Finite Group Databasebeta

← C3×#80450 #11887516

Subgroups — C3×#80450 #11887516

Order 1152.

900 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×#5727C211yesmaximal, characteristic, complemented
2C3×#5551C211yesmaximal, characteristic
3C32×#461C211yesmaximal, characteristic, complemented
4C3×#5171C212yesmaximal, complemented
5C3×#5170C212yesmaximal, complemented
6C3×#5079C211yesmaximal, characteristic, complemented
7C3×C4×#214C211yesmaximal, characteristic, complemented
8C3×#4851C211yesmaximal, characteristic, complemented
9C3×#4834C211yesmaximal, characteristic, complemented
10C3×#4805C212yesmaximal, complemented
11C3×D3×#103C212yesmaximal, complemented
12C3×#3298—31nomaximal
13#80450C311yesmaximal, characteristic, direct product factor
14C2×C3×D3×Q8C2212yes—
15C2×C3×#235—21no—
16C2×C32×SD16C2212yes—
17C32×C4⋊Q8C2211yescharacteristic
18C32×C4×D4C2211yescharacteristic
19C3×#946C2212yes—
20C3×#938C2212yes—
21C2×C3×#214C2211yescharacteristic
22C3×#924—21no—
23C2×C3×D3×C8C2212yes—
24C3×#896—22no—
25C3×#893—21no—
26C3×D3×C4⋊C4C2212yes—
27C3×#888—21no—
28C3×#883—22no—
29C3×C4×D12C2211yescharacteristic
30C3×#869C2211yescharacteristic
31C3×C4×Dic6C2211yescharacteristic
32C32×C4⋊C8C2211yescharacteristic
33C32×#103C2212yes—
34C3×#835C2212yes—
35C3×#835C2212yes—
36C3×#832C2211yescharacteristic
37C3×(C2×D3)⋊C8C2212yes—
38C3×#817C2212yes—
39C3×#816C2212yes—
40C3×#810C2211yescharacteristic
41C3×#808C2211yescharacteristic
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×D4×Q8—31no—
44C3×#547—31no—
45#5727C2×C311yescharacteristic, complemented
46#5551C2×C311yescharacteristic
47C3×#487—31no—
48C3×#483—32no—
49C3×#475—31no—
50C3×#465—32no—
51C3×#461—21no—
52C3×#461D311yescharacteristic, complemented
53C3×#461C2×C311yescharacteristic, complemented
54C3×#448—32no—
55C3×C4×SD16—31no—
56C3×C8×D4—31no—
57C2×C3×#103—32no—
58#5171C2×C312yescomplemented
59#5170C2×C312yescomplemented
60#5079C2×C311yescharacteristic, complemented
61C4×#214C2×C311yescharacteristic, complemented
62#4851C2×C311yescharacteristic, complemented
63#4834C2×C311yescharacteristic, complemented
64#4805C2×C312yescomplemented
65D3×#103C2×C312yescomplemented
66C2×C32×Q8C2312yes—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C2×C3×#67—22no—
70C3×D3×Q8—24no—
71C3×D3×Q8—42no—
72C3×#235—44no—
73C2×C3×D12D411yescharacteristic
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3C2312yes—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6C2311yescharacteristic
80C32×SD16—42no—
81C2×C32×C8C2312yes—
82C32×C4⋊C4—21no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2311yescharacteristic
85C32×C22⋊C4—21no—
86C32×C42C2311yescharacteristic
87C3×#217—22no—
88C3×#214—22no—
89C3×(C2×D3)⋊C4—22no—
90C3×(C2×D3)⋊C4D412yes—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3D411yescharacteristic
93C3×C4⋊Dic3C2312yes—
94C3×C4⋊Dic3C2311yescharacteristic
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—22no—
97C3×Dic3⋊C4—22no—
98C3×C4×Dic3—21no—
99C2×C3×C3⋊C8C2311yescharacteristic
100C2×C3×C3⋊C8C2311yescharacteristic
101C3×D3×C8—42no—
102#3298—31noSylow
103C2×C3×#41—61no—
104C22×C3×Q8—32no—
105C2×D3×Q8C22×C312yes—
106C2×#235—21no—
107C2×C3×SD16—22no—
108C2×C3×SD16—31no—
109C2×C3×SD16C2×D312yes—
110C2×C3×SD16C22×C312yes—
111C22×C3×C8—32no—
112C3×C4⋊Q8—21no—
113C3×C4⋊Q8—31no—
114C3×C4⋊Q8—61no—
115C3×C4⋊Q8C2×D311yescharacteristic
116C3×C4⋊Q8C22×C311yescharacteristic
117C3×#123—62no—
118C3×C22⋊Q8—32no—
119C3×C22⋊Q8—62no—
120C3×C22⋊Q8—61no—
121C3×#121—32no—
122C3×C4×Q8—31no—
123C3×C4×D4—21no—
124C3×C4×D4—31no—
125C3×C4×D4—61no—
126C3×C4×D4C2×D311yescharacteristic
127C3×C4×D4C22×C311yescharacteristic
128C2×C3×C4⋊C4—32no—
129#946C22×C312yes—
130#938C22×C312yes—
131C2×#214C22×C311yescharacteristic
132#924—21no—
133C2×D3×C8C22×C312yes—
134#896—22no—
135#893—21no—
136D3×C4⋊C4C22×C312yes—
137#888—21no—
138#883—22no—
139C4×D12C22×C311yescharacteristic
140#869C22×C311yescharacteristic
141C4×Dic6C22×C311yescharacteristic
142C3×#106—32no—
143C3×#106—31no—
144C3×C4⋊C8—21no—
145C3×C4⋊C8C2×D311yescharacteristic
146C3×C4⋊C8C22×C311yescharacteristic
147C3×#103—22no—
148C3×#103—32no—
149C3×#103—32no—
150C3×#103—32no—
151C3×#103—31no—
152C3×#103C2×D312yes—
153C3×#103C22×C312yes—
154C3×#102—31no—
155C3×C22⋊C8—32no—
156C3×C4×C8—31no—
157#835C22×C312yes—
158#835C22×C312yes—
159#832C22×C311yescharacteristic
160(C2×D3)⋊C8C22×C312yes—
161#817C22×C312yes—
162#816C22×C312yes—
163#810C22×C311yescharacteristic
164#808C22×C311yescharacteristic
165C4×C3⋊C8C22×C311yescharacteristic
166C23×C32—21no—
167C22×C3×D3C2×D412yes—
168C32×Q8—22no—
169C32×Q8—22no—
170C32×D4—21no—
171C32×D4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4—42no—
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2411yescharacteristic
178C3×#67—44no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3C2×D412yes—
183C3×D12SD1614yescomplemented
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—44no—
187C3×C4×D3—44no—
188C3×Dic6—21no—
189C3×Dic6—44no—
190C3×Dic6—42no—
191C3×Dic6#4112yescomplemented
192C32×C8—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195D4×Q8—31no—
196#547—31no—
197#487—31no—
198#483—32no—
199#475—31no—
200#465—32no—
201#461C3×D311yescharacteristic, complemented
202#448—32no—
203C4×SD16—31no—
204C8×D4—31no—
205C2×#103—32no—
206C3×#41—124no—
207C2×C3×Q8—22no—
208C2×C3×Q8—31no—
209C2×C3×Q8—62no—
210C2×C3×Q8—61no—
211C2×C3×Q8—64no—
212C2×C3×Q8—122no—
213C2×C3×Q8C22×D312yes—
214C2×C3×Q8C23×C312yes—
215C2×C3×D4—21no—
216C2×C3×D4—31no—
217C2×C3×D4—62no—
218C2×C3×D4C22×D311yescharacteristic
219C2×C3×D4C23×C311yescharacteristic
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—32no—
223C22×C3×C4—41no—
224C22×C3×C4—62no—
225C22×C3×C4—62no—
226C2×#67—22no—
227D3×Q8—24no—
228D3×Q8—42no—
229#235—44no—
230C2×D12C3×D411yescharacteristic
231C2×C4×D3—22no—
232C2×C4×D3—22no—
233C2×C4×D3C23×C312yes—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6C23×C311yescharacteristic
237C3×SD16—42no—
238C3×SD16—42no—
239C3×SD16—44no—
240C3×SD16—62no—
241C2×C3×C8—22no—
242C2×C3×C8—31no—
243C2×C3×C8—31no—
244C2×C3×C8—122no—
245C2×C3×C8C22×D312yes—
246C2×C3×C8C23×C312yes—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—31no—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—62no—
258C3×C4⋊C4—62no—
259C3×C4⋊C4—61no—
260C3×C4⋊C4—61no—
261C3×C4⋊C4C22×D311yescharacteristic
262C3×C4⋊C4C23×C311yescharacteristic
263C3×C22⋊C4—21no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—32no—
266C3×C22⋊C4—41no—
267C3×C22⋊C4—62no—
268C3×C22⋊C4—62no—
269C3×C42—21no—
270C3×C42—61no—
271C3×C42C22×D311yescharacteristic
272C3×C42C23×C311yescharacteristic
273#217—22no—
274#214—22no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4C3×D412yes—
277C4⋊Dic3—21no—
278C4⋊Dic3C3×D411yescharacteristic
279C4⋊Dic3C23×C311yescharacteristic
280C4⋊Dic3C23×C312yes—
281Dic3⋊C4—21no—
282Dic3⋊C4—22no—
283Dic3⋊C4—22no—
284C4×Dic3—21no—
285C2×C3⋊C8C23×C311yescharacteristic
286C2×C3⋊C8C23×C311yescharacteristic
287D3×C8—42no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32C22×D411yescharacteristic
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3C2×SD1614yes—
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—22no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C2×#41—61no—
306C22×Q8—32no—
307C2×SD16—31no—
308C2×SD16C2×C3×D312yes—
309C22×C8—32no—
310C4⋊Q8—31no—
311C4⋊Q8—61no—
312C4⋊Q8C2×C3×D311yescharacteristic
313#123—62no—
314C22⋊Q8—32no—
315C22⋊Q8—62no—
316C22⋊Q8—61no—
317#121—32no—
318C4×Q8—31no—
319C4×D4—31no—
320C4×D4—61no—
321C4×D4C2×C3×D311yescharacteristic
322C2×C4⋊C4—32no—
323#106—32no—
324#106—31no—
325C4⋊C8C2×C3×D311yescharacteristic
326#103—32no—
327#103—32no—
328#103—32no—
329#103—31no—
330#103C2×C3×D312yes—
331#102—31no—
332C22⋊C8—32no—
333C4×C8—31no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—32no—
337C23×C3—41no—
338C22×D3C2×C3×D412yes—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—32no—
344C3×Q8—42no—
345C3×Q8—42no—
346C3×Q8—61no—
347C3×Q8—124no—
348C3×Q8—122no—
349C3×D4—22no—
350C3×D4—21no—
351C3×D4—21no—
352C3×D4—21no—
353C3×D4—21no—
354C3×D4—34no—
355C3×D4—41no—
356C3×D4—124no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—32no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—62no—
374C2×C3×C4—82no—
375C2×C3×C4—124no—
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—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3C2×C3×D412yes—
388D12C3×SD1614yescomplemented
389C4×D3—22no—
390C4×D3—22no—
391C4×D3—44no—
392C4×D3—44no—
393Dic6—21no—
394Dic6—44no—
395Dic6—42no—
396Dic6C3×#4112yescomplemented
397C3×C8—22no—
398C3×C8—22no—
399C3×C8—42no—
400C3×C8—61no—
401C3×C8—61no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C2×C32—81no—
405C2×C32#54611yescharacteristic
406C2×C32C22×SD1611yescharacteristic
407C2×C32#57811yescharacteristic
408C3×D3—24no—
409#41—124no—
410C2×Q8—31no—
411C2×Q8—62no—
412C2×Q8—61no—
413C2×Q8—64no—
414C2×Q8—122no—
415C2×Q8C22×C3×D312yes—
416C2×D4—31no—
417C2×D4—62no—
418C2×D4C22×C3×D311yescharacteristic
419C22×C4—21no—
420C22×C4—32no—
421C22×C4—62no—
422C22×C4—62no—
423SD16—42no—
424SD16—62no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—122no—
428C2×C8C22×C3×D312yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—31no—
432C4⋊C4—31no—
433C4⋊C4—32no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4C22×C3×D311yescharacteristic
439C22⋊C4—21no—
440C22⋊C4—32no—
441C22⋊C4—62no—
442C22⋊C4—62no—
443C42—61no—
444C42C22×C3×D311yescharacteristic
445C22×C3—21no—
446C22×C3—34no—
447C22×C3—42no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3C2×D3×D411yescharacteristic
459C22×C3C22×C3×D411yescharacteristic
460C2×D3—22no—
461C2×D3—22no—
462C2×D3C2×C3×SD1614yes—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—22no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—62no—
474C3×C4—62no—
475C3×C4—81no—
476C3×C4—82no—
477C3×C4—122no—
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—22no—
487Dic3—42no—
488Dic3—41no—
489C32#329811yescharacteristic, complemented, Sylow
490C23—21no—
491C23—32no—
492Q8—22no—
493Q8—22no—
494Q8—32no—
495Q8—61no—
496Q8—124no—
497Q8—122no—
498D4—21no—
499D4—21no—
500D4—34no—
501D4—124no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—32no—
505C2×C4—42no—
506C2×C4—61no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—124no—
512C2×C4—124no—
513C2×C4C3×D3×D411yescharacteristic
514C2×C4C3×D3×D411yescharacteristic
515C2×C4C23×C3×D311yescharacteristic
516C8—22no—
517C8—61no—
518C8—61no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—64no—
523C2×C3—82no—
524C2×C3—81no—
525C2×C3—81no—
526C2×C3#561211yescharacteristic
527C2×C3C2×D3×SD1611yescharacteristic
528C2×C3#577911yescharacteristic
529C2×C3C3×#54611yescharacteristic
530C2×C3C22×C3×SD1611yescharacteristic
531C2×C3C3×#57811yescharacteristic
532D3—24no—
533C22—34no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—62no—
538C22—62no—
539C22C2×C3×D3×D411yescharacteristic
540C4—21no—
541C4—41no—
542C4—42no—
543C4—62no—
544C4—62no—
545C4—122no—
546C4—121no—
547C4C3×D3×SD1612yes—
548C4C3×#100411yescharacteristic
549C4C3×D3×#4111yescharacteristic
550C3—21no—
551C3#8045011yescharacteristic, direct product factor
552C3C3×#329811yescharacteristic, complemented
553C2—64no—
554C2—81no—
555C2C3×#561211yescharacteristic
556C2C2×C3×D3×SD1611yescharacteristic
557C2C3×#577911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#461C2
socleC22×C32C22×D4
prime radicalC3×#5551C2
hypercenterC3×#461D3
hypocenterC3C3×#3298
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×#5727C2semidirect
3. C32×#461C2semidirect
4. C3×#5171C2semidirect
5. C3×#5170C2semidirect
6. C3×#5079C2semidirect
7. C3×C4×#214C2semidirect
8. C3×#4851C2semidirect
9. C3×#4834C2semidirect
10. C3×#4805C2semidirect
11. C3×D3×#103C2semidirect
13. #80450C3direct (N × H)
45. #5727C2×C3semidirect
52. C3×#461D3semidirect
53. C3×#461C2×C3semidirect
58. #5171C2×C3semidirect
59. #5170C2×C3semidirect
60. #5079C2×C3semidirect
61. C4×#214C2×C3semidirect
62. #4851C2×C3semidirect
63. #4834C2×C3semidirect
64. #4805C2×C3semidirect
65. D3×#103C2×C3semidirect
183. C3×D12SD16semidirect
191. C3×Dic6#41semidirect
201. #461C3×D3semidirect
388. D12C3×SD16semidirect
396. Dic6C3×#41semidirect
489. C32#3298semidirect
551. C3#80450direct (N × H)
552. C3C3×#3298semidirect

Maximal subgroups

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

subgroupindex
C3×#57272
C3×#55512
C32×#4612
C3×#51712
C3×#51702
C3×#50792
C3×C4×#2142
C3×#48512
C3×#48342
C3×#48052
C3×D3×#1032
C3×#32983
#804503