Finite Group Databasebeta

← C3×#80281 #11886854

Subgroups — C3×#80281 #11886854

Order 1152.

1004 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5836C212yesmaximal, complemented
2C3×#5775C212yesmaximal, complemented
3C32×#453C211yesmaximal, characteristic, complemented
4C3×#5200C212yesmaximal, complemented
5C3×#5077C211yesmaximal, characteristic, complemented
6C3×#5076C211yesmaximal, characteristic, complemented
7C3×#4826C212yesmaximal, complemented
8C3×#4824C212yesmaximal, complemented
9C3×#4748C212yesmaximal, complemented
10C3×#3291—31nomaximal
11#80281C311yesmaximal, characteristic, direct product factor
12C32×H(5,𝔽2)C2211yescharacteristic, complemented
13C3×#1010C2212yes—
14C3×#1004C2211yescharacteristic, complemented
15C32×AGL(1,ℤ/8)C2212yes—
16C32×#127C2211yescharacteristic
17C3×#951—22no—
18C3×#951C2212yescomplemented
19C3×#950—22no—
20C3×#950C2212yes—
21C3×#948C2212yescomplemented
22C2×C3×#214—22no—
23C2×C3×#214C2212yescomplemented
24C2×C3×#214C2212yescomplemented
25C3×#932—22no—
26C3×#932C2212yes—
27C3×#915—22no—
28C3×D3×SD16—22no—
29C3×#912—22no—
30C3×#911—22no—
31C3×#909C2212yescomplemented
32C3×#907C2212yescomplemented
33C3×#869C2211yescharacteristic, complemented
34C32×#104C2212yescomplemented
35C32×#100C2211yescharacteristic, complemented
36C3×#833C2211yescharacteristic, complemented
37C3×#824C2212yescomplemented
38C3×#822C2212yescomplemented
39C3×#805C2212yescomplemented
40C3×C4×C3⋊C8C2211yescharacteristic, complemented
41C3×#577—32no—
42C3×#577—32no—
43#5836C2×C312yescomplemented
44#5775C2×C312yescomplemented
45C3×#492—31no—
46C3×#471—32no—
47C3×#471—32no—
48C3×#456—32no—
49C3×#453—21no—
50C3×#453—31no—
51C3×#453D311yescharacteristic, complemented
52C3×#453C2×C311yescharacteristic, complemented
53C3×#443—32no—
54#5200C2×C312yescomplemented
55#5077C2×C311yescharacteristic, complemented
56#5076C2×C311yescharacteristic, complemented
57#4826C2×C312yescomplemented
58#4824C2×C312yescomplemented
59#4748C2×C312yescomplemented
60C32×#41—22no—
61C32×#41C2312yes—
62C2×C32×D4—22no—
63C2×C32×D4—41no—
64C2×C32×D4—41no—
65C2×C32×D4C2311yescharacteristic
66C2×C32×D4C2311yescharacteristic
67C2×C3×#67—41no—
68C3×D3×Q8—22no—
69C3×#237—22no—
70C3×#237—22no—
71C3×#237—42no—
72C3×D3×D4—22no—
73C3×#235—22no—
74C3×#235C2312yes—
75C2×C3×Dic6—22no—
76C2×C3×Dic6C2312yes—
77C32×SD16—22no—
78C32×D8—22no—
79C32×M16C2312yes—
80C32×C42C2311yescharacteristic
81C3×#216—22no—
82C3×#216D412yescomplemented
83C3×#215D412yescomplemented
84C3×#214—22no—
85C3×#214—22no—
86C3×#214—22no—
87C3×#214—22no—
88C3×#214—22no—
89C3×#214—22no—
90C3×#214D412yescomplemented
91C3×#213—22no—
92C3×#213D412yescomplemented
93C3×C4⋊Dic3—41no—
94C3×#208—22no—
95C3×#208C2312yes—
96C2×C3×C3⋊C8—22no—
97C2×C3×C3⋊C8C2312yes—
98C3×Dic12—22no—
99C3×#204—22no—
100C3×ΓL(1,𝔽25)—22no—
101C3×D3×C8—22no—
102#3291—31noSylow
103C3×2−1+4—32no—
104C3×H(5,𝔽2)—21no—
105C3×H(5,𝔽2)—31no—
106C3×H(5,𝔽2)C2×D311yescharacteristic, complemented
107C3×H(5,𝔽2)C22×C311yescharacteristic, complemented
108#1010C22×C312yes—
109#1004C22×C311yescharacteristic, complemented
110C3×#137—32no—
111C3×#137—32no—
112C3×#137—62no—
113C3×#137—62no—
114C3×AGL(1,ℤ/8)—22no—
115C3×AGL(1,ℤ/8)—32no—
116C3×AGL(1,ℤ/8)—62no—
117C3×AGL(1,ℤ/8)—62no—
118C3×AGL(1,ℤ/8)C2×D312yes—
119C3×AGL(1,ℤ/8)C22×C312yes—
120C3×#135—32no—
121C3×#135—62no—
122C3×#135—62no—
123C2×C3×SD16—32no—
124C2×C3×SD16—32no—
125C2×C3×SD16—62no—
126C2×C3×SD16—62no—
127C3×#131—32no—
128C3×#131—32no—
129C3×C4⋊Q8—31no—
130C3×#127—21no—
131C3×#127C2×D311yescharacteristic
132C3×#127C22×C311yescharacteristic
133#951—22no—
134#951C22×C312yescomplemented
135#950—22no—
136#950C22×C312yes—
137#948C22×C312yescomplemented
138C2×#214—22no—
139C2×#214C22×C312yescomplemented
140C2×#214C22×C312yescomplemented
141#932—22no—
142#932C22×C312yes—
143#915—22no—
144D3×SD16—22no—
145#912—22no—
146#911—22no—
147#909C22×C312yescomplemented
148#907C22×C312yescomplemented
149#869C22×C311yescharacteristic, complemented
150C3×#108—32no—
151C3×#104—22no—
152C3×#104—32no—
153C3×#104C2×D312yescomplemented
154C3×#104C22×C312yescomplemented
155C3×#101—32no—
156C3×#100—21no—
157C3×#100—31no—
158C3×#100C2×D311yescharacteristic, complemented
159C3×#100C22×C311yescharacteristic, complemented
160C3×C4×C8—31no—
161#833C22×C311yescharacteristic, complemented
162#824C22×C312yescomplemented
163#822C22×C312yescomplemented
164#805C22×C312yescomplemented
165C4×C3⋊C8C22×C311yescharacteristic, complemented
166C23×C32—21no—
167C23×C32—21no—
168C23×C32—41no—
169C22×C3×D3—41no—
170C32×Q8C2×D412yes—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—22no—
174C32×D4—42no—
175C32×D4—82no—
176C32×D4C2×D412yes—
177C2×C32×C4—21no—
178C2×C32×C4—22no—
179C2×C32×C4—41no—
180C2×C32×C4C2411yescharacteristic
181C3×#67—22no—
182C3×#67—42no—
183C3×#67—42no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—42no—
187C3×D12C2×D412yes—
188C3×C4×D3—22no—
189C3×C4×D3—42no—
190C3×Dic6—22no—
191C3×Dic6—22no—
192C3×Dic6—42no—
193C3×Dic6C2×D412yes—
194C32×C8—22no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8C2×D412yes—
197C3×C3⋊C8C2×D412yes—
198#577—32no—
199#577—32no—
200#492—31no—
201#471—32no—
202#471—32no—
203#456—32no—
204#453—31no—
205#453C3×D311yescharacteristic, complemented
206#443—32no—
207C3×#41—22no—
208C3×#41—22no—
209C3×#41—22no—
210C3×#41—32no—
211C3×#41—42no—
212C3×#41—62no—
213C3×#41—62no—
214C3×#41—62no—
215C3×#41—122no—
216C3×#41C22×D312yes—
217C3×#41C23×C312yes—
218C2×C3×Q8—32no—
219C2×C3×Q8—62no—
220C2×C3×Q8—62no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—42no—
226C2×C3×D4—41no—
227C2×C3×D4—41no—
228C2×C3×D4—41no—
229C2×C3×D4—41no—
230C2×C3×D4—42no—
231C2×C3×D4—42no—
232C2×C3×D4—62no—
233C2×C3×D4—121no—
234C2×C3×D4C22×D311yescharacteristic
235C2×C3×D4C22×D311yescharacteristic
236C2×C3×D4C23×C311yescharacteristic
237C2×C3×D4C23×C311yescharacteristic
238C2×#67—41no—
239D3×Q8—22no—
240#237—22no—
241#237—22no—
242#237—42no—
243D3×D4—22no—
244#235—22no—
245#235C23×C312yes—
246C2×Dic6—22no—
247C2×Dic6C23×C312yes—
248C3×Q16—32no—
249C3×Q16—62no—
250C3×Q16—62no—
251C3×SD16—22no—
252C3×SD16—22no—
253C3×SD16—32no—
254C3×SD16—32no—
255C3×SD16—42no—
256C3×SD16—62no—
257C3×SD16—62no—
258C3×SD16—62no—
259C3×SD16—62no—
260C3×SD16—62no—
261C3×SD16—62no—
262C3×SD16—62no—
263C3×D8—22no—
264C3×D8—22no—
265C3×D8—32no—
266C3×D8—42no—
267C3×D8—62no—
268C3×M16—22no—
269C3×M16—32no—
270C3×M16—62no—
271C3×M16—62no—
272C3×M16C22×D312yes—
273C3×M16C23×C312yes—
274C2×C3×C8—32no—
275C2×C3×C8—62no—
276C2×C3×C8—62no—
277C3×C4⋊C4—121no—
278C3×C42—21no—
279C3×C42C22×D311yescharacteristic
280C3×C42C23×C311yescharacteristic
281#216—22no—
282#216C3×D412yescomplemented
283#215C3×D412yescomplemented
284#214—22no—
285#214—22no—
286#214—22no—
287#214—22no—
288#214—22no—
289#214—22no—
290#214C3×D412yescomplemented
291#213—22no—
292#213C3×D412yescomplemented
293C4⋊Dic3—41no—
294#208—22no—
295#208C23×C312yes—
296C2×C3⋊C8—22no—
297C2×C3⋊C8C23×C312yes—
298Dic12—22no—
299#204—22no—
300ΓL(1,𝔽25)—22no—
301D3×C8—22no—
302C22×C32—22no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—81no—
306C22×C32—81no—
307C22×C32—161no—
308C22×C32H(5,𝔽2)11yescharacteristic
309C2×C3×D3—22no—
310C2×C3×D3—161no—
311C32×C4—22no—
312C32×C4—22no—
313C32×C4C22×D412yes—
314C3×Dic3—22no—
315C3×Dic3—42no—
3162−1+4—32no—
317H(5,𝔽2)—31no—
318H(5,𝔽2)C2×C3×D311yescharacteristic, complemented
319#137—32no—
320#137—32no—
321#137—62no—
322#137—62no—
323AGL(1,ℤ/8)—32no—
324AGL(1,ℤ/8)—62no—
325AGL(1,ℤ/8)—62no—
326AGL(1,ℤ/8)C2×C3×D312yes—
327#135—32no—
328#135—62no—
329#135—62no—
330C2×SD16—32no—
331C2×SD16—32no—
332C2×SD16—62no—
333C2×SD16—62no—
334#131—32no—
335#131—32no—
336C4⋊Q8—31no—
337#127C2×C3×D311yescharacteristic
338#108—32no—
339#104—32no—
340#104C2×C3×D312yescomplemented
341#101—32no—
342#100—31no—
343#100C2×C3×D311yescharacteristic, complemented
344C4×C8—31no—
345C23×C3—22no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—42no—
354C23×C3—121no—
355C22×D3—41no—
356C3×Q8—22no—
357C3×Q8—32no—
358C3×Q8—62no—
359C3×Q8—62no—
360C3×Q8—122no—
361C3×Q8D3×D412yes—
362C3×Q8C2×C3×D412yes—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—32no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—44no—
377C3×D4—62no—
378C3×D4—82no—
379C3×D4—82no—
380C3×D4—84no—
381C3×D4—122no—
382C3×D4—122no—
383C3×D4D3×D412yes—
384C3×D4C2×C3×D412yes—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—42no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—122no—
398C2×C3×C4—121no—
399C2×C3×C4—122no—
400C2×C3×C4C23×D311yescharacteristic
401C2×C3×C4C24×C311yescharacteristic
402#67—22no—
403#67—42no—
404#67—42no—
405C2×Dic3—22no—
406C2×Dic3—41no—
407C2×Dic3—42no—
408D12C2×C3×D412yes—
409C4×D3—22no—
410C4×D3—42no—
411Dic6—22no—
412Dic6—22no—
413Dic6—42no—
414Dic6C2×C3×D412yes—
415C3×C8—22no—
416C3×C8—22no—
417C3×C8—32no—
418C3×C8—32no—
419C3×C8—42no—
420C3×C8—62no—
421C3⋊C8—22no—
422C3⋊C8C2×C3×D412yes—
423C3⋊C8C2×C3×D412yes—
424C2×C32—21no—
425C2×C32—42no—
426C2×C32—81no—
427C2×C32—81no—
428C2×C32D4211yescharacteristic
429C3×D3—42no—
430#41—22no—
431#41—32no—
432#41—62no—
433#41—62no—
434#41—62no—
435#41—122no—
436#41C22×C3×D312yes—
437C2×Q8—32no—
438C2×Q8—62no—
439C2×Q8—62no—
440C2×D4—22no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—62no—
444C2×D4—121no—
445C2×D4C22×C3×D311yescharacteristic
446C2×D4C22×C3×D311yescharacteristic
447Q16—32no—
448Q16—62no—
449Q16—62no—
450SD16—22no—
451SD16—32no—
452SD16—32no—
453SD16—62no—
454SD16—62no—
455SD16—62no—
456SD16—62no—
457SD16—62no—
458SD16—62no—
459SD16—62no—
460D8—22no—
461D8—32no—
462D8—62no—
463M16—32no—
464M16—62no—
465M16—62no—
466M16C22×C3×D312yes—
467C2×C8—32no—
468C2×C8—62no—
469C2×C8—62no—
470C4⋊C4—121no—
471C42C22×C3×D311yescharacteristic
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—62no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—161no—
490C22×C3—161no—
491C22×C3—162no—
492C22×C3—481no—
493C22×C3#100411yescharacteristic
494C22×C3C3×H(5,𝔽2)11yescharacteristic
495C2×D3—22no—
496C2×D3—161no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—62no—
505C3×C4—122no—
506C3×C4C2×D3×D412yes—
507C3×C4C22×C3×D412yes—
508Dic3—22no—
509Dic3—42no—
510C32#329111yescharacteristic, complemented, Sylow
511C23—21no—
512C23—21no—
513C23—41no—
514C23—121no—
515Q8—32no—
516Q8—62no—
517Q8—62no—
518Q8—122no—
519Q8C3×D3×D412yes—
520D4—22no—
521D4—22no—
522D4—22no—
523D4—32no—
524D4—42no—
525D4—62no—
526D4—82no—
527D4—122no—
528D4—122no—
529D4C3×D3×D412yes—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—122no—
536C2×C4—121no—
537C2×C4—122no—
538C2×C4C23×C3×D311yescharacteristic
539C8—22no—
540C8—32no—
541C8—32no—
542C8—62no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—41no—
547C2×C3—42no—
548C2×C3—42no—
549C2×C3—82no—
550C2×C3—82no—
551C2×C3—81no—
552C2×C3—81no—
553C2×C3—81no—
554C2×C3—81no—
555C2×C3—122no—
556C2×C3—161no—
557C2×C3#559011yescharacteristic
558C2×C3C3×D4211yescharacteristic
559D3—42no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—62no—
564C22—81no—
565C22—81no—
566C22—161no—
567C22—481no—
568C22C3×#100411yescharacteristic
569C4—22no—
570C4—22no—
571C4—62no—
572C4—122no—
573C4C2×C3×D3×D412yes—
574C3—21no—
575C3#8028111yescharacteristic, direct product factor
576C3C3×#329111yescharacteristic, complemented
577C2—21no—
578C2—42no—
579C2—81no—
580C2—81no—
581C2—122no—
582C2C3×#559011yescharacteristic
subgroupG/H
centerC2×C3#5590
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#453C2
socleC2×C32D42
hypercenterC3×#453D3
hypocenterC3C3×#3291
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#453

Complements

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

NH ≅ G/Ntype
1. C3×#5836C2semidirect
2. C3×#5775C2semidirect
3. C32×#453C2semidirect
4. C3×#5200C2semidirect
5. C3×#5077C2semidirect
6. C3×#5076C2semidirect
7. C3×#4826C2semidirect
8. C3×#4824C2semidirect
9. C3×#4748C2semidirect
11. #80281C3direct (N × H)
12. C32×H(5,𝔽2)C22semidirect
14. C3×#1004C22semidirect
18. C3×#951C22semidirect
21. C3×#948C22semidirect
23. C2×C3×#214C22semidirect
24. C2×C3×#214C22semidirect
31. C3×#909C22semidirect
32. C3×#907C22semidirect
33. C3×#869C22semidirect
34. C32×#104C22semidirect
35. C32×#100C22semidirect
36. C3×#833C22semidirect
37. C3×#824C22semidirect
38. C3×#822C22semidirect
39. C3×#805C22semidirect
40. C3×C4×C3⋊C8C22semidirect
43. #5836C2×C3semidirect
44. #5775C2×C3semidirect
51. C3×#453D3semidirect
52. C3×#453C2×C3semidirect
54. #5200C2×C3semidirect
55. #5077C2×C3semidirect
56. #5076C2×C3semidirect
57. #4826C2×C3semidirect
58. #4824C2×C3semidirect
59. #4748C2×C3semidirect
82. C3×#216D4semidirect
83. C3×#215D4semidirect
90. C3×#214D4semidirect
92. C3×#213D4semidirect
106. C3×H(5,𝔽2)C2×D3semidirect
107. C3×H(5,𝔽2)C22×C3semidirect
109. #1004C22×C3semidirect
134. #951C22×C3semidirect
137. #948C22×C3semidirect
139. C2×#214C22×C3semidirect
140. C2×#214C22×C3semidirect
147. #909C22×C3semidirect
148. #907C22×C3semidirect
149. #869C22×C3semidirect
153. C3×#104C2×D3semidirect
154. C3×#104C22×C3semidirect
158. C3×#100C2×D3semidirect
159. C3×#100C22×C3semidirect
161. #833C22×C3semidirect
162. #824C22×C3semidirect
163. #822C22×C3semidirect
164. #805C22×C3semidirect
165. C4×C3⋊C8C22×C3semidirect
205. #453C3×D3semidirect
282. #216C3×D4semidirect
283. #215C3×D4semidirect
290. #214C3×D4semidirect
292. #213C3×D4semidirect
318. H(5,𝔽2)C2×C3×D3semidirect
340. #104C2×C3×D3semidirect
343. #100C2×C3×D3semidirect
510. C32#3291semidirect
575. C3#80281direct (N × H)
576. C3C3×#3291semidirect

Maximal subgroups

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

subgroupindex
C3×#58362
C3×#57752
C32×#4532
C3×#52002
C3×#50772
C3×#50762
C3×#48262
C3×#48242
C3×#47482
C3×#32913
#802813