Finite Group Databasebeta

← C3×#81071 #11882316

Subgroups — C3×#81071 #11882316

Order 1152.

850 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5725C211yesmaximal, characteristic, complemented
2C3×#5557C211yesmaximal, characteristic, complemented
3C32×#476C211yesmaximal, characteristic, complemented
4C3×#5091C211yesmaximal, characteristic, complemented
5C3×#5015C211yesmaximal, characteristic, complemented
6C3×#4871C212yesmaximal, complemented
7C3×#4864C212yesmaximal, complemented
8C3×#4848C211yesmaximal, characteristic, complemented
9C3×D3×C4⋊C8C211yesmaximal, characteristic, complemented
10C3×#4787C212yesmaximal, complemented
11C3×#4775C212yesmaximal, complemented
12C3×#3247—31nomaximal
13#81071C311yesmaximal, characteristic, direct product factor
14C2×C3×#239—21no—
15C32×C4⋊Q8C2211yescharacteristic
16C32×C4×D4C2211yescharacteristic
17C3×#947—21no—
18C3×#938C2211yescharacteristic
19C3×#936C2211yescharacteristic
20C2×C3×#213C2212yes—
21C3×C4×#67—21no—
22C3×#926—21no—
23C2×C3×#204C2212yes—
24C2×C3×D3×C8C2212yes—
25C3×#897C2211yescharacteristic
26C3×#895—21no—
27C3×#895C2212yes—
28C3×#893—21no—
29C3×#893C2212yes—
30C3×#890C2211yescharacteristic
31C3×#883—21no—
32C3×#879—21no—
33C3×#874C2211yescharacteristic
34C3×C42×D3C2211yescharacteristic
35C3×C4×Dic6C2211yescharacteristic
36C32×#106C2212yes—
37C32×C4⋊C8C2211yescharacteristic
38C32×#102C2212yes—
39C3×#821C2212yes—
40C3×Dic3⋊C8C2212yes—
41C3×#810C2212yes—
42C3×#809C2212yes—
43C3×#807C2212yes—
44C3×C4⋊(C3⋊C8)C2211yescharacteristic
45C3×#532—31no—
46C3×#529—31no—
47#5725C2×C311yescharacteristic, complemented
48#5557C2×C311yescharacteristic, complemented
49C3×#478—32no—
50C3×#477—31no—
51C3×#476—21no—
52C3×#476D311yescharacteristic, complemented
53C3×#476C2×C311yescharacteristic, complemented
54C3×#466—32no—
55C3×#465—32no—
56C3×#463—32no—
57C3×#460—31no—
58C3×#459—31no—
59C2×C3×C4⋊C8—31no—
60#5091C2×C311yescharacteristic, complemented
61#5015C2×C311yescharacteristic, complemented
62#4871C2×C312yescomplemented
63#4864C2×C312yescomplemented
64#4848C2×C311yescharacteristic, complemented
65D3×C4⋊C8C2×C311yescharacteristic, complemented
66#4787C2×C312yescomplemented
67#4775C2×C312yescomplemented
68C2×C32×Q8—21no—
69C2×C32×D4C2311yescharacteristic
70C22×C32×C4—21no—
71C2×C3×#67—21no—
72C3×#239—44no—
73C2×C3×D12—22no—
74C2×C3×D12—21no—
75C2×C3×D12—21no—
76C2×C3×D12C2312yes—
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3D411yescharacteristic
79C2×C3×C4×D3D411yescharacteristic
80C2×C3×C4×D3C2311yescharacteristic
81C2×C3×Dic6C2311yescharacteristic
82C2×C32×C8C2312yes—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2311yescharacteristic
85C32×C4⋊C4C2312yes—
86C32×C22⋊C4—21no—
87C32×C42C2311yescharacteristic
88C3×#217—21no—
89C3×#217—21no—
90C3×#217—21no—
91C3×#213—42no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—22no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3C2311yescharacteristic
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3D411yescharacteristic
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3C2311yescharacteristic
104C2×C3×C3⋊C8C2312yes—
105C3×#204—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—31no—
124C3×#121—32no—
125C3×#121—61no—
126C3×C4×Q8—31no—
127C3×C4×D4—21no—
128C3×C4×D4—32no—
129C3×C4×D4—61no—
130C3×C4×D4—61no—
131C3×C4×D4C2×D311yescharacteristic
132C3×C4×D4C22×C311yescharacteristic
133C3×#117—61no—
134C2×C3×C42—31no—
135#947—21no—
136#938C22×C311yescharacteristic
137#936C22×C311yescharacteristic
138C2×#213C22×C312yes—
139C4×#67—21no—
140#926—21no—
141C2×#204C22×C312yes—
142C2×D3×C8C22×C312yes—
143#897C22×C311yescharacteristic
144#895—21no—
145#895C22×C312yes—
146#893—21no—
147#893C22×C312yes—
148#890C22×C311yescharacteristic
149#883—21no—
150#879—21no—
151#874C22×C311yescharacteristic
152C42×D3C22×C311yescharacteristic
153C4×Dic6C22×C311yescharacteristic
154C3×#107—32no—
155C3×#106—22no—
156C3×#106C2×D312yes—
157C3×#106C22×C312yes—
158C3×C4⋊C8—21no—
159C3×C4⋊C8—32no—
160C3×C4⋊C8—31no—
161C3×C4⋊C8C2×D311yescharacteristic
162C3×C4⋊C8C22×C311yescharacteristic
163C3×#103—32no—
164C3×#102—22no—
165C3×#102—32no—
166C3×#102—32no—
167C3×#102C2×D312yes—
168C3×#102C22×C312yes—
169#821C22×C312yes—
170Dic3⋊C8C22×C312yes—
171#810C22×C312yes—
172#809C22×C312yes—
173#807C22×C312yes—
174C4⋊(C3⋊C8)C22×C311yescharacteristic
175C23×C32—21no—
176C22×C3×D3—22no—
177C22×C3×D3C2×D411yescharacteristic
178C32×Q8—42no—
179C32×D4—21no—
180C32×D4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—22no—
183C2×C32×C4—42no—
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2411yescharacteristic
187C3×#67—42no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3C2×D411yescharacteristic
191C2×C3×Dic3C2×D411yescharacteristic
192C2×C3×Dic3C2×D411yescharacteristic
193C3×D12—22no—
194C3×D12—22no—
195C3×D12—42no—
196C3×D12—44no—
197C3×D12—42no—
198C3×C4×D3—22no—
199C3×C4×D3—22no—
200C3×C4×D3—21no—
201C3×C4×D3—21no—
202C3×C4×D3—44no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C32×C8—22no—
206C3×C3⋊C8—22no—
207#532—31no—
208#529—31no—
209#478—32no—
210#477—31no—
211#476C3×D311yescharacteristic, complemented
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—31no—
222C2×C3×Q8—41no—
223C2×C3×D4—21no—
224C2×C3×D4—32no—
225C2×C3×D4—62no—
226C2×C3×D4—61no—
227C2×C3×D4—61no—
228C2×C3×D4—61no—
229C2×C3×D4C22×D311yescharacteristic
230C2×C3×D4C23×C311yescharacteristic
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—41no—
237C22×C3×C4—62no—
238C2×#67—21no—
239#239—44no—
240C2×D12—22no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12C23×C312yes—
244C2×C4×D3—22no—
245C2×C4×D3C3×D411yescharacteristic
246C2×C4×D3C3×D411yescharacteristic
247C2×C4×D3C23×C311yescharacteristic
248C2×Dic6C23×C311yescharacteristic
249C3×SD16—122no—
250C3×D8—122no—
251C2×C3×C8—22no—
252C2×C3×C8—32no—
253C2×C3×C8—122no—
254C2×C3×C8C22×D312yes—
255C2×C3×C8C23×C312yes—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4—61no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4C22×D311yescharacteristic
267C3×C4⋊C4C22×D312yes—
268C3×C4⋊C4C23×C312yes—
269C3×C4⋊C4C23×C311yescharacteristic
270C3×C22⋊C4—21no—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—62no—
274C3×C22⋊C4—62no—
275C3×C22⋊C4—61no—
276C3×C22⋊C4—61no—
277C3×C22⋊C4—61no—
278C3×C22⋊C4—61no—
279C3×C42—21no—
280C3×C42—31no—
281C3×C42—31no—
282C3×C42—31no—
283C3×C42—61no—
284C3×C42C22×D311yescharacteristic
285C3×C42C23×C311yescharacteristic
286#217—21no—
287#217—21no—
288#217—21no—
289#213—42no—
290(C2×D3)⋊C4—21no—
291(C2×D3)⋊C4—22no—
292(C2×D3)⋊C4—22no—
293C4⋊Dic3—21no—
294C4⋊Dic3—21no—
295C4⋊Dic3C23×C311yescharacteristic
296Dic3⋊C4—21no—
297Dic3⋊C4—21no—
298C4×Dic3—21no—
299C4×Dic3C3×D411yescharacteristic
300C4×Dic3C3×D411yescharacteristic
301C4×Dic3C23×C311yescharacteristic
302C2×C3⋊C8C23×C312yes—
303#204—42no—
304D3×C8—42no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32C22×D411yescharacteristic
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3—42no—
312C2×C3×D3—42no—
313C2×C3×D3—42no—
314C2×C3×D3AGL(1,ℤ/8)12yes—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4#13512yes—
319C32×C4H(5,𝔽2)11yescharacteristic
320C32×C42−1+411yescharacteristic
321C3×Dic3—21no—
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—41no—
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×Q8—31no—
342C4×D4—32no—
343C4×D4—61no—
344C4×D4—61no—
345C4×D4C2×C3×D311yescharacteristic
346#117—61no—
347C2×C42—31no—
348#107—32no—
349#106C2×C3×D312yes—
350C4⋊C8—32no—
351C4⋊C8—31no—
352C4⋊C8C2×C3×D311yescharacteristic
353#103—32no—
354#102—32no—
355#102—32no—
356#102C2×C3×D312yes—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—31no—
360C23×C3—41no—
361C23×C3—62no—
362C22×D3—22no—
363C22×D3C2×C3×D411yescharacteristic
364C3×Q8—42no—
365C3×Q8—42no—
366C3×Q8—61no—
367C3×Q8—61no—
368C3×Q8—82no—
369C3×D4—22no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—41no—
375C3×D4—62no—
376C3×D4—62no—
377C3×D4—122no—
378C3×D4—124no—
379C3×D4—122no—
380C3×D4—122no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—41no—
392C2×C3×C4—42no—
393C2×C3×C4—42no—
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—61no—
401C2×C3×C4—82no—
402C2×C3×C4—124no—
403C2×C3×C4D3×D411yescharacteristic
404C2×C3×C4D3×D411yescharacteristic
405C2×C3×C4C2×C3×D411yescharacteristic
406C2×C3×C4C2×C3×D411yescharacteristic
407C2×C3×C4C23×D311yescharacteristic
408C2×C3×C4C24×C311yescharacteristic
409#67—42no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3C2×C3×D411yescharacteristic
413C2×Dic3C2×C3×D411yescharacteristic
414C2×Dic3C2×C3×D411yescharacteristic
415D12—22no—
416D12—22no—
417D12—42no—
418D12—44no—
419D12—42no—
420C4×D3—22no—
421C4×D3—22no—
422C4×D3—21no—
423C4×D3—21no—
424C4×D3—44no—
425Dic6—21no—
426Dic6—21no—
427C3×C8—22no—
428C3×C8—22no—
429C3×C8—42no—
430C3×C8—62no—
431C3⋊C8—22no—
432C2×C32—81no—
433C2×C32#53711yescharacteristic
434C2×C32C2×#13511yescharacteristic
435C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
436C3×D3—22no—
437C3×D3—82no—
438#41—124no—
439C2×Q8—21no—
440C2×Q8—31no—
441C2×D4—32no—
442C2×D4—62no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4C22×C3×D311yescharacteristic
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—62no—
452SD16—122no—
453D8—122no—
454C2×C8—32no—
455C2×C8—122no—
456C2×C8C22×C3×D312yes—
457C4⋊C4—21no—
458C4⋊C4—31no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4C22×C3×D311yescharacteristic
464C4⋊C4C22×C3×D312yes—
465C22⋊C4—21no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C42—31no—
473C42—31no—
474C42—31no—
475C42—61no—
476C42C22×C3×D311yescharacteristic
477C22×C3—21no—
478C22×C3—32no—
479C22×C3—42no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—122no—
491C22×C3—122no—
492C22×C3—122no—
493C22×C3C2×D3×D411yescharacteristic
494C22×C3C22×C3×D411yescharacteristic
495C2×D3—21no—
496C2×D3—21no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3C3×AGL(1,ℤ/8)12yes—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—22no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—61no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—81no—
515C3×C4—82no—
516C3×C4—121no—
517C3×C4—121no—
518C3×C4#91612yes—
519C3×C4C3×#13512yes—
520C3×C4#100711yescharacteristic
521C3×C4#100911yescharacteristic
522C3×C4C3×H(5,𝔽2)11yescharacteristic
523C3×C4C3×2−1+411yescharacteristic
524Dic3—21no—
525Dic3—21no—
526Dic3—21no—
527Dic3—41no—
528Dic3—41no—
529C32#324711yescharacteristic, complemented, Sylow
530C23—21no—
531C23—31no—
532C23—62no—
533Q8—42no—
534Q8—61no—
535Q8—61no—
536D4—21no—
537D4—21no—
538D4—62no—
539D4—62no—
540D4—122no—
541D4—124no—
542D4—122no—
543D4—122no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—42no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—124no—
557C2×C4C3×D3×D411yescharacteristic
558C2×C4C3×D3×D411yescharacteristic
559C2×C4C23×C3×D311yescharacteristic
560C8—22no—
561C8—62no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—62no—
566C2×C3—82no—
567C2×C3—81no—
568C2×C3—81no—
569C2×C3—242no—
570C2×C3#562911yescharacteristic
571C2×C3C2×#91611yescharacteristic
572C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
573C2×C3C3×#53711yescharacteristic
574C2×C3C2×C3×#13511yescharacteristic
575C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
576D3—22no—
577D3—82no—
578C22—32no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—61no—
583C22—61no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22C2×C3×D3×D411yescharacteristic
588C4—21no—
589C4—41no—
590C4—42no—
591C4—61no—
592C4—61no—
593C4—61no—
594C4—121no—
595C4—121no—
596C4C3×#91612yes—
597C4C3×#100711yescharacteristic
598C4C3×#100911yescharacteristic
599C3—21no—
600C3#8107111yescharacteristic, direct product factor
601C3C3×#324711yescharacteristic, complemented
602C2—62no—
603C2—81no—
604C2—242no—
605C2C3×#562911yescharacteristic
606C2C2×C3×#91611yescharacteristic
607C2C3×D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#476C2
socleC22×C32C22×D4
hypercenterC3×#476D3
hypocenterC3C3×#3247
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#476

Complements

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

NH ≅ G/Ntype
1. C3×#5725C2semidirect
2. C3×#5557C2semidirect
3. C32×#476C2semidirect
4. C3×#5091C2semidirect
5. C3×#5015C2semidirect
6. C3×#4871C2semidirect
7. C3×#4864C2semidirect
8. C3×#4848C2semidirect
9. C3×D3×C4⋊C8C2semidirect
10. C3×#4787C2semidirect
11. C3×#4775C2semidirect
13. #81071C3direct (N × H)
47. #5725C2×C3semidirect
48. #5557C2×C3semidirect
52. C3×#476D3semidirect
53. C3×#476C2×C3semidirect
60. #5091C2×C3semidirect
61. #5015C2×C3semidirect
62. #4871C2×C3semidirect
63. #4864C2×C3semidirect
64. #4848C2×C3semidirect
65. D3×C4⋊C8C2×C3semidirect
66. #4787C2×C3semidirect
67. #4775C2×C3semidirect
211. #476C3×D3semidirect
529. C32#3247semidirect
600. C3#81071direct (N × H)
601. 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×#55572
C32×#4762
C3×#50912
C3×#50152
C3×#48712
C3×#48642
C3×#48482
C3×D3×C4⋊C82
C3×#47872
C3×#47752
C3×#32473
#810713