Finite Group Databasebeta

← C2×C3×#5664 #11903809

Subgroups — C2×C3×#5664 #11903809

Order 1152.

2651 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#123C211yesmaximal, characteristic, complemented
2C2×C3×#940C211yesmaximal, characteristic
3C2×C3×#935C211yesmaximal, characteristic, complemented
4C2×C3×C4×#67C212yesmaximal, complemented
5C3×#5664C2116yesmaximal, direct product factor
6C2×C3×#894C212yesmaximal, complemented
7C2×C3×#887C212yesmaximal, complemented
8C2×C3×#885C212yesmaximal, complemented
9C2×C3×#885C212yesmaximal, complemented
10C2×C3×#67⋊C4C212yesmaximal, complemented
11C2×C3×#559—31nomaximal
12C2×#5664C311yesmaximal, characteristic, direct product factor
13C22×C32×D4C2211yescharacteristic
14C23×C32×C4C2211yescharacteristic
15C22×C3×#67C2212yes—
16C22×C3×#67C2212yes—
17C23×C3×Dic3C2211yescharacteristic
18C22×C3×D12C2211yescharacteristic
19C22×C3×C4×D3C2212yes—
20C22×C3×Dic6C2212yes—
21C32×#123C2218yescomplemented
22C2×C32×C4⋊C4C2212yes—
23C2×C32×C22⋊C4C2211yescharacteristic
24C2×C32×C22⋊C4C2212yes—
25C2×C3×#217C2211yescharacteristic
26C2×C3×#217C2212yes—
27C3×#940C2218yes—
28C3×#935C2218yescomplemented
29C3×C4×#67C22116yescomplemented
30C2×C3×(C2×D3)⋊C4C2212yes—
31C2×C3×(C2×D3)⋊C4C2212yes—
32C2×C3×(C2×D3)⋊C4C2212yes—
33C2×C3×Dic3⋊C4C2212yes—
34C2×C3×Dic3⋊C4C2212yes—
35C2×C3×C4×Dic3C2212yes—
36C2×C3×C4×Dic3C2212yes—
37C2×C3×C4×Dic3C2211yescharacteristic
38C3×#894C22116yescomplemented
39C3×#887C22116yescomplemented
40C3×#885C22116yescomplemented
41C3×#885C22116yescomplemented
42C3×#67⋊C4C22116yescomplemented
43C3×#559—316no—
44C2×C3×#127—31no—
45C2×C3×#124—32no—
46C2×C3×#124—32no—
47C2×C3×#123—21no—
48C2×C3×#123—31no—
49C2×C3×#123—32no—
50C2×C3×#123D311yescharacteristic, complemented
51C2×C3×#123C2×C311yescharacteristic, complemented
52C2×C3×C4×Q8—32no—
53C2×C3×C4×D4—32no—
54C2×C3×C4×D4—32no—
55C2×#940C2×C311yescharacteristic
56C2×#935C2×C311yescharacteristic, complemented
57C2×C4×#67C2×C312yescomplemented
58#5664C2×C3116yesdirect product factor
59C2×#894C2×C312yescomplemented
60C2×#887C2×C312yescomplemented
61C2×#885C2×C312yescomplemented
62C2×#885C2×C312yescomplemented
63C2×#67⋊C4C2×C312yescomplemented
64C24×C32C2311yescharacteristic
65C24×C32C2311yescharacteristic
66C23×C3×D3C2312yes—
67C2×C32×D4—42no—
68C2×C32×D4C2314yes—
69C22×C32×C4—42no—
70C22×C32×C4C2311yescharacteristic
71C22×C32×C4C2312yes—
72C22×C32×C4C2312yes—
73C22×C32×C4C2314yes—
74C2×C3×#67—28no—
75C2×C3×#67—28no—
76C2×C3×#67C2318yes—
77C2×C3×#67C2318yes—
78C22×C3×Dic3—42no—
79C22×C3×Dic3C2312yes—
80C22×C3×Dic3C2312yes—
81C22×C3×Dic3C2314yes—
82C22×C3×Dic3C2312yes—
83C2×C3×D12—42no—
84C2×C3×D12C2314yes—
85C2×C3×C4×D3—44no—
86C2×C3×C4×D3C2318yes—
87C2×C3×Dic6—28no—
88C2×C3×Dic6C2318yes—
89C32×C4⋊C4C2318yes—
90C32×C22⋊C4C2318yes—
91C32×C22⋊C4C2314yes—
92C3×#217C2314yes—
93C3×#217C2318yes—
94C3×(C2×D3)⋊C4C2318yes—
95C3×(C2×D3)⋊C4C2318yes—
96C3×(C2×D3)⋊C4C2318yes—
97C3×Dic3⋊C4C2318yes—
98C3×Dic3⋊C4C2318yes—
99C3×C4×Dic3C2318yes—
100C3×C4×Dic3C2318yes—
101C3×C4×Dic3C2314yes—
102C2×#559—31noSylow
103C22×C3×Q8—32no—
104C22×C3×D4—21no—
105C22×C3×D4—32no—
106C22×C3×D4—32no—
107C22×C3×D4—31no—
108C22×C3×D4C2×D311yescharacteristic
109C22×C3×D4C22×C311yescharacteristic
110C23×C3×C4—21no—
111C23×C3×C4—31no—
112C23×C3×C4—32no—
113C23×C3×C4C2×D311yescharacteristic
114C23×C3×C4C22×C311yescharacteristic
115C22×#67C22×C312yes—
116C22×#67C22×C312yes—
117C23×Dic3C22×C311yescharacteristic
118C22×D12C22×C311yescharacteristic
119C22×C4×D3C22×C312yes—
120C22×Dic6C22×C312yes—
121C3×#127—38no—
122C3×#124—316no—
123C3×#124—316no—
124C3×#123—28no—
125C3×#123—38no—
126C3×#123—316no—
127C3×#123C2×D318yescomplemented
128C3×#123C22×C318yescomplemented
129C3×C4×Q8—316no—
130C3×C4×D4—316no—
131C3×C4×D4—316no—
132C2×C3×C4⋊C4—22no—
133C2×C3×C4⋊C4—32no—
134C2×C3×C4⋊C4—32no—
135C2×C3×C4⋊C4C2×D312yes—
136C2×C3×C4⋊C4C22×C312yes—
137C2×C3×C22⋊C4—22no—
138C2×C3×C22⋊C4—21no—
139C2×C3×C22⋊C4—32no—
140C2×C3×C22⋊C4—31no—
141C2×C3×C22⋊C4—32no—
142C2×C3×C22⋊C4—32no—
143C2×C3×C22⋊C4—32no—
144C2×C3×C22⋊C4C2×D312yes—
145C2×C3×C22⋊C4C2×D311yescharacteristic
146C2×C3×C22⋊C4C22×C312yes—
147C2×C3×C22⋊C4C22×C311yescharacteristic
148C2×C3×C42—32no—
149C2×C3×C42—32no—
150C2×C3×C42—31no—
151C2×#217C22×C311yescharacteristic
152C2×#217C22×C312yes—
153#940C22×C318yes—
154#935C22×C318yescomplemented
155C4×#67C22×C3116yescomplemented
156C2×(C2×D3)⋊C4C22×C312yes—
157C2×(C2×D3)⋊C4C22×C312yes—
158C2×(C2×D3)⋊C4C22×C312yes—
159C2×Dic3⋊C4C22×C312yes—
160C2×Dic3⋊C4C22×C312yes—
161C2×C4×Dic3C22×C312yes—
162C2×C4×Dic3C22×C312yes—
163C2×C4×Dic3C22×C311yescharacteristic
164#894C22×C3116yescomplemented
165#887C22×C3116yescomplemented
166#885C22×C3116yescomplemented
167#885C22×C3116yescomplemented
168#67⋊C4C22×C3116yescomplemented
169C23×C32—22no—
170C23×C32—24no—
171C23×C32—24no—
172C23×C32—22no—
173C23×C32C2412yes—
174C23×C32C2412yes—
175C23×C32C2411yescharacteristic
176C22×C3×D3—24no—
177C22×C3×D3—24no—
178C22×C3×D3—24no—
179C22×C3×D3C2414yes—
180C32×D4—44no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—22no—
184C2×C32×C4—44no—
185C2×C32×C4C2412yes—
186C2×C32×C4C2414yes—
187C2×C32×C4C2414yes—
188C3×#67—216no—
189C3×#67—216no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—24no—
192C2×C3×Dic3—44no—
193C2×C3×Dic3#4118yes—
194C2×C3×Dic3C2414yes—
195C2×C3×Dic3C2414yes—
196C2×C3×Dic3C2414yes—
197C3×D12—44no—
198C3×C4×D3—48no—
199C3×Dic6—216no—
200#559—316no—
201C2×#127—31no—
202C2×#124—32no—
203C2×#124—32no—
204C2×#123—31no—
205C2×#123—32no—
206C2×#123C3×D311yescharacteristic, complemented
207C2×C4×Q8—32no—
208C2×C4×D4—32no—
209C2×C4×D4—32no—
210C24×C3—21no—
211C24×C3—21no—
212C24×C3—32no—
213C24×C3C22×D311yescharacteristic
214C24×C3C22×D311yescharacteristic
215C24×C3C23×C311yescharacteristic
216C24×C3C23×C311yescharacteristic
217C23×D3C23×C312yes—
218C2×C3×Q8—38no—
219C2×C3×Q8—68no—
220C2×C3×D4—24no—
221C2×C3×D4—38no—
222C2×C3×D4—38no—
223C2×C3×D4—34no—
224C2×C3×D4—42no—
225C2×C3×D4—42no—
226C2×C3×D4—44no—
227C2×C3×D4—68no—
228C2×C3×D4—68no—
229C2×C3×D4—122no—
230C2×C3×D4C22×D314yes—
231C2×C3×D4C23×C314yes—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—24no—
236C22×C3×C4—32no—
237C22×C3×C4—32no—
238C22×C3×C4—32no—
239C22×C3×C4—34no—
240C22×C3×C4—38no—
241C22×C3×C4—42no—
242C22×C3×C4—42no—
243C22×C3×C4—44no—
244C22×C3×C4—124no—
245C22×C3×C4—122no—
246C22×C3×C4C22×D312yes—
247C22×C3×C4C22×D312yes—
248C22×C3×C4C22×D311yescharacteristic
249C22×C3×C4C22×D314yes—
250C22×C3×C4C23×C311yescharacteristic
251C22×C3×C4C23×C312yes—
252C22×C3×C4C23×C312yes—
253C22×C3×C4C23×C314yes—
254C2×#67—28no—
255C2×#67—28no—
256C2×#67C23×C318yes—
257C2×#67C23×C318yes—
258C22×Dic3—42no—
259C22×Dic3C23×C312yes—
260C22×Dic3C23×C312yes—
261C22×Dic3C23×C314yes—
262C22×Dic3C23×C312yes—
263C2×D12—42no—
264C2×D12C23×C314yes—
265C2×C4×D3—44no—
266C2×C4×D3C23×C318yes—
267C2×Dic6—28no—
268C2×Dic6C23×C318yes—
269C3×C4⋊C4—28no—
270C3×C4⋊C4—38no—
271C3×C4⋊C4—38no—
272C3×C4⋊C4C22×D318yes—
273C3×C4⋊C4C23×C318yes—
274C3×C22⋊C4—28no—
275C3×C22⋊C4—24no—
276C3×C22⋊C4—38no—
277C3×C22⋊C4—34no—
278C3×C22⋊C4—38no—
279C3×C22⋊C4—38no—
280C3×C22⋊C4—38no—
281C3×C22⋊C4C22×D318yes—
282C3×C22⋊C4C22×D314yes—
283C3×C22⋊C4C23×C318yes—
284C3×C22⋊C4C23×C314yes—
285C3×C42—38no—
286C3×C42—38no—
287C3×C42—34no—
288#217C23×C314yes—
289#217C23×C318yes—
290(C2×D3)⋊C4C23×C318yes—
291(C2×D3)⋊C4C23×C318yes—
292(C2×D3)⋊C4C23×C318yes—
293Dic3⋊C4C23×C318yes—
294Dic3⋊C4C23×C318yes—
295C4×Dic3C23×C318yes—
296C4×Dic3C23×C318yes—
297C4×Dic3C23×C314yes—
298C22×C32—22no—
299C22×C32—24no—
300C22×C32—24no—
301C22×C32—22no—
302C22×C32—44no—
303C22×C32—44no—
304C22×C32C2×#4114yes—
305C22×C32H(5,𝔽2)12yes—
306C22×C32C2511yescharacteristic
307C2×C3×D3—24no—
308C2×C3×D3—24no—
309C2×C3×D3—24no—
310C2×C3×D3—48no—
311C32×C4—24no—
312C32×C4—24no—
313C32×C4—22no—
314C3×Dic3—24no—
315C3×Dic3—24no—
316C3×Dic3C2×#4118yes—
317C22×Q8—32no—
318C22×D4—32no—
319C22×D4—32no—
320C22×D4—31no—
321C22×D4C2×C3×D311yescharacteristic
322C23×C4—31no—
323C23×C4—32no—
324C23×C4C2×C3×D311yescharacteristic
325#127—38no—
326#124—316no—
327#124—316no—
328#123—38no—
329#123—316no—
330#123C2×C3×D318yescomplemented
331C4×Q8—316no—
332C4×D4—316no—
333C4×D4—316no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4C2×C3×D312yes—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4C2×C3×D312yes—
343C2×C22⋊C4C2×C3×D311yescharacteristic
344C2×C42—32no—
345C2×C42—32no—
346C2×C42—31no—
347C23×C3—24no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—24no—
351C23×C3—24no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—34no—
360C23×C3—44no—
361C23×C3—42no—
362C23×C3—44no—
363C23×C3—64no—
364C23×C3—64no—
365C23×C3—64no—
366C23×C3C23×D312yes—
367C23×C3C23×D312yes—
368C23×C3C23×D311yescharacteristic
369C23×C3C24×C312yes—
370C23×C3C24×C312yes—
371C23×C3C24×C311yescharacteristic
372C22×D3—24no—
373C22×D3—24no—
374C22×D3—24no—
375C22×D3C24×C314yes—
376C3×Q8—616no—
377C3×D4—44no—
378C3×D4—44no—
379C3×D4—48no—
380C3×D4—616no—
381C3×D4—616no—
382C3×D4—124no—
383C2×C3×C4—24no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—24no—
391C2×C3×C4—22no—
392C2×C3×C4—38no—
393C2×C3×C4—34no—
394C2×C3×C4—34no—
395C2×C3×C4—34no—
396C2×C3×C4—44no—
397C2×C3×C4—44no—
398C2×C3×C4—42no—
399C2×C3×C4—44no—
400C2×C3×C4—44no—
401C2×C3×C4—48no—
402C2×C3×C4—64no—
403C2×C3×C4—64no—
404C2×C3×C4—128no—
405C2×C3×C4—124no—
406C2×C3×C4C23×D314yes—
407C2×C3×C4C23×D314yes—
408C2×C3×C4C23×D312yes—
409C2×C3×C4C24×C312yes—
410C2×C3×C4C24×C314yes—
411C2×C3×C4C24×C314yes—
412#67—216no—
413#67—216no—
414C2×Dic3—24no—
415C2×Dic3—24no—
416C2×Dic3—44no—
417C2×Dic3C3×#4118yes—
418C2×Dic3C24×C314yes—
419C2×Dic3C24×C314yes—
420C2×Dic3C24×C314yes—
421D12—44no—
422C4×D3—48no—
423Dic6—216no—
424C2×C32—42no—
425C2×C32—42no—
426C2×C32#55914yescomplemented
427C2×C32C22×#4112yes—
428C2×C32C2×H(5,𝔽2)11yescharacteristic
429C3×D3—44no—
430C24—32no—
431C24C22×C3×D311yescharacteristic
432C24C22×C3×D311yescharacteristic
433C2×Q8—38no—
434C2×Q8—68no—
435C2×D4—38no—
436C2×D4—38no—
437C2×D4—34no—
438C2×D4—42no—
439C2×D4—68no—
440C2×D4—68no—
441C2×D4—122no—
442C2×D4C22×C3×D314yes—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—38no—
448C22×C4—42no—
449C22×C4—124no—
450C22×C4—122no—
451C22×C4C22×C3×D312yes—
452C22×C4C22×C3×D312yes—
453C22×C4C22×C3×D311yescharacteristic
454C22×C4C22×C3×D314yes—
455C4⋊C4—38no—
456C4⋊C4—38no—
457C4⋊C4C22×C3×D318yes—
458C22⋊C4—38no—
459C22⋊C4—34no—
460C22⋊C4—38no—
461C22⋊C4—38no—
462C22⋊C4—38no—
463C22⋊C4C22×C3×D318yes—
464C22⋊C4C22×C3×D314yes—
465C42—38no—
466C42—38no—
467C42—34no—
468C22×C3—24no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—24no—
479C22×C3—21no—
480C22×C3—44no—
481C22×C3—42no—
482C22×C3—44no—
483C22×C3—48no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—64no—
489C22×C3—64no—
490C22×C3—64no—
491C22×C3—84no—
492C22×C3—128no—
493C22×C3#100412yes—
494C22×C3D3×#4114yes—
495C22×C3C2×C3×#4114yes—
496C22×C3C3×H(5,𝔽2)12yes—
497C22×C3C24×D311yescharacteristic
498C22×C3C25×C311yescharacteristic
499C2×D3—24no—
500C2×D3—24no—
501C2×D3—24no—
502C2×D3—48no—
503C3×C4—24no—
504C3×C4—22no—
505C3×C4—24no—
506C3×C4—24no—
507C3×C4—24no—
508C3×C4—22no—
509C3×C4—38no—
510C3×C4—44no—
511C3×C4—42no—
512C3×C4—44no—
513C3×C4—64no—
514C3×C4—64no—
515Dic3—24no—
516Dic3—24no—
517Dic3C2×C3×#4118yes—
518C32C2×#55911yescharacteristic, complemented, Sylow
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—34no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23C23×C3×D312yes—
528C23C23×C3×D312yes—
529C23C23×C3×D311yescharacteristic
530Q8—616no—
531D4—44no—
532D4—616no—
533D4—616no—
534D4—124no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—38no—
539C2×C4—34no—
540C2×C4—34no—
541C2×C4—34no—
542C2×C4—44no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—128no—
546C2×C4—124no—
547C2×C4C23×C3×D314yes—
548C2×C4C23×C3×D314yes—
549C2×C4C23×C3×D312yes—
550C2×C3—21no—
551C2×C3—22no—
552C2×C3—24no—
553C2×C3—44no—
554C2×C3—42no—
555C2×C3—42no—
556C2×C3—42no—
557C2×C3—42no—
558C2×C3—82no—
559C2×C3—124no—
560C2×C3#566414yesdirect product factor
561C2×C3C3×#55914yescomplemented
562C2×C3C2×#100411yescharacteristic
563C2×C3C2×D3×#4112yes—
564C2×C3C22×C3×#4112yes—
565C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
566D3—44no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—44no—
572C22—44no—
573C22—64no—
574C22—64no—
575C22—64no—
576C22—128no—
577C22C3×#100412yes—
578C22C3×D3×#4114yes—
579C22C24×C3×D311yescharacteristic
580C4—24no—
581C4—22no—
582C4—24no—
583C4—38no—
584C4—64no—
585C4—64no—
586C3—21no—
587C3C2×#566411yescharacteristic, direct product factor
588C3C2×C3×#55911yescharacteristic, complemented
589C2—42no—
590C2—42no—
591C2—124no—
592C2C3×#566414yesdirect product factor
593C2C2×C3×#100411yescharacteristic
594C2C2×C3×D3×#4112yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#123C2
socleC23×C32C24
prime radicalC2×C3×#940C2
hypercenterC2×C3×#123D3
hypocenterC3C2×C3×#559
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#123

Complements

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

NH ≅ G/Ntype
1. C2×C32×#123C2semidirect
3. C2×C3×#935C2semidirect
4. C2×C3×C4×#67C2semidirect
5. C3×#5664C2direct (N × H)
6. C2×C3×#894C2semidirect
7. C2×C3×#887C2semidirect
8. C2×C3×#885C2semidirect
9. C2×C3×#885C2semidirect
10. C2×C3×#67⋊C4C2semidirect
12. C2×#5664C3direct (N × H)
21. C32×#123C22semidirect
28. C3×#935C22semidirect
29. C3×C4×#67C22semidirect
38. C3×#894C22semidirect
39. C3×#887C22semidirect
40. C3×#885C22semidirect
41. C3×#885C22semidirect
42. C3×#67⋊C4C22semidirect
50. C2×C3×#123D3semidirect
51. C2×C3×#123C2×C3semidirect
56. C2×#935C2×C3semidirect
57. C2×C4×#67C2×C3semidirect
58. #5664C2×C3direct (N × H)
59. C2×#894C2×C3semidirect
60. C2×#887C2×C3semidirect
61. C2×#885C2×C3semidirect
62. C2×#885C2×C3semidirect
63. C2×#67⋊C4C2×C3semidirect
127. C3×#123C2×D3semidirect
128. C3×#123C22×C3semidirect
154. #935C22×C3semidirect
155. C4×#67C22×C3semidirect
164. #894C22×C3semidirect
165. #887C22×C3semidirect
166. #885C22×C3semidirect
167. #885C22×C3semidirect
168. #67⋊C4C22×C3semidirect
206. C2×#123C3×D3semidirect
330. #123C2×C3×D3semidirect
426. C2×C32#559semidirect
518. C32C2×#559semidirect
560. C2×C3#5664direct (N × H)
561. C2×C3C3×#559semidirect
587. C3C2×#5664direct (N × H)
588. C3C2×C3×#559semidirect
592. C2C3×#5664direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1232
C2×C3×#9402
C2×C3×#9352
C2×C3×C4×#672
C3×#56642
C2×C3×#8942
C2×C3×#8872
C2×C3×#8852
C2×C3×#67⋊C42
C2×C3×#5593
C2×#56643