Finite Group Databasebeta

← C3×#81131 #11882319

Subgroups — C3×#81131 #11882319

Order 1152.

844 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5725C211yesmaximal, characteristic, complemented
2C3×#5580C211yesmaximal, characteristic, complemented
3C32×#477C211yesmaximal, characteristic, complemented
4C3×#5084C211yesmaximal, characteristic, complemented
5C3×#5027C211yesmaximal, characteristic, complemented
6C3×#4891C212yesmaximal, complemented
7C3×#4883C212yesmaximal, complemented
8C3×#4843C211yesmaximal, characteristic, complemented
9C3×D3×C4⋊C8C211yesmaximal, characteristic, complemented
10C3×#4819C212yesmaximal, complemented
11C3×#4807C212yesmaximal, complemented
12C3×#3247—31nomaximal
13#81131C311yesmaximal, characteristic, direct product factor
14C2×C3×#239—21no—
15C32×C4⋊Q8C2211yescharacteristic
16C32×C4×Q8C2211yescharacteristic
17C3×#947—21no—
18C3×#947C2211yescharacteristic
19C3×#946C2211yescharacteristic
20C2×C3×#215C2212yes—
21C2×C3×D24C2212yes—
22C2×C3×D3×C8C2212yes—
23C3×#898—21no—
24C3×#895—21no—
25C3×#895C2211yescharacteristic
26C3×#895C2212yes—
27C3×#894—21no—
28C3×#893—21no—
29C3×#893C2212yes—
30C3×#890C2211yescharacteristic
31C3×#874C2211yescharacteristic
32C3×C4×D12—21no—
33C3×C4×D12C2211yescharacteristic
34C3×#872—21no—
35C3×C42×D3C2211yescharacteristic
36C32×#107C2212yes—
37C32×C4⋊C8C2211yescharacteristic
38C32×#103C2212yes—
39C3×#821C2212yes—
40C3×Dic3⋊C8C2212yes—
41C3×#809C2212yes—
42C3×#809C2212yes—
43C3×#808C2212yes—
44C3×C4⋊(C3⋊C8)C2211yescharacteristic
45C3×#532—31no—
46C3×#529—31no—
47#5725C2×C311yescharacteristic, complemented
48#5580C2×C311yescharacteristic, complemented
49C3×#478—32no—
50C3×#477—21no—
51C3×#477D311yescharacteristic, complemented
52C3×#477C2×C311yescharacteristic, complemented
53C3×#476—31no—
54C3×#466—32no—
55C3×#465—32no—
56C3×#463—32no—
57C3×#460—31no—
58C3×#459—31no—
59C2×C3×C4⋊C8—31no—
60#5084C2×C311yescharacteristic, complemented
61#5027C2×C311yescharacteristic, complemented
62#4891C2×C312yescomplemented
63#4883C2×C312yescomplemented
64#4843C2×C311yescharacteristic, complemented
65D3×C4⋊C8C2×C311yescharacteristic, complemented
66#4819C2×C312yescomplemented
67#4807C2×C312yescomplemented
68C2×C32×Q8—21no—
69C2×C32×Q8C2311yescharacteristic
70C3×#239—44no—
71C2×C3×D12—21no—
72C2×C3×D12—22no—
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12C2312yes—
76C2×C3×D12C2311yescharacteristic
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3D411yescharacteristic
80C2×C3×C4×D3D411yescharacteristic
81C2×C3×C4×D3C2311yescharacteristic
82C2×C32×C8C2312yes—
83C32×C4⋊C4—21no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2311yescharacteristic
86C32×C4⋊C4C2312yes—
87C32×C42—21no—
88C32×C42C2311yescharacteristic
89C3×#215—42no—
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4—22no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—21no—
95C3×(C2×D3)⋊C4—21no—
96C3×(C2×D3)⋊C4—21no—
97C3×C4⋊Dic3—21no—
98C3×C4⋊Dic3C2311yescharacteristic
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3D411yescharacteristic
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3C2311yescharacteristic
104C2×C3×C3⋊C8C2312yes—
105C3×D24—42no—
106C3×D3×C8—42no—
107#3247—31noSylow
108C2×C3×#41—61no—
109C2×#239—21no—
110C2×C3×SD16—32no—
111C2×C3×D8—32no—
112C22×C3×C8—32no—
113C3×C4⋊Q8—21no—
114C3×C4⋊Q8C2×D311yescharacteristic
115C3×C4⋊Q8C22×C311yescharacteristic
116C3×#127—31no—
117C3×#126—61no—
118C3×#125—31no—
119C3×#124—31no—
120C3×#124—61no—
121C3×#123—61no—
122C3×C22⋊Q8—31no—
123C3×#121—32no—
124C3×#121—31no—
125C3×#121—61no—
126C3×C4×Q8—21no—
127C3×C4×Q8C2×D311yescharacteristic
128C3×C4×Q8C22×C311yescharacteristic
129C3×C4×D4—32no—
130C3×C4×D4—31no—
131C3×C4×D4—61no—
132C3×C4×D4—61no—
133C3×#117—61no—
134C2×C3×C42—31no—
135#947—21no—
136#947C22×C311yescharacteristic
137#946C22×C311yescharacteristic
138C2×#215C22×C312yes—
139C2×D24C22×C312yes—
140C2×D3×C8C22×C312yes—
141#898—21no—
142#895—21no—
143#895C22×C311yescharacteristic
144#895C22×C312yes—
145#894—21no—
146#893—21no—
147#893C22×C312yes—
148#890C22×C311yescharacteristic
149#874C22×C311yescharacteristic
150C4×D12—21no—
151C4×D12C22×C311yescharacteristic
152#872—21no—
153C42×D3C22×C311yescharacteristic
154C3×#107—22no—
155C3×#107C2×D312yes—
156C3×#107C22×C312yes—
157C3×#106—32no—
158C3×C4⋊C8—21no—
159C3×C4⋊C8—32no—
160C3×C4⋊C8—31no—
161C3×C4⋊C8C2×D311yescharacteristic
162C3×C4⋊C8C22×C311yescharacteristic
163C3×#103—22no—
164C3×#103C2×D312yes—
165C3×#103C22×C312yes—
166C3×#102—32no—
167C3×#102—32no—
168C3×#102—32no—
169#821C22×C312yes—
170Dic3⋊C8C22×C312yes—
171#809C22×C312yes—
172#809C22×C312yes—
173#808C22×C312yes—
174C4⋊(C3⋊C8)C22×C311yescharacteristic
175C22×C3×D3—21no—
176C22×C3×D3—22no—
177C22×C3×D3C2×D411yescharacteristic
178C32×Q8—21no—
179C32×Q8—21no—
180C32×Q8—42no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2411yescharacteristic
187C2×C3×Dic3—21no—
188C2×C3×Dic3C2×D411yescharacteristic
189C2×C3×Dic3C2×D411yescharacteristic
190C2×C3×Dic3C2×D411yescharacteristic
191C3×D12—21no—
192C3×D12—22no—
193C3×D12—22no—
194C3×D12—21no—
195C3×D12—42no—
196C3×D12—42no—
197C3×D12—44no—
198C3×D12—42no—
199C3×C4×D3—22no—
200C3×C4×D3—22no—
201C3×C4×D3—21no—
202C3×C4×D3—21no—
203C3×C4×D3—44no—
204C3×C4×D3—42no—
205C32×C8—22no—
206C3×C3⋊C8—22no—
207#532—31no—
208#529—31no—
209#478—32no—
210#477C3×D311yescharacteristic, complemented
211#476—31no—
212#466—32no—
213#465—32no—
214#463—32no—
215#460—31no—
216#459—31no—
217C2×C4⋊C8—31no—
218C3×#41—124no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—41no—
223C2×C3×Q8C22×D311yescharacteristic
224C2×C3×Q8C23×C311yescharacteristic
225C2×C3×D4—31no—
226C2×C3×D4—32no—
227C2×C3×D4—61no—
228C2×C3×D4—62no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—62no—
235C22×C3×C4—61no—
236#239—44no—
237C2×D12—21no—
238C2×D12—22no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12C23×C311yescharacteristic
242C2×D12C23×C312yes—
243C2×C4×D3—21no—
244C2×C4×D3—22no—
245C2×C4×D3C3×D411yescharacteristic
246C2×C4×D3C3×D411yescharacteristic
247C2×C4×D3C23×C311yescharacteristic
248C3×SD16—122no—
249C3×D8—122no—
250C2×C3×C8—22no—
251C2×C3×C8—32no—
252C2×C3×C8—122no—
253C2×C3×C8C22×D312yes—
254C2×C3×C8C23×C312yes—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4—41no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4C22×D311yescharacteristic
268C3×C4⋊C4C22×D312yes—
269C3×C4⋊C4C23×C312yes—
270C3×C4⋊C4C23×C311yescharacteristic
271C3×C22⋊C4—61no—
272C3×C22⋊C4—61no—
273C3×C22⋊C4—62no—
274C3×C22⋊C4—62no—
275C3×C22⋊C4—61no—
276C3×C22⋊C4—61no—
277C3×C22⋊C4—61no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—21no—
281C3×C42—31no—
282C3×C42—31no—
283C3×C42—31no—
284C3×C42—41no—
285C3×C42C22×D311yescharacteristic
286C3×C42C23×C311yescharacteristic
287#215—42no—
288(C2×D3)⋊C4—21no—
289(C2×D3)⋊C4—21no—
290(C2×D3)⋊C4—21no—
291(C2×D3)⋊C4—21no—
292(C2×D3)⋊C4—21no—
293(C2×D3)⋊C4—22no—
294(C2×D3)⋊C4—22no—
295C4⋊Dic3—21no—
296C4⋊Dic3C23×C311yescharacteristic
297Dic3⋊C4—21no—
298Dic3⋊C4—21no—
299C4×Dic3C3×D411yescharacteristic
300C4×Dic3C3×D411yescharacteristic
301C4×Dic3C23×C311yescharacteristic
302C2×C3⋊C8C23×C312yes—
303D24—42no—
304D3×C8—42no—
305C22×C32C22×D411yescharacteristic
306C2×C3×D3—21no—
307C2×C3×D3—21no—
308C2×C3×D3—41no—
309C2×C3×D3—42no—
310C2×C3×D3—41no—
311C2×C3×D3—42no—
312C2×C3×D3—42no—
313C2×C3×D3—41no—
314C2×C3×D3AGL(1,ℤ/8)12yes—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4—41no—
319C32×C4#13512yes—
320C32×C4H(5,𝔽2)11yescharacteristic
321C32×C42−1+411yescharacteristic
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—41no—
326C2×#41—61no—
327C2×SD16—32no—
328C2×D8—32no—
329C22×C8—32no—
330C4⋊Q8C2×C3×D311yescharacteristic
331#127—31no—
332#126—61no—
333#125—31no—
334#124—31no—
335#124—61no—
336#123—61no—
337C22⋊Q8—31no—
338#121—31no—
339#121—32no—
340#121—61no—
341C4×Q8C2×C3×D311yescharacteristic
342C4×D4—32no—
343C4×D4—31no—
344C4×D4—61no—
345C4×D4—61no—
346#117—61no—
347C2×C42—31no—
348#107C2×C3×D312yes—
349#106—32no—
350C4⋊C8—32no—
351C4⋊C8—31no—
352C4⋊C8C2×C3×D311yescharacteristic
353#103C2×C3×D312yes—
354#102—32no—
355#102—32no—
356#102—32no—
357C23×C3—31no—
358C23×C3—61no—
359C23×C3—62no—
360C22×D3—21no—
361C22×D3—22no—
362C22×D3C2×C3×D411yescharacteristic
363C3×Q8—22no—
364C3×Q8—21no—
365C3×Q8—21no—
366C3×Q8—21no—
367C3×Q8—21no—
368C3×Q8—41no—
369C3×Q8—42no—
370C3×Q8—42no—
371C3×Q8—82no—
372C3×D4—61no—
373C3×D4—62no—
374C3×D4—61no—
375C3×D4—62no—
376C3×D4—122no—
377C3×D4—122no—
378C3×D4—124no—
379C3×D4—122no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—61no—
400C2×C3×C4—124no—
401C2×C3×C4—122no—
402C2×C3×C4D3×D411yescharacteristic
403C2×C3×C4D3×D411yescharacteristic
404C2×C3×C4C2×C3×D411yescharacteristic
405C2×C3×C4C2×C3×D411yescharacteristic
406C2×C3×C4C23×D311yescharacteristic
407C2×C3×C4C24×C311yescharacteristic
408C2×Dic3—21no—
409C2×Dic3C2×C3×D411yescharacteristic
410C2×Dic3C2×C3×D411yescharacteristic
411C2×Dic3C2×C3×D411yescharacteristic
412D12—21no—
413D12—22no—
414D12—21no—
415D12—22no—
416D12—42no—
417D12—42no—
418D12—44no—
419D12—42no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—21no—
423C4×D3—21no—
424C4×D3—44no—
425C4×D3—42no—
426C3×C8—22no—
427C3×C8—22no—
428C3×C8—42no—
429C3×C8—62no—
430C3⋊C8—22no—
431C2×C32#53711yescharacteristic
432C2×C32C2×#13511yescharacteristic
433C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
434C3×D3—22no—
435C3×D3—81no—
436C3×D3—82no—
437#41—124no—
438C2×Q8—21no—
439C2×Q8C22×C3×D311yescharacteristic
440C2×D4—31no—
441C2×D4—32no—
442C2×D4—61no—
443C2×D4—62no—
444C2×D4—61no—
445C2×D4—61no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—62no—
450C22×C4—61no—
451SD16—122no—
452D8—122no—
453C2×C8—32no—
454C2×C8—122no—
455C2×C8C22×C3×D312yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—31no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4C22×C3×D311yescharacteristic
463C4⋊C4C22×C3×D312yes—
464C22⋊C4—61no—
465C22⋊C4—61no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C42—21no—
472C42—31no—
473C42—31no—
474C42—31no—
475C42C22×C3×D311yescharacteristic
476C22×C3—21no—
477C22×C3—32no—
478C22×C3—61no—
479C22×C3—61no—
480C22×C3—121no—
481C22×C3—122no—
482C22×C3—122no—
483C22×C3—121no—
484C22×C3—121no—
485C22×C3—122no—
486C22×C3C2×D3×D411yescharacteristic
487C22×C3C22×C3×D411yescharacteristic
488C2×D3—21no—
489C2×D3—21no—
490C2×D3—41no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—41no—
494C2×D3—41no—
495C2×D3—42no—
496C2×D3C3×AGL(1,ℤ/8)12yes—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—22no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—41no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—41no—
506C3×C4—42no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—61no—
511C3×C4—61no—
512C3×C4—61no—
513C3×C4—82no—
514C3×C4—81no—
515C3×C4—121no—
516C3×C4#91912yes—
517C3×C4C3×#13512yes—
518C3×C4#100711yescharacteristic
519C3×C4#100911yescharacteristic
520C3×C4C3×H(5,𝔽2)11yescharacteristic
521C3×C4C3×2−1+411yescharacteristic
522Dic3—21no—
523Dic3—21no—
524Dic3—21no—
525Dic3—41no—
526C32#324711yescharacteristic, complemented, Sylow
527C23—31no—
528C23—61no—
529C23—62no—
530Q8—21no—
531Q8—21no—
532Q8—42no—
533D4—61no—
534D4—62no—
535D4—61no—
536D4—62no—
537D4—122no—
538D4—124no—
539D4—122no—
540D4—122no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—124no—
553C2×C4—122no—
554C2×C4C3×D3×D411yescharacteristic
555C2×C4C3×D3×D411yescharacteristic
556C2×C4C23×C3×D311yescharacteristic
557C8—22no—
558C8—62no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—62no—
563C2×C3—241no—
564C2×C3—242no—
565C2×C3#562911yescharacteristic
566C2×C3C2×#91911yescharacteristic
567C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
568C2×C3C3×#53711yescharacteristic
569C2×C3C2×C3×#13511yescharacteristic
570C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
571D3—22no—
572D3—81no—
573D3—82no—
574C22—32no—
575C22—61no—
576C22—61no—
577C22—121no—
578C22—122no—
579C22—122no—
580C22—121no—
581C22—121no—
582C22—122no—
583C22C2×C3×D3×D411yescharacteristic
584C4—21no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—61no—
589C4—61no—
590C4—61no—
591C4—121no—
592C4C3×#91912yes—
593C4C3×#100711yescharacteristic
594C4C3×#100911yescharacteristic
595C3—21no—
596C3#8113111yescharacteristic, direct product factor
597C3C3×#324711yescharacteristic, complemented
598C2—62no—
599C2—241no—
600C2—242no—
601C2C3×#562911yescharacteristic
602C2C2×C3×#91911yescharacteristic
603C2C3×D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#477C2
socleC22×C32C22×D4
hypercenterC3×#477D3
hypocenterC3C3×#3247
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#477

Complements

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

NH ≅ G/Ntype
1. C3×#5725C2semidirect
2. C3×#5580C2semidirect
3. C32×#477C2semidirect
4. C3×#5084C2semidirect
5. C3×#5027C2semidirect
6. C3×#4891C2semidirect
7. C3×#4883C2semidirect
8. C3×#4843C2semidirect
9. C3×D3×C4⋊C8C2semidirect
10. C3×#4819C2semidirect
11. C3×#4807C2semidirect
13. #81131C3direct (N × H)
47. #5725C2×C3semidirect
48. #5580C2×C3semidirect
51. C3×#477D3semidirect
52. C3×#477C2×C3semidirect
60. #5084C2×C3semidirect
61. #5027C2×C3semidirect
62. #4891C2×C3semidirect
63. #4883C2×C3semidirect
64. #4843C2×C3semidirect
65. D3×C4⋊C8C2×C3semidirect
66. #4819C2×C3semidirect
67. #4807C2×C3semidirect
210. #477C3×D3semidirect
526. C32#3247semidirect
596. C3#81131direct (N × H)
597. C3C3×#3247semidirect

Maximal subgroups

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

subgroupindex
C3×#57252
C3×#55802
C32×#4772
C3×#50842
C3×#50272
C3×#48912
C3×#48832
C3×#48432
C3×D3×C4⋊C82
C3×#48192
C3×#48072
C3×#32473
#811313