Finite Group Databasebeta

← C3×#86221 #11904231

Subgroups — C3×#86221 #11904231

Order 1152.

2559 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#532C211yesmaximal, characteristic, complemented
2C3×#5832C214yesmaximal, complemented
3C3×#5734C211yesmaximal, characteristic, complemented
4C3×#5722C211yesmaximal, characteristic, complemented
5C3×#5718C212yesmaximal, complemented
6C3×#5670C214yesmaximal, complemented
7C3×#5616C214yesmaximal, complemented
8C3×#5599C214yesmaximal, complemented
9C3×#5548C214yesmaximal, complemented
10C3×C4×#237C214yesmaximal, complemented
11C3×C4×#235C212yesmaximal, complemented
12C3×#3483—31nomaximal
13#86221C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2212yes—
15C2×C3×#237C2214yes—
16C2×C3×#235C2212yes—
17C2×C3×#235C2214yes—
18C32×C4⋊Q8C2211yescharacteristic
19C32×#127C2211yescharacteristic
20C32×#124C2212yes—
21C32×#121C2214yes—
22C32×C4×D4C2214yes—
23C2×C32×C42C2211yescharacteristic
24C3×#947C2214yes—
25C3×#944C2214yes—
26C3×#940C2214yes—
27C3×#940C2214yes—
28C3×#939C2218yes—
29C3×#939C2218yes—
30C3×#938C2214yes—
31C3×#936C2214yes—
32C3×D4×Dic3C2214yes—
33C3×C4×#67C2218yes—
34C3×C4×#67C2214yes—
35C3×C4×#67C2214yes—
36C3×#926C2212yes—
37C2×C3×C4×Dic3C2214yes—
38C3×#895C2214yes—
39C3×#892C2214yes—
40C3×#888C2214yes—
41C3×#887C2214yes—
42C3×#885C2218yes—
43C3×#884C2218yes—
44C3×#67⋊C4C2218yes—
45C3×#877C2218yes—
46C3×C4×D12C2212yes—
47C3×C4×D12C2212yes—
48C3×#872C2214yes—
49C3×C42×D3C2214yes—
50C3×C4×Dic6C2212yes—
51C3×C4×Dic6C2212yes—
52C3×Q82—31no—
53C3×#555—31no—
54C3×#555—34no—
55C3×#555—34no—
56C3×#550—32no—
57C3×#550—34no—
58C3×#532—21no—
59C3×#532—34no—
60C3×#532—34no—
61C3×#532D311yescharacteristic, complemented
62C3×#532C2×C311yescharacteristic, complemented
63C3×C4×#41—34no—
64C3×C4×#41—32no—
65#5832C2×C314yescomplemented
66#5734C2×C311yescharacteristic, complemented
67#5722C2×C311yescharacteristic, complemented
68#5718C2×C312yescomplemented
69#5670C2×C314yescomplemented
70#5616C2×C314yescomplemented
71#5599C2×C314yescomplemented
72#5548C2×C314yescomplemented
73C4×#237C2×C314yescomplemented
74C4×#235C2×C312yescomplemented
75C32×#41—44no—
76C2×C32×Q8C2312yes—
77C2×C32×D4C2314yes—
78C2×C32×D4C2314yes—
79C2×C32×D4C2312yes—
80C22×C32×C4C2314yes—
81C22×C32×C4C2312yes—
82C22×C32×C4C2311yescharacteristic
83C2×C3×#67C2318yes—
84C2×C3×#67C2314yes—
85C2×C3×#67C2318yes—
86C22×C3×Dic3C2318yes—
87C3×#237—216no—
88C3×#235—28no—
89C3×#235—48no—
90C2×C3×D12C2312yes—
91C2×C3×D12C2314yes—
92C2×C3×C4×D3C2314yes—
93C2×C3×C4×D3C2314yes—
94C2×C3×C4×D3C2314yes—
95C2×C3×Dic6C2314yes—
96C2×C3×Dic6C2312yes—
97C32×C4⋊C4C2314yes—
98C32×C22⋊C4C2318yes—
99C32×C42C2312yes—
100C32×C42C2311yescharacteristic
101C32×C42C2311yescharacteristic
102C3×#217C2318yes—
103C3×#217C2314yes—
104C3×(C2×D3)⋊C4C2318yes—
105C3×(C2×D3)⋊C4C2314yes—
106C3×(C2×D3)⋊C4C2314yes—
107C3×C4⋊Dic3C2312yes—
108C3×C4⋊Dic3C2312yes—
109C3×Dic3⋊C4C2314yes—
110C3×Dic3⋊C4C2318yes—
111C3×Dic3⋊C4C2314yes—
112C3×C4×Dic3C2314yes—
113C3×C4×Dic3C2314yes—
114C3×C4×Dic3C2314yes—
115C3×C4×Dic3C2314yes—
116C3×C4×Dic3C2314yes—
117#3483—31noSylow
118C2×C3×#41—22no—
119C2×C3×#41—34no—
120C2×C3×#41—32no—
121C2×C3×#41—34no—
122C2×C3×#41C2×D312yes—
123C2×C3×#41C22×C312yes—
124C2×#237C22×C314yes—
125C2×#235C22×C312yes—
126C2×#235C22×C314yes—
127C3×C4⋊Q8—21no—
128C3×C4⋊Q8—34no—
129C3×C4⋊Q8—34no—
130C3×C4⋊Q8C2×D311yescharacteristic
131C3×C4⋊Q8C22×C311yescharacteristic
132C3×#127—21no—
133C3×#127—34no—
134C3×#127—34no—
135C3×#127C2×D311yescharacteristic
136C3×#127C22×C311yescharacteristic
137C3×#124—22no—
138C3×#124—34no—
139C3×#124—34no—
140C3×#124—38no—
141C3×#124C2×D312yes—
142C3×#124C22×C312yes—
143C3×#121—24no—
144C3×#121—34no—
145C3×#121—38no—
146C3×#121—38no—
147C3×#121—34no—
148C3×#121—38no—
149C3×#121C2×D314yes—
150C3×#121C22×C314yes—
151C3×C4×Q8—34no—
152C3×C4×Q8—32no—
153C3×C4×Q8—32no—
154C3×C4×D4—24no—
155C3×C4×D4—34no—
156C3×C4×D4—38no—
157C3×C4×D4—34no—
158C3×C4×D4—34no—
159C3×C4×D4—38no—
160C3×C4×D4—32no—
161C3×C4×D4—32no—
162C3×C4×D4C2×D314yes—
163C3×C4×D4C22×C314yes—
164C3×#117—32no—
165C3×#117—34no—
166C3×#117—38no—
167C3×#117—34no—
168C2×C3×C42—21no—
169C2×C3×C42—34no—
170C2×C3×C42—34no—
171C2×C3×C42C2×D311yescharacteristic
172C2×C3×C42C22×C311yescharacteristic
173#947C22×C314yes—
174#944C22×C314yes—
175#940C22×C314yes—
176#940C22×C314yes—
177#939C22×C318yes—
178#939C22×C318yes—
179#938C22×C314yes—
180#936C22×C314yes—
181D4×Dic3C22×C314yes—
182C4×#67C22×C314yes—
183C4×#67C22×C314yes—
184C4×#67C22×C318yes—
185#926C22×C312yes—
186C2×C4×Dic3C22×C314yes—
187#895C22×C314yes—
188#892C22×C314yes—
189#888C22×C314yes—
190#887C22×C314yes—
191#885C22×C318yes—
192#884C22×C318yes—
193#67⋊C4C22×C318yes—
194#877C22×C318yes—
195C4×D12C22×C312yes—
196C4×D12C22×C312yes—
197#872C22×C314yes—
198C42×D3C22×C314yes—
199C4×Dic6C22×C312yes—
200C4×Dic6C22×C312yes—
201C23×C32C2411yescharacteristic
202C23×C32C2414yes—
203C22×C3×D3C2414yes—
204C32×Q8—42no—
205C32×D4—28no—
206C32×D4—42no—
207C32×D4—44no—
208C2×C32×C4—22no—
209C2×C32×C4—24no—
210C2×C32×C4—28no—
211C2×C32×C4C2412yes—
212C2×C32×C4C2414yes—
213C2×C32×C4C2411yescharacteristic
214C2×C32×C4C2412yes—
215C2×C32×C4C2411yescharacteristic
216C3×#67—28no—
217C3×#67—216no—
218C3×#67—48no—
219C2×C3×Dic3—216no—
220C2×C3×Dic3C2414yes—
221C2×C3×Dic3C2414yes—
222C2×C3×Dic3C2414yes—
223C3×D12—24no—
224C3×D12—44no—
225C3×C4×D3—28no—
226C3×C4×D3—28no—
227C3×C4×D3—28no—
228C3×Dic6—44no—
229C3×Dic6#4118yescomplemented
230Q82—31no—
231#555—31no—
232#555—34no—
233#555—34no—
234#550—32no—
235#550—34no—
236#532—34no—
237#532—34no—
238#532C3×D311yescharacteristic, complemented
239C4×#41—34no—
240C4×#41—32no—
241C3×#41—44no—
242C3×#41—44no—
243C3×#41—48no—
244C3×#41—68no—
245C3×#41—616no—
246C3×#41—128no—
247C2×C3×Q8—22no—
248C2×C3×Q8—32no—
249C2×C3×Q8—34no—
250C2×C3×Q8C22×D312yes—
251C2×C3×Q8C23×C312yes—
252C2×C3×D4—24no—
253C2×C3×D4—24no—
254C2×C3×D4—22no—
255C2×C3×D4—34no—
256C2×C3×D4—38no—
257C2×C3×D4—38no—
258C2×C3×D4—32no—
259C2×C3×D4—34no—
260C2×C3×D4C22×D314yes—
261C2×C3×D4C22×D312yes—
262C2×C3×D4C22×D314yes—
263C2×C3×D4C23×C314yes—
264C2×C3×D4C23×C314yes—
265C2×C3×D4C23×C312yes—
266C22×C3×C4—21no—
267C22×C3×C4—22no—
268C22×C3×C4—24no—
269C22×C3×C4—38no—
270C22×C3×C4—34no—
271C22×C3×C4—34no—
272C22×C3×C4—34no—
273C22×C3×C4C22×D312yes—
274C22×C3×C4C22×D314yes—
275C22×C3×C4C22×D311yescharacteristic
276C22×C3×C4C23×C314yes—
277C22×C3×C4C23×C312yes—
278C22×C3×C4C23×C311yescharacteristic
279C2×#67C23×C318yes—
280C2×#67C23×C314yes—
281C2×#67C23×C318yes—
282C22×Dic3C23×C318yes—
283#237—216no—
284#235—28no—
285#235—48no—
286C2×D12C23×C312yes—
287C2×D12C23×C314yes—
288C2×C4×D3C23×C314yes—
289C2×C4×D3C23×C314yes—
290C2×C4×D3C23×C314yes—
291C2×Dic6C23×C314yes—
292C2×Dic6C23×C312yes—
293C3×C4⋊C4—24no—
294C3×C4⋊C4—34no—
295C3×C4⋊C4—38no—
296C3×C4⋊C4—34no—
297C3×C4⋊C4—32no—
298C3×C4⋊C4—32no—
299C3×C4⋊C4C22×D314yes—
300C3×C4⋊C4C23×C314yes—
301C3×C22⋊C4—28no—
302C3×C22⋊C4—34no—
303C3×C22⋊C4—38no—
304C3×C22⋊C4—38no—
305C3×C22⋊C4—34no—
306C3×C22⋊C4—34no—
307C3×C22⋊C4C22×D318yes—
308C3×C22⋊C4C23×C318yes—
309C3×C42—22no—
310C3×C42—21no—
311C3×C42—21no—
312C3×C42—34no—
313C3×C42—34no—
314C3×C42—34no—
315C3×C42—34no—
316C3×C42—34no—
317C3×C42C22×D311yescharacteristic
318C3×C42C22×D311yescharacteristic
319C3×C42C22×D312yes—
320C3×C42C23×C311yescharacteristic
321C3×C42C23×C312yes—
322C3×C42C23×C311yescharacteristic
323#217C23×C318yes—
324#217C23×C314yes—
325(C2×D3)⋊C4C23×C314yes—
326(C2×D3)⋊C4C23×C314yes—
327(C2×D3)⋊C4C23×C318yes—
328C4⋊Dic3C23×C312yes—
329C4⋊Dic3C23×C312yes—
330Dic3⋊C4C23×C314yes—
331Dic3⋊C4C23×C314yes—
332Dic3⋊C4C23×C318yes—
333C4×Dic3C23×C314yes—
334C4×Dic3C23×C314yes—
335C4×Dic3C23×C314yes—
336C4×Dic3C23×C314yes—
337C4×Dic3C23×C314yes—
338C22×C32—21no—
339C22×C32—28no—
340C22×C32—22no—
341C22×C32—24no—
342C22×C32C2511yescharacteristic
343C2×C3×D3—24no—
344C2×C3×D3—24no—
345C2×C3×D3—24no—
346C32×C4—21no—
347C32×C4—22no—
348C32×C4—24no—
349C32×C4C2×#4114yes—
350C32×C4H(5,𝔽2)12yes—
351C3×Dic3—24no—
352C3×Dic3—24no—
353C3×Dic3C2×#4118yes—
354C2×#41—34no—
355C2×#41—32no—
356C2×#41—34no—
357C2×#41C2×C3×D312yes—
358C4⋊Q8—34no—
359C4⋊Q8—34no—
360C4⋊Q8C2×C3×D311yescharacteristic
361#127—34no—
362#127—34no—
363#127C2×C3×D311yescharacteristic
364#124—34no—
365#124—34no—
366#124—38no—
367#124C2×C3×D312yes—
368#121—34no—
369#121—38no—
370#121—38no—
371#121—34no—
372#121—38no—
373#121C2×C3×D314yes—
374C4×Q8—34no—
375C4×Q8—32no—
376C4×Q8—32no—
377C4×D4—34no—
378C4×D4—38no—
379C4×D4—34no—
380C4×D4—34no—
381C4×D4—38no—
382C4×D4—32no—
383C4×D4—32no—
384C4×D4C2×C3×D314yes—
385#117—32no—
386#117—34no—
387#117—38no—
388#117—34no—
389C2×C42—34no—
390C2×C42—34no—
391C2×C42C2×C3×D311yescharacteristic
392C23×C3—21no—
393C23×C3—24no—
394C23×C3—34no—
395C23×C3C23×D311yescharacteristic
396C23×C3C23×D314yes—
397C23×C3C24×C311yescharacteristic
398C23×C3C24×C314yes—
399C22×D3C24×C314yes—
400C3×Q8—38no—
401C3×Q8—42no—
402C3×Q8—42no—
403C3×Q8—44no—
404C3×Q8—124no—
405C3×D4—28no—
406C3×D4—28no—
407C3×D4—48no—
408C3×D4—42no—
409C3×D4—44no—
410C3×D4—44no—
411C3×D4—42no—
412C3×D4—44no—
413C3×D4—48no—
414C3×D4—68no—
415C3×D4—616no—
416C3×D4—64no—
417C3×D4—124no—
418C3×D4—128no—
419C2×C3×C4—24no—
420C2×C3×C4—22no—
421C2×C3×C4—28no—
422C2×C3×C4—24no—
423C2×C3×C4—24no—
424C2×C3×C4—28no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—22no—
430C2×C3×C4—24no—
431C2×C3×C4—34no—
432C2×C3×C4—34no—
433C2×C3×C4—34no—
434C2×C3×C4—44no—
435C2×C3×C4—48no—
436C2×C3×C4—68no—
437C2×C3×C4—616no—
438C2×C3×C4—68no—
439C2×C3×C4—68no—
440C2×C3×C4C23×D312yes—
441C2×C3×C4C23×D314yes—
442C2×C3×C4C23×D311yescharacteristic
443C2×C3×C4C23×D312yes—
444C2×C3×C4C23×D311yescharacteristic
445C2×C3×C4C24×C312yes—
446C2×C3×C4C24×C314yes—
447C2×C3×C4C24×C311yescharacteristic
448C2×C3×C4C24×C312yes—
449C2×C3×C4C24×C311yescharacteristic
450#67—28no—
451#67—216no—
452#67—48no—
453C2×Dic3—216no—
454C2×Dic3C24×C314yes—
455C2×Dic3C24×C314yes—
456C2×Dic3C24×C314yes—
457D12—24no—
458D12—44no—
459C4×D3—28no—
460C4×D3—28no—
461C4×D3—28no—
462Dic6—44no—
463Dic6C3×#4118yescomplemented
464C2×C32—41no—
465C2×C32—44no—
466C2×C32C22×#4112yes—
467C2×C32C2×H(5,𝔽2)11yescharacteristic
468C3×D3—44no—
469#41—44no—
470#41—68no—
471#41—616no—
472#41—128no—
473C2×Q8—32no—
474C2×Q8—34no—
475C2×Q8C22×C3×D312yes—
476C2×D4—34no—
477C2×D4—38no—
478C2×D4—38no—
479C2×D4—32no—
480C2×D4—34no—
481C2×D4C22×C3×D314yes—
482C2×D4C22×C3×D312yes—
483C2×D4C22×C3×D314yes—
484C22×C4—38no—
485C22×C4—34no—
486C22×C4—34no—
487C22×C4—34no—
488C22×C4C22×C3×D314yes—
489C22×C4C22×C3×D312yes—
490C22×C4C22×C3×D311yescharacteristic
491C4⋊C4—34no—
492C4⋊C4—32no—
493C4⋊C4—38no—
494C4⋊C4—34no—
495C4⋊C4—32no—
496C4⋊C4C22×C3×D314yes—
497C22⋊C4—34no—
498C22⋊C4—38no—
499C22⋊C4—38no—
500C22⋊C4—34no—
501C22⋊C4—34no—
502C22⋊C4C22×C3×D318yes—
503C42—34no—
504C42—34no—
505C42—34no—
506C42—34no—
507C42—34no—
508C42C22×C3×D312yes—
509C42C22×C3×D311yescharacteristic
510C42C22×C3×D311yescharacteristic
511C22×C3—22no—
512C22×C3—21no—
513C22×C3—24no—
514C22×C3—22no—
515C22×C3—28no—
516C22×C3—28no—
517C22×C3—22no—
518C22×C3—24no—
519C22×C3—21no—
520C22×C3—21no—
521C22×C3—42no—
522C22×C3—44no—
523C22×C3—48no—
524C22×C3—64no—
525C22×C3—64no—
526C22×C3—64no—
527C22×C3C24×D311yescharacteristic
528C22×C3C25×C311yescharacteristic
529C2×D3—24no—
530C2×D3—24no—
531C2×D3—24no—
532C3×C4—22no—
533C3×C4—21no—
534C3×C4—24no—
535C3×C4—22no—
536C3×C4—22no—
537C3×C4—24no—
538C3×C4—21no—
539C3×C4—22no—
540C3×C4—24no—
541C3×C4—38no—
542C3×C4—44no—
543C3×C4—42no—
544C3×C4—64no—
545C3×C4—64no—
546C3×C4#100412yes—
547C3×C4D3×#4114yes—
548C3×C4C2×C3×#4114yes—
549C3×C4C3×H(5,𝔽2)12yes—
550Dic3—24no—
551Dic3—24no—
552Dic3C2×C3×#4118yes—
553C32#348311yescharacteristic, complemented, Sylow
554C23—34no—
555C23C23×C3×D311yescharacteristic
556C23C23×C3×D314yes—
557Q8—38no—
558Q8—42no—
559Q8—124no—
560D4—28no—
561D4—42no—
562D4—44no—
563D4—68no—
564D4—616no—
565D4—64no—
566D4—124no—
567D4—128no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—34no—
572C2×C4—34no—
573C2×C4—34no—
574C2×C4—68no—
575C2×C4—616no—
576C2×C4—68no—
577C2×C4—68no—
578C2×C4C23×C3×D312yes—
579C2×C4C23×C3×D314yes—
580C2×C4C23×C3×D311yescharacteristic
581C2×C4C23×C3×D311yescharacteristic
582C2×C4C23×C3×D312yes—
583C2×C3—21no—
584C2×C3—22no—
585C2×C3—42no—
586C2×C3—41no—
587C2×C3—44no—
588C2×C3—41no—
589C2×C3—44no—
590C2×C3—84no—
591C2×C3—124no—
592C2×C3C2×#100411yescharacteristic
593C2×C3C2×D3×#4112yes—
594C2×C3C22×C3×#4112yes—
595C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
596D3—44no—
597C22—21no—
598C22—24no—
599C22—22no—
600C22—28no—
601C22—64no—
602C22—64no—
603C22—64no—
604C22C24×C3×D311yescharacteristic
605C4—21no—
606C4—24no—
607C4—22no—
608C4—38no—
609C4—64no—
610C4—64no—
611C4C3×#100412yes—
612C4C3×D3×#4114yes—
613C3—21no—
614C3#8622111yescharacteristic, direct product factor
615C3C3×#348311yescharacteristic, complemented
616C2—41no—
617C2—44no—
618C2—124no—
619C2C2×C3×#100411yescharacteristic
620C2C2×C3×D3×#4112yes—
subgroupG/H
centerC2×C3×C4C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#532C2
socleC22×C32C25
hypercenterC3×#532D3
hypocenterC3C3×#3483
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×#532

Complements

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

NH ≅ G/Ntype
1. C32×#532C2semidirect
2. C3×#5832C2semidirect
3. C3×#5734C2semidirect
4. C3×#5722C2semidirect
5. C3×#5718C2semidirect
6. C3×#5670C2semidirect
7. C3×#5616C2semidirect
8. C3×#5599C2semidirect
9. C3×#5548C2semidirect
10. C3×C4×#237C2semidirect
11. C3×C4×#235C2semidirect
13. #86221C3direct (N × H)
61. C3×#532D3semidirect
62. C3×#532C2×C3semidirect
65. #5832C2×C3semidirect
66. #5734C2×C3semidirect
67. #5722C2×C3semidirect
68. #5718C2×C3semidirect
69. #5670C2×C3semidirect
70. #5616C2×C3semidirect
71. #5599C2×C3semidirect
72. #5548C2×C3semidirect
73. C4×#237C2×C3semidirect
74. C4×#235C2×C3semidirect
229. C3×Dic6#41semidirect
238. #532C3×D3semidirect
463. Dic6C3×#41semidirect
553. C32#3483semidirect
614. C3#86221direct (N × H)
615. C3C3×#3483semidirect

Maximal subgroups

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

subgroupindex
C32×#5322
C3×#58322
C3×#57342
C3×#57222
C3×#57182
C3×#56702
C3×#56162
C3×#55992
C3×#55482
C3×C4×#2372
C3×C4×#2352
C3×#34833
#862213