Finite Group Databasebeta

← C2×C3×#5200 #11867847

Subgroups — C2×C3×#5200 #11867847

Order 1152.

1085 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic, complemented
3C2×C3×#948C211yesmaximal, characteristic, complemented
4C22×C3×#214C211yesmaximal, characteristic, complemented
5C2×C3×#833C211yesmaximal, characteristic, complemented
6C3×#5200C218yesmaximal, direct product factor
7C2×C3×#824C211yesmaximal, characteristic, complemented
8C2×C3×#822C211yesmaximal, characteristic, complemented
9C2×C3×#471—31nomaximal
10C2×#5200C311yesmaximal, characteristic, direct product factor
11C2×C32×#41C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C32×AGL(1,ℤ/8)—22no—
15C32×AGL(1,ℤ/8)C2214yes—
16C2×C32×SD16—21no—
17C2×C32×D8—21no—
18C2×C32×M16C2211yescharacteristic
19C3×#951—22no—
20C3×#951C2214yescomplemented
21C3×#948—22no—
22C3×#948C2214yescomplemented
23C2×C3×#216—21no—
24C2×C3×#214—21no—
25C2×C3×#214—21no—
26C2×C3×#214—21no—
27C2×C3×#214—22no—
28C2×C3×#214C2214yescomplemented
29C2×C3×#208—21no—
30C2×C3×#208C2211yescharacteristic
31C2×C3×#208C2211yescharacteristic
32C22×C3×C3⋊C8—21no—
33C22×C3×C3⋊C8C2211yescharacteristic
34C3×#833C2214yescomplemented
35C3×#824C2214yescomplemented
36C3×#822C2214yescomplemented
37C2×C3×#137—31no—
38C2×C3×AGL(1,ℤ/8)—21no—
39C2×C3×AGL(1,ℤ/8)D311yescharacteristic
40C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic
41C22×C3×SD16—31no—
42C2×C3×#131—31no—
43C2×#951C2×C311yescharacteristic, complemented
44C2×#948C2×C311yescharacteristic, complemented
45C22×#214C2×C311yescharacteristic, complemented
46C3×#471—38no—
47C2×C3×#108—31no—
48C2×C3×#101—31no—
49C2×C3×#100—31no—
50C2×#833C2×C311yescharacteristic, complemented
51#5200C2×C318yesdirect product factor
52C2×#824C2×C311yescharacteristic, complemented
53C2×#822C2×C311yescharacteristic, complemented
54C24×C32—21no—
55C32×#41—21no—
56C32×#41—21no—
57C32×#41D412yes—
58C32×#41C2312yes—
59C2×C32×Q8D411yescharacteristic
60C2×C32×D4—21no—
61C2×C32×D4—21no—
62C2×C32×D4—21no—
63C2×C32×D4—21no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4D411yescharacteristic
67C2×C32×D4C2312yes—
68C22×C32×C4—21no—
69C22×C32×C4C2311yescharacteristic
70C22×C3×Dic3—21no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6C2312yes—
77C32×SD16—24no—
78C32×D8—24no—
79C32×M16—21no—
80C32×M16C2312yes—
81C2×C32×C8—21no—
82C3×#216—24no—
83C3×#214—24no—
84C3×#214—24no—
85C3×#214—24no—
86C3×#208—21no—
87C3×#208—21no—
88C3×#208—24no—
89C3×#208C2312yes—
90C3×#208C2312yes—
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8—24no—
94C2×C3×C3⋊C8D411yescharacteristic
95C2×C3×C3⋊C8D411yescharacteristic
96C2×C3×C3⋊C8D412yes—
97C2×C3×C3⋊C8C2312yes—
98C2×#471—31noSylow
99C2×C3×#41—21no—
100C2×C3×#41C2×D311yescharacteristic
101C2×C3×#41C22×C311yescharacteristic
102C22×C3×Q8—31no—
103C22×C3×D4—21no—
104C22×C3×D4C2×D311yescharacteristic
105C22×C3×D4C22×C311yescharacteristic
106C22×Dic6C22×C311yescharacteristic
107C3×#137—34no—
108C3×#137—62no—
109C3×AGL(1,ℤ/8)—24no—
110C3×AGL(1,ℤ/8)—22no—
111C3×AGL(1,ℤ/8)—22no—
112C3×AGL(1,ℤ/8)—24no—
113C3×AGL(1,ℤ/8)C2×D314yes—
114C3×AGL(1,ℤ/8)C22×C314yes—
115C2×C3×Q16—61no—
116C2×C3×SD16—22no—
117C2×C3×SD16—21no—
118C2×C3×SD16—21no—
119C2×C3×SD16—34no—
120C2×C3×SD16—61no—
121C2×C3×SD16—61no—
122C2×C3×SD16—62no—
123C2×C3×SD16—61no—
124C2×C3×D8—22no—
125C2×C3×D8—21no—
126C2×C3×D8—21no—
127C3×#131—34no—
128C3×#131—62no—
129C2×C3×M16—21no—
130C2×C3×M16—31no—
131C2×C3×M16—31no—
132C2×C3×M16—61no—
133C2×C3×M16C2×D311yescharacteristic
134C2×C3×M16C22×C311yescharacteristic
135C22×C3×C8—31no—
136C22×C3×C8—61no—
137#951—22no—
138#951C22×C314yescomplemented
139#948—22no—
140#948C22×C314yescomplemented
141C2×#216—21no—
142C2×#214—21no—
143C2×#214—21no—
144C2×#214—21no—
145C2×#214—22no—
146C2×#214C22×C314yescomplemented
147C2×#208—21no—
148C2×#208C22×C311yescharacteristic
149C2×#208C22×C311yescharacteristic
150C22×C3⋊C8—21no—
151C22×C3⋊C8C22×C311yescharacteristic
152C3×#108—34no—
153C3×#101—34no—
154C3×#100—34no—
155#833C22×C314yescomplemented
156#824C22×C314yescomplemented
157#822C22×C314yescomplemented
158C23×C32—21no—
159C23×C32—22no—
160C23×C32—41no—
161C23×C32—41no—
162C23×C32—82no—
163C23×C32#4111yescharacteristic
164C32×Q8—21no—
165C32×Q8C2×D412yes—
166C32×D4—22no—
167C32×D4—22no—
168C32×D4—22no—
169C32×D4—21no—
170C32×D4—22no—
171C32×D4—41no—
172C32×D4—41no—
173C32×D4C2×D412yes—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—22no—
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4#4111yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C2×C3×Dic3—22no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—41no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—41no—
189C3×Dic6—41no—
190C32×C8—22no—
191C3×C3⋊C8—22no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8C2×D412yes—
194C3×C3⋊C8C2×D412yes—
195C2×#137—31no—
196C2×AGL(1,ℤ/8)C3×D311yescharacteristic
197C22×SD16—31no—
198C2×#131—31no—
199#471—38no—
200C2×#108—31no—
201C2×#101—31no—
202C2×#100—31no—
203C24×C3—21no—
204C24×C3—21no—
205C24×C3—41no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C3×#41—22no—
211C3×#41—22no—
212C3×#41—41no—
213C3×#41—41no—
214C3×#41#6712yes—
215C3×#41C3×D412yes—
216C3×#41C22×D312yes—
217C3×#41C23×C312yes—
218C2×C3×Q8—21no—
219C2×C3×Q8—32no—
220C2×C3×Q8—61no—
221C2×C3×Q8—61no—
222C2×C3×Q8—61no—
223C2×C3×Q8—61no—
224C2×C3×Q8—61no—
225C2×C3×Q8#6711yescharacteristic
226C2×C3×Q8C3×D411yescharacteristic
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—22no—
230C2×C3×D4—22no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—22no—
243C2×C3×D4—41no—
244C2×C3×D4—41no—
245C2×C3×D4—41no—
246C2×C3×D4—41no—
247C2×C3×D4#6711yescharacteristic
248C2×C3×D4C3×D411yescharacteristic
249C2×C3×D4C22×D312yes—
250C2×C3×D4C23×C312yes—
251C22×C3×C4—21no—
252C22×C3×C4—21no—
253C22×C3×C4—21no—
254C22×C3×C4—41no—
255C22×C3×C4—61no—
256C22×C3×C4C22×D311yescharacteristic
257C22×C3×C4C23×C311yescharacteristic
258C22×Dic3—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—21no—
262C2×Dic6—21no—
263C2×Dic6—21no—
264C2×Dic6C23×C312yes—
265C3×Q16—64no—
266C3×SD16—28no—
267C3×SD16—24no—
268C3×SD16—24no—
269C3×SD16—64no—
270C3×SD16—64no—
271C3×SD16—64no—
272C3×D8—28no—
273C3×D8—24no—
274C3×D8—24no—
275C3×M16—22no—
276C3×M16—21no—
277C3×M16—21no—
278C3×M16—22no—
279C3×M16—32no—
280C3×M16—32no—
281C3×M16—64no—
282C3×M16—61no—
283C3×M16—61no—
284C3×M16C22×D312yes—
285C3×M16C23×C312yes—
286C2×C3×C8—22no—
287C2×C3×C8—21no—
288C2×C3×C8—21no—
289C2×C3×C8—31no—
290C2×C3×C8—31no—
291C2×C3×C8—32no—
292C2×C3×C8—32no—
293C2×C3×C8—64no—
294C2×C3×C8—61no—
295C2×C3×C8—61no—
296#216—24no—
297#214—24no—
298#214—24no—
299#214—24no—
300#208—21no—
301#208—21no—
302#208—24no—
303#208C23×C312yes—
304#208C23×C312yes—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—24no—
308C2×C3⋊C8C3×D411yescharacteristic
309C2×C3⋊C8C3×D411yescharacteristic
310C2×C3⋊C8C3×D412yes—
311C2×C3⋊C8C23×C312yes—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—42no—
315C22×C32—42no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32—82no—
319C22×C32#12111yescharacteristic
320C22×C32#12111yescharacteristic
321C22×C32C2×#4111yescharacteristic
322C32×C4—22no—
323C32×C4#12111yescharacteristic
324C32×C4#12111yescharacteristic
325C32×C4C22×D411yescharacteristic
326C32×C4C22×D411yescharacteristic
327C3×Dic3—42no—
328C2×#41C2×C3×D311yescharacteristic
329C22×Q8—31no—
330C22×D4C2×C3×D311yescharacteristic
331#137—34no—
332#137—62no—
333AGL(1,ℤ/8)—22no—
334AGL(1,ℤ/8)C2×C3×D314yes—
335C2×Q16—61no—
336C2×SD16—21no—
337C2×SD16—34no—
338C2×SD16—61no—
339C2×SD16—61no—
340C2×SD16—62no—
341C2×SD16—61no—
342C2×D8—21no—
343#131—34no—
344#131—62no—
345C2×M16—31no—
346C2×M16—31no—
347C2×M16—61no—
348C2×M16C2×C3×D311yescharacteristic
349C22×C8—31no—
350C22×C8—61no—
351#108—34no—
352#101—34no—
353#100—34no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3—42no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—82no—
368C23×C3—82no—
369C23×C3—84no—
370C23×C3#23711yescharacteristic
371C23×C3C3×#4111yescharacteristic
372C3×Q8—21no—
373C3×Q8—21no—
374C3×Q8—22no—
375C3×Q8—41no—
376C3×Q8—62no—
377C3×Q8—62no—
378C3×Q8—121no—
379C3×Q8—121no—
380C3×Q8C2×#6712yes—
381C3×Q8C2×C3×D412yes—
382C3×D4—22no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—24no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—42no—
395C3×D4—41no—
396C3×D4—42no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—42no—
400C3×D4—41no—
401C3×D4—41no—
402C3×D4—41no—
403C3×D4—41no—
404C3×D4C2×#6712yes—
405C3×D4C2×C3×D412yes—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—42no—
419C2×C3×C4—42no—
420C2×C3×C4—41no—
421C2×C3×C4—62no—
422C2×C3×C4—121no—
423C2×C3×C4—121no—
424C2×C3×C4D3×D411yescharacteristic
425C2×C3×C4D3×D411yescharacteristic
426C2×C3×C4#23711yescharacteristic
427C2×C3×C4C2×#6711yescharacteristic
428C2×C3×C4C2×#6711yescharacteristic
429C2×C3×C4C2×C3×D411yescharacteristic
430C2×C3×C4C2×C3×D411yescharacteristic
431C2×C3×C4C2×C3×D411yescharacteristic
432C2×C3×C4C2×C3×D411yescharacteristic
433C2×C3×C4C3×#4111yescharacteristic
434C2×C3×C4C23×D311yescharacteristic
435C2×C3×C4C24×C311yescharacteristic
436C2×Dic3—22no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439Dic6—22no—
440Dic6—22no—
441Dic6—41no—
442Dic6—41no—
443C3×C8—24no—
444C3×C8—22no—
445C3×C8—22no—
446C3×C8—32no—
447C3×C8—32no—
448C3×C8—62no—
449C3×C8—62no—
450C3⋊C8—22no—
451C3⋊C8—22no—
452C3⋊C8C2×C3×D412yes—
453C3⋊C8C2×C3×D412yes—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32—42no—
457C2×C32—82no—
458C2×C32#47112yescomplemented
459C2×C32C2×#12111yescharacteristic
460C24—21no—
461#41—21no—
462#41—21no—
463#41C3×#6712yes—
464#41C22×C3×D312yes—
465C2×Q8—32no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8C3×#6711yescharacteristic
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C3×#6711yescharacteristic
479C2×D4C22×C3×D312yes—
480C22×C4—21no—
481C22×C4—61no—
482C22×C4C22×C3×D311yescharacteristic
483Q16—64no—
484SD16—24no—
485SD16—64no—
486SD16—64no—
487SD16—64no—
488D8—24no—
489M16—21no—
490M16—32no—
491M16—32no—
492M16—64no—
493M16—61no—
494M16—61no—
495M16C22×C3×D312yes—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—31no—
499C2×C8—31no—
500C2×C8—32no—
501C2×C8—64no—
502C2×C8—61no—
503C2×C8—61no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—21no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—44no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—82no—
519C22×C3—82no—
520C22×C3—82no—
521C22×C3—82no—
522C22×C3—82no—
523C22×C3—84no—
524C22×C3—84no—
525C22×C3—84no—
526C22×C3—82no—
527C22×C3—82no—
528C22×C3#93911yescharacteristic
529C22×C3#93911yescharacteristic
530C22×C3C3×#12111yescharacteristic
531C22×C3C3×#12111yescharacteristic
532C22×C3C2×#23711yescharacteristic
533C22×C3C2×C3×#4111yescharacteristic
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—42no—
541C3×C4—122no—
542C3×C4#93911yescharacteristic
543C3×C4#93911yescharacteristic
544C3×C4C3×#12111yescharacteristic
545C3×C4C3×#12111yescharacteristic
546C3×C4C2×D3×D411yescharacteristic
547C3×C4C22×#6711yescharacteristic
548C3×C4C22×C3×D411yescharacteristic
549C3×C4C22×C3×D411yescharacteristic
550Dic3—42no—
551C32C2×#47111yescharacteristic, complemented, Sylow
552C23—21no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—82no—
557C23C3×#23711yescharacteristic
558Q8—21no—
559Q8—62no—
560Q8—62no—
561Q8—121no—
562Q8—121no—
563Q8C2×C3×#6712yes—
564D4—22no—
565D4—22no—
566D4—21no—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4C2×C3×#6712yes—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—62no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4C3×D3×D411yescharacteristic
579C2×C4C3×D3×D411yescharacteristic
580C2×C4C3×#23711yescharacteristic
581C2×C4C2×C3×#6711yescharacteristic
582C2×C4C2×C3×#6711yescharacteristic
583C2×C4C23×C3×D311yescharacteristic
584C8—22no—
585C8—32no—
586C8—32no—
587C8—62no—
588C8—62no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—22no—
594C2×C3—21no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—42no—
598C2×C3—42no—
599C2×C3—82no—
600C2×C3—84no—
601C2×C3—82no—
602C2×C3—82no—
603C2×C3#520012yesdirect product factor
604C2×C3C3×#47112yescomplemented
605C2×C3C2×#93911yescharacteristic
606C2×C3C2×C3×#12111yescharacteristic
607C22—22no—
608C22—22no—
609C22—42no—
610C22—42no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22C3×#93911yescharacteristic
615C22C3×#93911yescharacteristic
616C22C2×C3×#23711yescharacteristic
617C4—22no—
618C4—122no—
619C4C3×#93911yescharacteristic
620C4C3×#93911yescharacteristic
621C4C2×C3×D3×D411yescharacteristic
622C4C22×C3×#6711yescharacteristic
623C3—21no—
624C3C2×#520011yescharacteristic, direct product factor
625C3C2×C3×#47111yescharacteristic, complemented
626C2—21no—
627C2—21no—
628C2—42no—
629C2—82no—
630C2C3×#520012yesdirect product factor
631C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC22×C32#121
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C2×C3×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C2×C3×#951C2semidirect
3. C2×C3×#948C2semidirect
4. C22×C3×#214C2semidirect
5. C2×C3×#833C2semidirect
6. C3×#5200C2direct (N × H)
7. C2×C3×#824C2semidirect
8. C2×C3×#822C2semidirect
10. C2×#5200C3direct (N × H)
20. C3×#951C22semidirect
22. C3×#948C22semidirect
28. C2×C3×#214C22semidirect
34. C3×#833C22semidirect
35. C3×#824C22semidirect
36. C3×#822C22semidirect
43. C2×#951C2×C3semidirect
44. C2×#948C2×C3semidirect
45. C22×#214C2×C3semidirect
50. C2×#833C2×C3semidirect
51. #5200C2×C3direct (N × H)
52. C2×#824C2×C3semidirect
53. C2×#822C2×C3semidirect
138. #951C22×C3semidirect
140. #948C22×C3semidirect
146. C2×#214C22×C3semidirect
155. #833C22×C3semidirect
156. #824C22×C3semidirect
157. #822C22×C3semidirect
458. C2×C32#471semidirect
551. C32C2×#471semidirect
603. C2×C3#5200direct (N × H)
604. C2×C3C3×#471semidirect
624. C3C2×#5200direct (N × H)
625. C3C2×C3×#471semidirect
630. C2C3×#5200direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
C2×C3×#9512
C2×C3×#9482
C22×C3×#2142
C2×C3×#8332
C3×#52002
C2×C3×#8242
C2×C3×#8222
C2×C3×#4713
C2×#52003