Finite Group Databasebeta

← C3×#81566 #11878645

Subgroups — C3×#81566 #11878645

Order 1152.

1123 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5725C211yesmaximal, characteristic, complemented
2C3×D3×#127C211yesmaximal, characteristic, complemented
3C32×#486C211yesmaximal, characteristic, complemented
4C3×#5088C211yesmaximal, characteristic, complemented
5C3×#5073C211yesmaximal, characteristic, complemented
6C3×#4786C214yesmaximal, complemented
7C3×#4773C214yesmaximal, complemented
8C3×#4695C211yesmaximal, characteristic, complemented
9C3×#4685C211yesmaximal, characteristic, complemented
10C3×#3205—31nomaximal
11#81566C311yesmaximal, characteristic, direct product factor
12C2×C3×#239—21no—
13C2×C3×D3×D4—21no—
14C2×C3×D3×D4—21no—
15C2×C3×D3×D4C2212yes—
16C32×C4⋊Q8C2211yescharacteristic, complemented
17C32×#127C2211yescharacteristic
18C3×#947—21no—
19C3×#940—21no—
20C3×#940—21no—
21C3×#940C2212yes—
22C2×C3×#213C2214yescomplemented
23C2×C3×D24C2214yescomplemented
24C3×#895—21no—
25C3×#895C2212yes—
26C3×#893—21no—
27C3×#893C2212yescomplemented
28C3×#874C2211yescharacteristic
29C3×#874C2211yescharacteristic, complemented
30C3×C42×D3C2211yescharacteristic
31C32×#102C2214yescomplemented
32C32×C4×C8C2211yescharacteristic, complemented
33C3×(C2×D3)⋊C8C2212yes—
34C3×Dic3⋊C8C2212yescomplemented
35C3×#809C2214yescomplemented
36C3×C4×C3⋊C8C2211yescharacteristic, complemented
37C3×#532—31no—
38C2×C3×#127—31no—
39#5725C2×C311yescharacteristic, complemented
40D3×#127C2×C311yescharacteristic, complemented
41C3×#493—31no—
42C3×#493—31no—
43C3×#486—21no—
44C3×#486—31no—
45C3×#486D311yescharacteristic, complemented
46C3×#486C2×C311yescharacteristic, complemented
47C3×#459—34no—
48C3×#447—34no—
49C3×#431—31no—
50#5088C2×C311yescharacteristic, complemented
51#5073C2×C311yescharacteristic, complemented
52#4786C2×C314yescomplemented
53#4773C2×C314yescomplemented
54#4695C2×C311yescharacteristic, complemented
55#4685C2×C311yescharacteristic, complemented
56C23×C3×D3—22no—
57C2×C32×Q8—21no—
58C2×C32×D4—21no—
59C2×C32×D4—21no—
60C2×C32×D4C2312yes—
61C2×C3×#67—22no—
62C2×C3×#67—22no—
63C2×C3×#67—22no—
64C3×#239—44no—
65C3×D3×D4—44no—
66C3×D3×D4—44no—
67C3×D3×D4—44no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12C2312yes—
74C2×C3×D12C2312yes—
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3D411yescharacteristic
77C2×C3×C4×D3D411yescharacteristic
78C2×C3×C4×D3C2311yescharacteristic
79C2×C32×C8C2312yes—
80C32×C4⋊C4—21no—
81C32×C4⋊C4C2312yes—
82C32×C42C2311yescharacteristic
83C3×#213—44no—
84C3×(C2×D3)⋊C4—22no—
85C3×(C2×D3)⋊C4—22no—
86C3×C4×Dic3D411yescharacteristic
87C3×C4×Dic3D411yescharacteristic
88C3×C4×Dic3C2311yescharacteristic
89C2×C3×C3⋊C8C2312yes—
90C3×D24—44no—
91#3205—31noSylow
92C2×C3×#41—61no—
93C22×C3×D4—32no—
94C22×C3×D4—61no—
95C22×C3×D4—61no—
96C2×#239—21no—
97C2×D3×D4—21no—
98C2×D3×D4—21no—
99C2×D3×D4C22×C312yes—
100C2×C3×D8—34no—
101C2×C3×D8—34no—
102C3×C4⋊Q8—21no—
103C3×C4⋊Q8C2×D311yescharacteristic, complemented
104C3×C4⋊Q8C22×C311yescharacteristic, complemented
105C3×#127—21no—
106C3×#127—32no—
107C3×#127—31no—
108C3×#127—31no—
109C3×#127—61no—
110C3×#127—61no—
111C3×#127C2×D311yescharacteristic
112C3×#127C22×C311yescharacteristic
113C3×#124—61no—
114C3×#121—32no—
115C3×#121—61no—
116C3×C4×D4—32no—
117C3×C4×D4—61no—
118C2×C3×C42—31no—
119#947—21no—
120#940—21no—
121#940—21no—
122#940C22×C312yes—
123C2×#213C22×C314yescomplemented
124C2×D24C22×C314yescomplemented
125#895—21no—
126#895C22×C312yes—
127#893—21no—
128#893C22×C312yescomplemented
129#874C22×C311yescharacteristic
130#874C22×C311yescharacteristic, complemented
131C42×D3C22×C311yescharacteristic
132C3×C4⋊C8—32no—
133C3×#102—24no—
134C3×#102—34no—
135C3×#102C2×D314yescomplemented
136C3×#102C22×C314yescomplemented
137C3×C22⋊C8—32no—
138C3×C4×C8—21no—
139C3×C4×C8—31no—
140C3×C4×C8C2×D311yescharacteristic, complemented
141C3×C4×C8C22×C311yescharacteristic, complemented
142(C2×D3)⋊C8C22×C312yes—
143Dic3⋊C8C22×C312yescomplemented
144#809C22×C314yescomplemented
145C4×C3⋊C8C22×C311yescharacteristic, complemented
146C23×C32—22no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3—44no—
150C22×C3×D3—44no—
151C22×C3×D3—44no—
152C22×C3×D3C2×D411yescharacteristic
153C32×Q8—42no—
154C32×D4—24no—
155C32×D4—42no—
156C32×D4—42no—
157C2×C32×C4—22no—
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2411yescharacteristic
161C3×#67—44no—
162C3×#67—44no—
163C3×#67—44no—
164C2×C3×Dic3C2×D411yescharacteristic
165C2×C3×Dic3C2×D411yescharacteristic
166C2×C3×Dic3C2×D411yescharacteristic
167C3×D12—24no—
168C3×D12—24no—
169C3×D12—44no—
170C3×D12—42no—
171C3×D12—42no—
172C3×D12—42no—
173C3×D12—42no—
174C3×C4×D3—22no—
175C3×C4×D3—22no—
176C3×C4×D3—44no—
177C3×C4×D3D814yescomplemented
178C32×C8—22no—
179C3×C3⋊C8—22no—
180#532—31no—
181C2×#127—31no—
182#493—31no—
183#493—31no—
184#486—31no—
185#486C3×D311yescharacteristic, complemented
186#459—34no—
187#447—34no—
188#431—31no—
189C24×C3—62no—
190C23×D3—22no—
191C3×#41—124no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—41no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4—32no—
201C2×C3×D4—32no—
202C2×C3×D4—41no—
203C2×C3×D4—41no—
204C2×C3×D4—62no—
205C2×C3×D4—62no—
206C2×C3×D4—62no—
207C2×C3×D4—62no—
208C2×C3×D4—61no—
209C2×C3×D4—61no—
210C2×C3×D4—61no—
211C2×C3×D4—61no—
212C2×C3×D4—124no—
213C2×C3×D4—124no—
214C2×C3×D4—124no—
215C2×C3×D4C22×D312yes—
216C2×C3×D4C23×C312yes—
217C22×C3×C4—31no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—62no—
221C2×#67—22no—
222C2×#67—22no—
223C2×#67—22no—
224#239—44no—
225D3×D4—44no—
226D3×D4—44no—
227D3×D4—44no—
228C2×D12—22no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12C23×C312yes—
234C2×D12C23×C312yes—
235C2×C4×D3—22no—
236C2×C4×D3C3×D411yescharacteristic
237C2×C4×D3C3×D411yescharacteristic
238C2×C4×D3C23×C311yescharacteristic
239C3×D8—124no—
240C3×D8—124no—
241C2×C3×C8—22no—
242C2×C3×C8—32no—
243C2×C3×C8C22×D312yes—
244C2×C3×C8C23×C312yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4C22×D312yes—
250C3×C4⋊C4C23×C312yes—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—62no—
253C3×C42—21no—
254C3×C42—31no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42C22×D311yescharacteristic
258C3×C42C23×C311yescharacteristic
259#213—44no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—22no—
262C4×Dic3C3×D411yescharacteristic
263C4×Dic3C3×D411yescharacteristic
264C4×Dic3C23×C311yescharacteristic
265C2×C3⋊C8C23×C312yes—
266D24—44no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—42no—
270C22×C32C22×D411yescharacteristic
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—42no—
274C2×C3×D3—42no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C2×C3×D3—42no—
278C2×C3×D3—42no—
279C2×C3×D3—88no—
280C2×C3×D3C2×D812yes—
281C32×C4—42no—
282C32×C4C2×D812yes—
283C32×C4AGL(1,ℤ/8)12yes—
284C32×C4H(5,𝔽2)12yes—
285C3×Dic3—21no—
286C3×Dic3—21no—
287C3×Dic3C2×D812yes—
288C2×#41—61no—
289C22×D4—32no—
290C22×D4—61no—
291C22×D4—61no—
292C2×D8—34no—
293C2×D8—34no—
294C4⋊Q8C2×C3×D311yescharacteristic, complemented
295#127—32no—
296#127—31no—
297#127—31no—
298#127—61no—
299#127—61no—
300#127C2×C3×D311yescharacteristic
301#124—61no—
302#121—32no—
303#121—61no—
304C4×D4—32no—
305C4×D4—61no—
306C2×C42—31no—
307C4⋊C8—32no—
308#102—34no—
309#102C2×C3×D314yescomplemented
310C22⋊C8—32no—
311C4×C8—31no—
312C4×C8C2×C3×D311yescharacteristic, complemented
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—31no—
316C23×C3—42no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—124no—
320C23×C3—124no—
321C23×C3—124no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—44no—
325C22×D3—44no—
326C22×D3—44no—
327C22×D3C2×C3×D411yescharacteristic
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—82no—
331C3×D4—24no—
332C3×D4—24no—
333C3×D4—44no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—64no—
339C3×D4—64no—
340C3×D4—82no—
341C3×D4—82no—
342C3×D4—124no—
343C3×D4—124no—
344C3×D4—124no—
345C3×D4—124no—
346C3×D4—122no—
347C3×D4—122no—
348C3×D4—122no—
349C3×D4—122no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—31no—
356C2×C3×C4—34no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—42no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—124no—
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C2×C3×D411yescharacteristic
367C2×C3×C4C23×D311yescharacteristic
368C2×C3×C4C24×C311yescharacteristic
369#67—44no—
370#67—44no—
371#67—44no—
372C2×Dic3C2×C3×D411yescharacteristic
373C2×Dic3C2×C3×D411yescharacteristic
374C2×Dic3C2×C3×D411yescharacteristic
375D12—24no—
376D12—24no—
377D12—44no—
378D12—42no—
379D12—42no—
380D12—42no—
381D12—42no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—44no—
385C4×D3C3×D814yescomplemented
386C3×C8—22no—
387C3×C8—22no—
388C3×C8—42no—
389C3×C8—62no—
390C3⋊C8—22no—
391C2×C32—82no—
392C2×C32#53511yescharacteristic
393C2×C32C22×D811yescharacteristic
394C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
395C3×D3—22no—
396C3×D3—82no—
397C3×D3—82no—
398C24—62no—
399#41—124no—
400C2×Q8—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—32no—
404C2×D4—32no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—124no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4C22×C3×D312yes—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—62no—
421D8—124no—
422D8—124no—
423C2×C8—32no—
424C2×C8C22×C3×D312yes—
425C4⋊C4—21no—
426C4⋊C4C22×C3×D312yes—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C42—31no—
430C42—31no—
431C42—31no—
432C42C22×C3×D311yescharacteristic
433C22×C3—21no—
434C22×C3—32no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—61no—
442C22×C3—61no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—122no—
447C22×C3—122no—
448C22×C3—122no—
449C22×C3—122no—
450C22×C3—122no—
451C22×C3—122no—
452C22×C3—248no—
453C22×C3C2×D3×D411yescharacteristic
454C22×C3C22×C3×D411yescharacteristic
455C2×D3—21no—
456C2×D3—21no—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—88no—
464C2×D3C2×C3×D812yes—
465C3×C4—22no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—32no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—82no—
474C3×C4D3×D812yes—
475C3×C4#91412yes—
476C3×C4C2×C3×D812yes—
477C3×C4C3×AGL(1,ℤ/8)12yes—
478C3×C4#100912yes—
479C3×C4C3×H(5,𝔽2)12yes—
480Dic3—21no—
481Dic3—21no—
482Dic3C2×C3×D812yes—
483C32#320511yescharacteristic, complemented, Sylow
484C23—22no—
485C23—31no—
486C23—62no—
487C23—62no—
488C23—124no—
489C23—124no—
490C23—124no—
491Q8—42no—
492D4—24no—
493D4—42no—
494D4—42no—
495D4—64no—
496D4—64no—
497D4—124no—
498D4—124no—
499D4—124no—
500D4—124no—
501D4—122no—
502D4—122no—
503D4—122no—
504D4—122no—
505C2×C4—22no—
506C2×C4—31no—
507C2×C4—34no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—124no—
513C2×C4C3×D3×D411yescharacteristic
514C2×C4C3×D3×D411yescharacteristic
515C2×C4C23×C3×D311yescharacteristic
516C8—22no—
517C8—62no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—62no—
522C2×C3—82no—
523C2×C3—82no—
524C2×C3—162no—
525C2×C3—242no—
526C2×C3—242no—
527C2×C3#567411yescharacteristic
528C2×C3C2×D3×D811yescharacteristic
529C2×C3C2×#91411yescharacteristic
530C2×C3C3×#53511yescharacteristic
531C2×C3C22×C3×D811yescharacteristic
532C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
533D3—22no—
534D3—82no—
535D3—82no—
536C22—32no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—61no—
541C22—61no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22—248no—
549C22C2×C3×D3×D411yescharacteristic
550C4—32no—
551C4—42no—
552C4—61no—
553C4—61no—
554C4C3×D3×D812yes—
555C4C3×#91412yes—
556C4C3×#100912yes—
557C3—21no—
558C3#8156611yescharacteristic, direct product factor
559C3C3×#320511yescharacteristic, complemented
560C2—62no—
561C2—82no—
562C2—242no—
563C2—242no—
564C2C3×#567411yescharacteristic
565C2C2×C3×D3×D811yescharacteristic
566C2C2×C3×#91411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#486C2
socleC22×C32C22×D4
hypercenterC3×#486D3
hypocenterC3C3×#3205
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×#5725C2semidirect
2. C3×D3×#127C2semidirect
3. C32×#486C2semidirect
4. C3×#5088C2semidirect
5. C3×#5073C2semidirect
6. C3×#4786C2semidirect
7. C3×#4773C2semidirect
8. C3×#4695C2semidirect
9. C3×#4685C2semidirect
11. #81566C3direct (N × H)
16. C32×C4⋊Q8C22semidirect
22. C2×C3×#213C22semidirect
23. C2×C3×D24C22semidirect
27. C3×#893C22semidirect
29. C3×#874C22semidirect
31. C32×#102C22semidirect
32. C32×C4×C8C22semidirect
34. C3×Dic3⋊C8C22semidirect
35. C3×#809C22semidirect
36. C3×C4×C3⋊C8C22semidirect
39. #5725C2×C3semidirect
40. D3×#127C2×C3semidirect
45. C3×#486D3semidirect
46. C3×#486C2×C3semidirect
50. #5088C2×C3semidirect
51. #5073C2×C3semidirect
52. #4786C2×C3semidirect
53. #4773C2×C3semidirect
54. #4695C2×C3semidirect
55. #4685C2×C3semidirect
103. C3×C4⋊Q8C2×D3semidirect
104. C3×C4⋊Q8C22×C3semidirect
123. C2×#213C22×C3semidirect
124. C2×D24C22×C3semidirect
128. #893C22×C3semidirect
130. #874C22×C3semidirect
135. C3×#102C2×D3semidirect
136. C3×#102C22×C3semidirect
140. C3×C4×C8C2×D3semidirect
141. C3×C4×C8C22×C3semidirect
143. Dic3⋊C8C22×C3semidirect
144. #809C22×C3semidirect
145. C4×C3⋊C8C22×C3semidirect
177. C3×C4×D3D8semidirect
185. #486C3×D3semidirect
294. C4⋊Q8C2×C3×D3semidirect
309. #102C2×C3×D3semidirect
312. C4×C8C2×C3×D3semidirect
385. C4×D3C3×D8semidirect
483. C32#3205semidirect
558. C3#81566direct (N × H)
559. C3C3×#3205semidirect

Maximal subgroups

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

subgroupindex
C3×#57252
C3×D3×#1272
C32×#4862
C3×#50882
C3×#50732
C3×#47862
C3×#47732
C3×#46952
C3×#46852
C3×#32053
#815663