Finite Group Databasebeta

← C2×C3×#5203 #11867859

Subgroups — C2×C3×#5203 #11867859

Order 1152.

1065 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic, complemented
2C2×C3×#949C211yesmaximal, characteristic
3C2×C3×#948C211yesmaximal, characteristic, complemented
4C22×C3×#215C211yesmaximal, characteristic, complemented
5C2×C3×#836C211yesmaximal, characteristic, complemented
6C3×#5203C218yesmaximal, direct product factor
7C2×C3×#823C211yesmaximal, characteristic, complemented
8C2×C3×#822C211yesmaximal, characteristic, complemented
9C2×C3×#471—31nomaximal
10C2×#5203C311yesmaximal, characteristic, direct product factor
11C2×C32×#41C2211yescharacteristic
12C22×C32×Q8C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C32×#137—22no—
15C32×#137C2214yescomplemented
16C2×C32×Q16—21no—
17C2×C32×SD16—21no—
18C2×C32×M16C2211yescharacteristic
19C3×#949—22no—
20C3×#949C2214yes—
21C3×#948—22no—
22C3×#948C2214yescomplemented
23C2×C3×#215—21no—
24C2×C3×#215—21no—
25C2×C3×#215—21no—
26C2×C3×#215—22no—
27C2×C3×#215C2214yescomplemented
28C2×C3×#213—21no—
29C2×C3×#208—21no—
30C2×C3×#208C2211yescharacteristic
31C2×C3×#208C2211yescharacteristic
32C22×C3×C3⋊C8—21no—
33C22×C3×C3⋊C8C2211yescharacteristic
34C3×#836C2214yescomplemented
35C3×#823C2214yescomplemented
36C3×#822C2214yescomplemented
37C2×C3×#137—21no—
38C2×C3×#137D311yescharacteristic, complemented
39C2×C3×#137C2×C311yescharacteristic, complemented
40C2×C3×AGL(1,ℤ/8)—31no—
41C22×C3×SD16—31no—
42C2×C3×#131—31no—
43C2×#949C2×C311yescharacteristic
44C2×#948C2×C311yescharacteristic, complemented
45C22×#215C2×C311yescharacteristic, complemented
46C3×#471—38no—
47C2×C3×#108—31no—
48C2×C3×#101—31no—
49C2×C3×#100—31no—
50C2×#836C2×C311yescharacteristic, complemented
51#5203C2×C318yesdirect product factor
52C2×#823C2×C311yescharacteristic, complemented
53C2×#822C2×C311yescharacteristic, complemented
54C23×C3×D3—21no—
55C32×#41—21no—
56C32×#41—21no—
57C32×#41D412yes—
58C32×#41C2312yes—
59C2×C32×Q8—21no—
60C2×C32×Q8—21no—
61C2×C32×Q8—21no—
62C2×C32×Q8—21no—
63C2×C32×Q8—21no—
64C2×C32×Q8D411yescharacteristic
65C2×C32×Q8C2312yes—
66C2×C32×D4—21no—
67C2×C32×D4D411yescharacteristic
68C22×C32×C4—21no—
69C22×C32×C4—21no—
70C22×C32×C4C2311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12—21no—
76C2×C3×D12C2312yes—
77C32×Q16—24no—
78C32×SD16—24no—
79C32×M16—21no—
80C32×M16C2312yes—
81C2×C32×C8—21no—
82C3×#215—24no—
83C3×#215—24no—
84C3×#215—24no—
85C3×#213—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—21no—
103C22×C3×Q8C2×D311yescharacteristic
104C22×C3×Q8C22×C311yescharacteristic
105C22×C3×D4—31no—
106C22×D12C22×C311yescharacteristic
107C3×#137—24no—
108C3×#137—22no—
109C3×#137—22no—
110C3×#137—24no—
111C3×#137C2×D314yescomplemented
112C3×#137C22×C314yescomplemented
113C3×AGL(1,ℤ/8)—34no—
114C3×AGL(1,ℤ/8)—62no—
115C2×C3×Q16—22no—
116C2×C3×Q16—21no—
117C2×C3×Q16—21no—
118C2×C3×SD16—22no—
119C2×C3×SD16—21no—
120C2×C3×SD16—21no—
121C2×C3×SD16—34no—
122C2×C3×SD16—61no—
123C2×C3×SD16—61no—
124C2×C3×SD16—62no—
125C2×C3×SD16—61no—
126C2×C3×D8—61no—
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#949—22no—
138#949C22×C314yes—
139#948—22no—
140#948C22×C314yescomplemented
141C2×#215—21no—
142C2×#215—21no—
143C2×#215—21no—
144C2×#215—22no—
145C2×#215C22×C314yescomplemented
146C2×#213—21no—
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#836C22×C314yescomplemented
156#823C22×C314yescomplemented
157#822C22×C314yescomplemented
158C23×C32—21no—
159C23×C32#4111yescharacteristic
160C22×C3×D3—22no—
161C22×C3×D3—41no—
162C22×C3×D3—41no—
163C22×C3×D3—82no—
164C32×Q8—22no—
165C32×Q8—21no—
166C32×Q8—22no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8C2×D412yes—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—21no—
173C32×D4C2×D412yes—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—22no—
177C2×C32×C4—22no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2×D411yescharacteristic
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4#4111yescharacteristic
185C2×C32×C4C2411yescharacteristic
186C3×D12—22no—
187C3×D12—22no—
188C3×D12—41no—
189C3×D12—41no—
190C32×C8—22no—
191C3×C3⋊C8—22no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8C2×D412yes—
194C3×C3⋊C8C2×D412yes—
195C2×#137C3×D311yescharacteristic, complemented
196C2×AGL(1,ℤ/8)—31no—
197C22×SD16—31no—
198C2×#131—31no—
199#471—38no—
200C2×#108—31no—
201C2×#101—31no—
202C2×#100—31no—
203C24×C3—61no—
204C23×D3—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—22no—
210C3×#41—22no—
211C3×#41—41no—
212C3×#41—41no—
213C3×#41#6712yes—
214C3×#41C3×D412yes—
215C3×#41C22×D312yes—
216C3×#41C23×C312yes—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—22no—
220C2×C3×Q8—22no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—22no—
231C2×C3×Q8—41no—
232C2×C3×Q8—41no—
233C2×C3×Q8—41no—
234C2×C3×Q8#6711yescharacteristic
235C2×C3×Q8C3×D411yescharacteristic
236C2×C3×Q8C22×D312yes—
237C2×C3×Q8C23×C312yes—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—32no—
242C2×C3×D4—41no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—61no—
246C2×C3×D4—61no—
247C2×C3×D4—61no—
248C2×C3×D4#6711yescharacteristic
249C2×C3×D4C3×D411yescharacteristic
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—21no—
253C22×C3×C4—21no—
254C22×C3×C4—21no—
255C22×C3×C4—41no—
256C22×C3×C4—41no—
257C22×C3×C4C22×D311yescharacteristic
258C22×C3×C4C23×C311yescharacteristic
259C2×D12—21no—
260C2×D12—21no—
261C2×D12—21no—
262C2×D12—21no—
263C2×D12—21no—
264C2×D12C23×C312yes—
265C3×Q16—28no—
266C3×Q16—24no—
267C3×Q16—24no—
268C3×SD16—28no—
269C3×SD16—24no—
270C3×SD16—24no—
271C3×SD16—64no—
272C3×SD16—64no—
273C3×SD16—64no—
274C3×D8—64no—
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#215—24no—
297#215—24no—
298#215—24no—
299#213—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#12111yescharacteristic
316C22×C32#12111yescharacteristic
317C22×C32C2×#4111yescharacteristic
318C2×C3×D3—42no—
319C2×C3×D3—82no—
320C2×C3×D3—82no—
321C2×C3×D3—82no—
322C32×C4—22no—
323C32×C4—42no—
324C32×C4#12111yescharacteristic
325C32×C4#12111yescharacteristic
326C32×C4C22×D411yescharacteristic
327C32×C4C22×D411yescharacteristic
328C2×#41C2×C3×D311yescharacteristic
329C22×Q8C2×C3×D311yescharacteristic
330C22×D4—31no—
331#137—22no—
332#137C2×C3×D314yescomplemented
333AGL(1,ℤ/8)—34no—
334AGL(1,ℤ/8)—62no—
335C2×Q16—21no—
336C2×SD16—21no—
337C2×SD16—34no—
338C2×SD16—61no—
339C2×SD16—61no—
340C2×SD16—62no—
341C2×SD16—61no—
342C2×D8—61no—
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—21no—
357C23×C3—41no—
358C23×C3—62no—
359C23×C3—121no—
360C23×C3—121no—
361C23×C3—242no—
362C23×C3#23711yescharacteristic
363C23×C3C3×#4111yescharacteristic
364C22×D3—22no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—82no—
368C3×Q8—21no—
369C3×Q8—21no—
370C3×Q8—24no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8—42no—
377C3×Q8—41no—
378C3×Q8—42no—
379C3×Q8—42no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8C2×#6712yes—
385C3×Q8C2×C3×D412yes—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—22no—
393C3×D4—41no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—62no—
397C3×D4—62no—
398C3×D4—121no—
399C3×D4—121no—
400C3×D4C2×#6712yes—
401C3×D4C2×C3×D412yes—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—42no—
417C2×C3×C4—42no—
418C2×C3×C4—41no—
419C2×C3×C4—42no—
420C2×C3×C4—42no—
421C2×C3×C4—42no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4D3×D411yescharacteristic
427C2×C3×C4D3×D411yescharacteristic
428C2×C3×C4#23711yescharacteristic
429C2×C3×C4C2×#6711yescharacteristic
430C2×C3×C4C2×#6711yescharacteristic
431C2×C3×C4C2×C3×D411yescharacteristic
432C2×C3×C4C2×C3×D411yescharacteristic
433C2×C3×C4C2×C3×D411yescharacteristic
434C2×C3×C4C2×C3×D411yescharacteristic
435C2×C3×C4C3×#4111yescharacteristic
436C2×C3×C4C23×D311yescharacteristic
437C2×C3×C4C24×C311yescharacteristic
438D12—22no—
439D12—22no—
440D12—41no—
441D12—41no—
442C3×C8—24no—
443C3×C8—22no—
444C3×C8—22no—
445C3×C8—32no—
446C3×C8—32no—
447C3×C8—62no—
448C3×C8—62no—
449C3⋊C8—22no—
450C3⋊C8—22no—
451C3⋊C8C2×C3×D412yes—
452C3⋊C8C2×C3×D412yes—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—42no—
456C2×C32#47112yescomplemented
457C2×C32C2×#12111yescharacteristic
458C3×D3—82no—
459C24—61no—
460#41—21no—
461#41—21no—
462#41C3×#6712yes—
463#41C22×C3×D312yes—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8C3×#6711yescharacteristic
470C2×Q8C22×C3×D312yes—
471C2×D4—21no—
472C2×D4—32no—
473C2×D4—61no—
474C2×D4—61no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4C3×#6711yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4C22×C3×D311yescharacteristic
482Q16—24no—
483SD16—24no—
484SD16—64no—
485SD16—64no—
486SD16—64no—
487D8—64no—
488M16—21no—
489M16—32no—
490M16—32no—
491M16—64no—
492M16—61no—
493M16—61no—
494M16C22×C3×D312yes—
495C2×C8—21no—
496C2×C8—32no—
497C2×C8—31no—
498C2×C8—31no—
499C2×C8—32no—
500C2×C8—64no—
501C2×C8—61no—
502C2×C8—61no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—82no—
515C22×C3—122no—
516C22×C3—242no—
517C22×C3—242no—
518C22×C3—242no—
519C22×C3#93911yescharacteristic
520C22×C3#93911yescharacteristic
521C22×C3C3×#12111yescharacteristic
522C22×C3C3×#12111yescharacteristic
523C22×C3C2×#23711yescharacteristic
524C22×C3C2×C3×#4111yescharacteristic
525C2×D3—42no—
526C2×D3—82no—
527C2×D3—82no—
528C2×D3—82no—
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—42no—
536C3×C4—44no—
537C3×C4—42no—
538C3×C4—42no—
539C3×C4#93911yescharacteristic
540C3×C4#93911yescharacteristic
541C3×C4C3×#12111yescharacteristic
542C3×C4C3×#12111yescharacteristic
543C3×C4C2×D3×D411yescharacteristic
544C3×C4C22×#6711yescharacteristic
545C3×C4C22×C3×D411yescharacteristic
546C3×C4C22×C3×D411yescharacteristic
547C32C2×#47111yescharacteristic, complemented, Sylow
548C23—21no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—242no—
553C23C3×#23711yescharacteristic
554Q8—21no—
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8C2×C3×#6712yes—
560D4—22no—
561D4—22no—
562D4—21no—
563D4—62no—
564D4—62no—
565D4—121no—
566D4—121no—
567D4C2×C3×#6712yes—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4C3×D3×D411yescharacteristic
575C2×C4C3×D3×D411yescharacteristic
576C2×C4C3×#23711yescharacteristic
577C2×C4C2×C3×#6711yescharacteristic
578C2×C4C2×C3×#6711yescharacteristic
579C2×C4C23×C3×D311yescharacteristic
580C8—22no—
581C8—32no—
582C8—32no—
583C8—62no—
584C8—62no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—21no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—42no—
594C2×C3—42no—
595C2×C3—82no—
596C2×C3—242no—
597C2×C3#520312yesdirect product factor
598C2×C3C3×#47112yescomplemented
599C2×C3C2×#93911yescharacteristic
600C2×C3C2×C3×#12111yescharacteristic
601D3—82no—
602C22—22no—
603C22—22no—
604C22—42no—
605C22—122no—
606C22—242no—
607C22—242no—
608C22—242no—
609C22C3×#93911yescharacteristic
610C22C3×#93911yescharacteristic
611C22C2×C3×#23711yescharacteristic
612C4—22no—
613C4—42no—
614C4C3×#93911yescharacteristic
615C4C3×#93911yescharacteristic
616C4C2×C3×D3×D411yescharacteristic
617C4C22×C3×#6711yescharacteristic
618C3—21no—
619C3C2×#520311yescharacteristic, direct product factor
620C3C2×C3×#47111yescharacteristic, complemented
621C2—21no—
622C2—21no—
623C2—42no—
624C2—242no—
625C2C3×#520312yesdirect product factor
626C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#137C2
socleC22×C32#121
prime radicalC2×C3×#949C2
hypercenterC2×C3×#137D3
hypocenterC3C2×C3×#471
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
1. C2×C32×#137C2semidirect
3. C2×C3×#948C2semidirect
4. C22×C3×#215C2semidirect
5. C2×C3×#836C2semidirect
6. C3×#5203C2direct (N × H)
7. C2×C3×#823C2semidirect
8. C2×C3×#822C2semidirect
10. C2×#5203C3direct (N × H)
15. C32×#137C22semidirect
22. C3×#948C22semidirect
27. C2×C3×#215C22semidirect
34. C3×#836C22semidirect
35. C3×#823C22semidirect
36. C3×#822C22semidirect
38. C2×C3×#137D3semidirect
39. C2×C3×#137C2×C3semidirect
44. C2×#948C2×C3semidirect
45. C22×#215C2×C3semidirect
50. C2×#836C2×C3semidirect
51. #5203C2×C3direct (N × H)
52. C2×#823C2×C3semidirect
53. C2×#822C2×C3semidirect
111. C3×#137C2×D3semidirect
112. C3×#137C22×C3semidirect
140. #948C22×C3semidirect
145. C2×#215C22×C3semidirect
155. #836C22×C3semidirect
156. #823C22×C3semidirect
157. #822C22×C3semidirect
195. C2×#137C3×D3semidirect
332. #137C2×C3×D3semidirect
456. C2×C32#471semidirect
547. C32C2×#471semidirect
597. C2×C3#5203direct (N × H)
598. C2×C3C3×#471semidirect
619. C3C2×#5203direct (N × H)
620. C3C2×C3×#471semidirect
625. C2C3×#5203direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C2×C3×#9492
C2×C3×#9482
C22×C3×#2152
C2×C3×#8362
C3×#52032
C2×C3×#8232
C2×C3×#8222
C2×C3×#4713
C2×#52033