Finite Group Databasebeta

← C3×#86286 #11910445

Subgroups — C3×#86286 #11910445

Order 1152.

1971 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#533C211yesmaximal, characteristic, complemented
2C3×#5735C214yesmaximal, complemented
3C3×#5700C211yesmaximal, characteristic, complemented
4C3×#5699C212yesmaximal, complemented
5C3×C3⋊#564C211yesmaximal, characteristic, complemented
6C3×#5647C214yesmaximal, complemented
7C3×#5646C214yesmaximal, complemented
8C3×#5580C214yesmaximal, complemented
9C3×#5578C214yesmaximal, complemented
10C3×#5528C214yesmaximal, complemented
11C3×#5479C212yesmaximal, complemented
12C3×#3531—31nomaximal
13#86286C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8C2212yes—
15C32×#125C2211yescharacteristic
16C32×#125C2211yescharacteristic
17C32×C22⋊Q8C2214yes—
18C32×C4×Q8C2214yes—
19C32×#117C2212yes—
20C2×C32×C42C2211yescharacteristic
21C3×#947C2214yes—
22C3×#946C2214yes—
23C3×#946C2214yes—
24C3×#944C2214yes—
25C3×#930C2212yes—
26C3×#929C2214yes—
27C3×C4×#67C2214yes—
28C3×#924C2212yes—
29C3×#898C2214yes—
30C3×#898C2214yes—
31C3×#898C2214yes—
32C3×#897C2214yes—
33C3×#896C2214yes—
34C3×#896C2214yes—
35C3×#896C2214yes—
36C3×#895C2214yes—
37C3×#894C2214yes—
38C3×#894C2214yes—
39C3×#894C2214yes—
40C3×#890C2214yes—
41C3×#889C2214yes—
42C3×#889C2214yes—
43C3×#889C2214yes—
44C3×#888C2214yes—
45C3×#885C2214yes—
46C3×#884C2214yes—
47C3×#883C2214yes—
48C3×#879C2214yes—
49C3×#876C2214yes—
50C3×#875C2212yes—
51C3×C4×D12C2214yes—
52C3×C4×D12C2212yes—
53C3×#872C2214yes—
54C3×#872C2214yes—
55C3×C42×D3C2214yes—
56C3×#870C2212yes—
57C3×C4×Dic6C2214yes—
58C3×C4×Dic6C2212yes—
59C3×#564—31no—
60C3×#563—34no—
61C3×#563—32no—
62C3×#563—34no—
63C3×#562—31no—
64C3×#562—34no—
65C3×#533—21no—
66C3×#533—34no—
67C3×#533D311yescharacteristic, complemented
68C3×#533C2×C311yescharacteristic, complemented
69C3×#529—34no—
70C3×#529—34no—
71C3×#529—32no—
72#5735C2×C314yescomplemented
73#5700C2×C311yescharacteristic, complemented
74#5699C2×C312yescomplemented
75C3⋊#564C2×C311yescharacteristic, complemented
76#5647C2×C314yescomplemented
77#5646C2×C314yescomplemented
78#5580C2×C314yescomplemented
79#5578C2×C314yescomplemented
80#5528C2×C314yescomplemented
81#5479C2×C312yescomplemented
82C2×C32×Q8C2314yes—
83C22×C32×C4C2311yescharacteristic
84C22×C32×C4C2312yes—
85C2×C3×#67C2314yes—
86C2×C3×D12C2314yes—
87C2×C3×D12C2312yes—
88C2×C3×C4×D3C2314yes—
89C2×C3×C4×D3C2314yes—
90C2×C3×C4×D3C2314yes—
91C2×C3×Dic6C2314yes—
92C2×C3×Dic6C2312yes—
93C32×C4⋊C4C2314yes—
94C32×C4⋊C4C2312yes—
95C32×C4⋊C4C2314yes—
96C32×C4⋊C4C2314yes—
97C32×C4⋊C4C2312yes—
98C32×C22⋊C4C2314yes—
99C32×C42C2312yes—
100C32×C42C2311yescharacteristic
101C32×C42C2311yescharacteristic
102C32×C42C2314yes—
103C3×#217C2314yes—
104C3×(C2×D3)⋊C4C2314yes—
105C3×(C2×D3)⋊C4C2314yes—
106C3×(C2×D3)⋊C4C2314yes—
107C3×(C2×D3)⋊C4C2314yes—
108C3×(C2×D3)⋊C4C2314yes—
109C3×(C2×D3)⋊C4C2314yes—
110C3×(C2×D3)⋊C4C2314yes—
111C3×(C2×D3)⋊C4C2314yes—
112C3×C4⋊Dic3C2314yes—
113C3×C4⋊Dic3C2314yes—
114C3×C4⋊Dic3C2312yes—
115C3×C4⋊Dic3C2312yes—
116C3×Dic3⋊C4C2314yes—
117C3×Dic3⋊C4C2314yes—
118C3×Dic3⋊C4C2314yes—
119C3×Dic3⋊C4C2314yes—
120C3×Dic3⋊C4C2314yes—
121C3×Dic3⋊C4C2314yes—
122C3×Dic3⋊C4C2314yes—
123C3×Dic3⋊C4C2314yes—
124C3×C4×Dic3C2314yes—
125C3×C4×Dic3C2314yes—
126C3×C4×Dic3C2314yes—
127#3531—31noSylow
128C3×C4⋊Q8—22no—
129C3×C4⋊Q8—34no—
130C3×C4⋊Q8—34no—
131C3×C4⋊Q8C2×D312yes—
132C3×C4⋊Q8C22×C312yes—
133C3×#126—34no—
134C3×#126—34no—
135C3×#126—34no—
136C3×#126—34no—
137C3×#126—34no—
138C3×#125—21no—
139C3×#125—21no—
140C3×#125—34no—
141C3×#125—34no—
142C3×#125—34no—
143C3×#125—34no—
144C3×#125—32no—
145C3×#125C2×D311yescharacteristic
146C3×#125C2×D311yescharacteristic
147C3×#125C22×C311yescharacteristic
148C3×#125C22×C311yescharacteristic
149C3×#124—34no—
150C3×#124—34no—
151C3×#124—32no—
152C3×#123—34no—
153C3×#123—34no—
154C3×#123—34no—
155C3×#123—34no—
156C3×#123—34no—
157C3×C22⋊Q8—24no—
158C3×C22⋊Q8—34no—
159C3×C22⋊Q8—34no—
160C3×C22⋊Q8—32no—
161C3×C22⋊Q8—34no—
162C3×C22⋊Q8—34no—
163C3×C22⋊Q8—34no—
164C3×C22⋊Q8—34no—
165C3×C22⋊Q8C2×D314yes—
166C3×C22⋊Q8C22×C314yes—
167C3×#121—32no—
168C3×#121—34no—
169C3×#121—34no—
170C3×C4×Q8—24no—
171C3×C4×Q8—34no—
172C3×C4×Q8—32no—
173C3×C4×Q8C2×D314yes—
174C3×C4×Q8C22×C314yes—
175C3×C4×D4—34no—
176C3×C4×D4—34no—
177C3×C4×D4—32no—
178C3×#117—22no—
179C3×#117—34no—
180C3×#117—34no—
181C3×#117C2×D312yes—
182C3×#117C22×C312yes—
183C2×C3×C42—21no—
184C2×C3×C42—34no—
185C2×C3×C42C2×D311yescharacteristic
186C2×C3×C42C22×C311yescharacteristic
187#947C22×C314yes—
188#946C22×C314yes—
189#946C22×C314yes—
190#944C22×C314yes—
191#930C22×C312yes—
192#929C22×C314yes—
193C4×#67C22×C314yes—
194#924C22×C312yes—
195#898C22×C314yes—
196#898C22×C314yes—
197#898C22×C314yes—
198#897C22×C314yes—
199#896C22×C314yes—
200#896C22×C314yes—
201#896C22×C314yes—
202#895C22×C314yes—
203#894C22×C314yes—
204#894C22×C314yes—
205#894C22×C314yes—
206#890C22×C314yes—
207#889C22×C314yes—
208#889C22×C314yes—
209#889C22×C314yes—
210#888C22×C314yes—
211#885C22×C314yes—
212#884C22×C314yes—
213#883C22×C314yes—
214#879C22×C314yes—
215#876C22×C314yes—
216#875C22×C312yes—
217C4×D12C22×C312yes—
218C4×D12C22×C314yes—
219#872C22×C314yes—
220#872C22×C314yes—
221C42×D3C22×C314yes—
222#870C22×C312yes—
223C4×Dic6C22×C312yes—
224C4×Dic6C22×C314yes—
225C23×C32C2411yescharacteristic
226C22×C3×D3C2414yes—
227C32×Q8—44no—
228C2×C32×C4—22no—
229C2×C32×C4—42no—
230C2×C32×C4C2414yes—
231C2×C32×C4C2414yes—
232C2×C32×C4C2411yescharacteristic
233C2×C32×C4C2412yes—
234C2×C32×C4C2412yes—
235C2×C32×C4C2411yescharacteristic
236C3×#67—44no—
237C2×C3×Dic3C2414yes—
238C2×C3×Dic3C2414yes—
239C2×C3×Dic3C2414yes—
240C3×D12—42no—
241C3×D12—44no—
242C3×C4×D3—28no—
243C3×C4×D3—44no—
244C3×C4×D3—44no—
245C3×Dic6—42no—
246C3×Dic6—44no—
247#564—31no—
248#563—34no—
249#563—32no—
250#563—34no—
251#562—31no—
252#562—34no—
253#533—34no—
254#533C3×D311yescharacteristic, complemented
255#529—34no—
256#529—34no—
257#529—32no—
258C2×C3×Q8—24no—
259C2×C3×Q8—32no—
260C2×C3×Q8—34no—
261C2×C3×Q8C22×D314yes—
262C2×C3×Q8C23×C314yes—
263C2×C3×D4—34no—
264C2×C3×D4—34no—
265C2×C3×D4—32no—
266C22×C3×C4—21no—
267C22×C3×C4—22no—
268C22×C3×C4—34no—
269C22×C3×C4—34no—
270C22×C3×C4—34no—
271C22×C3×C4C22×D311yescharacteristic
272C22×C3×C4C22×D312yes—
273C22×C3×C4C23×C311yescharacteristic
274C22×C3×C4C23×C312yes—
275C2×#67C23×C314yes—
276C2×D12C23×C312yes—
277C2×D12C23×C314yes—
278C2×C4×D3C23×C314yes—
279C2×C4×D3C23×C314yes—
280C2×C4×D3C23×C314yes—
281C2×Dic6C23×C312yes—
282C2×Dic6C23×C314yes—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4—24no—
286C3×C4⋊C4—24no—
287C3×C4⋊C4—24no—
288C3×C4⋊C4—34no—
289C3×C4⋊C4—34no—
290C3×C4⋊C4—34no—
291C3×C4⋊C4—34no—
292C3×C4⋊C4—34no—
293C3×C4⋊C4—34no—
294C3×C4⋊C4—34no—
295C3×C4⋊C4—34no—
296C3×C4⋊C4—34no—
297C3×C4⋊C4—32no—
298C3×C4⋊C4—32no—
299C3×C4⋊C4—34no—
300C3×C4⋊C4C22×D312yes—
301C3×C4⋊C4C22×D312yes—
302C3×C4⋊C4C22×D314yes—
303C3×C4⋊C4C22×D314yes—
304C3×C4⋊C4C22×D314yes—
305C3×C4⋊C4C23×C314yes—
306C3×C4⋊C4C23×C312yes—
307C3×C4⋊C4C23×C314yes—
308C3×C4⋊C4C23×C314yes—
309C3×C4⋊C4C23×C312yes—
310C3×C22⋊C4—24no—
311C3×C22⋊C4—34no—
312C3×C22⋊C4—34no—
313C3×C22⋊C4—34no—
314C3×C22⋊C4—34no—
315C3×C22⋊C4—34no—
316C3×C22⋊C4—34no—
317C3×C22⋊C4—34no—
318C3×C22⋊C4—34no—
319C3×C22⋊C4—34no—
320C3×C22⋊C4C22×D314yes—
321C3×C22⋊C4C23×C314yes—
322C3×C42—22no—
323C3×C42—21no—
324C3×C42—21no—
325C3×C42—24no—
326C3×C42—34no—
327C3×C42—34no—
328C3×C42—34no—
329C3×C42C22×D311yescharacteristic
330C3×C42C22×D311yescharacteristic
331C3×C42C22×D314yes—
332C3×C42C22×D312yes—
333C3×C42C23×C314yes—
334C3×C42C23×C312yes—
335C3×C42C23×C311yescharacteristic
336C3×C42C23×C311yescharacteristic
337#217C23×C314yes—
338(C2×D3)⋊C4C23×C314yes—
339(C2×D3)⋊C4C23×C314yes—
340(C2×D3)⋊C4C23×C314yes—
341(C2×D3)⋊C4C23×C314yes—
342(C2×D3)⋊C4C23×C314yes—
343(C2×D3)⋊C4C23×C314yes—
344(C2×D3)⋊C4C23×C314yes—
345(C2×D3)⋊C4C23×C314yes—
346C4⋊Dic3C23×C312yes—
347C4⋊Dic3C23×C314yes—
348C4⋊Dic3C23×C312yes—
349C4⋊Dic3C23×C314yes—
350Dic3⋊C4C23×C314yes—
351Dic3⋊C4C23×C314yes—
352Dic3⋊C4C23×C314yes—
353Dic3⋊C4C23×C314yes—
354Dic3⋊C4C23×C314yes—
355Dic3⋊C4C23×C314yes—
356Dic3⋊C4C23×C314yes—
357Dic3⋊C4C23×C314yes—
358C4×Dic3C23×C314yes—
359C4×Dic3C23×C314yes—
360C4×Dic3C23×C314yes—
361C22×C32—21no—
362C22×C32—22no—
363C22×C32C2511yescharacteristic
364C2×C3×D3—24no—
365C2×C3×D3—24no—
366C2×C3×D3—24no—
367C32×C4—21no—
368C32×C4—22no—
369C32×C4—24no—
370C32×C4—22no—
371C32×C4—24no—
372C32×C42−1+412yes—
373C3×Dic3—24no—
374C3×Dic3—24no—
375C3×Dic3—24no—
376C4⋊Q8—34no—
377C4⋊Q8—34no—
378C4⋊Q8C2×C3×D312yes—
379#126—34no—
380#126—34no—
381#126—34no—
382#126—34no—
383#126—34no—
384#125—34no—
385#125—34no—
386#125—34no—
387#125—34no—
388#125—32no—
389#125C2×C3×D311yescharacteristic
390#125C2×C3×D311yescharacteristic
391#124—34no—
392#124—34no—
393#124—32no—
394#123—34no—
395#123—34no—
396#123—34no—
397#123—34no—
398#123—34no—
399C22⋊Q8—34no—
400C22⋊Q8—34no—
401C22⋊Q8—32no—
402C22⋊Q8—34no—
403C22⋊Q8—34no—
404C22⋊Q8—34no—
405C22⋊Q8—34no—
406C22⋊Q8C2×C3×D314yes—
407#121—32no—
408#121—34no—
409#121—34no—
410C4×Q8—34no—
411C4×Q8—32no—
412C4×Q8C2×C3×D314yes—
413C4×D4—34no—
414C4×D4—34no—
415C4×D4—32no—
416#117—34no—
417#117—34no—
418#117C2×C3×D312yes—
419C2×C42—34no—
420C2×C42C2×C3×D311yescharacteristic
421C23×C3—21no—
422C23×C3—34no—
423C23×C3C23×D311yescharacteristic
424C23×C3C24×C311yescharacteristic
425C22×D3C24×C314yes—
426C3×Q8—44no—
427C3×Q8—44no—
428C3×Q8—48no—
429C3×Q8—122no—
430C3×Q8—124no—
431C3×D4—122no—
432C3×D4—124no—
433C3×D4—124no—
434C2×C3×C4—24no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—21no—
438C2×C3×C4—21no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—42no—
447C2×C3×C4—42no—
448C2×C3×C4—44no—
449C2×C3×C4—68no—
450C2×C3×C4—124no—
451C2×C3×C4—124no—
452C2×C3×C4C23×D314yes—
453C2×C3×C4C23×D314yes—
454C2×C3×C4C23×D311yescharacteristic
455C2×C3×C4C23×D312yes—
456C2×C3×C4C23×D312yes—
457C2×C3×C4C23×D311yescharacteristic
458C2×C3×C4C24×C314yes—
459C2×C3×C4C24×C314yes—
460C2×C3×C4C24×C311yescharacteristic
461C2×C3×C4C24×C312yes—
462C2×C3×C4C24×C312yes—
463C2×C3×C4C24×C311yescharacteristic
464#67—44no—
465C2×Dic3C24×C314yes—
466C2×Dic3C24×C314yes—
467C2×Dic3C24×C314yes—
468D12—42no—
469D12—44no—
470C4×D3—28no—
471C4×D3—44no—
472C4×D3—44no—
473Dic6—42no—
474Dic6—44no—
475C2×C32—41no—
476C2×C32C2×2−1+411yescharacteristic
477C2×C32#58512yes—
478C3×D3—44no—
479C2×Q8—32no—
480C2×Q8—34no—
481C2×Q8C22×C3×D314yes—
482C2×D4—34no—
483C2×D4—34no—
484C2×D4—32no—
485C22×C4—34no—
486C22×C4—34no—
487C22×C4—34no—
488C22×C4C22×C3×D311yescharacteristic
489C22×C4C22×C3×D312yes—
490C4⋊C4—34no—
491C4⋊C4—34no—
492C4⋊C4—34no—
493C4⋊C4—34no—
494C4⋊C4—34no—
495C4⋊C4—34no—
496C4⋊C4—32no—
497C4⋊C4—34no—
498C4⋊C4—32no—
499C4⋊C4—34no—
500C4⋊C4—34no—
501C4⋊C4—34no—
502C4⋊C4C22×C3×D312yes—
503C4⋊C4C22×C3×D312yes—
504C4⋊C4C22×C3×D314yes—
505C4⋊C4C22×C3×D314yes—
506C4⋊C4C22×C3×D314yes—
507C22⋊C4—34no—
508C22⋊C4—34no—
509C22⋊C4—34no—
510C22⋊C4—34no—
511C22⋊C4—34no—
512C22⋊C4—34no—
513C22⋊C4—34no—
514C22⋊C4—34no—
515C22⋊C4—34no—
516C22⋊C4C22×C3×D314yes—
517C42—34no—
518C42—34no—
519C42—34no—
520C42C22×C3×D314yes—
521C42C22×C3×D312yes—
522C42C22×C3×D311yescharacteristic
523C42C22×C3×D311yescharacteristic
524C22×C3—22no—
525C22×C3—21no—
526C22×C3—22no—
527C22×C3—21no—
528C22×C3—22no—
529C22×C3—21no—
530C22×C3—42no—
531C22×C3—64no—
532C22×C3—64no—
533C22×C3—64no—
534C22×C3C24×D311yescharacteristic
535C22×C3C25×C311yescharacteristic
536C2×D3—24no—
537C2×D3—24no—
538C2×D3—24no—
539C3×C4—22no—
540C3×C4—24no—
541C3×C4—21no—
542C3×C4—22no—
543C3×C4—22no—
544C3×C4—24no—
545C3×C4—24no—
546C3×C4—21no—
547C3×C4—24no—
548C3×C4—22no—
549C3×C4—24no—
550C3×C4—22no—
551C3×C4—22no—
552C3×C4—42no—
553C3×C4—44no—
554C3×C4—44no—
555C3×C4—64no—
556C3×C4—64no—
557C3×C4—64no—
558C3×C4#100712yes—
559C3×C4C3×2−1+412yes—
560Dic3—24no—
561Dic3—24no—
562Dic3—24no—
563C32#353111yescharacteristic, complemented, Sylow
564C23—34no—
565C23C23×C3×D311yescharacteristic
566Q8—44no—
567Q8—122no—
568Q8—124no—
569D4—122no—
570D4—124no—
571D4—124no—
572C2×C4—22no—
573C2×C4—34no—
574C2×C4—34no—
575C2×C4—34no—
576C2×C4—42no—
577C2×C4—68no—
578C2×C4—124no—
579C2×C4—124no—
580C2×C4C23×C3×D314yes—
581C2×C4C23×C3×D314yes—
582C2×C4C23×C3×D311yescharacteristic
583C2×C4C23×C3×D312yes—
584C2×C4C23×C3×D312yes—
585C2×C4C23×C3×D311yescharacteristic
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—42no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—124no—
592C2×C3C2×#100711yescharacteristic
593C2×C3#596412yes—
594C2×C3C2×C3×2−1+411yescharacteristic
595C2×C3C3×#58512yes—
596D3—44no—
597C22—21no—
598C22—22no—
599C22—64no—
600C22—64no—
601C22—64no—
602C22C24×C3×D311yescharacteristic
603C4—21no—
604C4—22no—
605C4—22no—
606C4—24no—
607C4—24no—
608C4—64no—
609C4—64no—
610C4—64no—
611C4C3×#100712yes—
612C3—21no—
613C3#8628611yescharacteristic, direct product factor
614C3C3×#353111yescharacteristic, complemented
615C2—41no—
616C2—124no—
617C2C2×C3×#100711yescharacteristic
618C2C3×#596412yes—
subgroupG/H
centerC2×C3×C4C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#533C2
socleC22×C32C25
hypercenterC3×#533D3
hypocenterC3C3×#3531
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×#533

Complements

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

NH ≅ G/Ntype
1. C32×#533C2semidirect
2. C3×#5735C2semidirect
3. C3×#5700C2semidirect
4. C3×#5699C2semidirect
5. C3×C3⋊#564C2semidirect
6. C3×#5647C2semidirect
7. C3×#5646C2semidirect
8. C3×#5580C2semidirect
9. C3×#5578C2semidirect
10. C3×#5528C2semidirect
11. C3×#5479C2semidirect
13. #86286C3direct (N × H)
67. C3×#533D3semidirect
68. C3×#533C2×C3semidirect
72. #5735C2×C3semidirect
73. #5700C2×C3semidirect
74. #5699C2×C3semidirect
75. C3⋊#564C2×C3semidirect
76. #5647C2×C3semidirect
77. #5646C2×C3semidirect
78. #5580C2×C3semidirect
79. #5578C2×C3semidirect
80. #5528C2×C3semidirect
81. #5479C2×C3semidirect
254. #533C3×D3semidirect
563. C32#3531semidirect
613. C3#86286direct (N × H)
614. C3C3×#3531semidirect

Maximal subgroups

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

subgroupindex
C32×#5332
C3×#57352
C3×#57002
C3×#56992
C3×C3⋊#5642
C3×#56472
C3×#56462
C3×#55802
C3×#55782
C3×#55282
C3×#54792
C3×#35313
#862863