Finite Group Databasebeta

← C3×#80287 #11886579

Subgroups — C3×#80287 #11886579

Order 1152.

1001 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5837C212yesmaximal, complemented
2C3×#5774C212yesmaximal, complemented
3C32×#454C211yesmaximal, characteristic, complemented
4C3×#5199C212yesmaximal, complemented
5C3×#5061C211yesmaximal, characteristic, complemented
6C3×#5058C211yesmaximal, characteristic, complemented
7C3×#4823C212yesmaximal, complemented
8C3×#4822C212yesmaximal, complemented
9C3×#4754C212yesmaximal, complemented
10C3×#3288—31nomaximal
11#80287C311yesmaximal, characteristic, direct product factor
12C32×2−1+4C2211yescharacteristic, complemented
13C3×#1009C2212yescomplemented
14C3×#1004C2211yescharacteristic
15C32×AGL(1,ℤ/8)C2212yescomplemented
16C32×#124C2211yescharacteristic, complemented
17C3×#950—22no—
18C3×#950C2212yes—
19C3×#949—22no—
20C3×#949C2212yescomplemented
21C3×#948C2212yescomplemented
22C3×#942—22no—
23C3×#942C2212yescomplemented
24C2×C3×#215—22no—
25C2×C3×#215C2212yescomplemented
26C2×C3×#213C2212yescomplemented
27C3×#916—22no—
28C3×#914—22no—
29C3×#911—22no—
30C3×D3×D8—22no—
31C3×#908C2212yes—
32C3×#907C2212yescomplemented
33C3×#874C2211yescharacteristic, complemented
34C32×#104C2212yescomplemented
35C32×#101C2211yescharacteristic, complemented
36C3×#833C2211yescharacteristic, complemented
37C3×#823C2212yescomplemented
38C3×#822C2212yescomplemented
39C3×#805C2212yescomplemented
40C3×#803C2211yescharacteristic, complemented
41C3×#577—32no—
42C3×#576—32no—
43#5837C2×C312yescomplemented
44#5774C2×C312yescomplemented
45C3×#496—31no—
46C3×#472—32no—
47C3×#471—32no—
48C3×#455—31no—
49C3×#454—21no—
50C3×#454D311yescharacteristic, complemented
51C3×#454C2×C311yescharacteristic, complemented
52C3×#453—32no—
53C3×#444—32no—
54#5199C2×C312yescomplemented
55#5061C2×C311yescharacteristic, complemented
56#5058C2×C311yescharacteristic, complemented
57#4823C2×C312yescomplemented
58#4822C2×C312yescomplemented
59#4754C2×C312yescomplemented
60C32×#41—22no—
61C32×#41—41no—
62C32×#41C2312yes—
63C2×C32×Q8—22no—
64C2×C32×Q8C2311yescharacteristic
65C2×C32×D4C2311yescharacteristic
66C2×C3×#67—41no—
67C3×#239—22no—
68C3×#237—22no—
69C3×D3×D4—22no—
70C3×D3×D4—22no—
71C3×D3×D4—42no—
72C3×#235—22no—
73C3×#235C2312yes—
74C2×C3×D12—22no—
75C2×C3×D12—41no—
76C2×C3×D12C2312yes—
77C32×SD16—22no—
78C32×D8—22no—
79C32×M16C2312yes—
80C32×C22⋊C4—41no—
81C32×C42C2311yescharacteristic
82C3×#216—22no—
83C3×#216D412yescomplemented
84C3×#215—22no—
85C3×#215—22no—
86C3×#215—22no—
87C3×#215—22no—
88C3×#215D412yescomplemented
89C3×#214D412yescomplemented
90C3×#213—22no—
91C3×#213—22no—
92C3×#213—22no—
93C3×#213D412yescomplemented
94C3×#208—22no—
95C3×#208C2312yes—
96C2×C3×C3⋊C8—22no—
97C2×C3×C3⋊C8C2312yes—
98C3×D24—22no—
99C3×#204—22no—
100C3×ΓL(1,𝔽25)—22no—
101C3×D3×C8—22no—
102#3288—31noSylow
103C3×2−1+4—21no—
104C3×2−1+4C2×D311yescharacteristic, complemented
105C3×2−1+4C22×C311yescharacteristic, complemented
106C3×H(5,𝔽2)—32no—
107C3×H(5,𝔽2)—31no—
108#1009C22×C312yescomplemented
109#1004C22×C311yescharacteristic
110C3×#137—32no—
111C3×#137—62no—
112C3×AGL(1,ℤ/8)—22no—
113C3×AGL(1,ℤ/8)—32no—
114C3×AGL(1,ℤ/8)—32no—
115C3×AGL(1,ℤ/8)—62no—
116C3×AGL(1,ℤ/8)—62no—
117C3×AGL(1,ℤ/8)—62no—
118C3×AGL(1,ℤ/8)C2×D312yescomplemented
119C3×AGL(1,ℤ/8)C22×C312yescomplemented
120C3×#135—32no—
121C3×#135—62no—
122C3×#135—62no—
123C2×C3×SD16—32no—
124C2×C3×SD16—62no—
125C2×C3×D8—32no—
126C2×C3×D8—62no—
127C3×#131—32no—
128C3×#131—32no—
129C3×#127—31no—
130C3×#124—21no—
131C3×#124C2×D311yescharacteristic, complemented
132C3×#124C22×C311yescharacteristic, complemented
133#950—22no—
134#950C22×C312yes—
135#949—22no—
136#949C22×C312yescomplemented
137#948C22×C312yescomplemented
138#942—22no—
139#942C22×C312yescomplemented
140C2×#215—22no—
141C2×#215C22×C312yescomplemented
142C2×#213C22×C312yescomplemented
143#916—22no—
144#914—22no—
145#911—22no—
146D3×D8—22no—
147#908C22×C312yes—
148#907C22×C312yescomplemented
149#874C22×C311yescharacteristic, complemented
150C3×#108—32no—
151C3×#104—22no—
152C3×#104—32no—
153C3×#104C2×D312yescomplemented
154C3×#104C22×C312yescomplemented
155C3×#101—21no—
156C3×#101C2×D311yescharacteristic, complemented
157C3×#101C22×C311yescharacteristic, complemented
158C3×#100—31no—
159C3×#100—32no—
160C3×#97—31no—
161#833C22×C311yescharacteristic, complemented
162#823C22×C312yescomplemented
163#822C22×C312yescomplemented
164#805C22×C312yescomplemented
165#803C22×C311yescharacteristic, complemented
166C23×C32—21no—
167C22×C3×D3—22no—
168C22×C3×D3—41no—
169C22×C3×D3—42no—
170C32×Q8—22no—
171C32×Q8—41no—
172C32×Q8C2×D412yes—
173C32×D4—22no—
174C32×D4—22no—
175C32×D4—41no—
176C32×D4C2×D412yes—
177C2×C32×C4—21no—
178C2×C32×C4—22no—
179C2×C32×C4—21no—
180C2×C32×C4—42no—
181C2×C32×C4C2411yescharacteristic
182C3×#67—22no—
183C3×#67—42no—
184C3×#67—42no—
185C2×C3×Dic3—41no—
186C3×D12—22no—
187C3×D12—22no—
188C3×D12—42no—
189C3×D12—82no—
190C3×D12C2×D412yes—
191C3×C4×D3—22no—
192C3×C4×D3—42no—
193C3×Dic6C2×D412yes—
194C32×C8—22no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8C2×D412yes—
197C3×C3⋊C8C2×D412yes—
198#577—32no—
199#576—32no—
200#496—31no—
201#472—32no—
202#471—32no—
203#455—31no—
204#454C3×D311yescharacteristic, complemented
205#453—32no—
206#444—32no—
207C3×#41—22no—
208C3×#41—22no—
209C3×#41—22no—
210C3×#41—32no—
211C3×#41—42no—
212C3×#41—41no—
213C3×#41—41no—
214C3×#41—42no—
215C3×#41—62no—
216C3×#41—62no—
217C3×#41—62no—
218C3×#41C22×D312yes—
219C3×#41C23×C312yes—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8—21no—
223C2×C3×Q8—42no—
224C2×C3×Q8C22×D311yescharacteristic
225C2×C3×Q8C23×C311yescharacteristic
226C2×C3×D4—21no—
227C2×C3×D4—32no—
228C2×C3×D4—62no—
229C2×C3×D4—62no—
230C2×C3×D4—62no—
231C2×C3×D4—121no—
232C2×C3×D4—122no—
233C2×C3×D4—121no—
234C2×C3×D4C22×D311yescharacteristic
235C2×C3×D4C23×C311yescharacteristic
236C2×#67—41no—
237#239—22no—
238#237—22no—
239D3×D4—22no—
240D3×D4—22no—
241D3×D4—42no—
242#235—22no—
243#235C23×C312yes—
244C2×D12—22no—
245C2×D12—41no—
246C2×D12C23×C312yes—
247C3×Q16—32no—
248C3×Q16—62no—
249C3×SD16—22no—
250C3×SD16—22no—
251C3×SD16—32no—
252C3×SD16—32no—
253C3×SD16—42no—
254C3×SD16—62no—
255C3×SD16—62no—
256C3×SD16—62no—
257C3×SD16—62no—
258C3×SD16—62no—
259C3×D8—22no—
260C3×D8—22no—
261C3×D8—32no—
262C3×D8—42no—
263C3×D8—62no—
264C3×D8—62no—
265C3×D8—62no—
266C3×D8—62no—
267C3×M16—22no—
268C3×M16—32no—
269C3×M16—62no—
270C3×M16—62no—
271C3×M16C22×D312yes—
272C3×M16C23×C312yes—
273C2×C3×C8—32no—
274C2×C3×C8—62no—
275C2×C3×C8—62no—
276C3×C22⋊C4—41no—
277C3×C22⋊C4—41no—
278C3×C22⋊C4—42no—
279C3×C42—21no—
280C3×C42C22×D311yescharacteristic
281C3×C42C23×C311yescharacteristic
282#216—22no—
283#216C3×D412yescomplemented
284#215—22no—
285#215—22no—
286#215—22no—
287#215—22no—
288#215C3×D412yescomplemented
289#214C3×D412yescomplemented
290#213—22no—
291#213—22no—
292#213—22no—
293#213C3×D412yescomplemented
294#208—22no—
295#208C23×C312yes—
296C2×C3⋊C8—22no—
297C2×C3⋊C8C23×C312yes—
298D24—22no—
299#204—22no—
300ΓL(1,𝔽25)—22no—
301D3×C8—22no—
302C22×C32—22no—
303C22×C32—41no—
304C22×C32—81no—
305C22×C32H(5,𝔽2)11yescharacteristic
306C2×C3×D3—22no—
307C2×C3×D3—42no—
308C2×C3×D3—82no—
309C2×C3×D3—161no—
310C2×C3×D3—162no—
311C32×C4—22no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4C22×D412yes—
315C3×Dic3—22no—
3162−1+4C2×C3×D311yescharacteristic, complemented
317H(5,𝔽2)—32no—
318H(5,𝔽2)—31no—
319#137—32no—
320#137—62no—
321AGL(1,ℤ/8)—32no—
322AGL(1,ℤ/8)—32no—
323AGL(1,ℤ/8)—62no—
324AGL(1,ℤ/8)—62no—
325AGL(1,ℤ/8)—62no—
326AGL(1,ℤ/8)C2×C3×D312yescomplemented
327#135—32no—
328#135—62no—
329#135—62no—
330C2×SD16—32no—
331C2×SD16—62no—
332C2×D8—32no—
333C2×D8—62no—
334#131—32no—
335#131—32no—
336#127—31no—
337#124C2×C3×D311yescharacteristic, complemented
338#108—32no—
339#104—32no—
340#104C2×C3×D312yescomplemented
341#101C2×C3×D311yescharacteristic, complemented
342#100—31no—
343#100—32no—
344#97—31no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—41no—
348C23×C3—62no—
349C23×C3—122no—
350C23×C3—121no—
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—42no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—32no—
358C3×Q8—42no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—42no—
362C3×Q8D3×D412yes—
363C3×Q8C2×C3×D412yes—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—32no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—42no—
375C3×D4—62no—
376C3×D4—62no—
377C3×D4—62no—
378C3×D4—122no—
379C3×D4—122no—
380C3×D4—122no—
381C3×D4—242no—
382C3×D4D3×D412yes—
383C3×D4C2×C3×D412yes—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—42no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—44no—
397C2×C3×C4—62no—
398C2×C3×C4—122no—
399C2×C3×C4—121no—
400C2×C3×C4C23×D311yescharacteristic
401C2×C3×C4C24×C311yescharacteristic
402#67—22no—
403#67—42no—
404#67—42no—
405C2×Dic3—41no—
406D12—22no—
407D12—22no—
408D12—42no—
409D12—82no—
410D12C2×C3×D412yes—
411C4×D3—22no—
412C4×D3—42no—
413Dic6C2×C3×D412yes—
414C3×C8—22no—
415C3×C8—22no—
416C3×C8—32no—
417C3×C8—32no—
418C3×C8—42no—
419C3×C8—62no—
420C3⋊C8—22no—
421C3⋊C8C2×C3×D412yes—
422C3⋊C8C2×C3×D412yes—
423C2×C32—21no—
424C2×C32—42no—
425C2×C32—81no—
426C2×C32D4211yescharacteristic
427C3×D3—42no—
428C3×D3—82no—
429#41—22no—
430#41—32no—
431#41—41no—
432#41—62no—
433#41—62no—
434#41—62no—
435#41C22×C3×D312yes—
436C2×Q8—22no—
437C2×Q8C22×C3×D311yescharacteristic
438C2×D4—32no—
439C2×D4—62no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—121no—
443C2×D4—122no—
444C2×D4—121no—
445C2×D4C22×C3×D311yescharacteristic
446Q16—32no—
447Q16—62no—
448SD16—22no—
449SD16—32no—
450SD16—32no—
451SD16—62no—
452SD16—62no—
453SD16—62no—
454SD16—62no—
455SD16—62no—
456D8—22no—
457D8—32no—
458D8—62no—
459D8—62no—
460D8—62no—
461D8—62no—
462M16—32no—
463M16—62no—
464M16—62no—
465M16C22×C3×D312yes—
466C2×C8—32no—
467C2×C8—62no—
468C2×C8—62no—
469C22⋊C4—41no—
470C42C22×C3×D311yescharacteristic
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—42no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—62no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—82no—
482C22×C3—122no—
483C22×C3—242no—
484C22×C3—482no—
485C22×C3—481no—
486C22×C3#100411yescharacteristic
487C22×C3C3×H(5,𝔽2)11yescharacteristic
488C2×D3—22no—
489C2×D3—42no—
490C2×D3—82no—
491C2×D3—161no—
492C2×D3—162no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—62no—
503C3×C4—81no—
504C3×C4C2×D3×D412yes—
505C3×C4C22×C3×D412yes—
506Dic3—22no—
507C32#328811yescharacteristic, complemented, Sylow
508C23—21no—
509C23—62no—
510C23—122no—
511C23—121no—
512Q8—22no—
513Q8—32no—
514Q8—41no—
515Q8C3×D3×D412yes—
516D4—22no—
517D4—22no—
518D4—32no—
519D4—41no—
520D4—62no—
521D4—62no—
522D4—62no—
523D4—122no—
524D4—122no—
525D4—122no—
526D4—242no—
527D4C3×D3×D412yes—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—42no—
532C2×C4—62no—
533C2×C4—122no—
534C2×C4—121no—
535C2×C4C23×C3×D311yescharacteristic
536C8—22no—
537C8—32no—
538C8—32no—
539C8—62no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—41no—
544C2×C3—42no—
545C2×C3—42no—
546C2×C3—82no—
547C2×C3—81no—
548C2×C3—81no—
549C2×C3—122no—
550C2×C3—161no—
551C2×C3—242no—
552C2×C3#559011yescharacteristic
553C2×C3C3×D4211yescharacteristic
554D3—42no—
555D3—82no—
556C22—22no—
557C22—41no—
558C22—62no—
559C22—81no—
560C22—122no—
561C22—242no—
562C22—482no—
563C22—481no—
564C22C3×#100411yescharacteristic
565C4—22no—
566C4—41no—
567C4—41no—
568C4—62no—
569C4C2×C3×D3×D412yes—
570C3—21no—
571C3#8028711yescharacteristic, direct product factor
572C3C3×#328811yescharacteristic, complemented
573C2—21no—
574C2—42no—
575C2—81no—
576C2—122no—
577C2—242no—
578C2C3×#559011yescharacteristic
subgroupG/H
centerC2×C3#5590
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#454C2
socleC2×C32D42
hypercenterC3×#454D3
hypocenterC3C3×#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#454

Complements

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

NH ≅ G/Ntype
1. C3×#5837C2semidirect
2. C3×#5774C2semidirect
3. C32×#454C2semidirect
4. C3×#5199C2semidirect
5. C3×#5061C2semidirect
6. C3×#5058C2semidirect
7. C3×#4823C2semidirect
8. C3×#4822C2semidirect
9. C3×#4754C2semidirect
11. #80287C3direct (N × H)
12. C32×2−1+4C22semidirect
13. C3×#1009C22semidirect
15. C32×AGL(1,ℤ/8)C22semidirect
16. C32×#124C22semidirect
20. C3×#949C22semidirect
21. C3×#948C22semidirect
23. C3×#942C22semidirect
25. C2×C3×#215C22semidirect
26. C2×C3×#213C22semidirect
32. C3×#907C22semidirect
33. C3×#874C22semidirect
34. C32×#104C22semidirect
35. C32×#101C22semidirect
36. C3×#833C22semidirect
37. C3×#823C22semidirect
38. C3×#822C22semidirect
39. C3×#805C22semidirect
40. C3×#803C22semidirect
43. #5837C2×C3semidirect
44. #5774C2×C3semidirect
50. C3×#454D3semidirect
51. C3×#454C2×C3semidirect
54. #5199C2×C3semidirect
55. #5061C2×C3semidirect
56. #5058C2×C3semidirect
57. #4823C2×C3semidirect
58. #4822C2×C3semidirect
59. #4754C2×C3semidirect
83. C3×#216D4semidirect
88. C3×#215D4semidirect
89. C3×#214D4semidirect
93. C3×#213D4semidirect
104. C3×2−1+4C2×D3semidirect
105. C3×2−1+4C22×C3semidirect
108. #1009C22×C3semidirect
118. C3×AGL(1,ℤ/8)C2×D3semidirect
119. C3×AGL(1,ℤ/8)C22×C3semidirect
131. C3×#124C2×D3semidirect
132. C3×#124C22×C3semidirect
136. #949C22×C3semidirect
137. #948C22×C3semidirect
139. #942C22×C3semidirect
141. C2×#215C22×C3semidirect
142. C2×#213C22×C3semidirect
148. #907C22×C3semidirect
149. #874C22×C3semidirect
153. C3×#104C2×D3semidirect
154. C3×#104C22×C3semidirect
156. C3×#101C2×D3semidirect
157. C3×#101C22×C3semidirect
161. #833C22×C3semidirect
162. #823C22×C3semidirect
163. #822C22×C3semidirect
164. #805C22×C3semidirect
165. #803C22×C3semidirect
204. #454C3×D3semidirect
283. #216C3×D4semidirect
288. #215C3×D4semidirect
289. #214C3×D4semidirect
293. #213C3×D4semidirect
316. 2−1+4C2×C3×D3semidirect
326. AGL(1,ℤ/8)C2×C3×D3semidirect
337. #124C2×C3×D3semidirect
340. #104C2×C3×D3semidirect
341. #101C2×C3×D3semidirect
507. C32#3288semidirect
571. C3#80287direct (N × H)
572. C3C3×#3288semidirect

Maximal subgroups

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

subgroupindex
C3×#58372
C3×#57742
C32×#4542
C3×#51992
C3×#50612
C3×#50582
C3×#48232
C3×#48222
C3×#47542
C3×#32883
#802873