Finite Group Databasebeta

← C2×#74872 #11001079

Subgroups — C2×#74872 #11001079

Order 768.

4070 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#935C211yesmaximal, characteristic, complemented
2C22×#939C211yesmaximal, characteristic
3C23×C4×Dic3C211yesmaximal, characteristic, complemented
4C22×#886C211yesmaximal, characteristic, complemented
5C2×C3×#386C211yesmaximal, characteristic, complemented
6C2×#5213C211yesmaximal, characteristic, complemented
7C2×#4957C212yesmaximal, complemented
8C2×#4951C212yesmaximal, complemented
9C2×#4942C211yesmaximal, characteristic, complemented
10#74872C2116yesmaximal, direct product factor
11C2×#4647C212yesmaximal, complemented
12C2×#4669C212yesmaximal, complemented
13C2×#2854—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C23×#67C2211yescharacteristic
17C23×#67C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C24×Dic3C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2211yescharacteristic
24C22×#217C2212yes—
25C2×#939—28no—
26C2×#939C2218yes—
27C2×#935—28no—
28C2×#935C2218yescomplemented
29C22×(C2×D3)⋊C4C2212yes—
30C22×(C2×D3)⋊C4C2211yescharacteristic
31C22×(C2×D3)⋊C4C2212yes—
32C22×C4⋊Dic3C2212yes—
33C22×Dic3⋊C4C2212yes—
34C22×Dic3⋊C4C2211yescharacteristic
35C22×Dic3⋊C4C2212yes—
36C22×C4×Dic3—28no—
37C22×C4×Dic3C2212yes—
38C22×C4×Dic3C2212yes—
39C22×C4×Dic3C2218yescomplemented
40C2×#886—28no—
41C2×#886C2218yescomplemented
42C3×#386C2218yescomplemented
43C2×C3×#95C2212yes—
44#5213C2218yescomplemented
45C2×C3⋊#95C2212yes—
46C2×C3⋊#95C2212yes—
47#4957C22116yescomplemented
48#4951C22116yescomplemented
49#4942C2218yescomplemented
50#4669C22116yescomplemented
51#4647C22116yescomplemented
52C22×#123—31no—
53C22×#123—31no—
54C22×#121—31no—
55C23×C42—31no—
56#2854—316no—
57C2×#390—32no—
58C2×#389—32no—
59C2×#388—32no—
60C2×#388—32no—
61C2×#386—31no—
62C2×#386D311yescharacteristic, complemented
63C2×#380—31no—
64C25×C3C2311yescharacteristic
65C25×C3C2311yescharacteristic
66C24×D3C2311yescharacteristic
67C22×C3×D4—24no—
68C22×C3×D4—24no—
69C22×C3×D4—24no—
70C22×C3×D4C2314yes—
71C23×C3×C4—24no—
72C23×C3×C4—28no—
73C23×C3×C4C2314yes—
74C23×C3×C4C2311yescharacteristic
75C23×C3×C4C2312yes—
76C23×C3×C4C2312yes—
77C22×#67—24no—
78C22×#67—24no—
79C22×#67—24no—
80C22×#67—24no—
81C22×#67—28no—
82C22×#67C2314yes—
83C22×#67C2314yes—
84C23×Dic3—24no—
85C23×Dic3—24no—
86C23×Dic3—28no—
87C23×Dic3D418yes—
88C23×Dic3D418yescomplemented
89C23×Dic3C2311yescharacteristic
90C23×Dic3C2312yes—
91C23×Dic3C2314yes—
92C23×Dic3C2311yescharacteristic
93C23×Dic3C2312yes—
94C23×Dic3C2314yes—
95C23×Dic3C2311yescharacteristic
96C2×C3×C22⋊C4—24no—
97C2×C3×C22⋊C4—28no—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2318yes—
100C2×#217—24no—
101C2×#217—28no—
102C2×#217—24no—
103C2×#217C2314yes—
104C2×#217C2314yes—
105C2×#217C2318yes—
106#939—232no—
107#935—232no—
108C2×(C2×D3)⋊C4—28no—
109C2×(C2×D3)⋊C4—28no—
110C2×(C2×D3)⋊C4—24no—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2318yes—
113C2×(C2×D3)⋊C4C2318yes—
114C2×C4⋊Dic3—28no—
115C2×C4⋊Dic3C2318yes—
116C2×Dic3⋊C4—28no—
117C2×Dic3⋊C4—24no—
118C2×Dic3⋊C4—28no—
119C2×Dic3⋊C4C2314yes—
120C2×Dic3⋊C4C2318yes—
121C2×Dic3⋊C4C2318yes—
122C2×C4×Dic3—28no—
123C2×C4×Dic3—28no—
124C2×C4×Dic3—232no—
125C2×C4×Dic3C2318yes—
126C2×C4×Dic3C2318yes—
127#886—232no—
128C3×#95C2318yes—
129C3⋊#95C2318yes—
130C3⋊#95C2318yes—
131C23×D4—31no—
132C23×D4—31no—
133C23×D4C2×D311yescharacteristic
134C24×C4—31no—
135C24×C4—31no—
136C24×C4C2×D311yescharacteristic
137C2×#123—38no—
138C2×#123—38no—
139C2×#123—68no—
140C2×#123—68no—
141C2×#121—38no—
142C2×#121—68no—
143C22×C4⋊C4—31no—
144C22×C4⋊C4—32no—
145C22×C4⋊C4—32no—
146C22×C4⋊C4—32no—
147C22×C22⋊C4—31no—
148C22×C22⋊C4—31no—
149C22×C22⋊C4—32no—
150C22×C22⋊C4—31no—
151C22×C22⋊C4—32no—
152C22×C22⋊C4—32no—
153C22×C22⋊C4C2×D311yescharacteristic
154C22×C22⋊C4C2×D312yes—
155C22×C42—38no—
156C22×C42—32no—
157C22×C42—32no—
158C22×C42—68no—
159#390—316no—
160#389—316no—
161#388—316no—
162#388—316no—
163#386—38no—
164#386C2×D318yescomplemented
165#380—38no—
166C2×#95—32no—
167C2×#95—32no—
168C2×#95C2×D312yes—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—22no—
172C24×C3—22no—
173C24×C3—24no—
174C24×C3—24no—
175C24×C3—22no—
176C24×C3C2×D414yes—
177C24×C3C2×D414yes—
178C24×C3#4114yes—
179C24×C3C2412yes—
180C24×C3C2412yes—
181C24×C3C2411yescharacteristic
182C23×D3—22no—
183C23×D3—22no—
184C23×D3—24no—
185C23×D3—22no—
186C23×D3—24no—
187C23×D3C2412yes—
188C2×C3×D4—28no—
189C2×C3×D4—28no—
190C2×C3×D4—28no—
191C2×C3×D4—416no—
192C22×C3×C4—232no—
193C22×C3×C4—28no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—216no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—24no—
200C22×C3×C4—24no—
201C22×C3×C4—24no—
202C22×C3×C4—22no—
203C22×C3×C4#4118yes—
204C22×C3×C4C2414yes—
205C22×C3×C4C2412yes—
206C22×C3×C4C2414yes—
207C2×#67—28no—
208C2×#67—28no—
209C2×#67—28no—
210C2×#67—28no—
211C2×#67—216no—
212C2×#67—416no—
213C2×#67—416no—
214C22×Dic3—232no—
215C22×Dic3—24no—
216C22×Dic3—28no—
217C22×Dic3—24no—
218C22×Dic3—28no—
219C22×Dic3—24no—
220C22×Dic3—232no—
221C22×Dic3—216no—
222C22×Dic3—24no—
223C22×Dic3—22no—
224C22×Dic3—22no—
225C22×Dic3—24no—
226C22×Dic3—24no—
227C22×Dic3—22no—
228C22×Dic3C2×D414yes—
229C22×Dic3C2×D4116yes—
230C22×Dic3C2×D418yes—
231C22×Dic3C2×D414yes—
232C22×Dic3C2×D4116yescomplemented
233C22×Dic3#4118yes—
234C22×Dic3C2414yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C22×Dic3C2414yes—
238C22×Dic3C2412yes—
239C3×C22⋊C4—28no—
240C3×C22⋊C4—216no—
241#217—28no—
242#217—216no—
243#217—28no—
244(C2×D3)⋊C4—216no—
245(C2×D3)⋊C4—216no—
246(C2×D3)⋊C4—28no—
247C4⋊Dic3—216no—
248Dic3⋊C4—216no—
249Dic3⋊C4—28no—
250Dic3⋊C4—216no—
251C4×Dic3—216no—
252C4×Dic3—216no—
253C25—31no—
254C25C22×D311yescharacteristic
255C25C22×D311yescharacteristic
256C22×D4—24no—
257C22×D4—24no—
258C22×D4—24no—
259C22×D4—34no—
260C22×D4—34no—
261C22×D4—64no—
262C22×D4—68no—
263C22×D4—64no—
264C22×D4—64no—
265C22×D4—64no—
266C22×D4C22×D314yes—
267C23×C4—24no—
268C23×C4—28no—
269C23×C4—31no—
270C23×C4—38no—
271C23×C4—38no—
272C23×C4—34no—
273C23×C4—34no—
274C23×C4—32no—
275C23×C4—31no—
276C23×C4—32no—
277C23×C4—31no—
278C23×C4—68no—
279C23×C4—64no—
280C23×C4—64no—
281C23×C4C22×D314yes—
282C23×C4C22×D312yes—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D312yes—
285#123—632no—
286#123—632no—
287#121—632no—
288C2×C4⋊C4—34no—
289C2×C4⋊C4—38no—
290C2×C4⋊C4—38no—
291C2×C4⋊C4—38no—
292C2×C4⋊C4—64no—
293C2×C4⋊C4—68no—
294C2×C4⋊C4—68no—
295C2×C4⋊C4—68no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—34no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—38no—
304C2×C22⋊C4—64no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4—64no—
307C2×C22⋊C4—68no—
308C2×C22⋊C4—64no—
309C2×C22⋊C4—68no—
310C2×C22⋊C4C22×D314yes—
311C2×C22⋊C4C22×D318yes—
312C2×C42—38no—
313C2×C42—38no—
314C2×C42—68no—
315C2×C42—68no—
316C2×C42—632no—
317#95—38no—
318#95—38no—
319#95C22×D318yes—
320C23×C3—28no—
321C23×C3—28no—
322C23×C3—216no—
323C23×C3—216no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—22no—
331C23×C3—44no—
332C23×C3—44no—
333C23×C3—44no—
334C23×C3—48no—
335C23×C3—48no—
336C23×C3#121116yes—
337C23×C3C22×D414yes—
338C23×C3C22×D414yes—
339C23×C3C22×D412yes—
340C23×C3C22×D412yes—
341C23×C3C2×#4114yes—
342C23×C3C2×#4112yes—
343C23×C3C2×#4114yes—
344C23×C3C2×#4114yes—
345C23×C3C2511yescharacteristic
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—24no—
349C22×D3—22no—
350C22×D3—24no—
351C22×D3—44no—
352C22×D3—44no—
353C22×D3—44no—
354C22×D3—48no—
355C22×D3—48no—
356C3×D4—416no—
357C2×C3×C4—232no—
358C2×C3×C4—216no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—44no—
368C2×C3×C4—48no—
369C2×C3×C4C2×#4118yes—
370#67—416no—
371#67—416no—
372C2×Dic3—216no—
373C2×Dic3—232no—
374C2×Dic3—28no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—232no—
379C2×Dic3—24no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—22no—
385C2×Dic3—44no—
386C2×Dic3—48no—
387C2×Dic3#123116yescomplemented
388C2×Dic3C22×D414yes—
389C2×Dic3C22×D418yes—
390C2×Dic3C22×D414yes—
391C2×Dic3C2×#4118yes—
392C24—24no—
393C24—24no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—24no—
398C24—24no—
399C24—32no—
400C24—62no—
401C24—62no—
402C24—62no—
403C24—64no—
404C24—64no—
405C24#23514yes—
406C24D3×D414yes—
407C24D3×D414yes—
408C24C23×D312yes—
409C24C23×D312yes—
410C24C23×D311yescharacteristic
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—416no—
415C2×D4—68no—
416C2×D4—616no—
417C2×D4—68no—
418C2×D4—68no—
419C2×D4—68no—
420C2×D4—1216no—
421C2×D4—1216no—
422C22×C4—232no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—216no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—34no—
434C22×C4—38no—
435C22×C4—32no—
436C22×C4—34no—
437C22×C4—316no—
438C22×C4—38no—
439C22×C4—316no—
440C22×C4—32no—
441C22×C4—34no—
442C22×C4—34no—
443C22×C4—32no—
444C22×C4—632no—
445C22×C4—616no—
446C22×C4—632no—
447C22×C4—68no—
448C22×C4—64no—
449C22×C4—68no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—62no—
453C22×C4—64no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—64no—
457C22×C4—64no—
458C22×C4#23718yes—
459C22×C4C23×D314yes—
460C22×C4C23×D312yes—
461C22×C4C23×D314yes—
462C4⋊C4—68no—
463C4⋊C4—616no—
464C4⋊C4—616no—
465C4⋊C4—616no—
466C22⋊C4—28no—
467C22⋊C4—216no—
468C22⋊C4—68no—
469C22⋊C4—616no—
470C22⋊C4—68no—
471C22⋊C4—616no—
472C22⋊C4—68no—
473C22⋊C4—616no—
474C42—616no—
475C42—616no—
476C22×C3—232no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—28no—
480C22×C3—28no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—88no—
487C22×C3C2×#12118yes—
488C22×C3C2×#12114yes—
489C22×C3C2×#12314yes—
490C22×C3#52514yes—
491C22×C3#52918yes—
492C22×C3#53218yes—
493C22×C3C23×D411yescharacteristic
494C22×C3C23×D411yescharacteristic
495C22×C3C22×#4111yescharacteristic
496C22×C3C22×#4112yes—
497C22×C3C22×#4112yes—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—88no—
504C3×C4—28no—
505C3×C4—42no—
506C3×C4—44no—
507Dic3—28no—
508Dic3—24no—
509Dic3—42no—
510Dic3—44no—
511Dic3C2×#12318yescomplemented
512C23—28no—
513C23—24no—
514C23—24no—
515C23—216no—
516C23—28no—
517C23—216no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—44no—
524C23—44no—
525C23—48no—
526C23—44no—
527C23—48no—
528C23—62no—
529C23—62no—
530C23—62no—
531C23—64no—
532C23—64no—
533C23—124no—
534C23—124no—
535C23—128no—
536C23—124no—
537C23—128no—
538C23#884116yes—
539C23C2×#23514yes—
540C23C2×#23512yes—
541C23C2×D3×D414yes—
542C23C2×D3×D414yes—
543C23C2×D3×D412yes—
544C23C2×D3×D412yes—
545C23C2×#23714yes—
546C23D3×#4114yes—
547C23C24×D311yescharacteristic
548D4—416no—
549D4—1216no—
550D4—1216no—
551C2×C4—232no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—216no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—316no—
562C2×C4—34no—
563C2×C4—38no—
564C2×C4—34no—
565C2×C4—38no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—632no—
569C2×C4—68no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—64no—
573C2×C4—632no—
574C2×C4—616no—
575C2×C4—62no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—124no—
582C2×C4—128no—
583C2×C4C2×#23718yes—
584C2×C3—28no—
585C2×C3—82no—
586C2×C3#285418yescomplemented
587C2×C3C22×#12111yescharacteristic
588C2×C3C22×#12311yescharacteristic
589C2×C3C2×#52511yescharacteristic
590C2×C3C2×#52912yes—
591C2×C3C2×#53212yes—
592D3—82no—
593C22—232no—
594C22—24no—
595C22—28no—
596C22—24no—
597C22—28no—
598C22—42no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—88no—
604C22—122no—
605C22—124no—
606C22—122no—
607C22—122no—
608C22—124no—
609C22—248no—
610C22C2×#88418yes—
611C22C2×#88414yes—
612C22#553818yes—
613C22#558614yes—
614C22#559618yes—
615C22D3×#12314yes—
616C22C22×#23511yescharacteristic
617C22C22×D3×D411yescharacteristic
618C22C22×D3×D411yescharacteristic
619C22C22×#23712yes—
620C22C2×D3×#4112yes—
621C4—28no—
622C4—38no—
623C4—44no—
624C4—42no—
625C4—68no—
626C4—64no—
627C4—124no—
628C4—122no—
629C3C2×#285411yescharacteristic, complemented, Sylow
630C2—28no—
631C2—82no—
632C2—242no—
633C2#7487218yesdirect product factor
634C2C22×#88411yescharacteristic
635C2C2×#553812yes—
636C2C2×#558611yescharacteristic
637C2C2×#559612yes—
638C2C2×D3×#12311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#386C2
socleC24×C3C24
prime radicalC22×#939C2
hypercenterC2×#386D3
hypocenterC3C2×#2854
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C22×#935C2semidirect
3. C23×C4×Dic3C2semidirect
4. C22×#886C2semidirect
5. C2×C3×#386C2semidirect
6. C2×#5213C2semidirect
7. C2×#4957C2semidirect
8. C2×#4951C2semidirect
9. C2×#4942C2semidirect
10. #74872C2direct (N × H)
11. C2×#4647C2semidirect
12. C2×#4669C2semidirect
28. C2×#935C22semidirect
39. C22×C4×Dic3C22semidirect
41. C2×#886C22semidirect
42. C3×#386C22semidirect
44. #5213C22semidirect
47. #4957C22semidirect
48. #4951C22semidirect
49. #4942C22semidirect
50. #4669C22semidirect
51. #4647C22semidirect
62. C2×#386D3semidirect
88. C23×Dic3D4semidirect
164. #386C2×D3semidirect
232. C22×Dic3C2×D4semidirect
387. C2×Dic3#123semidirect
511. Dic3C2×#123semidirect
586. C2×C3#2854semidirect
629. C3C2×#2854semidirect
633. C2#74872direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9352
C22×#9392
C23×C4×Dic32
C22×#8862
C2×C3×#3862
C2×#52132
C2×#49572
C2×#49512
C2×#49422
#748722
C2×#46472
C2×#46692
C2×#28543