Finite Group Databasebeta

← C3×#80420 #11898280

Subgroups — C3×#80420 #11898280

Order 1152.

868 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5721C211yesmaximal, characteristic, complemented
2C3×#5553C211yesmaximal, characteristic
3C32×#461C211yesmaximal, characteristic, complemented
4C3×#5175C212yesmaximal, complemented
5C3×#5163C212yesmaximal, complemented
6C3×#5085C211yesmaximal, characteristic, complemented
7C3×C4×#214C211yesmaximal, characteristic, complemented
8C3×#4848C211yesmaximal, characteristic, complemented
9C3×#4830C211yesmaximal, characteristic, complemented
10C3×#4818C212yesmaximal, complemented
11C3×#4788C212yesmaximal, complemented
12C3×#3392—31nomaximal
13#80420C311yesmaximal, characteristic, direct product factor
14C2×C3×#235—21no—
15C2×C32×SD16C2212yes—
16C32×C4⋊Q8C2211yescharacteristic
17C32×C4×D4C2211yescharacteristic
18C3×Q8×Dic3C2212yes—
19C3×#944C2212yes—
20C2×C3×#215C2212yes—
21C2×C3×#215C2212yes—
22C3×#940C2212yes—
23C2×C3×#214C2211yescharacteristic
24C3×#930—21no—
25C2×C3×#204C2212yes—
26C3×#893C2212yes—
27C3×#890—21no—
28C3×#889—22no—
29C3×#887—21no—
30C3×#884—22no—
31C3×#874C2211yescharacteristic
32C3×C4×Dic6C2211yescharacteristic
33C3×C4×Dic6C2211yescharacteristic
34C32×C4⋊C8C2211yescharacteristic
35C32×#103C2212yes—
36C3×#832C2211yescharacteristic
37C3×#821C2212yes—
38C3×Dic3⋊C8C2212yes—
39C3×C8×Dic3C2212yes—
40C3×#810C2211yescharacteristic
41C3×#808C2211yescharacteristic
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×#557—31no—
44C3×#550—31no—
45#5721C2×C311yescharacteristic, complemented
46#5553C2×C311yescharacteristic
47C3×#492—32no—
48C3×#492—31no—
49C3×#461—21no—
50C3×#461—32no—
51C3×#461D311yescharacteristic, complemented
52C3×#461C2×C311yescharacteristic, complemented
53C3×#460—32no—
54C3×#460—31no—
55C3×C8×Q8—31no—
56C3×C4×SD16—31no—
57C3×C4×SD16—32no—
58#5175C2×C312yescomplemented
59#5163C2×C312yescomplemented
60#5085C2×C311yescharacteristic, complemented
61C4×#214C2×C311yescharacteristic, complemented
62#4848C2×C311yescharacteristic, complemented
63#4830C2×C311yescharacteristic, complemented
64#4818C2×C312yescomplemented
65#4788C2×C312yescomplemented
66C2×C32×Q8C2312yes—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C2×C3×#67—22no—
70C2×C3×#67—22no—
71C3×#235—44no—
72C2×C3×D12—21no—
73C2×C3×D12—21no—
74C2×C3×D12C2312yes—
75C2×C3×C4×D3—22no—
76C2×C3×Dic6D411yescharacteristic
77C2×C3×Dic6C2311yescharacteristic
78C32×SD16—42no—
79C2×C32×C8C2312yes—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4C2311yescharacteristic
83C32×C22⋊C4—21no—
84C32×C42C2311yescharacteristic
85C3×#215—24no—
86C3×#215—42no—
87C3×#214—22no—
88C3×(C2×D3)⋊C4—22no—
89C3×C4⋊Dic3—22no—
90C3×C4⋊Dic3D411yescharacteristic
91C3×C4⋊Dic3C2311yescharacteristic
92C3×Dic3⋊C4—21no—
93C3×Dic3⋊C4—22no—
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4D412yes—
96C3×C4×Dic3—21no—
97C3×C4×Dic3—22no—
98C3×C4×Dic3C2312yes—
99C2×C3×C3⋊C8C2311yescharacteristic
100C2×C3×C3⋊C8C2311yescharacteristic
101C3×#204—42no—
102#3392—31noSylow
103C2×C3×#41—61no—
104C2×#235—21no—
105C2×C3×SD16—22no—
106C2×C3×SD16—32no—
107C2×C3×SD16—31no—
108C2×C3×SD16—32no—
109C2×C3×SD16—32no—
110C2×C3×SD16C2×D312yes—
111C2×C3×SD16C22×C312yes—
112C3×C4⋊Q8—21no—
113C3×C4⋊Q8—32no—
114C3×C4⋊Q8C2×D311yescharacteristic
115C3×C4⋊Q8C22×C311yescharacteristic
116C3×#127—32no—
117C3×#127—31no—
118C3×#125—62no—
119C3×#125—61no—
120C3×#121—62no—
121C3×#121—61no—
122C3×C4×Q8—31no—
123C3×C4×Q8—32no—
124C3×C4×Q8—31no—
125C3×C4×Q8—61no—
126C3×C4×D4—21no—
127C3×C4×D4—32no—
128C3×C4×D4C2×D311yescharacteristic
129C3×C4×D4C22×C311yescharacteristic
130Q8×Dic3C22×C312yes—
131#944C22×C312yes—
132C2×#215C22×C312yes—
133C2×#215C22×C312yes—
134#940C22×C312yes—
135C2×#214C22×C311yescharacteristic
136#930—21no—
137C2×#204C22×C312yes—
138#893C22×C312yes—
139#890—21no—
140#889—22no—
141#887—21no—
142#884—22no—
143#874C22×C311yescharacteristic
144C4×Dic6C22×C311yescharacteristic
145C4×Dic6C22×C311yescharacteristic
146C3×#106—31no—
147C3×C4⋊C8—21no—
148C3×C4⋊C8—32no—
149C3×C4⋊C8C2×D311yescharacteristic
150C3×C4⋊C8C22×C311yescharacteristic
151C3×#103—22no—
152C3×#103—31no—
153C3×#103C2×D312yes—
154C3×#103C22×C312yes—
155C3×#102—31no—
156C3×#102—32no—
157C3×C4×C8—32no—
158C3×C4×C8—31no—
159#832C22×C311yescharacteristic
160#821C22×C312yes—
161Dic3⋊C8C22×C312yes—
162C8×Dic3C22×C312yes—
163#810C22×C311yescharacteristic
164#808C22×C311yescharacteristic
165C4×C3⋊C8C22×C311yescharacteristic
166C23×C32—21no—
167C22×C3×D3—22no—
168C32×Q8—22no—
169C32×Q8—22no—
170C32×D4—21no—
171C32×D4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—21no—
174C2×C32×C4—42no—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2411yescharacteristic
178C3×#67—44no—
179C3×#67—82no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3C2×D412yes—
183C2×C3×Dic3C2×D412yes—
184C3×D12—22no—
185C3×D12—22no—
186C3×D12—42no—
187C3×D12—81no—
188C3×C4×D3—44no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6SD1614yescomplemented
192C32×C8—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8#4112yescomplemented
195#557—31no—
196#550—31no—
197#492—32no—
198#492—31no—
199#461—32no—
200#461C3×D311yescharacteristic, complemented
201#460—32no—
202#460—31no—
203C8×Q8—31no—
204C4×SD16—31no—
205C4×SD16—32no—
206C3×#41—124no—
207C2×C3×Q8—22no—
208C2×C3×Q8—31no—
209C2×C3×Q8—31no—
210C2×C3×Q8C22×D312yes—
211C2×C3×Q8C23×C312yes—
212C2×C3×D4—21no—
213C2×C3×D4—32no—
214C2×C3×D4—62no—
215C2×C3×D4—62no—
216C2×C3×D4—61no—
217C2×C3×D4—61no—
218C2×C3×D4C22×D311yescharacteristic
219C2×C3×D4C23×C311yescharacteristic
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4—62no—
224C2×#67—22no—
225C2×#67—22no—
226#235—44no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12C23×C312yes—
230C2×C4×D3—22no—
231C2×Dic6C3×D411yescharacteristic
232C2×Dic6C23×C311yescharacteristic
233C3×SD16—42no—
234C3×SD16—42no—
235C3×SD16—44no—
236C3×SD16—62no—
237C3×SD16—64no—
238C3×SD16—122no—
239C3×SD16—122no—
240C2×C3×C8—22no—
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8C22×D312yes—
244C2×C3×C8C23×C312yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—32no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—62no—
256C3×C4⋊C4—62no—
257C3×C4⋊C4—62no—
258C3×C4⋊C4—61no—
259C3×C4⋊C4C22×D311yescharacteristic
260C3×C4⋊C4C23×C311yescharacteristic
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—41no—
264C3×C22⋊C4—62no—
265C3×C42—21no—
266C3×C42—32no—
267C3×C42—62no—
268C3×C42—61no—
269C3×C42C22×D311yescharacteristic
270C3×C42C23×C311yescharacteristic
271#215—24no—
272#215—42no—
273#214—22no—
274(C2×D3)⋊C4—22no—
275C4⋊Dic3—22no—
276C4⋊Dic3C3×D411yescharacteristic
277C4⋊Dic3C23×C311yescharacteristic
278Dic3⋊C4—21no—
279Dic3⋊C4—22no—
280Dic3⋊C4—22no—
281Dic3⋊C4C3×D412yes—
282C4×Dic3—21no—
283C4×Dic3—22no—
284C4×Dic3C23×C312yes—
285C2×C3⋊C8C23×C311yescharacteristic
286C2×C3⋊C8C23×C311yescharacteristic
287#204—42no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32C22×D411yescharacteristic
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C32×C4—21no—
296C32×C4—42no—
297C32×C4—41no—
298C32×C4C2×SD1612yes—
299C32×C4C2×#4111yescharacteristic
300C32×C4H(5,𝔽2)11yescharacteristic
301C3×Dic3—22no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3C2×SD1614yes—
305C2×#41—61no—
306C2×SD16—32no—
307C2×SD16—31no—
308C2×SD16—32no—
309C2×SD16—32no—
310C2×SD16C2×C3×D312yes—
311C4⋊Q8—32no—
312C4⋊Q8C2×C3×D311yescharacteristic
313#127—32no—
314#127—31no—
315#125—62no—
316#125—61no—
317#121—62no—
318#121—61no—
319C4×Q8—31no—
320C4×Q8—32no—
321C4×Q8—31no—
322C4×Q8—61no—
323C4×D4—32no—
324C4×D4C2×C3×D311yescharacteristic
325#106—31no—
326C4⋊C8—32no—
327C4⋊C8C2×C3×D311yescharacteristic
328#103—31no—
329#103C2×C3×D312yes—
330#102—31no—
331#102—32no—
332C4×C8—32no—
333C4×C8—31no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—41no—
337C23×C3—62no—
338C22×D3—22no—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—34no—
344C3×Q8—42no—
345C3×Q8—42no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×D4—22no—
349C3×D4—21no—
350C3×D4—21no—
351C3×D4—21no—
352C3×D4—21no—
353C3×D4—41no—
354C3×D4—62no—
355C3×D4—62no—
356C3×D4—124no—
357C3×D4—122no—
358C3×D4—241no—
359C3×D4—242no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—82no—
376C2×C3×C4—124no—
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4D3×D411yescharacteristic
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C2×C3×D411yescharacteristic
381C2×C3×C4C23×D311yescharacteristic
382C2×C3×C4C24×C311yescharacteristic
383#67—44no—
384#67—82no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3C2×C3×D412yes—
388C2×Dic3C2×C3×D412yes—
389D12—22no—
390D12—22no—
391D12—42no—
392D12—81no—
393C4×D3—44no—
394Dic6—21no—
395Dic6—21no—
396Dic6C3×SD1614yescomplemented
397C3×C8—22no—
398C3×C8—22no—
399C3×C8—32no—
400C3×C8—42no—
401C3×C8—61no—
402C3⋊C8—21no—
403C3⋊C8C3×#4112yescomplemented
404C2×C32—81no—
405C2×C32#55011yescharacteristic
406C2×C32C22×SD1611yescharacteristic
407C2×C32#57711yescharacteristic
408C3×D3—82no—
409#41—124no—
410C2×Q8—31no—
411C2×Q8—31no—
412C2×Q8C22×C3×D312yes—
413C2×D4—32no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4C22×C3×D311yescharacteristic
419C22×C4—21no—
420C22×C4—62no—
421SD16—42no—
422SD16—62no—
423SD16—64no—
424SD16—122no—
425SD16—122no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8C22×C3×D312yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—32no—
432C4⋊C4—31no—
433C4⋊C4—31no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—61no—
438C4⋊C4C22×C3×D311yescharacteristic
439C22⋊C4—21no—
440C22⋊C4—62no—
441C42—32no—
442C42—62no—
443C42—61no—
444C42C22×C3×D311yescharacteristic
445C22×C3—21no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—122no—
456C22×C3—122no—
457C22×C3—122no—
458C22×C3C2×D3×D411yescharacteristic
459C22×C3C22×C3×D411yescharacteristic
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—22no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—34no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—42no—
473C3×C4—42no—
474C3×C4—62no—
475C3×C4—81no—
476C3×C4—82no—
477C3×C4—121no—
478C3×C4—121no—
479C3×C4D3×SD1612yes—
480C3×C4C2×C3×SD1612yes—
481C3×C4#100411yescharacteristic
482C3×C4D3×#4111yescharacteristic
483C3×C4C2×C3×#4111yescharacteristic
484C3×C4C3×H(5,𝔽2)11yescharacteristic
485Dic3—22no—
486Dic3—41no—
487Dic3—41no—
488Dic3C2×C3×SD1614yes—
489C32#339211yescharacteristic, complemented, Sylow
490C23—21no—
491C23—62no—
492Q8—22no—
493Q8—22no—
494Q8—34no—
495Q8—61no—
496Q8—61no—
497D4—21no—
498D4—21no—
499D4—62no—
500D4—62no—
501D4—124no—
502D4—122no—
503D4—241no—
504D4—242no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—42no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—124no—
513C2×C4C3×D3×D411yescharacteristic
514C2×C4C3×D3×D411yescharacteristic
515C2×C4C23×C3×D311yescharacteristic
516C8—22no—
517C8—32no—
518C8—61no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—82no—
523C2×C3—81no—
524C2×C3—81no—
525C2×C3—242no—
526C2×C3#559911yescharacteristic
527C2×C3C2×D3×SD1611yescharacteristic
528C2×C3#577811yescharacteristic
529C2×C3C3×#55011yescharacteristic
530C2×C3C22×C3×SD1611yescharacteristic
531C2×C3C3×#57711yescharacteristic
532D3—82no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22C2×C3×D3×D411yescharacteristic
540C4—21no—
541C4—34no—
542C4—41no—
543C4—42no—
544C4—62no—
545C4—121no—
546C4—121no—
547C4C3×D3×SD1612yes—
548C4C3×#100411yescharacteristic
549C4C3×D3×#4111yescharacteristic
550C3—21no—
551C3#8042011yescharacteristic, direct product factor
552C3C3×#339211yescharacteristic, complemented
553C2—81no—
554C2—242no—
555C2C3×#559911yescharacteristic
556C2C2×C3×D3×SD1611yescharacteristic
557C2C3×#577811yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#461C2
socleC22×C32C22×D4
prime radicalC3×#5553C2
hypercenterC3×#461D3
hypocenterC3C3×#3392
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#461

Complements

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

NH ≅ G/Ntype
1. C3×#5721C2semidirect
3. C32×#461C2semidirect
4. C3×#5175C2semidirect
5. C3×#5163C2semidirect
6. C3×#5085C2semidirect
7. C3×C4×#214C2semidirect
8. C3×#4848C2semidirect
9. C3×#4830C2semidirect
10. C3×#4818C2semidirect
11. C3×#4788C2semidirect
13. #80420C3direct (N × H)
45. #5721C2×C3semidirect
51. C3×#461D3semidirect
52. C3×#461C2×C3semidirect
58. #5175C2×C3semidirect
59. #5163C2×C3semidirect
60. #5085C2×C3semidirect
61. C4×#214C2×C3semidirect
62. #4848C2×C3semidirect
63. #4830C2×C3semidirect
64. #4818C2×C3semidirect
65. #4788C2×C3semidirect
191. C3×Dic6SD16semidirect
194. C3×C3⋊C8#41semidirect
200. #461C3×D3semidirect
396. Dic6C3×SD16semidirect
403. C3⋊C8C3×#41semidirect
489. C32#3392semidirect
551. C3#80420direct (N × H)
552. C3C3×#3392semidirect

Maximal subgroups

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

subgroupindex
C3×#57212
C3×#55532
C32×#4612
C3×#51752
C3×#51632
C3×#50852
C3×C4×#2142
C3×#48482
C3×#48302
C3×#48182
C3×#47882
C3×#33923
#804203