Finite Group Databasebeta

← C3×#80282 #11886893

Subgroups — C3×#80282 #11886893

Order 1152.

1036 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5835C212yesmaximal, complemented
2C3×#5774C212yesmaximal, complemented
3C32×#453C211yesmaximal, characteristic, complemented
4C3×#5199C212yesmaximal, complemented
5C3×#5077C211yesmaximal, characteristic, complemented
6C3×#5073C211yesmaximal, characteristic, complemented
7C3×#4824C212yesmaximal, complemented
8C3×#4822C212yesmaximal, complemented
9C3×#4749C212yesmaximal, complemented
10C3×#3292—31nomaximal
11#80282C311yesmaximal, characteristic, direct product factor
12C32×H(5,𝔽2)C2211yescharacteristic, complemented
13C3×#1009C2212yescomplemented
14C3×#1004C2211yescharacteristic, complemented
15C32×AGL(1,ℤ/8)C2212yescomplemented
16C32×#127C2211yescharacteristic, complemented
17C3×#950—22no—
18C3×#950C2212yes—
19C3×#949—22no—
20C3×#949C2212yescomplemented
21C3×#948C2212yescomplemented
22C3×#932—22no—
23C3×#932C2212yescomplemented
24C2×C3×#213—22no—
25C2×C3×#213C2212yescomplemented
26C2×C3×#213C2212yescomplemented
27C3×#916—22no—
28C3×#914—22no—
29C3×#911—22no—
30C3×D3×D8—22no—
31C3×#908C2212yescomplemented
32C3×#907C2212yescomplemented
33C3×#874C2211yescharacteristic, complemented
34C32×#104C2212yescomplemented
35C32×#100C2211yescharacteristic, complemented
36C3×#833C2211yescharacteristic, complemented
37C3×#823C2212yescomplemented
38C3×#822C2212yescomplemented
39C3×#805C2212yescomplemented
40C3×C4×C3⋊C8C2211yescharacteristic, complemented
41C3×#576—32no—
42C3×#576—32no—
43#5835C2×C312yescomplemented
44#5774C2×C312yescomplemented
45C3×#493—31no—
46C3×#472—32no—
47C3×#472—32no—
48C3×#453—21no—
49C3×#453—31no—
50C3×#453—32no—
51C3×#453D311yescharacteristic, complemented
52C3×#453C2×C311yescharacteristic, complemented
53C3×#443—32no—
54#5199C2×C312yescomplemented
55#5077C2×C311yescharacteristic, complemented
56#5073C2×C311yescharacteristic, complemented
57#4824C2×C312yescomplemented
58#4822C2×C312yescomplemented
59#4749C2×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×#239—22no—
69C3×#237—22no—
70C3×D3×D4—22no—
71C3×D3×D4—22no—
72C3×D3×D4—42no—
73C3×#235—22no—
74C3×#235C2312yes—
75C2×C3×D12—22no—
76C2×C3×D12—41no—
77C2×C3×D12C2312yes—
78C32×SD16—22no—
79C32×D8—22no—
80C32×M16C2312yes—
81C32×C42C2311yescharacteristic
82C3×#216D412yescomplemented
83C3×#215—22no—
84C3×#215D412yescomplemented
85C3×#214—22no—
86C3×#214D412yescomplemented
87C3×#213—22no—
88C3×#213—22no—
89C3×#213—22no—
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#3292—31noSylow
103C3×H(5,𝔽2)—21no—
104C3×H(5,𝔽2)—32no—
105C3×H(5,𝔽2)—31no—
106C3×H(5,𝔽2)C2×D311yescharacteristic, complemented
107C3×H(5,𝔽2)C22×C311yescharacteristic, complemented
108#1009C22×C312yescomplemented
109#1004C22×C311yescharacteristic, complemented
110C3×AGL(1,ℤ/8)—22no—
111C3×AGL(1,ℤ/8)—32no—
112C3×AGL(1,ℤ/8)—32no—
113C3×AGL(1,ℤ/8)—32no—
114C3×AGL(1,ℤ/8)—62no—
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×D8—32no—
124C2×C3×D8—32no—
125C2×C3×D8—62no—
126C2×C3×D8—62no—
127C3×#131—32no—
128C3×#131—32no—
129C3×#127—21no—
130C3×#127—31no—
131C3×#127C2×D311yescharacteristic, complemented
132C3×#127C22×C311yescharacteristic, complemented
133#950—22no—
134#950C22×C312yes—
135#949—22no—
136#949C22×C312yescomplemented
137#948C22×C312yescomplemented
138#932—22no—
139#932C22×C312yescomplemented
140C2×#213—22no—
141C2×#213C22×C312yescomplemented
142C2×#213C22×C312yescomplemented
143#916—22no—
144#914—22no—
145#911—22no—
146D3×D8—22no—
147#908C22×C312yescomplemented
148#907C22×C312yescomplemented
149#874C22×C311yescharacteristic, complemented
150C3×#108—32no—
151C3×#104—22no—
152C3×#104—32no—
153C3×#104C2×D312yescomplemented
154C3×#104C22×C312yescomplemented
155C3×#100—21no—
156C3×#100—31no—
157C3×#100—32no—
158C3×#100C2×D311yescharacteristic, complemented
159C3×#100C22×C311yescharacteristic, complemented
160C3×C4×C8—31no—
161#833C22×C311yescharacteristic, complemented
162#823C22×C312yescomplemented
163#822C22×C312yescomplemented
164#805C22×C312yescomplemented
165C4×C3⋊C8C22×C311yescharacteristic, complemented
166C23×C32—21no—
167C23×C32—21no—
168C23×C32—41no—
169C22×C3×D3—22no—
170C22×C3×D3—41no—
171C22×C3×D3—42no—
172C32×Q8C2×D412yes—
173C32×D4—22no—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—42no—
177C32×D4—82no—
178C32×D4C2×D412yes—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4—41no—
182C2×C32×C4C2411yescharacteristic
183C3×#67—22no—
184C3×#67—42no—
185C3×#67—42no—
186C2×C3×Dic3—41no—
187C3×D12—22no—
188C3×D12—22no—
189C3×D12—42no—
190C3×D12—82no—
191C3×D12C2×D412yes—
192C3×C4×D3—22no—
193C3×C4×D3—42no—
194C3×Dic6C2×D412yes—
195C32×C8—22no—
196C3×C3⋊C8—22no—
197C3×C3⋊C8C2×D412yes—
198C3×C3⋊C8C2×D412yes—
199#576—32no—
200#576—32no—
201#493—31no—
202#472—32no—
203#472—32no—
204#453—31no—
205#453—32no—
206#453C3×D311yescharacteristic, complemented
207#443—32no—
208C3×#41—22no—
209C3×#41—22no—
210C3×#41—22no—
211C3×#41—32no—
212C3×#41—42no—
213C3×#41—62no—
214C3×#41—62no—
215C3×#41—62no—
216C3×#41C22×D312yes—
217C3×#41C23×C312yes—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—32no—
223C2×C3×D4—42no—
224C2×C3×D4—41no—
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4—41no—
228C2×C3×D4—42no—
229C2×C3×D4—42no—
230C2×C3×D4—62no—
231C2×C3×D4—62no—
232C2×C3×D4—62no—
233C2×C3×D4—121no—
234C2×C3×D4—122no—
235C2×C3×D4—121no—
236C2×C3×D4C22×D311yescharacteristic
237C2×C3×D4C22×D311yescharacteristic
238C2×C3×D4C23×C311yescharacteristic
239C2×C3×D4C23×C311yescharacteristic
240C2×#67—41no—
241#239—22no—
242#237—22no—
243D3×D4—22no—
244D3×D4—22no—
245D3×D4—42no—
246#235—22no—
247#235C23×C312yes—
248C2×D12—22no—
249C2×D12—41no—
250C2×D12C23×C312yes—
251C3×Q16—32no—
252C3×SD16—22no—
253C3×SD16—22no—
254C3×SD16—32no—
255C3×SD16—32no—
256C3×SD16—42no—
257C3×SD16—62no—
258C3×SD16—62no—
259C3×SD16—62no—
260C3×D8—22no—
261C3×D8—22no—
262C3×D8—32no—
263C3×D8—42no—
264C3×D8—62no—
265C3×D8—62no—
266C3×D8—62no—
267C3×D8—62no—
268C3×D8—62no—
269C3×D8—62no—
270C3×D8—62no—
271C3×M16—22no—
272C3×M16—32no—
273C3×M16—62no—
274C3×M16—62no—
275C3×M16C22×D312yes—
276C3×M16C23×C312yes—
277C2×C3×C8—32no—
278C2×C3×C8—62no—
279C2×C3×C8—62no—
280C3×C42—21no—
281C3×C42C22×D311yescharacteristic
282C3×C42C23×C311yescharacteristic
283#216C3×D412yescomplemented
284#215—22no—
285#215C3×D412yescomplemented
286#214—22no—
287#214C3×D412yescomplemented
288#213—22no—
289#213—22no—
290#213—22no—
291#213—22no—
292#213—22no—
293#213—22no—
294#213C3×D412yescomplemented
295#208—22no—
296#208C23×C312yes—
297C2×C3⋊C8—22no—
298C2×C3⋊C8C23×C312yes—
299D24—22no—
300#204—22no—
301ΓL(1,𝔽25)—22no—
302D3×C8—22no—
303C22×C32—22no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—81no—
307C22×C32—81no—
308C22×C32—161no—
309C22×C32H(5,𝔽2)11yescharacteristic
310C2×C3×D3—22no—
311C2×C3×D3—42no—
312C2×C3×D3—82no—
313C2×C3×D3—161no—
314C2×C3×D3—162no—
315C32×C4—22no—
316C32×C4—22no—
317C32×C4C22×D412yes—
318C3×Dic3—22no—
319H(5,𝔽2)—32no—
320H(5,𝔽2)—31no—
321H(5,𝔽2)C2×C3×D311yescharacteristic, complemented
322AGL(1,ℤ/8)—32no—
323AGL(1,ℤ/8)—32no—
324AGL(1,ℤ/8)—32no—
325AGL(1,ℤ/8)—62no—
326AGL(1,ℤ/8)—62no—
327AGL(1,ℤ/8)—62no—
328AGL(1,ℤ/8)—62no—
329AGL(1,ℤ/8)C2×C3×D312yescomplemented
330#135—32no—
331#135—62no—
332#135—62no—
333C2×D8—32no—
334C2×D8—32no—
335C2×D8—62no—
336C2×D8—62no—
337#131—32no—
338#131—32no—
339#127—31no—
340#127C2×C3×D311yescharacteristic, complemented
341#108—32no—
342#104—32no—
343#104C2×C3×D312yescomplemented
344#100—31no—
345#100—32no—
346#100C2×C3×D311yescharacteristic, complemented
347C4×C8—31no—
348C23×C3—22no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—42no—
357C23×C3—62no—
358C23×C3—122no—
359C23×C3—121no—
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—42no—
363C3×Q8—22no—
364C3×Q8—32no—
365C3×Q8D3×D412yes—
366C3×Q8C2×C3×D412yes—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—32no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—42no—
379C3×D4—42no—
380C3×D4—44no—
381C3×D4—62no—
382C3×D4—62no—
383C3×D4—62no—
384C3×D4—82no—
385C3×D4—82no—
386C3×D4—84no—
387C3×D4—122no—
388C3×D4—122no—
389C3×D4—122no—
390C3×D4—242no—
391C3×D4D3×D412yes—
392C3×D4C2×C3×D412yes—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—42no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—42no—
403C2×C3×C4—62no—
404C2×C3×C4—122no—
405C2×C3×C4—121no—
406C2×C3×C4C23×D311yescharacteristic
407C2×C3×C4C24×C311yescharacteristic
408#67—22no—
409#67—42no—
410#67—42no—
411C2×Dic3—41no—
412D12—22no—
413D12—22no—
414D12—42no—
415D12—82no—
416D12C2×C3×D412yes—
417C4×D3—22no—
418C4×D3—42no—
419Dic6C2×C3×D412yes—
420C3×C8—22no—
421C3×C8—22no—
422C3×C8—32no—
423C3×C8—32no—
424C3×C8—42no—
425C3×C8—62no—
426C3⋊C8—22no—
427C3⋊C8C2×C3×D412yes—
428C3⋊C8C2×C3×D412yes—
429C2×C32—21no—
430C2×C32—42no—
431C2×C32—81no—
432C2×C32—81no—
433C2×C32D4211yescharacteristic
434C3×D3—42no—
435C3×D3—82no—
436#41—22no—
437#41—32no—
438#41—62no—
439#41—62no—
440#41—62no—
441#41C22×C3×D312yes—
442C2×D4—22no—
443C2×D4—32no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—62no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—121no—
450C2×D4—122no—
451C2×D4—121no—
452C2×D4C22×C3×D311yescharacteristic
453C2×D4C22×C3×D311yescharacteristic
454Q16—32no—
455SD16—22no—
456SD16—32no—
457SD16—32no—
458SD16—62no—
459SD16—62no—
460SD16—62no—
461D8—22no—
462D8—32no—
463D8—62no—
464D8—62no—
465D8—62no—
466D8—62no—
467D8—62no—
468D8—62no—
469D8—62no—
470M16—32no—
471M16—62no—
472M16—62no—
473M16C22×C3×D312yes—
474C2×C8—32no—
475C2×C8—62no—
476C2×C8—62no—
477C42C22×C3×D311yescharacteristic
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—62no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—122no—
496C22×C3—161no—
497C22×C3—161no—
498C22×C3—162no—
499C22×C3—242no—
500C22×C3—482no—
501C22×C3—481no—
502C22×C3#100411yescharacteristic
503C22×C3C3×H(5,𝔽2)11yescharacteristic
504C2×D3—22no—
505C2×D3—42no—
506C2×D3—82no—
507C2×D3—161no—
508C2×D3—162no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—22no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—62no—
517C3×C4C2×D3×D412yes—
518C3×C4C22×C3×D412yes—
519Dic3—22no—
520C32#329211yescharacteristic, complemented, Sylow
521C23—21no—
522C23—21no—
523C23—41no—
524C23—62no—
525C23—122no—
526C23—121no—
527Q8—32no—
528Q8C3×D3×D412yes—
529D4—22no—
530D4—22no—
531D4—22no—
532D4—32no—
533D4—42no—
534D4—62no—
535D4—62no—
536D4—62no—
537D4—82no—
538D4—122no—
539D4—122no—
540D4—122no—
541D4—242no—
542D4C3×D3×D412yes—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—62no—
547C2×C4—122no—
548C2×C4—121no—
549C2×C4C23×C3×D311yescharacteristic
550C8—22no—
551C8—32no—
552C8—32no—
553C8—62no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—41no—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3—82no—
561C2×C3—82no—
562C2×C3—81no—
563C2×C3—81no—
564C2×C3—81no—
565C2×C3—81no—
566C2×C3—122no—
567C2×C3—161no—
568C2×C3—242no—
569C2×C3#559011yescharacteristic
570C2×C3C3×D4211yescharacteristic
571D3—42no—
572D3—82no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—62no—
577C22—81no—
578C22—81no—
579C22—122no—
580C22—161no—
581C22—242no—
582C22—482no—
583C22—481no—
584C22C3×#100411yescharacteristic
585C4—22no—
586C4—22no—
587C4—62no—
588C4C2×C3×D3×D412yes—
589C3—21no—
590C3#8028211yescharacteristic, direct product factor
591C3C3×#329211yescharacteristic, complemented
592C2—21no—
593C2—42no—
594C2—81no—
595C2—81no—
596C2—122no—
597C2—242no—
598C2C3×#559011yescharacteristic
subgroupG/H
centerC2×C3#5590
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#453C2
socleC2×C32D42
hypercenterC3×#453D3
hypocenterC3C3×#3292
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×#5835C2semidirect
2. C3×#5774C2semidirect
3. C32×#453C2semidirect
4. C3×#5199C2semidirect
5. C3×#5077C2semidirect
6. C3×#5073C2semidirect
7. C3×#4824C2semidirect
8. C3×#4822C2semidirect
9. C3×#4749C2semidirect
11. #80282C3direct (N × H)
12. C32×H(5,𝔽2)C22semidirect
13. C3×#1009C22semidirect
14. C3×#1004C22semidirect
15. C32×AGL(1,ℤ/8)C22semidirect
16. C32×#127C22semidirect
20. C3×#949C22semidirect
21. C3×#948C22semidirect
23. C3×#932C22semidirect
25. C2×C3×#213C22semidirect
26. C2×C3×#213C22semidirect
31. C3×#908C22semidirect
32. C3×#907C22semidirect
33. C3×#874C22semidirect
34. C32×#104C22semidirect
35. C32×#100C22semidirect
36. C3×#833C22semidirect
37. C3×#823C22semidirect
38. C3×#822C22semidirect
39. C3×#805C22semidirect
40. C3×C4×C3⋊C8C22semidirect
43. #5835C2×C3semidirect
44. #5774C2×C3semidirect
51. C3×#453D3semidirect
52. C3×#453C2×C3semidirect
54. #5199C2×C3semidirect
55. #5077C2×C3semidirect
56. #5073C2×C3semidirect
57. #4824C2×C3semidirect
58. #4822C2×C3semidirect
59. #4749C2×C3semidirect
82. C3×#216D4semidirect
84. C3×#215D4semidirect
86. C3×#214D4semidirect
93. C3×#213D4semidirect
106. C3×H(5,𝔽2)C2×D3semidirect
107. C3×H(5,𝔽2)C22×C3semidirect
108. #1009C22×C3semidirect
109. #1004C22×C3semidirect
118. C3×AGL(1,ℤ/8)C2×D3semidirect
119. C3×AGL(1,ℤ/8)C22×C3semidirect
131. C3×#127C2×D3semidirect
132. C3×#127C22×C3semidirect
136. #949C22×C3semidirect
137. #948C22×C3semidirect
139. #932C22×C3semidirect
141. C2×#213C22×C3semidirect
142. C2×#213C22×C3semidirect
147. #908C22×C3semidirect
148. #907C22×C3semidirect
149. #874C22×C3semidirect
153. C3×#104C2×D3semidirect
154. C3×#104C22×C3semidirect
158. C3×#100C2×D3semidirect
159. C3×#100C22×C3semidirect
161. #833C22×C3semidirect
162. #823C22×C3semidirect
163. #822C22×C3semidirect
164. #805C22×C3semidirect
165. C4×C3⋊C8C22×C3semidirect
206. #453C3×D3semidirect
283. #216C3×D4semidirect
285. #215C3×D4semidirect
287. #214C3×D4semidirect
294. #213C3×D4semidirect
321. H(5,𝔽2)C2×C3×D3semidirect
329. AGL(1,ℤ/8)C2×C3×D3semidirect
340. #127C2×C3×D3semidirect
343. #104C2×C3×D3semidirect
346. #100C2×C3×D3semidirect
520. C32#3292semidirect
590. C3#80282direct (N × H)
591. C3C3×#3292semidirect

Maximal subgroups

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

subgroupindex
C3×#58352
C3×#57742
C32×#4532
C3×#51992
C3×#50772
C3×#50732
C3×#48242
C3×#48222
C3×#47492
C3×#32923
#802823