Finite Group Databasebeta

← C2×C3×#5042 #11866889

Subgroups — C2×C3×#5042 #11866889

Order 1152.

1301 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#121C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic, complemented
3C2×C3×#924C211yesmaximal, characteristic, complemented
4C22×C3×#208C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic, complemented
6C3×#5042C218yesmaximal, direct product factor
7C2×C3×#810C211yesmaximal, characteristic, complemented
8C2×C3×#807C211yesmaximal, characteristic, complemented
9C2×C3×#468—31nomaximal
10C2×#5042C311yesmaximal, characteristic, direct product factor
11C22×C32×D4—21no—
12C22×C32×D4C2211yescharacteristic
13C23×C32×C4C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C32×#121—22no—
16C32×#121C2214yes—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4—21no—
19C2×C3×#217—21no—
20C2×C3×#214—24no—
21C2×C3×#214C2214yescomplemented
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C3×#924—22no—
24C3×#924C2214yescomplemented
25C2×C3×Dic3⋊C4—21no—
26C2×C3×#208—24no—
27C2×C3×#208C2214yescomplemented
28C22×C3×C3⋊C8C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#832C2214yescomplemented
31C3×#810C2214yescomplemented
32C3×#807C2214yescomplemented
33C22×C3×SD16—31no—
34C22×C3×M16—31no—
35C2×C3×C22⋊Q8—31no—
36C2×C3×#121—21no—
37C2×C3×#121D311yescharacteristic
38C2×C3×#121C2×C311yescharacteristic
39C22×#214C2×C311yescharacteristic, complemented
40C2×#924C2×C311yescharacteristic, complemented
41C22×#208C2×C311yescharacteristic, complemented
42C3×#468—38no—
43C2×C3×#107—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46C2×#832C2×C311yescharacteristic, complemented
47#5042C2×C318yesdirect product factor
48C2×#810C2×C311yescharacteristic, complemented
49C2×#807C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C24×C32D411yescharacteristic
52C2×C32×D4—24no—
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—22no—
56C2×C32×D4—44no—
57C2×C32×D4C2312yes—
58C22×C32×C4—22no—
59C22×C32×C4—22no—
60C22×C32×C4—21no—
61C22×C32×C4D411yescharacteristic
62C22×C32×C4D412yes—
63C22×C32×C4C2312yes—
64C22×C32×C4C2311yescharacteristic
65C22×C3×Dic3—21no—
66C22×C3×Dic3—21no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2312yes—
73C32×C22⋊C4—24no—
74C3×#217—24no—
75C3×#214—28no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3C2312yes—
78C3×Dic3⋊C4—24no—
79C3×#208—28no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8D414yescomplemented
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2312yes—
84C2×#468—31noSylow
85C22×C3×Q8—31no—
86C22×C3×D4—22no—
87C22×C3×D4—21no—
88C22×C3×D4—21no—
89C22×C3×D4—21no—
90C22×C3×D4C2×D311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C23×C3×C4—21no—
93C23×C3×C4C2×D311yescharacteristic
94C23×C3×C4C22×C311yescharacteristic
95C22×Dic6C22×C311yescharacteristic
96C2×C3×SD16—34no—
97C2×C3×SD16—64no—
98C2×C3×M16—34no—
99C2×C3×M16—64no—
100C22×C3×C8—31no—
101C22×C3×C8—31no—
102C3×C22⋊Q8—34no—
103C3×C22⋊Q8—62no—
104C3×#121—24no—
105C3×#121—22no—
106C3×#121—22no—
107C3×#121—24no—
108C3×#121C2×D314yes—
109C3×#121C22×C314yes—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C4⋊C4—61no—
113C2×C3×C4⋊C4C2×D311yescharacteristic
114C2×C3×C4⋊C4C22×C311yescharacteristic
115C2×C3×C22⋊C4—22no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—21no—
118C2×C3×C22⋊C4—61no—
119C2×#217—21no—
120C2×#214—24no—
121C2×#214C22×C314yescomplemented
122C2×C4⋊Dic3C22×C311yescharacteristic
123#924—22no—
124#924C22×C314yescomplemented
125C2×Dic3⋊C4—21no—
126C2×#208—24no—
127C2×#208C22×C314yescomplemented
128C22×C3⋊C8C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C3×#107—34no—
131C3×#103—34no—
132C3×#102—34no—
133#832C22×C314yescomplemented
134#810C22×C314yescomplemented
135#807C22×C314yescomplemented
136C23×C32—22no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—42no—
141C23×C32—42no—
142C23×C32—42no—
143C23×C32C2×D412yes—
144C23×C32C2×D411yescharacteristic
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—48no—
148C32×D4—42no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—24no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—42no—
155C2×C32×C4C2×D412yes—
156C2×C32×C4C2×D412yes—
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4#4112yes—
159C2×C32×C4C2411yescharacteristic
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—42no—
163C2×C3×Dic3—42no—
164C3×Dic6—22no—
165C3×Dic6—22no—
166C3×Dic6—42no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8C2×D414yescomplemented
169C22×SD16—31no—
170C22×M16—31no—
171C2×C22⋊Q8—31no—
172C2×#121C3×D311yescharacteristic
173#468—38no—
174C2×#107—31no—
175C2×#103—31no—
176C2×#102—31no—
177C24×C3—22no—
178C24×C3—21no—
179C24×C3—21no—
180C24×C3—21no—
181C24×C3#6711yescharacteristic
182C24×C3C3×D411yescharacteristic
183C2×C3×Q8—32no—
184C2×C3×Q8—61no—
185C2×C3×Q8—62no—
186C2×C3×Q8—62no—
187C2×C3×D4—28no—
188C2×C3×D4—22no—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4—48no—
201C2×C3×D4—44no—
202C2×C3×D4—44no—
203C2×C3×D4C22×D312yes—
204C2×C3×D4C23×C312yes—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—22no—
214C22×C3×C4—21no—
215C22×C3×C4—22no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4—61no—
219C22×C3×C4—61no—
220C22×C3×C4#6712yes—
221C22×C3×C4#6711yescharacteristic
222C22×C3×C4C3×D411yescharacteristic
223C22×C3×C4C3×D412yes—
224C22×C3×C4C22×D312yes—
225C22×C3×C4C22×D311yescharacteristic
226C22×C3×C4C23×C312yes—
227C22×C3×C4C23×C311yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—21no—
233C2×Dic6C23×C312yes—
234C3×SD16—68no—
235C3×M16—68no—
236C2×C3×C8—34no—
237C2×C3×C8—32no—
238C2×C3×C8—32no—
239C2×C3×C8—62no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—64no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4C22×D312yes—
248C3×C4⋊C4C23×C312yes—
249C3×C22⋊C4—28no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—64no—
253#217—24no—
254#214—28no—
255C4⋊Dic3—21no—
256C4⋊Dic3C23×C312yes—
257Dic3⋊C4—24no—
258#208—28no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8C3×D414yescomplemented
261C2×C3⋊C8C23×C312yes—
262C2×C3⋊C8C23×C312yes—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—44no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—42no—
270C22×C32—84no—
271C22×C32#12112yes—
272C22×C32AGL(1,ℤ/8)12yes—
273C22×C32#13712yes—
274C22×C32C22×D411yescharacteristic
275C32×C4—22no—
276C32×C4—42no—
277C32×C4#12112yes—
278C32×C4C22×D411yescharacteristic
279C32×C4C2×#4111yescharacteristic
280C3×Dic3—42no—
281C3×Dic3—42no—
282C22×Q8—31no—
283C22×D4—21no—
284C22×D4C2×C3×D311yescharacteristic
285C23×C4C2×C3×D311yescharacteristic
286C2×SD16—34no—
287C2×SD16—64no—
288C2×M16—34no—
289C2×M16—64no—
290C22×C8—31no—
291C22×C8—31no—
292C22⋊Q8—34no—
293C22⋊Q8—62no—
294#121—22no—
295#121C2×C3×D314yes—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4C2×C3×D311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—61no—
301#107—34no—
302#103—34no—
303#102—34no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3C2×#6712yes—
328C23×C3C2×#6711yescharacteristic
329C23×C3C2×C3×D412yes—
330C23×C3C2×C3×D411yescharacteristic
331C3×Q8—62no—
332C3×Q8—62no—
333C3×Q8—122no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—44no—
341C3×D4—416no—
342C3×D4—48no—
343C3×D4—48no—
344C3×D4—42no—
345C3×D4—42no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—44no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—44no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—122no—
372C2×C3×C4—122no—
373C2×C3×C4D3×D412yes—
374C2×C3×C4#23712yes—
375C2×C3×C4C2×#6712yes—
376C2×C3×C4C2×#6711yescharacteristic
377C2×C3×C4C2×C3×D412yes—
378C2×C3×C4C2×C3×D412yes—
379C2×C3×C4C2×C3×D411yescharacteristic
380C2×C3×C4C3×#4112yes—
381C2×C3×C4C23×D311yescharacteristic
382C2×C3×C4C24×C311yescharacteristic
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387Dic6—22no—
388Dic6—22no—
389Dic6—42no—
390C3×C8—34no—
391C3×C8—62no—
392C3⋊C8—22no—
393C3⋊C8C2×C3×D414yescomplemented
394C2×C32—42no—
395C2×C32—82no—
396C2×C32#46814yescomplemented
397C2×C32C2×#12111yescharacteristic
398C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
399C2×C32C2×#13711yescharacteristic
400C24—21no—
401C24C3×#6711yescharacteristic
402C2×Q8—32no—
403C2×Q8—61no—
404C2×Q8—62no—
405C2×Q8—62no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—44no—
411C2×D4C22×C3×D312yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4C3×#6711yescharacteristic
418C22×C4C3×#6712yes—
419C22×C4C22×C3×D312yes—
420C22×C4C22×C3×D311yescharacteristic
421SD16—68no—
422M16—68no—
423C2×C8—32no—
424C2×C8—34no—
425C2×C8—32no—
426C2×C8—62no—
427C4⋊C4—21no—
428C4⋊C4—32no—
429C4⋊C4—61no—
430C4⋊C4—64no—
431C4⋊C4C22×C3×D312yes—
432C22⋊C4—24no—
433C22⋊C4—64no—
434C22×C3—24no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—48no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—88no—
460C22×C3—84no—
461C22×C3—84no—
462C22×C3#93212yes—
463C22×C3#93912yes—
464C22×C3#95112yes—
465C22×C3C3×#12112yes—
466C22×C3C3×AGL(1,ℤ/8)12yes—
467C22×C3C3×#13712yes—
468C22×C3C22×#6711yescharacteristic
469C22×C3C22×C3×D411yescharacteristic
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—42no—
476C3×C4—44no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—122no—
480C3×C4—122no—
481C3×C4#93912yes—
482C3×C4C3×#12112yes—
483C3×C4C2×D3×D411yescharacteristic
484C3×C4C2×#23711yescharacteristic
485C3×C4C22×C3×D411yescharacteristic
486C3×C4C2×C3×#4111yescharacteristic
487Dic3—42no—
488Dic3—42no—
489C32C2×#46811yescharacteristic, complemented, Sylow
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C2×C3×#6712yes—
498C23C2×C3×#6711yescharacteristic
499Q8—62no—
500Q8—62no—
501Q8—122no—
502D4—22no—
503D4—22no—
504D4—48no—
505D4—42no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4C3×D3×D412yes—
517C2×C4C3×#23712yes—
518C2×C4C2×C3×#6712yes—
519C2×C4C2×C3×#6711yescharacteristic
520C2×C4C23×C3×D311yescharacteristic
521C8—34no—
522C8—62no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—24no—
527C2×C3—44no—
528C2×C3—42no—
529C2×C3—42no—
530C2×C3—84no—
531C2×C3—82no—
532C2×C3—82no—
533C2×C3#504214yesdirect product factor
534C2×C3C3×#46814yescomplemented
535C2×C3C2×#93211yescharacteristic
536C2×C3C2×#93911yescharacteristic
537C2×C3C2×#95111yescharacteristic
538C2×C3C2×C3×#12111yescharacteristic
539C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
540C2×C3C2×C3×#13711yescharacteristic
541C22—22no—
542C22—22no—
543C22—22no—
544C22—44no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—84no—
549C22C3×#93212yes—
550C22C3×#93912yes—
551C22C3×#95112yes—
552C22C22×C3×#6711yescharacteristic
553C4—22no—
554C4—42no—
555C4—122no—
556C4—122no—
557C4C3×#93912yes—
558C4C2×C3×D3×D411yescharacteristic
559C4C2×C3×#23711yescharacteristic
560C3—21no—
561C3C2×#504211yescharacteristic, direct product factor
562C3C2×C3×#46811yescharacteristic, complemented
563C2—42no—
564C2—82no—
565C2C3×#504214yesdirect product factor
566C2C2×C3×#93211yescharacteristic
567C2C2×C3×#93911yescharacteristic
568C2C2×C3×#95111yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×#121C2
socleC23×C32C2×D4
hypercenterC2×C3×#121D3
hypocenterC3C2×C3×#468
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
2. C22×C3×#214C2semidirect
3. C2×C3×#924C2semidirect
4. C22×C3×#208C2semidirect
5. C2×C3×#832C2semidirect
6. C3×#5042C2direct (N × H)
7. C2×C3×#810C2semidirect
8. C2×C3×#807C2semidirect
10. C2×#5042C3direct (N × H)
21. C2×C3×#214C22semidirect
24. C3×#924C22semidirect
27. C2×C3×#208C22semidirect
30. C3×#832C22semidirect
31. C3×#810C22semidirect
32. C3×#807C22semidirect
39. C22×#214C2×C3semidirect
40. C2×#924C2×C3semidirect
41. C22×#208C2×C3semidirect
46. C2×#832C2×C3semidirect
47. #5042C2×C3direct (N × H)
48. C2×#810C2×C3semidirect
49. C2×#807C2×C3semidirect
81. C2×C3×C3⋊C8D4semidirect
121. C2×#214C22×C3semidirect
124. #924C22×C3semidirect
127. C2×#208C22×C3semidirect
133. #832C22×C3semidirect
134. #810C22×C3semidirect
135. #807C22×C3semidirect
168. C3×C3⋊C8C2×D4semidirect
260. C2×C3⋊C8C3×D4semidirect
393. C3⋊C8C2×C3×D4semidirect
396. C2×C32#468semidirect
489. C32C2×#468semidirect
533. C2×C3#5042direct (N × H)
534. C2×C3C3×#468semidirect
561. C3C2×#5042direct (N × H)
562. C3C2×C3×#468semidirect
565. C2C3×#5042direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1212
C22×C3×#2142
C2×C3×#9242
C22×C3×#2082
C2×C3×#8322
C3×#50422
C2×C3×#8102
C2×C3×#8072
C2×C3×#4683
C2×#50423