacteristic
375C2×Dic3—21no—
376C2×Dic3C2×C3×D412yes—
377C2×Dic3C2×C3×D412yes—
378D12—21no—
379D12—22no—
380D12—21no—
381D12—22no—
382D12—42no—
383D12—81no—
384C4×D3—42no—
385C4×D3—82no—
386Dic6—22no—
387C3×C8—22no—
388C3×C8—22no—
389C3×C8—32no—
390C3×C8—42no—
391C3×C8—61no—
392C3⋊C8—21no—
393C3⋊C8C3×#4112yes—
394C2×C32#55011yescharacteristic
395C2×C32C2×#13511yescharacteristic
396C2×C32#57711yescharacteristic
397C3×D3—81no—
398C3×D3—82no—
399C2×Q8—31no—
400C2×Q8C22×C3×D311yescharacteristic
401C2×Q8C22×C3×D312yes—
402C2×D4—31no—
403C2×D4—32no—
404C2×D4—61no—
405C2×D4—61no—
406C22×C4—62no—
407C22×C4—61no—
408Q16—42no—
409SD16—64no—
410SD16—62no—
411SD16—122no—
412D8—122no—
413C2×C8—31no—
414C2×C8—31no—
415C2×C8C22×C3×D312yes—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—31no—
420C4⋊C4—32no—
421C4⋊C4—31no—
422C4⋊C4—62no—
423C4⋊C4C22×C3×D311yescharacteristic
424C22⋊C4—62no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C22⋊C4—61no—
430C42—21no—
431C42—32no—
432C42—62no—
433C42C22×C3×D311yescharacteristic
434C22×C3—21no—
435C22×C3—121no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—122no—
441C22×C3C2×D3×D411yescharacteristic
442C22×C3C22×C3×D411yescharacteristic
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—41no—
447C2×D3—41no—
448C2×D3—42no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—42no—
460C3×C4—41no—
461C3×C4—42no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—82no—
465C3×C4—81no—
466C3×C4—121no—
467C3×C4#91912yes—
468C3×C4C3×#13512yes—
469C3×C4#100411yescharacteristic
470C3×C4D3×#4111yescharacteristic
471C3×C4C2×C3×#4111yescharacteristic
472C3×C4C3×H(5,𝔽2)11yescharacteristic
473Dic3—22no—
474Dic3—22no—
475Dic3—41no—
476C32#339411yescharacteristic, complemented, Sylow
477C23—61no—
478C23—62no—
479Q8—22no—
480Q8—21no—
481Q8—21no—
482Q8—22no—
483Q8—62no—
484D4—61no—
485D4—62no—
486D4—61no—
487D4—62no—
488D4—122no—
489D4—241no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—32no—
494C2×C4—32no—
495C2×C4—61no—
496C2×C4—122no—
497C2×C4—242no—
498C2×C4C3×D3×D411yescharacteristic
499C2×C4C3×D3×D411yescharacteristic
500C2×C4C23×C3×D311yescharacteristic
501C8—22no—
502C8—32no—
503C8—61no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—241no—
508C2×C3—242no—
509C2×C3#559911yescharacteristic
510C2×C3C2×#91911yescharacteristic
511C2×C3#577811yescharacteristic
512C2×C3C3×#55011yescharacteristic
513C2×C3C2×C3×#13511yescharacteristic
514C2×C3C3×#57711yescharacteristic
515D3—81no—
516D3—82no—
517C22—121no—
518C22—122no—
519C22—122no—
520C22—121no—
521C22—121no—
522C22—122no—
523C22C2×C3×D3×D411yescharacteristic
524C4—21no—
525C4—41no—
526C4—41no—
527C4—42no—
528C4—62no—
529C4—62no—
530C4—121no—
531C4C3×#91912yes—
532C4C3×#100411yescharacteristic
533C4C3×D3×#4111yescharacteristic
534C3—21no—
535C3#8047211yescharacteristic, direct product factor
536C3C3×#339411yescharacteristic, complemented
537C2—241no—
538C2—242no—
539C2C3×#559911yescharacteristic
540C2C2×C3×#91911yescharacteristic
541C2C3×#577811yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#462C2
socleC22×C32C22×D4
prime radicalC3×#4843C2
hypercenterC3×#462D3
hypocenterC3C3×#3394
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#462

Complements

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

NH ≅ G/Ntype
1. C3×#5732C2semidirect
2. C3×#5585C2semidirect
3. C32×#462C2semidirect
4. C3×#5191C2semidirect
5. C3×#5185C2semidirect
6. C3×#5085C2semidirect
7. C3×C4×#215C2semidirect
9. C3×#4830C2semidirect
10. C3×#4819C2semidirect
11. C3×#4788C2semidirect
13. #80472C3direct (N × H)
45. #5732C2×C3semidirect
46. #5585C2×C3semidirect
52. C3×#462D3semidirect
53. C3×#462C2×C3semidirect
58. #5191C2×C3semidirect
59. #5185C2×C3semidirect
60. #5085C2×C3semidirect
61. C4×#215C2×C3semidirect
63. #4830C2×C3semidirect
64. #4819C2×C3semidirect
65. #4788C2×C3semidirect
198. #462C3×D3semidirect
476. C32#3394semidirect
535. C3#80472direct (N × H)
536. C3C3×#3394semidirect

Maximal subgroups

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

subgroupindex
C3×#57322
C3×#55852
C32×#4622
C3×#51912
C3×#51852
C3×#50852
C3×C4×#2152
C3×#48432
C3×#48302
C3×#48192
C3×#47882
C3×#33943
#804723