Finite Group Databasebeta

← C2×#10590615 #11863728

Subgroups — C2×#10590615 #11863728

Order 1152.

1725 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3799C211yesmaximal, characteristic, complemented
2C2×C3×#895C211yesmaximal, characteristic, complemented
3C22×#3715C211yesmaximal, characteristic, complemented
4C22×#3714C211yesmaximal, characteristic, complemented
5C2×#63636C211yesmaximal, characteristic, complemented
6C2×C3×#810C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590615C218yesmaximal, direct product factor
9C2×#5037—31nomaximal
10C2×#4810—31nomaximal
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×D12—21no—
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C22×#626—21no—
17C22×#624—21no—
18C2×#3799—24no—
19C2×#3799—22no—
20C2×#3799—24no—
21C2×#3799C2214yescomplemented
22C2×C32×C4⋊C4C2211yescharacteristic
23C22×C3⋊(C3⋊C8)C2211yescharacteristic
24C2×C3×(C2×D3)⋊C4—21no—
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#895—22no—
27C3×#895C2214yescomplemented
28C2×#3715—24no—
29C2×#3715C2214yescomplemented
30C2×#3714—24no—
31C2×#3714C2214yescomplemented
32#63636C2214yescomplemented
33C3×#810C2214yescomplemented
34(C2×D3)⋊(C3⋊C8)C2214yescomplemented
35C22×#239—31no—
36C22×#235—31no—
37C2×C3×#121—31no—
38C22×#215—31no—
39C22×#214—31no—
40C22×#213—31no—
41C22×D24—31no—
42C2×#895D311yescharacteristic, complemented
43C2×C3×#103—31no—
44C2×#830—31no—
45C2×(C2×D3)⋊C8—31no—
46#5037—38no—
47C2×#810D311yescharacteristic, complemented
48C2×#809—31no—
49C2×#809—31no—
50C2×#809—61no—
51#4810—38no—
52C23×C3⋊D3—21no—
53C23×C3×D3—21no—
54C23×C3×D3D411yescharacteristic
55C22×C32×C4—21no—
56C22×C32×C4C2311yescharacteristic
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)—22no—
59C2×C4⋊(C3⋊D3)D414yescomplemented
60C2×C4⋊(C3⋊D3)C2312yes—
61C22×C3×Dic3—21no—
62C22×C3×Dic3D411yescharacteristic
63C2×C3×D12—22no—
64C2×C3×D12—22no—
65C2×C3×D12—24no—
66C2×C3×D12—21no—
67C2×C3×D12—44no—
68C2×C3×D12C2312yes—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3—22no—
71C2×C3×C4×D3D412yes—
72C2×C3×C4×D3C2312yes—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—22no—
75C2×C3×Dic6D414yescomplemented
76C2×C3×Dic6C2312yes—
77C2×#626—28no—
78C2×#626—24no—
79C2×#624—28no—
80C2×#624—24no—
81#3799—216no—
82#3799—28no—
83#3799—28no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2312yes—
86C2×C3⋊(C3⋊C8)—22no—
87C2×C3⋊(C3⋊C8)C2312yes—
88C3×(C2×D3)⋊C4—24no—
89C2×C3×C3⋊C8—22no—
90C2×C3×C3⋊C8C2312yes—
91#3715—28no—
92#3714—28no—
93C2×#449—91noSylow
94C22×C3×Q8—31no—
95C22×C3×D4—31no—
96C22×C3×D4—61no—
97C23×C3×C4—31no—
98C22×#67—61no—
99C2×#239—34no—
100C2×#239—64no—
101C2×#239—64no—
102C2×#239—62no—
103C2×#235—34no—
104C2×#235—64no—
105C2×#235—62no—
106C2×#235—64no—
107C22×D12—21no—
108C22×D12—31no—
109C22×D12—31no—
110C22×D12—61no—
111C22×D12—61no—
112C22×D12C2×D311yescharacteristic
113C22×C4×D3—61no—
114C22×C4×D3—61no—
115C22×C4×D3C2×D311yescharacteristic
116C22×Dic6C2×D311yescharacteristic
117C22×C3×C8—31no—
118C3×#121—34no—
119C3×#121—62no—
120C2×C3×C4⋊C4—21no—
121C2×C3×C4⋊C4C2×D311yescharacteristic
122C2×C3×C4⋊C4C2×D311yescharacteristic
123C2×C3×C22⋊C4—61no—
124C2×#215—34no—
125C2×#215—64no—
126C2×#214—34no—
127C2×#214—64no—
128C2×#213—34no—
129C2×#213—64no—
130C2×(C2×D3)⋊C4—21no—
131C22×C3⋊C8—31no—
132C22×C3⋊C8—31no—
133C22×C3⋊C8—61no—
134C22×C3⋊C8C2×D311yescharacteristic
135C2×D24—34no—
136C2×D24—64no—
137#895—22no—
138#895C2×D314yescomplemented
139C3×#103—34no—
140#830—34no—
141(C2×D3)⋊C8—34no—
142#810C2×D314yescomplemented
143#809—34no—
144#809—34no—
145#809—64no—
146C23×C32C2×D411yescharacteristic
147C22×C3⋊D3—24no—
148C22×C3⋊D3—22no—
149C22×C3⋊D3—42no—
150C22×C3⋊D3—42no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—42no—
156C22×C3×D3—42no—
157C22×C3×D3—42no—
158C22×C3×D3C2×D412yes—
159C2×C32×C4—22no—
160C2×C32×C4—42no—
161C2×C32×C4C2×D412yes—
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2411yescharacteristic
165C4⋊(C3⋊D3)—24no—
166C4⋊(C3⋊D3)—22no—
167C4⋊(C3⋊D3)C2×D414yescomplemented
168C2×C3×Dic3—22no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3C2×D412yes—
172C3×D12—22no—
173C3×D12—22no—
174C3×D12—48no—
175C3×D12—42no—
176C3×C4×D3—24no—
177C3×C4×D3—22no—
178C3×C4×D3—22no—
179C3×Dic6—24no—
180C3×Dic6—22no—
181C3×Dic6C2×D414yescomplemented
182#626—216no—
183#624—216no—
184C3⋊(C3⋊C8)—22no—
185C3×C3⋊C8—22no—
186C22×#41—91no—
187C22×SD16—91no—
188C22×D8—91no—
189C2×#121—31no—
190#449—98no—
191C2×#103—31no—
192C2×#102—91no—
193C2×C22⋊C8—91no—
194C24×C3—31no—
195C24×C3—61no—
196C23×D3—21no—
197C23×D3—61no—
198C23×D3—61no—
199C23×D3—121no—
200C23×D3#6711yescharacteristic
201C2×C3×Q8—34no—
202C2×C3×Q8—32no—
203C2×C3×Q8—62no—
204C2×C3×Q8—61no—
205C2×C3×D4—32no—
206C2×C3×D4—64no—
207C2×C3×D4—62no—
208C2×C3×D4—62no—
209C2×C3×D4—61no—
210C2×C3×D4—124no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—31no—
215C22×C3×C4—32no—
216C22×C3×C4—32no—
217C22×C3×C4—41no—
218C22×C3×C4—61no—
219C22×C3×C4—62no—
220C22×C3×C4—62no—
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C22×D311yescharacteristic
223C2×#67—64no—
224C2×#67—68no—
225C22×Dic3—21no—
226C22×Dic3#6711yescharacteristic
227#239—68no—
228#239—68no—
229#239—616no—
230#235—68no—
231#235—616no—
232#235—68no—
233C2×D12—22no—
234C2×D12—22no—
235C2×D12—24no—
236C2×D12—21no—
237C2×D12—34no—
238C2×D12—34no—
239C2×D12—32no—
240C2×D12—32no—
241C2×D12—44no—
242C2×D12—62no—
243C2×D12—64no—
244C2×D12—68no—
245C2×D12—64no—
246C2×D12—62no—
247C2×D12—61no—
248C2×D12—62no—
249C2×D12—61no—
250C2×D12—122no—
251C2×D12—121no—
252C2×D12C22×D312yes—
253C2×C4×D3—22no—
254C2×C4×D3—22no—
255C2×C4×D3—64no—
256C2×C4×D3—68no—
257C2×C4×D3—68no—
258C2×C4×D3—64no—
259C2×C4×D3#6712yes—
260C2×C4×D3C22×D312yes—
261C2×Dic6—22no—
262C2×Dic6—21no—
263C2×Dic6D1214yescomplemented
264C2×Dic6C22×D312yes—
265C2×C3×C8—32no—
266C2×C3×C8—62no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—22no—
270C3×C4⋊C4—41no—
271C3×C4⋊C4C22×D312yes—
272C3×C4⋊C4C22×D312yes—
273C3×C22⋊C4—64no—
274#215—68no—
275#214—68no—
276#213—68no—
277(C2×D3)⋊C4—24no—
278C2×C3⋊C8—22no—
279C2×C3⋊C8—32no—
280C2×C3⋊C8—32no—
281C2×C3⋊C8—62no—
282C2×C3⋊C8—62no—
283C2×C3⋊C8—62no—
284C2×C3⋊C8—122no—
285C2×C3⋊C8C22×D312yes—
286D24—68no—
287C22×C32#12012yes—
288C22×C32#13512yes—
289C22×C32AGL(1,ℤ/8)12yes—
290C22×C32C22×D411yescharacteristic
291C2×C3⋊D3—24no—
292C2×C3⋊D3—42no—
293C2×C3⋊D3—42no—
294C2×C3⋊D3—48no—
295C2×C3×D3—22no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—44no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—84no—
303C32×C4—42no—
304C32×C4#12012yes—
305C32×C4C22×D411yescharacteristic
306C32×C4C22×D411yescharacteristic
307C3×Dic3—22no—
308C3×Dic3—24no—
309C2×#41—94no—
310C2×#41—182no—
311C2×#41—184no—
312C2×#41—184no—
313C22×Q8—31no—
314C22×D4—31no—
315C22×D4—61no—
316C22×D4—91no—
317C22×D4—181no—
318C23×C4—31no—
319C23×C4—181no—
320C2×SD16—94no—
321C2×SD16—184no—
322C2×D8—94no—
323C2×D8—184no—
324C22×C8—31no—
325C22×C8—91no—
326#121—34no—
327#121—62no—
328C2×C4⋊C4D3211yescharacteristic
329C2×C22⋊C4—61no—
330#103—34no—
331#102—94no—
332C22⋊C8—94no—
333C23×C3—21no—
334C23×C3—32no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—62no—
339C23×C3—122no—
340C23×C3—122no—
341C23×C3—122no—
342C23×C3D3×D411yescharacteristic
343C23×C3C2×#6711yescharacteristic
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—64no—
352C22×D3—64no—
353C22×D3—62no—
354C22×D3—62no—
355C22×D3—122no—
356C22×D3—124no—
357C22×D3—122no—
358C22×D3—122no—
359C22×D3—122no—
360C22×D3—122no—
361C22×D3—242no—
362C22×D3—242no—
363C22×D3C2×#6712yes—
364C3×Q8—34no—
365C3×Q8—64no—
366C3×Q8—62no—
367C3×D4—62no—
368C3×D4—62no—
369C3×D4—128no—
370C3×D4—122no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—32no—
378C2×C3×C4—42no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—62no—
382C2×C3×C4—64no—
383C2×C3×C4—62no—
384C2×C3×C4—64no—
385C2×C3×C4—62no—
386C2×C3×C4—62no—
387C2×C3×C4—82no—
388C2×C3×C4C2×D1212yes—
389C2×C3×C4D3×D412yes—
390C2×C3×C4D3×D412yes—
391C2×C3×C4D3×D412yes—
392C2×C3×C4D3×D411yescharacteristic
393C2×C3×C4C2×#6711yescharacteristic
394C2×C3×C4C23×D311yescharacteristic
395C2×C3×C4C23×D311yescharacteristic
396#67—616no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3C2×#6712yes—
401D12—22no—
402D12—22no—
403D12—34no—
404D12—34no—
405D12—48no—
406D12—42no—
407D12—64no—
408D12—616no—
409D12—64no—
410D12—64no—
411D12—62no—
412D12—62no—
413D12—122no—
414D12—124no—
415C4×D3—24no—
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—616no—
419C4×D3—616no—
420Dic6—24no—
421Dic6—22no—
422Dic6C2×D1214yescomplemented
423C3×C8—62no—
424C3⋊C8—22no—
425C3⋊C8—62no—
426C3⋊C8—62no—
427C3⋊C8—122no—
428C2×C32#44914yescomplemented
429C2×C32C2×#12011yescharacteristic
430C2×C32C2×#13511yescharacteristic
431C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
432C3⋊D3—44no—
433C3×D3—42no—
434C3×D3—82no—
435C24—31no—
436C24—61no—
437C24—181no—
438#41—188no—
439#41—188no—
440#41—1816no—
441C2×Q8—32no—
442C2×Q8—34no—
443C2×Q8—62no—
444C2×Q8—61no—
445C2×D4—32no—
446C2×D4—64no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—61no—
450C2×D4—92no—
451C2×D4—94no—
452C2×D4—124no—
453C2×D4—184no—
454C2×D4—188no—
455C2×D4—182no—
456C2×D4—181no—
457C22×C4—21no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—61no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—184no—
465C22×C4—188no—
466C22×C4C2×D3211yescharacteristic
467SD16—188no—
468D8—188no—
469C2×C8—32no—
470C2×C8—62no—
471C2×C8—92no—
472C2×C8—182no—
473C4⋊C4—21no—
474C4⋊C4C2×D3212yes—
475C22⋊C4—64no—
476C22×C3—21no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—124no—
484C22×C3—122no—
485C22×C3—122no—
486C22×C3—122no—
487C22×C3—244no—
488C22×C3#88212yes—
489C22×C3#91412yes—
490C22×C3#91912yes—
491C22×C3#93212yes—
492C22×C3#93712yes—
493C22×C3#95012yes—
494C22×C3C2×D3×D411yescharacteristic
495C22×C3C22×#6711yescharacteristic
496C2×D3—22no—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—44no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—64no—
504C2×D3—64no—
505C2×D3—84no—
506C2×D3—124no—
507C2×D3—122no—
508C2×D3—122no—
509C2×D3—122no—
510C2×D3—128no—
511C2×D3—122no—
512C2×D3—128no—
513C2×D3—242no—
514C2×D3—242no—
515C2×D3—248no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—62no—
522C3×C4—64no—
523C3×C4—82no—
524C3×C4#88212yes—
525C3×C4#93712yes—
526C3×C4C22×D1211yescharacteristic
527C3×C4C2×D3×D411yescharacteristic
528C3×C4C2×D3×D411yescharacteristic
529C3×C4C2×D3×D411yescharacteristic
530Dic3—24no—
531Dic3—22no—
532C32C2×#44911yescharacteristic, complemented, Sylow
533C23—32no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—122no—
539C23—122no—
540C23—122no—
541C23—182no—
542C23—184no—
543C23—362no—
544C23—362no—
545C23D3×#6711yescharacteristic
546Q8—34no—
547Q8—62no—
548Q8—64no—
549D4—62no—
550D4—62no—
551D4—94no—
552D4—128no—
553D4—122no—
554D4—1816no—
555D4—182no—
556D4—184no—
557C2×C4—22no—
558C2×C4—32no—
559C2×C4—42no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—1816no—
567C2×C4D3×D1212yes—
568C2×C4#380212yes—
569C2×C4D3×#6711yescharacteristic
570C2×C4C22×D3211yescharacteristic
571C8—62no—
572C8—182no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—24no—
576C2×C3—21no—
577C2×C3—122no—
578C2×C3—242no—
579C2×C3#481014yescomplemented
580C2×C3#503714yescomplemented
581C2×C3C2×#88211yescharacteristic
582C2×C3C2×#91411yescharacteristic
583C2×C3C2×#91911yescharacteristic
584C2×C3C2×#93211yescharacteristic
585C2×C3C2×#93711yescharacteristic
586C2×C3C2×#95011yescharacteristic
587D3—42no—
588D3—82no—
589D3—124no—
590D3—124no—
591D3—244no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22—124no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—184no—
600C22—244no—
601C22—362no—
602C22—362no—
603C22—368no—
604C22#6392212yes—
605C22#6393412yes—
606C22#6394912yes—
607C22C2×D3×#6711yescharacteristic
608C4—42no—
609C4—62no—
610C4—64no—
611C4#6392212yes—
612C4C2×D3×D1211yescharacteristic
613C4C2×#380211yescharacteristic
614C3—21no—
615C3C2×#481011yescharacteristic, complemented
616C3C2×#503711yescharacteristic, complemented
617C2—122no—
618C2—242no—
619C2—364no—
620C2#1059061514yesdirect product factor
621C2C2×#6392211yescharacteristic
622C2C2×#6393411yescharacteristic
623C2C2×#6394911yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×C4⋊C4C22
socleC23×C32C2×D4
hypercenterC2×C4⋊C4D32
hypocenterC32C2×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#3799C2semidirect
2. C2×C3×#895C2semidirect
3. C22×#3715C2semidirect
4. C22×#3714C2semidirect
5. C2×#63636C2semidirect
6. C2×C3×#810C2semidirect
7. C2×(C2×D3)⋊(C3⋊C8)C2semidirect
8. #10590615C2direct (N × H)
21. C2×#3799C22semidirect
27. C3×#895C22semidirect
29. C2×#3715C22semidirect
31. C2×#3714C22semidirect
32. #63636C22semidirect
33. C3×#810C22semidirect
34. (C2×D3)⋊(C3⋊C8)C22semidirect
42. C2×#895D3semidirect
47. C2×#810D3semidirect
59. C2×C4⋊(C3⋊D3)D4semidirect
75. C2×C3×Dic6D4semidirect
138. #895C2×D3semidirect
142. #810C2×D3semidirect
167. C4⋊(C3⋊D3)C2×D4semidirect
181. C3×Dic6C2×D4semidirect
263. C2×Dic6D12semidirect
422. Dic6C2×D12semidirect
428. C2×C32#449semidirect
532. C32C2×#449semidirect
579. C2×C3#4810semidirect
580. C2×C3#5037semidirect
615. C3C2×#4810semidirect
616. C3C2×#5037semidirect
620. C2#10590615direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37992
C2×C3×#8952
C22×#37152
C22×#37142
C2×#636362
C2×C3×#8102
C2×(C2×D3)⋊(C3⋊C8)2
#105906152
C2×#50373
C2×#48103