Finite Group Databasebeta

← C2×C3×#5232 #11857900

Subgroups — C2×C3×#5232 #11857900

Order 1152.

2077 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C211yesmaximal, characteristic
2C22×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C22×C3×#208C211yesmaximal, characteristic, complemented
4C3×#5232C218yesmaximal, direct product factor
5C2×C3×#835C212yesmaximal, complemented
6C2×C3×#832C212yesmaximal, complemented
7C2×C3×#417—31nomaximal
8C2×#5232C311yesmaximal, characteristic, direct product factor
9C2×C32×#41—24no—
10C2×C32×#41—22no—
11C2×C32×#41C418yescomplemented
12C2×C32×#41C2214yes—
13C22×C32×Q8C2211yescharacteristic
14C22×C32×D4—21no—
15C22×C32×D4C2211yescharacteristic
16C23×C32×C4—21no—
17C23×C32×C4C2211yescharacteristic
18C23×C3×Dic3—21no—
19C2×C3×C4⋊Dic3—22no—
20C2×C3×C4⋊Dic3—21no—
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×C4⋊Dic3C2214yescomplemented
23C2×C3×#208—24no—
24C2×C3×#208C2214yescomplemented
25C22×C3×C3⋊C8C2212yes—
26C3×#835C2218yescomplemented
27C3×#832C2218yescomplemented
28C22×C3×#41—21no—
29C22×C3×#41D311yescharacteristic
30C22×C3×#41C2×C311yescharacteristic
31C22×C3×M16—31no—
32C22×C3×C4⋊C4—31no—
33C22×C4⋊Dic3C2×C311yescharacteristic, complemented
34C22×#208C2×C311yescharacteristic, complemented
35C3×#417—38no—
36C2×C3×#103—32no—
37C2×C3×#102—32no—
38#5232C2×C318yesdirect product factor
39C2×#835C2×C312yescomplemented
40C2×#832C2×C312yescomplemented
41C24×C32—21no—
42C24×C32D411yescharacteristic
43C32×#41—216no—
44C32×#41—28no—
45C32×#41C2×C4116yescomplemented
46C2×C32×Q8—22no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×C414yescomplemented
49C2×C32×Q8C2312yes—
50C2×C32×D4—28no—
51C2×C32×D4—24no—
52C2×C32×D4—21no—
53C2×C32×D4—22no—
54C2×C32×D4C2×C414yescomplemented
55C2×C32×D4C2312yes—
56C22×C32×C4—21no—
57C22×C32×C4—28no—
58C22×C32×C4—24no—
59C22×C32×C4C2×C414yes—
60C22×C32×C4D411yescharacteristic
61C22×C32×C4D412yes—
62C22×C32×C4D414yes—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2312yes—
65C22×C3×Dic3—22no—
66C22×C3×Dic3—24no—
67C22×C3×Dic3—44no—
68C3×C4⋊Dic3—22no—
69C3×C4⋊Dic3—24no—
70C3×C4⋊Dic3C2314yes—
71C3×#208—28no—
72C2×C3×C3⋊C8—24no—
73C2×C3×C3⋊C8C2314yes—
74C2×#417—31noSylow
75C2×C3×#41—28no—
76C2×C3×#41—24no—
77C2×C3×#41—24no—
78C2×C3×#41—22no—
79C2×C3×#41—24no—
80C2×C3×#41—22no—
81C2×C3×#41—28no—
82C2×C3×#41—24no—
83C2×C3×#41Dic318yescomplemented
84C2×C3×#41C3×C418yescomplemented
85C2×C3×#41C2×D314yes—
86C2×C3×#41C22×C314yes—
87C22×C3×Q8—21no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—22no—
91C22×C3×D4—21no—
92C22×C3×D4—21no—
93C22×C3×D4—21no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C23×C3×C4—22no—
97C23×C3×C4—21no—
98C23×C3×C4—21no—
99C23×C3×C4—21no—
100C23×C3×C4—61no—
101C23×C3×C4C2×D311yescharacteristic
102C23×C3×C4C22×C311yescharacteristic
103C23×Dic3—21no—
104C2×C3×M16—34no—
105C2×C3×M16—64no—
106C22×C3×C8—32no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C4⋊C4—34no—
109C2×C3×C4⋊C4—62no—
110C2×C3×C4⋊C4—61no—
111C2×C4⋊Dic3—22no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C22×C312yes—
114C2×C4⋊Dic3C22×C314yescomplemented
115C2×#208—24no—
116C2×#208C22×C314yescomplemented
117C22×C3⋊C8C22×C312yes—
118C3×#103—38no—
119C3×#102—38no—
120#835C22×C318yescomplemented
121#832C22×C318yescomplemented
122C23×C32—24no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—42no—
127C23×C32—42no—
128C23×C32C22⋊C414yes—
129C23×C32C2×D412yes—
130C23×C32C2×D411yescharacteristic
131C32×Q8—24no—
132C32×Q8—22no—
133C32×Q8C22×C414yescomplemented
134C32×D4—216no—
135C32×D4—24no—
136C32×D4—22no—
137C32×D4C22×C414yescomplemented
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4—216no—
141C2×C32×C4C22⋊C414yes—
142C2×C32×C4C22⋊C418yes—
143C2×C32×C4C22×C412yes—
144C2×C32×C4C22×C414yes—
145C2×C32×C4C2×D412yes—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D414yes—
149C2×C32×C4C2411yescharacteristic
150C2×C3×Dic3—24no—
151C2×C3×Dic3—48no—
152C2×C3×Dic3—44no—
153C3×C3⋊C8—24no—
154C22×#41C3×D311yescharacteristic
155C22×M16—31no—
156C22×C4⋊C4—31no—
157#417—38no—
158C2×#103—32no—
159C2×#102—32no—
160C24×C3—22no—
161C24×C3—21no—
162C24×C3—21no—
163C24×C3—21no—
164C24×C3#6711yescharacteristic
165C24×C3C3×D411yescharacteristic
166C3×#41—232no—
167C3×#41—216no—
168C3×#41—216no—
169C3×#41—216no—
170C3×#41—28no—
171C3×#41—28no—
172C3×#41—216no—
173C3×#41C2×Dic3116yescomplemented
174C3×#41C2×C3×C4116yescomplemented
175C2×C3×Q8—24no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—22no—
183C2×C3×Q8C2×Dic314yescomplemented
184C2×C3×Q8C2×C3×C414yescomplemented
185C2×C3×Q8C22×D312yes—
186C2×C3×Q8C23×C312yes—
187C2×C3×D4—216no—
188C2×C3×D4—28no—
189C2×C3×D4—22no—
190C2×C3×D4—24no—
191C2×C3×D4—28no—
192C2×C3×D4—28no—
193C2×C3×D4—24no—
194C2×C3×D4—24no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4C2×Dic314yescomplemented
202C2×C3×D4C2×C3×C414yescomplemented
203C2×C3×D4C22×D312yes—
204C2×C3×D4C23×C312yes—
205C22×C3×C4—22no—
206C22×C3×C4—216no—
207C22×C3×C4—28no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—28no—
211C22×C3×C4—28no—
212C22×C3×C4—24no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—21no—
216C22×C3×C4—22no—
217C22×C3×C4—24no—
218C22×C3×C4—21no—
219C22×C3×C4—22no—
220C22×C3×C4—64no—
221C22×C3×C4—62no—
222C22×C3×C4—124no—
223C22×C3×C4C2×Dic314yes—
224C22×C3×C4#6712yes—
225C22×C3×C4#6711yescharacteristic
226C22×C3×C4#6714yes—
227C22×C3×C4C2×C3×C414yes—
228C22×C3×C4C3×D411yescharacteristic
229C22×C3×C4C3×D412yes—
230C22×C3×C4C3×D414yes—
231C22×C3×C4C22×D312yes—
232C22×C3×C4C22×D311yescharacteristic
233C22×C3×C4C23×C311yescharacteristic
234C22×C3×C4C23×C312yes—
235C22×Dic3—22no—
236C22×Dic3—24no—
237C22×Dic3—44no—
238C3×M16—68no—
239C2×C3×C8—34no—
240C2×C3×C8—64no—
241C3×C4⋊C4—34no—
242C3×C4⋊C4—64no—
243C3×C4⋊C4—62no—
244C4⋊Dic3—22no—
245C4⋊Dic3—24no—
246C4⋊Dic3C23×C314yes—
247#208—28no—
248C2×C3⋊C8—24no—
249C2×C3⋊C8C23×C314yes—
250C22×C32—24no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—28no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—48no—
257C22×C32C2×C22⋊C414yes—
258C22×C32C2×C22⋊C412yes—
259C22×C32AGL(1,ℤ/8)12yes—
260C22×C32#13712yes—
261C22×C32C22×D411yescharacteristic
262C32×C4—24no—
263C32×C4C2×C22⋊C414yes—
264C32×C4C2×C22⋊C412yes—
265C32×C4C23×C411yescharacteristic
266C32×C4C22×D411yescharacteristic
267C3×Dic3—44no—
268C2×#41—24no—
269C2×#41—22no—
270C2×#41C3×Dic318yescomplemented
271C2×#41C2×C3×D314yes—
272C22×Q8C2×C3×D311yescharacteristic
273C22×D4—21no—
274C22×D4C2×C3×D311yescharacteristic
275C23×C4—21no—
276C23×C4—61no—
277C23×C4C2×C3×D311yescharacteristic
278C2×M16—34no—
279C2×M16—64no—
280C22×C8—32no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—34no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—62no—
285#103—38no—
286#102—38no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—28no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—21no—
299C23×C3—22no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3—44no—
303C23×C3—44no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3#21714yes—
309C23×C3C3×C22⋊C414yes—
310C23×C3C2×#6712yes—
311C23×C3C2×#6711yescharacteristic
312C23×C3C2×C3×D412yes—
313C23×C3C2×C3×D411yescharacteristic
314C3×Q8—28no—
315C3×Q8—24no—
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—22no—
319C3×Q8—22no—
320C3×Q8—24no—
321C3×Q8C22×Dic314yescomplemented
322C3×Q8C22×C3×C414yescomplemented
323C3×D4—232no—
324C3×D4—28no—
325C3×D4—24no—
326C3×D4—216no—
327C3×D4—216no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—24no—
333C3×D4C22×Dic314yescomplemented
334C3×D4C22×C3×C414yescomplemented
335C2×C3×C4—28no—
336C2×C3×C4—24no—
337C2×C3×C4—232no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—216no—
343C2×C3×C4—216no—
344C2×C3×C4—21no—
345C2×C3×C4—24no—
346C2×C3×C4—28no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—64no—
354C2×C3×C4—128no—
355C2×C3×C4—124no—
356C2×C3×C4#21714yes—
357C2×C3×C4#21718yes—
358C2×C3×C4C3×C22⋊C414yes—
359C2×C3×C4C3×C22⋊C418yes—
360C2×C3×C4C22×Dic312yes—
361C2×C3×C4C22×Dic314yes—
362C2×C3×C4C2×#6712yes—
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C2×#6712yes—
365C2×C3×C4C2×#6714yes—
366C2×C3×C4C22×C3×C412yes—
367C2×C3×C4C22×C3×C414yes—
368C2×C3×C4C2×C3×D412yes—
369C2×C3×C4C2×C3×D411yescharacteristic
370C2×C3×C4C2×C3×D412yes—
371C2×C3×C4C2×C3×D414yes—
372C2×C3×C4C23×D311yescharacteristic
373C2×C3×C4C24×C311yescharacteristic
374C2×Dic3—24no—
375C2×Dic3—48no—
376C2×Dic3—44no—
377C3×C8—64no—
378C3⋊C8—24no—
379C2×C32—24no—
380C2×C32—44no—
381C2×C32#41714yescomplemented
382C2×C32C22×C22⋊C411yescharacteristic
383C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C32C2×#13711yescharacteristic
385C24—21no—
386C24C3×#6711yescharacteristic
387#41—216no—
388#41—28no—
389#41C2×C3×Dic3116yescomplemented
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8C2×C3×Dic314yescomplemented
393C2×Q8C22×C3×D312yes—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4C2×C3×Dic314yescomplemented
399C2×D4C22×C3×D312yes—
400C22×C4—21no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—64no—
404C22×C4—62no—
405C22×C4—124no—
406C22×C4C2×C3×Dic314yes—
407C22×C4C3×#6711yescharacteristic
408C22×C4C3×#6712yes—
409C22×C4C3×#6714yes—
410C22×C4C22×C3×D311yescharacteristic
411C22×C4C22×C3×D312yes—
412M16—68no—
413C2×C8—34no—
414C2×C8—64no—
415C4⋊C4—34no—
416C4⋊C4—64no—
417C4⋊C4—62no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—28no—
421C22×C3—28no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—28no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—24no—
431C22×C3—21no—
432C22×C3—48no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—416no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—48no—
441C22×C3—48no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3#94912yes—
445C22×C3#95112yes—
446C22×C3C2×#21714yes—
447C22×C3C2×#21712yes—
448C22×C3C2×C3×C22⋊C414yes—
449C22×C3C2×C3×C22⋊C412yes—
450C22×C3C3×AGL(1,ℤ/8)12yes—
451C22×C3C3×#13712yes—
452C22×C3C22×#6711yescharacteristic
453C22×C3C22×C3×D411yescharacteristic
454C3×C4—28no—
455C3×C4—24no—
456C3×C4—24no—
457C3×C4—24no—
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—124no—
462C3×C4C2×#21714yes—
463C3×C4C2×#21712yes—
464C3×C4C2×C3×C22⋊C414yes—
465C3×C4C2×C3×C22⋊C412yes—
466C3×C4C23×Dic311yescharacteristic
467C3×C4C22×#6711yescharacteristic
468C3×C4C23×C3×C411yescharacteristic
469C3×C4C22×C3×D411yescharacteristic
470Dic3—44no—
471C32C2×#41711yescharacteristic, complemented, Sylow
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23C3×#21714yes—
479C23C2×C3×#6712yes—
480C23C2×C3×#6711yescharacteristic
481Q8—24no—
482Q8—22no—
483Q8C22×C3×Dic314yescomplemented
484D4—216no—
485D4—24no—
486D4—22no—
487D4C22×C3×Dic314yescomplemented
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—216no—
491C2×C4—64no—
492C2×C4—128no—
493C2×C4—124no—
494C2×C4C3×#21714yes—
495C2×C4C3×#21718yes—
496C2×C4C22×C3×Dic312yes—
497C2×C4C22×C3×Dic314yes—
498C2×C4C2×C3×#6712yes—
499C2×C4C2×C3×#6711yescharacteristic
500C2×C4C2×C3×#6712yes—
501C2×C4C2×C3×#6714yes—
502C2×C4C23×C3×D311yescharacteristic
503C8—64no—
504C2×C3—24no—
505C2×C3—24no—
506C2×C3—24no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—44no—
511C2×C3—48no—
512C2×C3—44no—
513C2×C3—44no—
514C2×C3#523214yesdirect product factor
515C2×C3C3×#41714yescomplemented
516C2×C3C2×#94911yescharacteristic
517C2×C3C2×#95111yescharacteristic
518C2×C3C22×#21711yescharacteristic
519C2×C3C22×C3×C22⋊C411yescharacteristic
520C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
521C2×C3C2×C3×#13711yescharacteristic
522C22—28no—
523C22—22no—
524C22—22no—
525C22—24no—
526C22—48no—
527C22—42no—
528C22—42no—
529C22C3×#94912yes—
530C22C3×#95112yes—
531C22C2×C3×#21714yes—
532C22C2×C3×#21712yes—
533C22C22×C3×#6711yescharacteristic
534C4—24no—
535C4—124no—
536C4C2×C3×#21714yes—
537C4C2×C3×#21712yes—
538C4C23×C3×Dic311yescharacteristic
539C4C22×C3×#6711yescharacteristic
540C3—21no—
541C3C2×#523211yescharacteristic, direct product factor
542C3C2×C3×#41711yescharacteristic, complemented
543C2—24no—
544C2—44no—
545C2C3×#523214yesdirect product factor
546C2C2×C3×#94911yescharacteristic
547C2C2×C3×#95111yescharacteristic
548C2C22×C3×#21711yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC22×C32×#41C2
socleC23×C32C2×D4
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×#41D3
hypocenterC3C2×C3×#417
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×#41

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4⋊Dic3C2semidirect
3. C22×C3×#208C2semidirect
4. C3×#5232C2direct (N × H)
5. C2×C3×#835C2semidirect
6. C2×C3×#832C2semidirect
8. C2×#5232C3direct (N × H)
11. C2×C32×#41C4semidirect
22. C2×C3×C4⋊Dic3C22semidirect
24. C2×C3×#208C22semidirect
26. C3×#835C22semidirect
27. C3×#832C22semidirect
33. C22×C4⋊Dic3C2×C3semidirect
34. C22×#208C2×C3semidirect
38. #5232C2×C3direct (N × H)
39. C2×#835C2×C3semidirect
40. C2×#832C2×C3semidirect
45. C32×#41C2×C4semidirect
48. C2×C32×Q8C2×C4semidirect
54. C2×C32×D4C2×C4semidirect
83. C2×C3×#41Dic3semidirect
84. C2×C3×#41C3×C4semidirect
114. C2×C4⋊Dic3C22×C3semidirect
116. C2×#208C22×C3semidirect
120. #835C22×C3semidirect
121. #832C22×C3semidirect
133. C32×Q8C22×C4semidirect
137. C32×D4C22×C4semidirect
173. C3×#41C2×Dic3semidirect
174. C3×#41C2×C3×C4semidirect
183. C2×C3×Q8C2×Dic3semidirect
184. C2×C3×Q8C2×C3×C4semidirect
201. C2×C3×D4C2×Dic3semidirect
202. C2×C3×D4C2×C3×C4semidirect
270. C2×#41C3×Dic3semidirect
321. C3×Q8C22×Dic3semidirect
322. C3×Q8C22×C3×C4semidirect
333. C3×D4C22×Dic3semidirect
334. C3×D4C22×C3×C4semidirect
381. C2×C32#417semidirect
389. #41C2×C3×Dic3semidirect
392. C2×Q8C2×C3×Dic3semidirect
398. C2×D4C2×C3×Dic3semidirect
471. C32C2×#417semidirect
483. Q8C22×C3×Dic3semidirect
487. D4C22×C3×Dic3semidirect
514. C2×C3#5232direct (N × H)
515. C2×C3C3×#417semidirect
541. C3C2×#5232direct (N × H)
542. C3C2×C3×#417semidirect
545. C2C3×#5232direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×#412
C22×C3×C4⋊Dic32
C22×C3×#2082
C3×#52322
C2×C3×#8352
C2×C3×#8322
C2×C3×#4173
C2×#52323