Finite Group Databasebeta

← C3×#80311 #11865340

Subgroups — C3×#80311 #11865340

Order 1152.

1680 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#5653C213yesmaximal, complemented
3C32×#458C211yesmaximal, characteristic, complemented
4C3×#5244C211yesmaximal, characteristic, complemented
5C3×#5053C213yesmaximal, complemented
6C3×D3×#99C213yesmaximal, complemented
7C3×#4741C213yesmaximal, complemented
8C3×#3026—31nomaximal
9#80311C311yesmaximal, characteristic, direct product factor
10C32×H(5,𝔽2)C2211yescharacteristic
11C3×#1009—23no—
12C3×D3×#41—23no—
13C3×#1004—23no—
14C3×#1004C2213yes—
15C2×C3×D3×D4—23no—
16C2×C3×D3×D4C2213yes—
17C22×C3×D12C2211yescharacteristic, complemented
18C32×#123C2213yescomplemented
19C3×#937C2213yescomplemented
20C3×#929C2213yescomplemented
21C3×#895—23no—
22C3×#894—23no—
23C3×#884—23no—
24C3×#882—23no—
25C3×#882C2213yescomplemented
26C3×D3×C22⋊C4C2213yescomplemented
27C32×#99C2213yescomplemented
28C3×#834C2213yescomplemented
29C3×#806C2216yescomplemented
30C2×C3×H(5,𝔽2)—31no—
31D3×H(5,𝔽2)C2×C311yescharacteristic, complemented
32C3×#534—33no—
33#5653C2×C313yescomplemented
34C3×#458—21no—
35C3×#458—33no—
36C3×#458D311yescharacteristic, complemented
37C3×#458C2×C311yescharacteristic, complemented
38C3×B(4,𝔽2)—31no—
39C3×B(4,𝔽2)—33no—
40C2×C3×#99—33no—
41#5244C2×C311yescharacteristic, complemented
42#5053C2×C313yescomplemented
43D3×#99C2×C313yescomplemented
44#4741C2×C313yescomplemented
45C23×C3×D3—21no—
46C23×C3×D3D413yescomplemented
47C23×C3×D3C2311yescharacteristic
48C23×C3×D3C2311yescharacteristic, complemented
49C32×#41—23no—
50C2×C32×D4—23no—
51C2×C32×D4C2313yes—
52C22×C32×C4C2311yescharacteristic, complemented
53C2×C3×#67—23no—
54C2×C3×#67—23no—
55C2×C3×#67—23no—
56C2×C3×#67D413yescomplemented
57C2×C3×#67C2313yes—
58C3×#239—23no—
59C3×D3×Q8—21no—
60C3×#237—23no—
61C3×#237—26no—
62C3×D3×D4—23no—
63C3×D3×D4—26no—
64C3×D3×D4—26no—
65C3×D3×D4—26no—
66C3×D3×D4—26no—
67C3×#235—26no—
68C3×#235—23no—
69C2×C3×D12—23no—
70C2×C3×D12—23no—
71C2×C3×D12—23no—
72C2×C3×D12—23no—
73C2×C3×D12D413yescomplemented
74C2×C3×C4×D3—23no—
75C2×C3×C4×D3—23no—
76C2×C3×C4×D3D413yescomplemented
77C32×C4⋊C4—23no—
78C32×C22⋊C4—23no—
79C32×C22⋊C4C2313yescomplemented
80C3×#217C2313yescomplemented
81C3×(C2×D3)⋊C4—23no—
82C3×(C2×D3)⋊C4—23no—
83C3×(C2×D3)⋊C4—23no—
84C3×(C2×D3)⋊C4—23no—
85C3×Dic3⋊C4—23no—
86#3026—31noSylow
87C3×H(5,𝔽2)—21no—
88C3×H(5,𝔽2)—33no—
89C3×H(5,𝔽2)—63no—
90C3×H(5,𝔽2)—63no—
91C3×H(5,𝔽2)C2×D311yescharacteristic
92C3×H(5,𝔽2)C22×C311yescharacteristic
93C2×C3×#41—63no—
94C22×C3×D4—33no—
95C22×C3×D4—31no—
96C22×C3×D4—63no—
97#1009—23no—
98D3×#41—23no—
99#1004—23no—
100#1004C22×C313yes—
101C2×D3×D4—23no—
102C2×D3×D4C22×C313yes—
103C22×D12C22×C311yescharacteristic, complemented
104C3×#123—23no—
105C3×#123—33no—
106C3×#123—63no—
107C3×#123C2×D313yescomplemented
108C3×#123C22×C313yescomplemented
109C3×#121—63no—
110C3×#121—63no—
111C3×#120—33no—
112C3×#120—33no—
113C3×#120—63no—
114C2×C3×C22⋊C4—33no—
115#937C22×C313yescomplemented
116#929C22×C313yescomplemented
117#895—23no—
118#894—23no—
119#884—23no—
120#882—23no—
121#882C22×C313yescomplemented
122D3×C22⋊C4C22×C313yescomplemented
123C3×#99—23no—
124C3×#99—33no—
125C3×#99—36no—
126C3×#99C2×D313yescomplemented
127C3×#99C22×C313yescomplemented
128#834C22×C313yescomplemented
129#806C22×C316yescomplemented
130C23×C32—21no—
131C23×C32C2×D413yes—
132C23×C32C2411yescharacteristic
133C22×C3×D3—23no—
134C22×C3×D3—23no—
135C22×C3×D3—23no—
136C22×C3×D3—23no—
137C22×C3×D3—81no—
138C22×C3×D3—81no—
139C22×C3×D3—83no—
140C22×C3×D3—82no—
141C22×C3×D3C2×D413yes—
142C22×C3×D3C2×D413yes—
143C22×C3×D3C2×D413yes—
144C32×Q8—21no—
145C32×D4—23no—
146C32×D4—26no—
147C2×C32×C4—23no—
148C2×C32×C4—23no—
149C2×C32×C4—23no—
150C2×C32×C4C2×D413yes—
151C3×#67—26no—
152C3×#67—26no—
153C3×#67—26no—
154C3×#67—83no—
155C2×C3×Dic3—23no—
156C2×C3×Dic3—23no—
157C2×C3×Dic3C2×D413yes—
158C3×D12—23no—
159C3×D12—26no—
160C3×D12—41no—
161C3×D12—43no—
162C3×D12—83no—
163C3×C4×D3—26no—
164C3×C4×D3—23no—
165C3×C4×D3—23no—
166C3×C4×D3—83no—
167C3×Dic6—23no—
168C2×H(5,𝔽2)—31no—
169#534—33no—
170#458—33no—
171#458C3×D311yescharacteristic, complemented
172B(4,𝔽2)—31no—
173B(4,𝔽2)—33no—
174C2×#99—33no—
175C24×C3—31no—
176C24×C3—33no—
177C24×C3—31no—
178C24×C3—61no—
179C23×D3—21no—
180C23×D3C3×D413yescomplemented
181C23×D3C23×C311yescharacteristic
182C23×D3C23×C311yescharacteristic, complemented
183C3×#41—23no—
184C3×#41—23no—
185C3×#41—43no—
186C3×#41—66no—
187C3×#41—66no—
188C3×#41—63no—
189C3×#41—63no—
190C3×#41—63no—
191C2×C3×Q8—61no—
192C2×C3×D4—23no—
193C2×C3×D4—23no—
194C2×C3×D4—23no—
195C2×C3×D4—33no—
196C2×C3×D4—33no—
197C2×C3×D4—33no—
198C2×C3×D4—43no—
199C2×C3×D4—63no—
200C2×C3×D4—63no—
201C2×C3×D4—63no—
202C2×C3×D4—63no—
203C2×C3×D4—66no—
204C2×C3×D4—66no—
205C2×C3×D4—66no—
206C2×C3×D4—63no—
207C2×C3×D4—66no—
208C2×C3×D4—63no—
209C2×C3×D4—63no—
210C2×C3×D4—63no—
211C2×C3×D4C22×D313yes—
212C2×C3×D4C23×C313yes—
213C22×C3×C4—21no—
214C22×C3×C4—33no—
215C22×C3×C4—63no—
216C22×C3×C4—63no—
217C22×C3×C4C22×D311yescharacteristic, complemented
218C22×C3×C4C23×C311yescharacteristic, complemented
219C2×#67—23no—
220C2×#67—23no—
221C2×#67—23no—
222C2×#67C3×D413yescomplemented
223C2×#67C23×C313yes—
224#239—23no—
225D3×Q8—21no—
226#237—23no—
227#237—26no—
228D3×D4—23no—
229D3×D4—26no—
230D3×D4—26no—
231D3×D4—26no—
232D3×D4—26no—
233#235—26no—
234#235—23no—
235C2×D12—23no—
236C2×D12—23no—
237C2×D12—23no—
238C2×D12—23no—
239C2×D12C3×D413yescomplemented
240C2×C4×D3—23no—
241C2×C4×D3—23no—
242C2×C4×D3C3×D413yescomplemented
243C3×C4⋊C4—23no—
244C3×C4⋊C4—23no—
245C3×C4⋊C4—43no—
246C3×C4⋊C4—63no—
247C3×C22⋊C4—23no—
248C3×C22⋊C4—23no—
249C3×C22⋊C4—23no—
250C3×C22⋊C4—33no—
251C3×C22⋊C4—43no—
252C3×C22⋊C4—63no—
253C3×C22⋊C4—63no—
254C3×C22⋊C4—63no—
255C3×C22⋊C4—63no—
256C3×C22⋊C4C22×D313yescomplemented
257C3×C22⋊C4C23×C313yescomplemented
258#217C23×C313yescomplemented
259(C2×D3)⋊C4—23no—
260(C2×D3)⋊C4—23no—
261(C2×D3)⋊C4—23no—
262(C2×D3)⋊C4—23no—
263Dic3⋊C4—23no—
264C22×C32—23no—
265C22×C32—41no—
266C22×C32—43no—
267C22×C32—81no—
268C22×C32C22×D413yes—
269C2×C3×D3—23no—
270C2×C3×D3—43no—
271C2×C3×D3—43no—
272C2×C3×D3—46no—
273C2×C3×D3—43no—
274C2×C3×D3—43no—
275C2×C3×D3—43no—
276C2×C3×D3—41no—
277C2×C3×D3—83no—
278C2×C3×D3—83no—
279C2×C3×D3#12011yescharacteristic
280C2×C3×D3#12013yes—
281C32×C4—23no—
282C32×C4—43no—
283C32×C4—41no—
284C3×Dic3—23no—
285C3×Dic3—43no—
286H(5,𝔽2)—33no—
287H(5,𝔽2)—63no—
288H(5,𝔽2)—63no—
289H(5,𝔽2)C2×C3×D311yescharacteristic
290C2×#41—63no—
291C22×D4—33no—
292C22×D4—31no—
293C22×D4—63no—
294#123—33no—
295#123—63no—
296#123C2×C3×D313yescomplemented
297#121—63no—
298#121—63no—
299#120—33no—
300#120—33no—
301#120—63no—
302C2×C22⋊C4—33no—
303#99—33no—
304#99—36no—
305#99C2×C3×D313yescomplemented
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—23no—
310C23×C3—33no—
311C23×C3—33no—
312C23×C3—33no—
313C23×C3—41no—
314C23×C3—63no—
315C23×C3—63no—
316C23×C3—63no—
317C23×C3—63no—
318C23×C3—241no—
319C23×C3—241no—
320C23×C3—243no—
321C23×C3—242no—
322C23×C3D3×D413yes—
323C23×C3C2×C3×D413yes—
324C23×C3C23×D311yescharacteristic
325C23×C3C24×C311yescharacteristic
326C22×D3—23no—
327C22×D3—23no—
328C22×D3—23no—
329C22×D3—23no—
330C22×D3—81no—
331C22×D3—81no—
332C22×D3—83no—
333C22×D3—82no—
334C22×D3C2×C3×D413yes—
335C22×D3C2×C3×D413yes—
336C22×D3C2×C3×D413yes—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8—41no—
340C3×Q8—63no—
341C3×D4—23no—
342C3×D4—23no—
343C3×D4—26no—
344C3×D4—26no—
345C3×D4—46no—
346C3×D4—43no—
347C3×D4—66no—
348C3×D4—63no—
349C3×D4—66no—
350C3×D4—66no—
351C3×D4—66no—
352C3×D4—121no—
353C3×D4—123no—
354C3×D4—243no—
355C3×D4—243no—
356C2×C3×C4—23no—
357C2×C3×C4—23no—
358C2×C3×C4—23no—
359C2×C3×C4—23no—
360C2×C3×C4—23no—
361C2×C3×C4—23no—
362C2×C3×C4—23no—
363C2×C3×C4—33no—
364C2×C3×C4—43no—
365C2×C3×C4—43no—
366C2×C3×C4—43no—
367C2×C3×C4—63no—
368C2×C3×C4—63no—
369C2×C3×C4—66no—
370C2×C3×C4—63no—
371C2×C3×C4—63no—
372C2×C3×C4—243no—
373C2×C3×C4D3×D413yes—
374C2×C3×C4C2×C3×D413yes—
375#67—26no—
376#67—26no—
377#67—26no—
378#67—83no—
379C2×Dic3—23no—
380C2×Dic3—23no—
381C2×Dic3C2×C3×D413yes—
382D12—23no—
383D12—26no—
384D12—41no—
385D12—43no—
386D12—83no—
387C4×D3—26no—
388C4×D3—23no—
389C4×D3—23no—
390C4×D3—83no—
391Dic6—23no—
392C2×C32—23no—
393C2×C32—43no—
394C2×C32C2×#12011yescharacteristic
395C3×D3—21no—
396C3×D3—23no—
397C3×D3—43no—
398C3×D3—81no—
399C24—31no—
400C24—33no—
401C24—31no—
402C24—61no—
403#41—23no—
404#41—66no—
405#41—66no—
406#41—63no—
407#41—63no—
408#41—63no—
409C2×Q8—61no—
410C2×D4—23no—
411C2×D4—33no—
412C2×D4—33no—
413C2×D4—33no—
414C2×D4—63no—
415C2×D4—63no—
416C2×D4—63no—
417C2×D4—63no—
418C2×D4—66no—
419C2×D4—66no—
420C2×D4—66no—
421C2×D4—63no—
422C2×D4—66no—
423C2×D4—63no—
424C2×D4—63no—
425C2×D4—63no—
426C2×D4C22×C3×D313yes—
427C22×C4—33no—
428C22×C4—63no—
429C22×C4—63no—
430C22×C4C22×C3×D311yescharacteristic, complemented
431C4⋊C4—23no—
432C4⋊C4—63no—
433C22⋊C4—23no—
434C22⋊C4—33no—
435C22⋊C4—63no—
436C22⋊C4—63no—
437C22⋊C4—63no—
438C22⋊C4—63no—
439C22⋊C4C22×C3×D313yescomplemented
440C22×C3—23no—
441C22×C3—23no—
442C22×C3—23no—
443C22×C3—31no—
444C22×C3—33no—
445C22×C3—43no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—43no—
449C22×C3—43no—
450C22×C3—63no—
451C22×C3—81no—
452C22×C3—83no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—123no—
456C22×C3—123no—
457C22×C3—123no—
458C22×C3—123no—
459C22×C3—123no—
460C22×C3—126no—
461C22×C3—121no—
462C22×C3—161no—
463C22×C3—243no—
464C22×C3—243no—
465C22×C3C2×D3×D413yes—
466C22×C3C22×C3×D413yes—
467C2×D3—23no—
468C2×D3—43no—
469C2×D3—43no—
470C2×D3—46no—
471C2×D3—43no—
472C2×D3—43no—
473C2×D3—43no—
474C2×D3—41no—
475C2×D3—83no—
476C2×D3—83no—
477C2×D3C3×#12011yescharacteristic
478C2×D3C3×#12013yes—
479C3×C4—23no—
480C3×C4—23no—
481C3×C4—43no—
482C3×C4—43no—
483C3×C4—43no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—63no—
487C3×C4—81no—
488C3×C4—83no—
489C3×C4—123no—
490Dic3—23no—
491Dic3—43no—
492C32#302611yescharacteristic, complemented, Sylow
493C23—21no—
494C23—33no—
495C23—33no—
496C23—33no—
497C23—63no—
498C23—63no—
499C23—63no—
500C23—63no—
501C23—241no—
502C23—241no—
503C23—243no—
504C23—242no—
505C23C3×D3×D413yes—
506C23C23×C3×D311yescharacteristic
507Q8—21no—
508Q8—63no—
509D4—23no—
510D4—26no—
511D4—66no—
512D4—63no—
513D4—66no—
514D4—66no—
515D4—66no—
516D4—121no—
517D4—123no—
518D4—243no—
519D4—243no—
520C2×C4—23no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—33no—
524C2×C4—63no—
525C2×C4—63no—
526C2×C4—66no—
527C2×C4—63no—
528C2×C4—63no—
529C2×C4—243no—
530C2×C4C3×D3×D413yes—
531C2×C3—23no—
532C2×C3—23no—
533C2×C3—21no—
534C2×C3—43no—
535C2×C3—43no—
536C2×C3—43no—
537C2×C3—61no—
538C2×C3—63no—
539C2×C3—83no—
540C2×C3—123no—
541C2×C3—241no—
542C2×C3D3×#12011yescharacteristic
543C2×C3C2×C3×#12011yescharacteristic
544D3—21no—
545D3—23no—
546D3—43no—
547D3—81no—
548C22—23no—
549C22—31no—
550C22—33no—
551C22—41no—
552C22—43no—
553C22—63no—
554C22—81no—
555C22—123no—
556C22—123no—
557C22—123no—
558C22—123no—
559C22—123no—
560C22—126no—
561C22—121no—
562C22—243no—
563C22—243no—
564C22C2×C3×D3×D413yes—
565C4—23no—
566C4—43no—
567C4—41no—
568C4—63no—
569C4—123no—
570C3—21no—
571C3#8031111yescharacteristic, direct product factor
572C3C3×#302611yescharacteristic, complemented
573C2—23no—
574C2—43no—
575C2—61no—
576C2—63no—
577C2—123no—
578C2—241no—
579C2C3×D3×#12011yescharacteristic
subgroupG/H
centerC2×C3D3×#120
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC32×#458C2
socleC2×C32C2×#120
hypercenterC3×#458D3
hypocenterC3C3×#3026
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#458

Complements

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

NH ≅ G/Ntype
1. C3×D3×H(5,𝔽2)C2semidirect
2. C3×#5653C2semidirect
3. C32×#458C2semidirect
4. C3×#5244C2semidirect
5. C3×#5053C2semidirect
6. C3×D3×#99C2semidirect
7. C3×#4741C2semidirect
9. #80311C3direct (N × H)
17. C22×C3×D12C22semidirect
18. C32×#123C22semidirect
19. C3×#937C22semidirect
20. C3×#929C22semidirect
25. C3×#882C22semidirect
26. C3×D3×C22⋊C4C22semidirect
27. C32×#99C22semidirect
28. C3×#834C22semidirect
29. C3×#806C22semidirect
31. D3×H(5,𝔽2)C2×C3semidirect
33. #5653C2×C3semidirect
36. C3×#458D3semidirect
37. C3×#458C2×C3semidirect
41. #5244C2×C3semidirect
42. #5053C2×C3semidirect
43. D3×#99C2×C3semidirect
44. #4741C2×C3semidirect
46. C23×C3×D3D4semidirect
48. C23×C3×D3C23semidirect
52. C22×C32×C4C23semidirect
56. C2×C3×#67D4semidirect
73. C2×C3×D12D4semidirect
76. C2×C3×C4×D3D4semidirect
79. C32×C22⋊C4C23semidirect
80. C3×#217C23semidirect
103. C22×D12C22×C3semidirect
107. C3×#123C2×D3semidirect
108. C3×#123C22×C3semidirect
115. #937C22×C3semidirect
116. #929C22×C3semidirect
121. #882C22×C3semidirect
122. D3×C22⋊C4C22×C3semidirect
126. C3×#99C2×D3semidirect
127. C3×#99C22×C3semidirect
128. #834C22×C3semidirect
129. #806C22×C3semidirect
171. #458C3×D3semidirect
180. C23×D3C3×D4semidirect
182. C23×D3C23×C3semidirect
217. C22×C3×C4C22×D3semidirect
218. C22×C3×C4C23×C3semidirect
222. C2×#67C3×D4semidirect
239. C2×D12C3×D4semidirect
242. C2×C4×D3C3×D4semidirect
256. C3×C22⋊C4C22×D3semidirect
257. C3×C22⋊C4C23×C3semidirect
258. #217C23×C3semidirect
296. #123C2×C3×D3semidirect
305. #99C2×C3×D3semidirect
430. C22×C4C22×C3×D3semidirect
439. C22⋊C4C22×C3×D3semidirect
492. C32#3026semidirect
571. C3#80311direct (N × H)
572. C3C3×#3026semidirect

Maximal subgroups

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

subgroupindex
C3×D3×H(5,𝔽2)2
C3×#56532
C32×#4582
C3×#52442
C3×#50532
C3×D3×#992
C3×#47412
C3×#30263
#803113