Finite Group Databasebeta

← C3×#81585 #11895326

Subgroups — C3×#81585 #11895326

Order 1152.

783 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5730C211yesmaximal, characteristic, complemented
2C3×#5713C211yesmaximal, characteristic, complemented
3C32×#486C211yesmaximal, characteristic, complemented
4C3×#5091C211yesmaximal, characteristic, complemented
5C3×#5072C211yesmaximal, characteristic
6C3×#4787C212yesmaximal, complemented
7C3×#4768C212yesmaximal, complemented
8C3×#4763C212yesmaximal, complemented
9C3×#4756C212yesmaximal, complemented
10C3×C4×#204C211yesmaximal, characteristic, complemented
11C3×C8×Dic6C211yesmaximal, characteristic, complemented
12C3×#3361—31nomaximal
13#81585C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8C2211yescharacteristic
15C32×#127C2211yescharacteristic
16C3×#946—22no—
17C3×#936—21no—
18C3×#936C2212yes—
19C3×#935—22no—
20C3×D4×Dic3—21no—
21C3×D4×Dic3C2212yes—
22C2×C3×#213C2212yes—
23C2×C3×#204C2211yescharacteristic
24C3×#897—21no—
25C3×#893C2212yes—
26C3×D3×C4⋊C4—21no—
27C3×#890C2212yes—
28C3×C4×D12C2211yescharacteristic
29C3×C4×Dic6C2211yescharacteristic
30C3×C4×Dic6C2211yescharacteristic
31C32×#102C2212yes—
32C32×#102C2212yes—
33C32×C4×C8C2211yescharacteristic
34C3×#832C2212yes—
35C3×#821C2211yescharacteristic
36C3×#817C2211yescharacteristic
37C3×#816C2211yescharacteristic
38C3×Dic3⋊C8C2212yes—
39C3×C8×Dic3C2212yes—
40C3×#810C2212yes—
41C3×#807C2212yes—
42C3×C4⋊(C3⋊C8)C2211yescharacteristic
43C3×#559—31no—
44C3×#554—31no—
45#5730C2×C311yescharacteristic, complemented
46#5713C2×C311yescharacteristic, complemented
47C3×#488—32no—
48C3×#486—21no—
49C3×#486D311yescharacteristic, complemented
50C3×#486C2×C311yescharacteristic, complemented
51C3×#478—32no—
52C3×#477—31no—
53C3×#463—32no—
54C3×#459—31no—
55C3×C8×Q8—31no—
56C3×C4×SD16—31no—
57C3×C4×D8—32no—
58#5091C2×C311yescharacteristic, complemented
59#5072C2×C311yescharacteristic
60#4787C2×C312yescomplemented
61#4768C2×C312yescomplemented
62#4763C2×C312yescomplemented
63#4756C2×C312yescomplemented
64C4×#204C2×C311yescharacteristic, complemented
65C8×Dic6C2×C311yescharacteristic, complemented
66C2×C32×Q8—21no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4C2312yes—
70C22×C3×Dic3—22no—
71C2×C3×D12C2311yescharacteristic
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3—22no—
74C2×C3×Dic6D411yescharacteristic
75C2×C3×Dic6C2311yescharacteristic
76C2×C32×C8C2311yescharacteristic
77C2×C32×C8C2311yescharacteristic
78C32×C4⋊C4—21no—
79C32×C4⋊C4C2312yes—
80C32×C42C2311yescharacteristic
81C3×#217—22no—
82C3×#217—22no—
83C3×#217—22no—
84C3×#213—24no—
85C3×(C2×D3)⋊C4—21no—
86C3×(C2×D3)⋊C4—22no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3D411yescharacteristic
89C3×C4⋊Dic3C2311yescharacteristic
90C3×C4⋊Dic3C2311yescharacteristic
91C3×Dic3⋊C4—21no—
92C3×Dic3⋊C4—22no—
93C3×Dic3⋊C4D412yes—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2312yes—
96C2×C3×C3⋊C8C2312yes—
97C3×#204—22no—
98#3361—31noSylow
99C2×C3×SD16—31no—
100C2×C3×D8—32no—
101C3×C4⋊Q8—21no—
102C3×C4⋊Q8C2×D311yescharacteristic
103C3×C4⋊Q8C22×C311yescharacteristic
104C3×#127—21no—
105C3×#127C2×D311yescharacteristic
106C3×#127C22×C311yescharacteristic
107C3×#125—32no—
108C3×#124—32no—
109C3×#124—61no—
110C3×#123—62no—
111C3×C22⋊Q8—61no—
112C3×C22⋊Q8—62no—
113C3×C4×Q8—31no—
114C3×C4×Q8—31no—
115C3×C4×D4—32no—
116C3×C4×D4—32no—
117C3×C4×D4—31no—
118C3×C4×D4—61no—
119C2×C3×C4⋊C4—61no—
120#946—22no—
121#936—21no—
122#936C22×C312yes—
123#935—22no—
124D4×Dic3—21no—
125D4×Dic3C22×C312yes—
126C2×#213C22×C312yes—
127C2×#204C22×C311yescharacteristic
128#897—21no—
129#893C22×C312yes—
130D3×C4⋊C4—21no—
131#890C22×C312yes—
132C4×D12C22×C311yescharacteristic
133C4×Dic6C22×C311yescharacteristic
134C4×Dic6C22×C311yescharacteristic
135C3×#107—32no—
136C3×#106—31no—
137C3×C4⋊C8—32no—
138C3×C4⋊C8—31no—
139C3×#103—31no—
140C3×#103—32no—
141C3×#102—22no—
142C3×#102—22no—
143C3×#102—32no—
144C3×#102—31no—
145C3×#102C2×D312yes—
146C3×#102C2×D312yes—
147C3×#102C22×C312yes—
148C3×#102C22×C312yes—
149C3×C4×C8—21no—
150C3×C4×C8—32no—
151C3×C4×C8C2×D311yescharacteristic
152C3×C4×C8C22×C311yescharacteristic
153#832C22×C312yes—
154#821C22×C311yescharacteristic
155#817C22×C311yescharacteristic
156#816C22×C311yescharacteristic
157Dic3⋊C8C22×C312yes—
158C8×Dic3C22×C312yes—
159#810C22×C312yes—
160#807C22×C312yes—
161C4⋊(C3⋊C8)C22×C311yescharacteristic
162C23×C32—22no—
163C22×C3×D3—21no—
164C32×Q8—81no—
165C32×D4—22no—
166C32×D4—22no—
167C32×D4—42no—
168C32×D4—81no—
169C2×C32×C4—22no—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2411yescharacteristic
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—82no—
177C2×C3×Dic3C2×D412yes—
178C2×C3×Dic3C2×D412yes—
179C3×D12—21no—
180C3×D12#4112yes—
181C3×C4×D3—42no—
182C3×C4×D3—82no—
183C3×Dic6—22no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C32×C8—21no—
187C32×C8—21no—
188C3×C3⋊C8—22no—
189#559—31no—
190#554—31no—
191#488—32no—
192#486C3×D311yescharacteristic, complemented
193#478—32no—
194#477—31no—
195#463—32no—
196#459—31no—
197C8×Q8—31no—
198C4×SD16—31no—
199C4×D8—32no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—31no—
203C2×C3×Q8—31no—
204C2×C3×Q8—41no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—31no—
211C2×C3×D4—41no—
212C2×C3×D4—41no—
213C2×C3×D4C22×D312yes—
214C2×C3×D4C23×C312yes—
215C22×C3×C4—62no—
216C22×C3×C4—62no—
217C22×C3×C4—61no—
218C22×Dic3—22no—
219C2×D12C23×C311yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—22no—
222C2×Dic6C3×D411yescharacteristic
223C2×Dic6C23×C311yescharacteristic
224C3×SD16—62no—
225C3×D8—64no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—32no—
229C2×C3×C8C22×D311yescharacteristic
230C2×C3×C8C22×D311yescharacteristic
231C2×C3×C8C23×C311yescharacteristic
232C2×C3×C8C23×C311yescharacteristic
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—62no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4C22×D312yes—
245C3×C4⋊C4C23×C312yes—
246C3×C22⋊C4—62no—
247C3×C22⋊C4—62no—
248C3×C22⋊C4—62no—
249C3×C22⋊C4—62no—
250C3×C22⋊C4—61no—
251C3×C42—21no—
252C3×C42—32no—
253C3×C42—61no—
254C3×C42C22×D311yescharacteristic
255C3×C42C23×C311yescharacteristic
256#217—22no—
257#217—22no—
258#217—22no—
259#213—24no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—22no—
262C4⋊Dic3—22no—
263C4⋊Dic3C3×D411yescharacteristic
264C4⋊Dic3C23×C311yescharacteristic
265C4⋊Dic3C23×C311yescharacteristic
266Dic3⋊C4—21no—
267Dic3⋊C4—22no—
268Dic3⋊C4C3×D412yes—
269C4×Dic3—21no—
270C4×Dic3C23×C312yes—
271C2×C3⋊C8C23×C312yes—
272#204—22no—
273C22×C32—42no—
274C22×C32—42no—
275C22×C32—42no—
276C22×C32C22×D411yescharacteristic
277C2×C3×D3—22no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C32×C4—21no—
281C32×C4—42no—
282C32×C4#13512yes—
283C32×C4C2×#4111yescharacteristic
284C32×C42−1+411yescharacteristic
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—41no—
290C2×SD16—31no—
291C2×D8—32no—
292C4⋊Q8C2×C3×D311yescharacteristic
293#127C2×C3×D311yescharacteristic
294#125—32no—
295#124—32no—
296#124—61no—
297#123—62no—
298C22⋊Q8—61no—
299C22⋊Q8—62no—
300C4×Q8—31no—
301C4×Q8—31no—
302C4×D4—32no—
303C4×D4—32no—
304C4×D4—31no—
305C4×D4—61no—
306C2×C4⋊C4—61no—
307#107—32no—
308#106—31no—
309C4⋊C8—32no—
310C4⋊C8—31no—
311#103—31no—
312#103—32no—
313#102—32no—
314#102—31no—
315#102C2×C3×D312yes—
316#102C2×C3×D312yes—
317C4×C8—32no—
318C4×C8C2×C3×D311yescharacteristic
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—42no—
322C23×C3—61no—
323C22×D3—21no—
324C3×Q8—62no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—81no—
328C3×Q8—81no—
329C3×Q8—82no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—32no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—61no—
340C3×D4—82no—
341C3×D4—81no—
342C3×D4—81no—
343C3×D4—82no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—32no—
350C2×C3×C4—32no—
351C2×C3×C4—42no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—122no—
356C2×C3×C4—242no—
357C2×C3×C4—242no—
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4C2×C3×D411yescharacteristic
361C2×C3×C4C2×C3×D411yescharacteristic
362C2×C3×C4C23×D311yescharacteristic
363C2×C3×C4C24×C311yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—82no—
368C2×Dic3C2×C3×D412yes—
369C2×Dic3C2×C3×D412yes—
370D12—21no—
371D12C3×#4112yes—
372C4×D3—42no—
373C4×D3—82no—
374Dic6—22no—
375Dic6—21no—
376Dic6—21no—
377C3×C8—21no—
378C3×C8—21no—
379C3×C8—21no—
380C3×C8—21no—
381C3×C8—41no—
382C3×C8—41no—
383C3×C8—62no—
384C3⋊C8—22no—
385C2×C32—82no—
386C2×C32#54811yescharacteristic
387C2×C32C2×#13511yescharacteristic
388C2×C32#57611yescharacteristic
389C3×D3—42no—
390C2×Q8—21no—
391C2×Q8—31no—
392C2×Q8—31no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—31no—
396C2×D4C22×C3×D312yes—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—61no—
400SD16—62no—
401D8—64no—
402C2×C8—32no—
403C2×C8C22×C3×D311yescharacteristic
404C2×C8C22×C3×D311yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—31no—
407C4⋊C4—32no—
408C4⋊C4—31no—
409C4⋊C4—31no—
410C4⋊C4—62no—
411C4⋊C4—62no—
412C4⋊C4—61no—
413C4⋊C4C22×C3×D312yes—
414C22⋊C4—62no—
415C22⋊C4—62no—
416C22⋊C4—62no—
417C22⋊C4—62no—
418C22⋊C4—61no—
419C42—32no—
420C42—61no—
421C42C22×C3×D311yescharacteristic
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—62no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—121no—
434C22×C3—121no—
435C22×C3C2×D3×D411yescharacteristic
436C22×C3C22×C3×D411yescharacteristic
437C2×D3—22no—
438C2×D3—41no—
439C2×D3—41no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—82no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4#91612yes—
455C3×C4C3×#13512yes—
456C3×C4D3×#4111yescharacteristic
457C3×C4#101011yescharacteristic
458C3×C4C2×C3×#4111yescharacteristic
459C3×C4C3×2−1+411yescharacteristic
460Dic3—22no—
461Dic3—22no—
462Dic3—41no—
463Dic3—41no—
464Dic3—41no—
465C32#336111yescharacteristic, complemented, Sylow
466C23—22no—
467C23—61no—
468Q8—62no—
469Q8—61no—
470Q8—61no—
471Q8—81no—
472D4—22no—
473D4—22no—
474D4—32no—
475D4—42no—
476D4—61no—
477D4—81no—
478C2×C4—22no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—122no—
485C2×C4—242no—
486C2×C4—242no—
487C2×C4C3×D3×D411yescharacteristic
488C2×C4C3×D3×D411yescharacteristic
489C2×C4C23×C3×D311yescharacteristic
490C8—21no—
491C8—21no—
492C8—62no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—82no—
497C2×C3—82no—
498C2×C3—122no—
499C2×C3—162no—
500C2×C3#567711yescharacteristic
501C2×C3#575711yescharacteristic
502C2×C3C2×#91611yescharacteristic
503C2×C3C3×#54811yescharacteristic
504C2×C3C2×C3×#13511yescharacteristic
505C2×C3C3×#57611yescharacteristic
506D3—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—62no—
511C22—121no—
512C22—121no—
513C22C2×C3×D3×D411yescharacteristic
514C4—21no—
515C4—42no—
516C4—62no—
517C4—62no—
518C4—121no—
519C4—121no—
520C4—121no—
521C4C3×#91612yes—
522C4C3×D3×#4111yescharacteristic
523C4C3×#101011yescharacteristic
524C3—21no—
525C3#8158511yescharacteristic, direct product factor
526C3C3×#336111yescharacteristic, complemented
527C2—82no—
528C2—122no—
529C2C3×#567711yescharacteristic
530C2C3×#575711yescharacteristic
531C2C2×C3×#91611yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#486C2
socleC22×C32C22×D4
prime radicalC3×#5072C2
hypercenterC3×#486D3
hypocenterC3C3×#3361
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#486

Complements

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

NH ≅ G/Ntype
1. C3×#5730C2semidirect
2. C3×#5713C2semidirect
3. C32×#486C2semidirect
4. C3×#5091C2semidirect
6. C3×#4787C2semidirect
7. C3×#4768C2semidirect
8. C3×#4763C2semidirect
9. C3×#4756C2semidirect
10. C3×C4×#204C2semidirect
11. C3×C8×Dic6C2semidirect
13. #81585C3direct (N × H)
45. #5730C2×C3semidirect
46. #5713C2×C3semidirect
49. C3×#486D3semidirect
50. C3×#486C2×C3semidirect
58. #5091C2×C3semidirect
60. #4787C2×C3semidirect
61. #4768C2×C3semidirect
62. #4763C2×C3semidirect
63. #4756C2×C3semidirect
64. C4×#204C2×C3semidirect
65. C8×Dic6C2×C3semidirect
192. #486C3×D3semidirect
465. C32#3361semidirect
525. C3#81585direct (N × H)
526. C3C3×#3361semidirect

Maximal subgroups

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

subgroupindex
C3×#57302
C3×#57132
C32×#4862
C3×#50912
C3×#50722
C3×#47872
C3×#47682
C3×#47632
C3×#47562
C3×C4×#2042
C3×C8×Dic62
C3×#33613
#815853