Finite Group Databasebeta

← C2×C3×#5150 #11865425

Subgroups — C2×C3×#5150 #11865425

Order 1152.

1465 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D8C211yesmaximal, characteristic, complemented
2C2×C3×#940C211yesmaximal, characteristic, complemented
3C2×C3×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×C3×#213C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic, complemented
6C3×#5150C218yesmaximal, direct product factor
7C2×C3×#821C211yesmaximal, characteristic, complemented
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#459—31nomaximal
10C2×#5150C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67—21no—
14C22×C3×#67—21no—
15C23×C3×Dic3—21no—
16C22×C3×D12C2211yescharacteristic
17C2×C32×D8—24no—
18C2×C32×D8C2214yescomplemented
19C22×C32×C8C2211yescharacteristic
20C2×C3×#217—21no—
21C3×#940—22no—
22C3×#940C2214yescomplemented
23C3×D4×Dic3—22no—
24C3×D4×Dic3C2214yescomplemented
25C2×C3×#213—24no—
26C2×C3×#213C2214yescomplemented
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#832C2214yescomplemented
31C3×#821C2214yescomplemented
32C3×Dic3⋊C8C2214yescomplemented
33C22×C3×D8—21no—
34C22×C3×D8—31no—
35C22×C3×D8D311yescharacteristic, complemented
36C22×C3×D8C2×C311yescharacteristic, complemented
37C2×C3×#127—31no—
38C2×C3×C4×D4—31no—
39C2×#940C2×C311yescharacteristic, complemented
40C2×D4×Dic3C2×C311yescharacteristic, complemented
41C22×#213C2×C311yescharacteristic, complemented
42C3×#459—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#102—31no—
45C2×C3×#102—31no—
46C2×#832C2×C311yescharacteristic, complemented
47#5150C2×C318yesdirect product factor
48C2×#821C2×C311yescharacteristic, complemented
49C2×Dic3⋊C8C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C24×C32—21no—
52C23×C3×D3—21no—
53C2×C32×D4—21no—
54C2×C32×D4—22no—
55C2×C32×D4—21no—
56C2×C32×D4—22no—
57C2×C32×D4—22no—
58C2×C32×D4D414yescomplemented
59C2×C32×D4C2312yes—
60C2×C32×D4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67—24no—
63C2×C3×#67—24no—
64C2×C3×#67—44no—
65C2×C3×#67—44no—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—24no—
68C22×C3×Dic3—44no—
69C22×C3×Dic3D411yescharacteristic
70C22×C3×Dic3D411yescharacteristic
71C2×C3×D12—22no—
72C2×C3×D12—22no—
73C2×C3×D12—21no—
74C2×C3×D12C2312yes—
75C32×D8—28no—
76C2×C32×C8—22no—
77C2×C32×C8C2312yes—
78C3×#217—24no—
79C3×#213—28no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3C2312yes—
82C3×C4×Dic3D412yes—
83C3×C4×Dic3C2312yes—
84C2×C3×C3⋊C8—22no—
85C2×C3×C3⋊C8C2312yes—
86C2×#459—31noSylow
87C22×C3×D4—21no—
88C22×C3×D4—21no—
89C22×C3×D4—31no—
90C22×C3×D4—61no—
91C22×C3×D4—61no—
92C22×C3×D4C2×D311yescharacteristic
93C22×C3×D4C2×D311yescharacteristic
94C22×C3×D4C22×C311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C23×C3×C4—61no—
97C22×#67—21no—
98C22×#67—21no—
99C23×Dic3—21no—
100C22×D12C22×C311yescharacteristic
101C2×C3×D8—28no—
102C2×C3×D8—24no—
103C2×C3×D8—24no—
104C2×C3×D8—24no—
105C2×C3×D8—34no—
106C2×C3×D8—64no—
107C2×C3×D8C2×D314yescomplemented
108C2×C3×D8C22×C314yescomplemented
109C22×C3×C8—21no—
110C22×C3×C8—31no—
111C22×C3×C8C2×D311yescharacteristic
112C22×C3×C8C22×C311yescharacteristic
113C3×#127—34no—
114C3×#127—62no—
115C3×C4×D4—34no—
116C3×C4×D4—62no—
117C2×C3×C4⋊C4—31no—
118C2×C3×C22⋊C4—61no—
119C2×C3×C42—31no—
120C2×#217—21no—
121#940—22no—
122#940C22×C314yescomplemented
123D4×Dic3—22no—
124D4×Dic3C22×C314yescomplemented
125C2×#213—24no—
126C2×#213C22×C314yescomplemented
127C2×C4⋊Dic3C22×C311yescharacteristic
128C2×C4×Dic3C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C3×C4⋊C8—34no—
131C3×#102—34no—
132C3×#102—34no—
133#832C22×C314yescomplemented
134#821C22×C314yescomplemented
135Dic3⋊C8C22×C314yescomplemented
136C23×C32—24no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—42no—
140C23×C32—42no—
141C23×C32—42no—
142C23×C32—42no—
143C23×C32—42no—
144C23×C32C2×D411yescharacteristic
145C22×C3×D3—22no—
146C22×C3×D3—42no—
147C22×C3×D3—42no—
148C22×C3×D3—42no—
149C32×D4—22no—
150C32×D4—24no—
151C32×D4—22no—
152C32×D4—22no—
153C32×D4—42no—
154C32×D4C2×D414yescomplemented
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D412yes—
157C2×C32×C4#4112yes—
158C2×C32×C4C2411yescharacteristic
159C3×#67—48no—
160C3×#67—48no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—42no—
164C2×C3×Dic3—48no—
165C2×C3×Dic3D814yescomplemented
166C2×C3×Dic3C2×D412yes—
167C2×C3×Dic3C2×D412yes—
168C3×D12—22no—
169C3×D12—22no—
170C3×D12—42no—
171C32×C8—22no—
172C3×C3⋊C8—22no—
173C22×D8—31no—
174C22×D8C3×D311yescharacteristic, complemented
175C2×#127—31no—
176C2×C4×D4—31no—
177#459—38no—
178C2×C4⋊C8—31no—
179C2×#102—31no—
180C2×#102—31no—
181C24×C3—21no—
182C24×C3—21no—
183C24×C3—21no—
184C24×C3—21no—
185C24×C3—41no—
186C24×C3—41no—
187C24×C3—61no—
188C23×D3—21no—
189C2×C3×D4—24no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—24no—
201C2×C3×D4—22no—
202C2×C3×D4—22no—
203C2×C3×D4—32no—
204C2×C3×D4—41no—
205C2×C3×D4—41no—
206C2×C3×D4—42no—
207C2×C3×D4—42no—
208C2×C3×D4—64no—
209C2×C3×D4—64no—
210C2×C3×D4—61no—
211C2×C3×D4—62no—
212C2×C3×D4—62no—
213C2×C3×D4—124no—
214C2×C3×D4—124no—
215C2×C3×D4#6714yescomplemented
216C2×C3×D4C3×D414yescomplemented
217C2×C3×D4C22×D312yes—
218C2×C3×D4C22×D312yes—
219C2×C3×D4C23×C312yes—
220C2×C3×D4C23×C312yes—
221C22×C3×C4—21no—
222C22×C3×C4—31no—
223C22×C3×C4—31no—
224C22×C3×C4—64no—
225C22×C3×C4—61no—
226C22×C3×C4—124no—
227C22×C3×C4C22×D311yescharacteristic
228C22×C3×C4C23×C311yescharacteristic
229C2×#67—24no—
230C2×#67—24no—
231C2×#67—44no—
232C2×#67—44no—
233C22×Dic3—21no—
234C22×Dic3—24no—
235C22×Dic3—44no—
236C22×Dic3C3×D411yescharacteristic
237C22×Dic3C3×D411yescharacteristic
238C2×D12—22no—
239C2×D12—22no—
240C2×D12—21no—
241C2×D12C23×C312yes—
242C3×D8—216no—
243C3×D8—28no—
244C3×D8—28no—
245C3×D8—68no—
246C2×C3×C8—24no—
247C2×C3×C8—22no—
248C2×C3×C8—22no—
249C2×C3×C8—22no—
250C2×C3×C8—32no—
251C2×C3×C8—62no—
252C2×C3×C8C22×D312yes—
253C2×C3×C8C23×C312yes—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—61no—
256C3×C22⋊C4—64no—
257C3×C42—32no—
258C3×C42—32no—
259#217—24no—
260#213—28no—
261C4⋊Dic3—21no—
262C4⋊Dic3C23×C312yes—
263C4×Dic3C3×D412yes—
264C4×Dic3C23×C312yes—
265C2×C3⋊C8—22no—
266C2×C3⋊C8C23×C312yes—
267C22×C32—24no—
268C22×C32—42no—
269C22×C32—42no—
270C22×C32—42no—
271C22×C32—48no—
272C22×C32—42no—
273C22×C32—42no—
274C22×C32—84no—
275C22×C32#12112yes—
276C22×C32C2×D812yes—
277C22×C32AGL(1,ℤ/8)12yes—
278C22×C32C22×D411yescharacteristic
279C2×C3×D3—42no—
280C2×C3×D3—42no—
281C2×C3×D3—42no—
282C2×C3×D3—84no—
283C32×C4#12112yes—
284C32×C4C22×D411yescharacteristic
285C32×C4C2×#4111yescharacteristic
286C3×Dic3—22no—
287C3×Dic3—42no—
288C3×Dic3C2×D814yescomplemented
289C22×D4—31no—
290C22×D4—61no—
291C22×D4—61no—
292C22×D4C2×C3×D311yescharacteristic
293C22×D4C2×C3×D311yescharacteristic
294C23×C4—61no—
295C2×D8—24no—
296C2×D8—34no—
297C2×D8—64no—
298C2×D8C2×C3×D314yescomplemented
299C22×C8—31no—
300C22×C8C2×C3×D311yescharacteristic
301#127—34no—
302#127—62no—
303C4×D4—34no—
304C4×D4—62no—
305C2×C4⋊C4—31no—
306C2×C22⋊C4—61no—
307C2×C42—31no—
308C4⋊C8—34no—
309#102—34no—
310#102—34no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3—44no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—44no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—62no—
333C23×C3—82no—
334C23×C3—82no—
335C23×C3—82no—
336C23×C3—82no—
337C23×C3—122no—
338C23×C3—122no—
339C23×C3—122no—
340C23×C3D3×D411yescharacteristic
341C23×C3C2×C3×D411yescharacteristic
342C22×D3—22no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C3×D4—24no—
347C3×D4—24no—
348C3×D4—24no—
349C3×D4—22no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—24no—
356C3×D4—44no—
357C3×D4—42no—
358C3×D4—42no—
359C3×D4—44no—
360C3×D4—42no—
361C3×D4—42no—
362C3×D4—62no—
363C3×D4—62no—
364C3×D4—128no—
365C3×D4—128no—
366C3×D4—122no—
367C3×D4C2×#6714yescomplemented
368C3×D4C2×C3×D414yescomplemented
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—34no—
374C2×C3×C4—32no—
375C2×C3×C4—32no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—128no—
379C2×C3×C4—122no—
380C2×C3×C4D3×D411yescharacteristic
381C2×C3×C4#23712yes—
382C2×C3×C4C2×#6712yes—
383C2×C3×C4C2×C3×D411yescharacteristic
384C2×C3×C4C2×C3×D412yes—
385C2×C3×C4C3×#4112yes—
386C2×C3×C4C23×D311yescharacteristic
387C2×C3×C4C24×C311yescharacteristic
388#67—48no—
389#67—48no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—42no—
393C2×Dic3—48no—
394C2×Dic3C3×D814yescomplemented
395C2×Dic3C2×C3×D412yes—
396C2×Dic3C2×C3×D412yes—
397D12—22no—
398D12—22no—
399D12—42no—
400C3×C8—24no—
401C3×C8—22no—
402C3×C8—22no—
403C3×C8—62no—
404C3⋊C8—22no—
405C2×C32—44no—
406C2×C32—82no—
407C2×C32#45914yescomplemented
408C2×C32C2×#12111yescharacteristic
409C2×C32C22×D811yescharacteristic
410C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
411C3×D3—82no—
412C24—21no—
413C24—21no—
414C24—61no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—32no—
421C2×D4—64no—
422C2×D4—64no—
423C2×D4—61no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—124no—
427C2×D4—124no—
428C2×D4C3×#6714yescomplemented
429C2×D4C22×C3×D312yes—
430C2×D4C22×C3×D312yes—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—64no—
434C22×C4—61no—
435C22×C4—124no—
436C22×C4C22×C3×D311yescharacteristic
437D8—28no—
438D8—68no—
439C2×C8—22no—
440C2×C8—32no—
441C2×C8—62no—
442C2×C8C22×C3×D312yes—
443C4⋊C4—32no—
444C4⋊C4—61no—
445C22⋊C4—64no—
446C42—32no—
447C42—32no—
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—21no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—48no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—48no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3—88no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3—88no—
474C22×C3—84no—
475C22×C3—84no—
476C22×C3—122no—
477C22×C3—122no—
478C22×C3—122no—
479C22×C3—244no—
480C22×C3D3×D812yes—
481C22×C3#91112yes—
482C22×C3#93912yes—
483C22×C3C3×#12112yes—
484C22×C3C2×C3×D812yes—
485C22×C3C3×AGL(1,ℤ/8)12yes—
486C22×C3C2×D3×D411yescharacteristic
487C22×C3C22×C3×D411yescharacteristic
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—84no—
492C3×C4—22no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—34no—
496C3×C4—62no—
497C3×C4—122no—
498C3×C4#93912yes—
499C3×C4C3×#12112yes—
500C3×C4C2×#23711yescharacteristic
501C3×C4C22×#6711yescharacteristic
502C3×C4C22×C3×D411yescharacteristic
503C3×C4C2×C3×#4111yescharacteristic
504Dic3—22no—
505Dic3—42no—
506Dic3C2×C3×D814yescomplemented
507C32C2×#45911yescharacteristic, complemented, Sylow
508C23—24no—
509C23—22no—
510C23—22no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—62no—
517C23—122no—
518C23—122no—
519C23—122no—
520C23C3×D3×D411yescharacteristic
521D4—24no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4—42no—
526D4—62no—
527D4—62no—
528D4—128no—
529D4—128no—
530D4—122no—
531D4C2×C3×#6714yescomplemented
532C2×C4—34no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—128no—
538C2×C4—122no—
539C2×C4C3×D3×D411yescharacteristic
540C2×C4C3×#23712yes—
541C2×C4C2×C3×#6712yes—
542C2×C4C23×C3×D311yescharacteristic
543C8—22no—
544C8—62no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—24no—
549C2×C3—44no—
550C2×C3—44no—
551C2×C3—84no—
552C2×C3—84no—
553C2×C3—82no—
554C2×C3—82no—
555C2×C3—242no—
556C2×C3#515014yesdirect product factor
557C2×C3C3×#45914yescomplemented
558C2×C3C2×D3×D811yescharacteristic
559C2×C3C2×#91111yescharacteristic
560C2×C3C2×#93911yescharacteristic
561C2×C3C2×C3×#12111yescharacteristic
562C2×C3C22×C3×D811yescharacteristic
563C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
564D3—82no—
565C22—24no—
566C22—42no—
567C22—42no—
568C22—48no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—84no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—244no—
577C22C3×D3×D812yes—
578C22C3×#91112yes—
579C22C3×#93912yes—
580C22C2×C3×D3×D411yescharacteristic
581C4—34no—
582C4—62no—
583C4—122no—
584C4C3×#93912yes—
585C4C2×C3×#23711yescharacteristic
586C4C22×C3×#6711yescharacteristic
587C3—21no—
588C3C2×#515011yescharacteristic, direct product factor
589C3C2×C3×#45911yescharacteristic, complemented
590C2—44no—
591C2—82no—
592C2—242no—
593C2C3×#515014yesdirect product factor
594C2C2×C3×D3×D811yescharacteristic
595C2C2×C3×#91111yescharacteristic
596C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×D8C2
socleC23×C32C2×D4
hypercenterC22×C3×D8D3
hypocenterC3C2×C3×#459
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×D8

Complements

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

NH ≅ G/Ntype
1. C22×C32×D8C2semidirect
2. C2×C3×#940C2semidirect
3. C2×C3×D4×Dic3C2semidirect
4. C22×C3×#213C2semidirect
5. C2×C3×#832C2semidirect
6. C3×#5150C2direct (N × H)
7. C2×C3×#821C2semidirect
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5150C3direct (N × H)
18. C2×C32×D8C22semidirect
22. C3×#940C22semidirect
24. C3×D4×Dic3C22semidirect
26. C2×C3×#213C22semidirect
30. C3×#832C22semidirect
31. C3×#821C22semidirect
32. C3×Dic3⋊C8C22semidirect
35. C22×C3×D8D3semidirect
36. C22×C3×D8C2×C3semidirect
39. C2×#940C2×C3semidirect
40. C2×D4×Dic3C2×C3semidirect
41. C22×#213C2×C3semidirect
46. C2×#832C2×C3semidirect
47. #5150C2×C3direct (N × H)
48. C2×#821C2×C3semidirect
49. C2×Dic3⋊C8C2×C3semidirect
58. C2×C32×D4D4semidirect
107. C2×C3×D8C2×D3semidirect
108. C2×C3×D8C22×C3semidirect
122. #940C22×C3semidirect
124. D4×Dic3C22×C3semidirect
126. C2×#213C22×C3semidirect
133. #832C22×C3semidirect
134. #821C22×C3semidirect
135. Dic3⋊C8C22×C3semidirect
154. C32×D4C2×D4semidirect
165. C2×C3×Dic3D8semidirect
174. C22×D8C3×D3semidirect
215. C2×C3×D4#67semidirect
216. C2×C3×D4C3×D4semidirect
288. C3×Dic3C2×D8semidirect
298. C2×D8C2×C3×D3semidirect
367. C3×D4C2×#67semidirect
368. C3×D4C2×C3×D4semidirect
394. C2×Dic3C3×D8semidirect
407. C2×C32#459semidirect
428. C2×D4C3×#67semidirect
506. Dic3C2×C3×D8semidirect
507. C32C2×#459semidirect
531. D4C2×C3×#67semidirect
556. C2×C3#5150direct (N × H)
557. C2×C3C3×#459semidirect
588. C3C2×#5150direct (N × H)
589. C3C2×C3×#459semidirect
593. C2C3×#5150direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×D82
C2×C3×#9402
C2×C3×D4×Dic32
C22×C3×#2132
C2×C3×#8322
C3×#51502
C2×C3×#8212
C2×C3×Dic3⋊C82
C2×C3×#4593
C2×#51503