Finite Group Databasebeta

← C2×#10591588 #11863731

Subgroups — C2×#10591588 #11863731

Order 1152.

1773 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#239C211yesmaximal, characteristic, complemented
2C22×#3755C211yesmaximal, characteristic, complemented
3C2×#63909C211yesmaximal, characteristic, complemented
4C22×#3714C211yesmaximal, characteristic, complemented
5C2×C3×#835C211yesmaximal, characteristic, complemented
6#10591588C218yesmaximal, direct product factor
7C2×#63562C211yesmaximal, characteristic, complemented
8C2×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×#5238—31nomaximal
10C2×#4810—31nomaximal
11C22×C32×Q8C2211yescharacteristic
12C22×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#239—24no—
14C2×C3×#239—24no—
15C2×C3×#239—22no—
16C2×C3×#239C2214yescomplemented
17C22×C3×D12—21no—
18C22×C3×D12C2211yescharacteristic
19C22×C3×C4×D3—21no—
20C22×C3×C4×D3C2211yescharacteristic
21C22×#626—21no—
22C2×#3755—24no—
23C2×#3755C2214yescomplemented
24C22×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27C2×#3722—21no—
28#63909—22no—
29#63909C2214yescomplemented
30C2×#3714—24no—
31C2×#3714C2214yescomplemented
32C3×#835C2214yescomplemented
33#63562C2214yescomplemented
34(C2×D3)⋊(C3⋊C8)C2214yescomplemented
35C22×C3×#41—31no—
36C22×#239D311yescharacteristic, complemented
37C22×#215—31no—
38C22×#215—31no—
39C22×#215—61no—
40C22×#213—31no—
41C2×#930—31no—
42C22×D24—31no—
43C2×#895—31no—
44C2×C3×#103—31no—
45#5238—38no—
46C2×#835D311yescharacteristic, complemented
47C2×#832—31no—
48C2×#830—31no—
49C2×(C2×D3)⋊C8—31no—
50C2×#809—31no—
51#4810—38no—
52C23×C3⋊D3—21no—
53C23×C3×D3—21no—
54C23×C3×D3D411yescharacteristic
55C2×C32×Q8—22no—
56C2×C32×Q8—21no—
57C2×C32×Q8D414yescomplemented
58C2×C32×Q8C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C2×C4⋊(C3⋊D3)—22no—
62C2×C4⋊(C3⋊D3)—22no—
63C2×C4⋊(C3⋊D3)—21no—
64C2×C4⋊(C3⋊D3)C2312yes—
65C22×C3×Dic3—21no—
66C22×C3×Dic3D411yescharacteristic
67C3×#239—216no—
68C3×#239—28no—
69C3×#239—28no—
70C2×C3×D12—28no—
71C2×C3×D12—24no—
72C2×C3×D12—21no—
73C2×C3×D12—22no—
74C2×C3×D12D414yescomplemented
75C2×C3×D12C2312yes—
76C2×C3×C4×D3—28no—
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3—24no—
80C2×C3×C4×D3D412yes—
81C2×C3×C4×D3C2312yes—
82C2×#626—24no—
83C2×#626—44no—
84#3755—28no—
85C2×C3⋊(C3⋊C8)—22no—
86C2×C3⋊(C3⋊C8)C2312yes—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2312yes—
89C2×C3×C3⋊C8—22no—
90C2×C3×C3⋊C8C2312yes—
91#3722—24no—
92#3714—28no—
93C2×#449—91noSylow
94C2×C3×#41—34no—
95C2×C3×#41—62no—
96C2×C3×#41—64no—
97C2×C3×#41—64no—
98C22×C3×Q8—21no—
99C22×C3×Q8C2×D311yescharacteristic
100C22×C3×Q8C2×D311yescharacteristic
101C22×C3×D4—31no—
102C22×C3×D4—61no—
103C23×C3×C4—31no—
104C23×C3×C4—61no—
105C22×#67—61no—
106C2×#239—24no—
107C2×#239—24no—
108C2×#239—22no—
109C2×#239C2×D314yescomplemented
110C22×D12—21no—
111C22×D12—31no—
112C22×D12—31no—
113C22×D12—61no—
114C22×D12—61no—
115C22×D12C2×D311yescharacteristic
116C22×C4×D3—21no—
117C22×C4×D3C2×D311yescharacteristic
118C22×C3×C8—31no—
119C2×C3×C4⋊C4—31no—
120C2×#215—34no—
121C2×#215—34no—
122C2×#215—64no—
123C2×#215—68no—
124C2×#215—64no—
125C2×#215—64no—
126C2×#213—34no—
127C2×#213—64no—
128#930—34no—
129#930—62no—
130C2×(C2×D3)⋊C4—61no—
131C2×(C2×D3)⋊C4—61no—
132C2×C4⋊Dic3C2×D311yescharacteristic
133C22×C3⋊C8—31no—
134C22×C3⋊C8—31no—
135C22×C3⋊C8—61no—
136C22×C3⋊C8C2×D311yescharacteristic
137C2×D24—34no—
138C2×D24—64no—
139#895—34no—
140#895—62no—
141C3×#103—34no—
142#835C2×D314yescomplemented
143#832—34no—
144#830—34no—
145(C2×D3)⋊C8—34no—
146#809—34no—
147C23×C32C2×D411yescharacteristic
148C22×C3⋊D3—22no—
149C22×C3⋊D3—42no—
150C22×C3⋊D3—42no—
151C22×C3⋊D3—42no—
152C22×C3×D3—24no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—22no—
156C22×C3×D3—22no—
157C22×C3×D3—42no—
158C22×C3×D3—42no—
159C22×C3×D3C2×D412yes—
160C32×Q8—24no—
161C32×Q8—22no—
162C32×Q8C2×D414yescomplemented
163C2×C32×C4—24no—
164C2×C32×C4—22no—
165C2×C32×C4C2×D412yes—
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D412yes—
168C2×C32×C4C2411yescharacteristic
169C4⋊(C3⋊D3)—22no—
170C4⋊(C3⋊D3)—22no—
171C4⋊(C3⋊D3)—42no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—42no—
175C2×C3×Dic3C2×D412yes—
176C3×D12—216no—
177C3×D12—24no—
178C3×D12—22no—
179C3×D12C2×D414yescomplemented
180C3×C4×D3—216no—
181C3×C4×D3—24no—
182C3×C4×D3—22no—
183C3×C4×D3—22no—
184#626—48no—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—22no—
187C22×#41—31no—
188C22×SD16—91no—
189C22×D8—91no—
190C2×#121—91no—
191#449—98no—
192C2×#103—31no—
193C2×#102—91no—
194C2×C22⋊C8—91no—
195C24×C3—31no—
196C24×C3—61no—
197C23×D3—21no—
198C23×D3—61no—
199C23×D3—61no—
200C23×D3—121no—
201C23×D3#6711yescharacteristic
202C3×#41—68no—
203C3×#41—616no—
204C3×#41—68no—
205C2×C3×Q8—22no—
206C2×C3×Q8—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—21no—
209C2×C3×Q8—24no—
210C2×C3×Q8—22no—
211C2×C3×Q8—41no—
212C2×C3×Q8—42no—
213C2×C3×Q8D1214yescomplemented
214C2×C3×Q8#6714yescomplemented
215C2×C3×Q8C22×D312yes—
216C2×C3×Q8C22×D312yes—
217C2×C3×D4—34no—
218C2×C3×D4—32no—
219C2×C3×D4—64no—
220C2×C3×D4—68no—
221C2×C3×D4—62no—
222C2×C3×D4—61no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—31no—
227C22×C3×C4—32no—
228C22×C3×C4—32no—
229C22×C3×C4—41no—
230C22×C3×C4—64no—
231C22×C3×C4—68no—
232C22×C3×C4—62no—
233C22×C3×C4—62no—
234C22×C3×C4—61no—
235C22×C3×C4C22×D311yescharacteristic
236C22×C3×C4C22×D311yescharacteristic
237C2×#67—64no—
238C2×#67—124no—
239C22×Dic3—21no—
240C22×Dic3#6711yescharacteristic
241#239—216no—
242#239—28no—
243#239—28no—
244C2×D12—28no—
245C2×D12—24no—
246C2×D12—22no—
247C2×D12—21no—
248C2×D12—32no—
249C2×D12—32no—
250C2×D12—62no—
251C2×D12—64no—
252C2×D12—62no—
253C2×D12—62no—
254C2×D12—62no—
255C2×D12—62no—
256C2×D12—64no—
257C2×D12—61no—
258C2×D12—61no—
259C2×D12—122no—
260C2×D12—121no—
261C2×D12—124no—
262C2×D12#6714yescomplemented
263C2×D12C22×D312yes—
264C2×C4×D3—28no—
265C2×C4×D3—22no—
266C2×C4×D3—22no—
267C2×C4×D3—24no—
268C2×C4×D3#6712yes—
269C2×C4×D3C22×D312yes—
270C2×C3×C8—32no—
271C2×C3×C8—62no—
272C3×C4⋊C4—32no—
273C3×C4⋊C4—61no—
274#215—616no—
275#215—68no—
276#215—68no—
277#213—68no—
278(C2×D3)⋊C4—64no—
279(C2×D3)⋊C4—64no—
280C4⋊Dic3—21no—
281C4⋊Dic3C22×D312yes—
282C2×C3⋊C8—22no—
283C2×C3⋊C8—32no—
284C2×C3⋊C8—32no—
285C2×C3⋊C8—62no—
286C2×C3⋊C8—64no—
287C2×C3⋊C8—62no—
288C2×C3⋊C8—62no—
289C2×C3⋊C8C22×D312yes—
290D24—68no—
291C22×C32#12012yes—
292C22×C32#13512yes—
293C22×C32AGL(1,ℤ/8)12yes—
294C22×C32C22×D411yescharacteristic
295C2×C3⋊D3—42no—
296C2×C3⋊D3—42no—
297C2×C3⋊D3—42no—
298C2×C3⋊D3—84no—
299C2×C3×D3—22no—
300C2×C3×D3—24no—
301C2×C3×D3—22no—
302C2×C3×D3—22no—
303C2×C3×D3—42no—
304C2×C3×D3—42no—
305C2×C3×D3—44no—
306C2×C3×D3—48no—
307C32×C4—24no—
308C32×C4#12012yes—
309C32×C4C22×D411yescharacteristic
310C32×C4C22×D411yescharacteristic
311C3×Dic3—22no—
312C3×Dic3—42no—
313C2×#41—34no—
314C2×#41—62no—
315C2×#41—64no—
316C2×#41—64no—
317C22×Q8D3211yescharacteristic
318C22×D4—31no—
319C22×D4—61no—
320C22×D4—91no—
321C22×D4—181no—
322C23×C4—31no—
323C23×C4—61no—
324C2×SD16—94no—
325C2×SD16—184no—
326C2×D8—94no—
327C2×D8—184no—
328C22×C8—31no—
329C22×C8—91no—
330#121—94no—
331#121—182no—
332C2×C4⋊C4—31no—
333C2×C22⋊C4—181no—
334#103—34no—
335#102—94no—
336C22⋊C8—94no—
337C23×C3—21no—
338C23×C3—32no—
339C23×C3—64no—
340C23×C3—62no—
341C23×C3—62no—
342C23×C3—62no—
343C23×C3—62no—
344C23×C3—122no—
345C23×C3—122no—
346C23×C3D3×D411yescharacteristic
347C23×C3C2×#6711yescharacteristic
348C22×D3—22no—
349C22×D3—22no—
350C22×D3—22no—
351C22×D3—24no—
352C22×D3—22no—
353C22×D3—42no—
354C22×D3—42no—
355C22×D3—62no—
356C22×D3—62no—
357C22×D3—122no—
358C22×D3—124no—
359C22×D3—122no—
360C22×D3—122no—
361C22×D3—122no—
362C22×D3—122no—
363C22×D3—122no—
364C22×D3—122no—
365C22×D3—242no—
366C22×D3—242no—
367C22×D3C2×#6712yes—
368C3×Q8—24no—
369C3×Q8—24no—
370C3×Q8—22no—
371C3×Q8—22no—
372C3×Q8—24no—
373C3×Q8—42no—
374C3×Q8—44no—
375C3×Q8C2×D1214yescomplemented
376C3×Q8C2×#6714yescomplemented
377C3×D4—34no—
378C3×D4—616no—
379C3×D4—64no—
380C3×D4—62no—
381C2×C3×C4—24no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—32no—
390C2×C3×C4—42no—
391C2×C3×C4—44no—
392C2×C3×C4—616no—
393C2×C3×C4—62no—
394C2×C3×C4—64no—
395C2×C3×C4—62no—
396C2×C3×C4—62no—
397C2×C3×C4—62no—
398C2×C3×C4—122no—
399C2×C3×C4C2×D1212yes—
400C2×C3×C4D3×D411yescharacteristic
401C2×C3×C4D3×D412yes—
402C2×C3×C4C2×#6712yes—
403C2×C3×C4C2×#6711yescharacteristic
404C2×C3×C4C2×#6712yes—
405C2×C3×C4C23×D311yescharacteristic
406C2×C3×C4C23×D311yescharacteristic
407#67—128no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—42no—
411C2×Dic3C2×#6712yes—
412D12—216no—
413D12—24no—
414D12—22no—
415D12—64no—
416D12—62no—
417D12—62no—
418D12—62no—
419D12—62no—
420D12—122no—
421D12—124no—
422D12—128no—
423D12—122no—
424D12—122no—
425D12C2×#6714yescomplemented
426C4×D3—216no—
427C4×D3—24no—
428C4×D3—22no—
429C4×D3—22no—
430C3×C8—62no—
431C3⋊C8—22no—
432C3⋊C8—64no—
433C3⋊C8—62no—
434C3⋊C8—62no—
435C2×C32#44914yescomplemented
436C2×C32C2×#12011yescharacteristic
437C2×C32C2×#13511yescharacteristic
438C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
439C3⋊D3—82no—
440C3×D3—42no—
441C3×D3—44no—
442C24—31no—
443C24—61no—
444C24—181no—
445#41—68no—
446#41—616no—
447#41—68no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8#62614yescomplemented
451C2×Q8C2×D3212yes—
452C2×D4—32no—
453C2×D4—34no—
454C2×D4—64no—
455C2×D4—68no—
456C2×D4—62no—
457C2×D4—61no—
458C2×D4—92no—
459C2×D4—184no—
460C2×D4—182no—
461C2×D4—182no—
462C2×D4—181no—
463C2×D4—364no—
464C22×C4—21no—
465C22×C4—31no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—64no—
469C22×C4—68no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—61no—
473C22×C4C2×D3211yescharacteristic
474SD16—188no—
475D8—188no—
476C2×C8—32no—
477C2×C8—62no—
478C2×C8—92no—
479C2×C8—182no—
480C4⋊C4—32no—
481C4⋊C4—61no—
482C22⋊C4—184no—
483C22×C3—21no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—64no—
488C22×C3—62no—
489C22×C3—62no—
490C22×C3—62no—
491C22×C3—122no—
492C22×C3—122no—
493C22×C3—124no—
494C22×C3—128no—
495C22×C3#88212yes—
496C22×C3#91412yes—
497C22×C3#91912yes—
498C22×C3#94912yes—
499C22×C3#95012yes—
500C22×C3#95312yes—
501C22×C3C2×D3×D411yescharacteristic
502C22×C3C22×#6711yescharacteristic
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—24no—
507C2×D3—44no—
508C2×D3—48no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—124no—
512C2×D3—122no—
513C2×D3—122no—
514C2×D3—122no—
515C2×D3—122no—
516C2×D3—122no—
517C2×D3—122no—
518C2×D3—242no—
519C2×D3—242no—
520C2×D3—248no—
521C2×D3—244no—
522C2×D3—244no—
523C3×C4—24no—
524C3×C4—24no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—22no—
528C3×C4—44no—
529C3×C4—62no—
530C3×C4—122no—
531C3×C4#88212yes—
532C3×C4#95312yes—
533C3×C4C22×D1211yescharacteristic
534C3×C4C2×D3×D411yescharacteristic
535C3×C4C22×#6711yescharacteristic
536C3×C4C22×#6711yescharacteristic
537Dic3—22no—
538Dic3—42no—
539C32C2×#44911yescharacteristic, complemented, Sylow
540C23—32no—
541C23—62no—
542C23—62no—
543C23—62no—
544C23—64no—
545C23—62no—
546C23—122no—
547C23—122no—
548C23—182no—
549C23—362no—
550C23—362no—
551C23—362no—
552C23D3×#6711yescharacteristic
553Q8—22no—
554Q8—24no—
555Q8C2×#62614yescomplemented
556D4—34no—
557D4—616no—
558D4—62no—
559D4—64no—
560D4—182no—
561D4—182no—
562D4—368no—
563D4—362no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—32no—
567C2×C4—616no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—122no—
574C2×C4C2×#62612yes—
575C2×C4D3×#6712yes—
576C2×C4D3×#6711yescharacteristic
577C2×C4C22×D3211yescharacteristic
578C8—62no—
579C8—182no—
580C2×C3—21no—
581C2×C3—24no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—122no—
585C2×C3—124no—
586C2×C3#481014yescomplemented
587C2×C3#523814yescomplemented
588C2×C3C2×#88211yescharacteristic
589C2×C3C2×#91411yescharacteristic
590C2×C3C2×#91911yescharacteristic
591C2×C3C2×#94911yescharacteristic
592C2×C3C2×#95011yescharacteristic
593C2×C3C2×#95311yescharacteristic
594D3—42no—
595D3—44no—
596D3—244no—
597D3—242no—
598D3—242no—
599C22—62no—
600C22—64no—
601C22—62no—
602C22—62no—
603C22—122no—
604C22—124no—
605C22—128no—
606C22—122no—
607C22—362no—
608C22—362no—
609C22—362no—
610C22—724no—
611C22#6393912yes—
612C22#6394912yes—
613C22#6397712yes—
614C22C2×D3×#6711yescharacteristic
615C4—24no—
616C4—62no—
617C4—122no—
618C4#6397712yes—
619C4C22×#62611yescharacteristic
620C4C2×D3×#6711yescharacteristic
621C3—21no—
622C3C2×#481011yescharacteristic, complemented
623C3C2×#523811yescharacteristic, complemented
624C2—122no—
625C2—124no—
626C2—722no—
627C2#1059158814yesdirect product factor
628C2C2×#6393911yescharacteristic
629C2C2×#6394911yescharacteristic
630C2C2×#6397711yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC23×C32C2×D4
hypercenterC22×Q8D32
hypocenterC32C2×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#239C2semidirect
2. C22×#3755C2semidirect
3. C2×#63909C2semidirect
4. C22×#3714C2semidirect
5. C2×C3×#835C2semidirect
6. #10591588C2direct (N × H)
7. C2×#63562C2semidirect
8. C2×(C2×D3)⋊(C3⋊C8)C2semidirect
16. C2×C3×#239C22semidirect
23. C2×#3755C22semidirect
29. #63909C22semidirect
31. C2×#3714C22semidirect
32. C3×#835C22semidirect
33. #63562C22semidirect
34. (C2×D3)⋊(C3⋊C8)C22semidirect
36. C22×#239D3semidirect
46. C2×#835D3semidirect
57. C2×C32×Q8D4semidirect
74. C2×C3×D12D4semidirect
109. C2×#239C2×D3semidirect
142. #835C2×D3semidirect
162. C32×Q8C2×D4semidirect
179. C3×D12C2×D4semidirect
213. C2×C3×Q8D12semidirect
214. C2×C3×Q8#67semidirect
262. C2×D12#67semidirect
375. C3×Q8C2×D12semidirect
376. C3×Q8C2×#67semidirect
425. D12C2×#67semidirect
435. C2×C32#449semidirect
450. C2×Q8#626semidirect
539. C32C2×#449semidirect
555. Q8C2×#626semidirect
586. C2×C3#4810semidirect
587. C2×C3#5238semidirect
622. C3C2×#4810semidirect
623. C3C2×#5238semidirect
627. C2#10591588direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C22×#37552
C2×#639092
C22×#37142
C2×C3×#8352
#105915882
C2×#635622
C2×(C2×D3)⋊(C3⋊C8)2
C2×#52383
C2×#48103