Finite Group Databasebeta

← C2×#76475 #11001081

Subgroups — C2×#76475 #11001081

Order 768.

4070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#938C211yesmaximal, characteristic
2C22×#883C212yesmaximal, complemented
3C22×C42×D3C211yesmaximal, characteristic, complemented
4C2×C3×#390C211yesmaximal, characteristic, complemented
5C2×#5221C211yesmaximal, characteristic, complemented
6C2×#4987C211yesmaximal, characteristic, complemented
7C2×#4984C211yesmaximal, characteristic, complemented
8C2×#4956C212yesmaximal, complemented
9C2×#4945C212yesmaximal, complemented
10C2×#4951C212yesmaximal, complemented
11C2×#4928C211yesmaximal, characteristic, complemented
12#76475C2116yesmaximal, direct product factor
13C2×#2854—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×#67C2212yes—
16C23×C4×D3C2211yescharacteristic
17C23×C4×D3C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C2×#938—28no—
27C2×#938C2218yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×C4⋊Dic3C2211yescharacteristic
30C22×C4⋊Dic3C2211yescharacteristic
31C22×C4⋊Dic3C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C22×C4×Dic3C2211yescharacteristic
35C22×C4×Dic3C2211yescharacteristic
36C2×#883—216no—
37C2×#883C22116yescomplemented
38C2×C42×D3—28no—
39C2×C42×D3C2218yescomplemented
40C3×#390C2218yescomplemented
41#5221C2218yescomplemented
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45#4987C2218yescomplemented
46#4984C2218yescomplemented
47#4956C22116yescomplemented
48#4951C22116yescomplemented
49#4945C22116yescomplemented
50#4928C2218yescomplemented
51C22×#123—32no—
52C22×#121—31no—
53C23×C42—31no—
54#2854—316no—
55C2×#390—31no—
56C2×#390D311yescharacteristic, complemented
57C2×#389—31no—
58C2×#389—31no—
59C2×#388—32no—
60C2×#388—32no—
61C2×#386—32no—
62C2×#380—31no—
63C25×C3C2312yes—
64C24×D3C2311yescharacteristic
65C22×C3×D4—24no—
66C22×C3×D4—24no—
67C22×C3×D4—24no—
68C22×C3×D4C2314yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×#67—28no—
74C22×#67—28no—
75C22×#67—28no—
76C22×#67C2318yes—
77C23×Dic3C2312yes—
78C23×Dic3C2312yes—
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2311yescharacteristic
81C23×Dic3C2311yescharacteristic
82C22×C4×D3—24no—
83C22×C4×D3—24no—
84C22×C4×D3—24no—
85C22×C4×D3—24no—
86C22×C4×D3—24no—
87C22×C4×D3—24no—
88C22×C4×D3—24no—
89C22×C4×D3D418yes—
90C22×C4×D3D418yescomplemented
91C22×C4×D3C2314yes—
92C22×C4×D3C2314yes—
93C22×C4×D3C2314yes—
94C2×C3×C4⋊C4—24no—
95C2×C3×C4⋊C4C2314yes—
96C2×C3×C22⋊C4—28no—
97C2×C3×C22⋊C4—28no—
98C2×C3×C22⋊C4C2318yes—
99C2×C3×C22⋊C4C2318yes—
100C2×C3×C42—24no—
101C2×C3×C42C2314yes—
102C2×#217—28no—
103C2×#217—28no—
104C2×#217—28no—
105C2×#217C2318yes—
106C2×#217C2318yes—
107C2×#217C2318yes—
108#938—232no—
109C2×(C2×D3)⋊C4—28no—
110C2×(C2×D3)⋊C4C2318yes—
111C2×C4⋊Dic3—24no—
112C2×C4⋊Dic3—24no—
113C2×C4⋊Dic3—28no—
114C2×C4⋊Dic3C2314yes—
115C2×C4⋊Dic3C2314yes—
116C2×C4⋊Dic3C2318yes—
117C2×Dic3⋊C4—28no—
118C2×Dic3⋊C4C2318yes—
119C2×C4×Dic3—24no—
120C2×C4×Dic3—24no—
121C2×C4×Dic3—24no—
122C2×C4×Dic3C2314yes—
123C2×C4×Dic3C2314yes—
124C2×C4×Dic3C2314yes—
125#883—264no—
126C42×D3—232no—
127C3⋊#95C2318yes—
128C3⋊#95C2318yes—
129C3⋊#95C2318yes—
130C23×D4—32no—
131C23×D4C2×D311yescharacteristic
132C24×C4—31no—
133C24×C4—31no—
134C24×C4—31no—
135C2×#123—316no—
136C2×#123—616no—
137C2×#121—38no—
138C2×#121—68no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—32no—
141C22×C4⋊C4—32no—
142C22×C4⋊C4—31no—
143C22×C4⋊C4C2×D311yescharacteristic
144C22×C22⋊C4—32no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—32no—
148C22×C22⋊C4C2×D312yes—
149C22×C22⋊C4C2×D312yes—
150C22×C42—31no—
151C22×C42—31no—
152C22×C42—31no—
153C22×C42—38no—
154C22×C42—68no—
155C22×C42C2×D311yescharacteristic
156#390—38no—
157#390C2×D318yescomplemented
158#389—38no—
159#389—38no—
160#388—316no—
161#388—316no—
162#386—316no—
163#380—38no—
164C2×#95—32no—
165C2×#95—32no—
166C2×#95—32no—
167C24×C3—24no—
168C24×C3—24no—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3—24no—
173C24×C3—24no—
174C24×C3C2414yes—
175C24×C3C2411yescharacteristic
176C23×D3—22no—
177C23×D3—22no—
178C23×D3—22no—
179C23×D3—22no—
180C23×D3C2×D414yes—
181C23×D3C2×D414yes—
182C23×D3#4114yes—
183C23×D3C2412yes—
184C2×C3×D4—28no—
185C2×C3×D4—28no—
186C2×C3×D4—28no—
187C2×C3×D4—416no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D414yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4114yes—
199C22×C3×C4#4114yes—
200C22×C3×C4C2414yes—
201C22×C3×C4C2412yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2412yes—
204C2×#67—216no—
205C2×#67—216no—
206C2×#67—216no—
207C2×#67—432no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3—24no—
217C22×Dic3—24no—
218C22×Dic3—24no—
219C22×Dic3—24no—
220C22×Dic3C2×D414yes—
221C22×Dic3C2×D414yes—
222C22×Dic3#4114yes—
223C22×Dic3C2414yes—
224C22×Dic3C2414yes—
225C22×Dic3C2412yes—
226C22×Dic3C2412yes—
227C22×Dic3C2412yes—
228C2×C4×D3—232no—
229C2×C4×D3—232no—
230C2×C4×D3—28no—
231C2×C4×D3—28no—
232C2×C4×D3—28no—
233C2×C4×D3—232no—
234C2×C4×D3—28no—
235C2×C4×D3—28no—
236C2×C4×D3—28no—
237C2×C4×D3—28no—
238C2×C4×D3C2×D4116yes—
239C2×C4×D3C2×D4116yescomplemented
240C3×C4⋊C4—28no—
241C3×C22⋊C4—216no—
242C3×C22⋊C4—216no—
243C3×C42—28no—
244#217—216no—
245#217—216no—
246#217—216no—
247(C2×D3)⋊C4—216no—
248C4⋊Dic3—28no—
249C4⋊Dic3—28no—
250C4⋊Dic3—216no—
251Dic3⋊C4—216no—
252C4×Dic3—28no—
253C4×Dic3—28no—
254C4×Dic3—28no—
255C25—31no—
256C25C22×D312yes—
257C22×D4—24no—
258C22×D4—24no—
259C22×D4—24no—
260C22×D4—38no—
261C22×D4—68no—
262C22×D4—68no—
263C22×D4—68no—
264C22×D4C22×D314yes—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—38no—
269C23×C4—38no—
270C23×C4—34no—
271C23×C4—34no—
272C23×C4—32no—
273C23×C4—32no—
274C23×C4—34no—
275C23×C4—64no—
276C23×C4—64no—
277C23×C4—64no—
278C23×C4—64no—
279C23×C4—64no—
280C23×C4—64no—
281C23×C4—64no—
282C23×C4C22×D312yes—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C23×C4C22×D311yescharacteristic
286#123—664no—
287#121—632no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—34no—
290C2×C4⋊C4—38no—
291C2×C4⋊C4—38no—
292C2×C4⋊C4—34no—
293C2×C4⋊C4—68no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—64no—
296C2×C4⋊C4—68no—
297C2×C4⋊C4C22×D314yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—38no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—38no—
304C2×C22⋊C4—68no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4—68no—
307C2×C22⋊C4—68no—
308C2×C22⋊C4C22×D318yes—
309C2×C22⋊C4C22×D318yes—
310C2×C42—24no—
311C2×C42—34no—
312C2×C42—34no—
313C2×C42—34no—
314C2×C42—632no—
315C2×C42—64no—
316C2×C42—64no—
317C2×C42—64no—
318C2×C42C22×D314yes—
319#95—38no—
320#95—38no—
321#95—38no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—48no—
330C23×C3—48no—
331C23×C3—48no—
332C23×C3—48no—
333C23×C3—48no—
334C23×C3—48no—
335C23×C3—48no—
336C23×C3C22×D412yes—
337C23×C3C22×D412yes—
338C23×C3C2×#4112yes—
339C23×C3C2×#4112yes—
340C23×C3C2×#4112yes—
341C23×C3C2×#4112yes—
342C23×C3C2×#4112yes—
343C23×C3C2511yescharacteristic
344C22×D3—28no—
345C22×D3—28no—
346C22×D3—28no—
347C22×D3—28no—
348C22×D3—28no—
349C22×D3—28no—
350C22×D3—22no—
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3#121116yes—
355C22×D3C22×D414yes—
356C22×D3C22×D414yes—
357C22×D3C2×#4114yes—
358C3×D4—416no—
359C2×C3×C4—28no—
360C2×C3×C4—28no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—48no—
369C2×C3×C4#123116yescomplemented
370C2×C3×C4C22×D414yes—
371C2×C3×C4C22×D414yes—
372C2×C3×C4C2×#4114yes—
373C2×C3×C4C2×#4114yes—
374C2×C3×C4C2×#4114yes—
375#67—432no—
376C2×Dic3—28no—
377C2×Dic3—28no—
378C2×Dic3—28no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—48no—
392C2×Dic3—48no—
393C2×Dic3C22×D414yes—
394C2×Dic3C22×D414yes—
395C2×Dic3C2×#4114yes—
396C4×D3—232no—
397C4×D3—232no—
398C4×D3—232no—
399C24—24no—
400C24—24no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—24no—
406C24—34no—
407C24—34no—
408C24—34no—
409C24—32no—
410C24—62no—
411C24—62no—
412C24—62no—
413C24—62no—
414C24C23×D314yes—
415C24C23×D311yescharacteristic
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—416no—
420C2×D4—616no—
421C2×D4—616no—
422C2×D4—616no—
423C2×D4—1232no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—34no—
432C22×C4—34no—
433C22×C4—34no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—316no—
438C22×C4—316no—
439C22×C4—34no—
440C22×C4—34no—
441C22×C4—68no—
442C22×C4—68no—
443C22×C4—68no—
444C22×C4—68no—
445C22×C4—632no—
446C22×C4—632no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—632no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—68no—
457C22×C4—64no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4#23514yes—
464C22×C4D3×D414yes—
465C22×C4D3×D414yes—
466C22×C4#23714yes—
467C22×C4#23714yes—
468C22×C4C23×D314yes—
469C22×C4C23×D312yes—
470C22×C4C23×D312yes—
471C22×C4C23×D312yes—
472C4⋊C4—28no—
473C4⋊C4—616no—
474C4⋊C4—68no—
475C4⋊C4—68no—
476C4⋊C4—616no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—616no—
480C22⋊C4—616no—
481C22⋊C4—616no—
482C22⋊C4—616no—
483C42—28no—
484C42—68no—
485C42—68no—
486C42—68no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—816no—
495C22×C3C2×#12114yes—
496C22×C3C2×#12314yes—
497C22×C3#52514yes—
498C22×C3#52914yes—
499C22×C3#52914yes—
500C22×C3#53214yes—
501C22×C3#53214yes—
502C22×C3C23×D411yescharacteristic
503C22×C3C23×D411yescharacteristic
504C22×C3C22×#4111yescharacteristic
505C22×C3C22×#4111yescharacteristic
506C22×C3C22×#4111yescharacteristic
507C22×C3C22×#4111yescharacteristic
508C22×C3C22×#4111yescharacteristic
509C2×D3—232no—
510C2×D3—24no—
511C2×D3—24no—
512C2×D3—24no—
513C2×D3—24no—
514C2×D3—24no—
515C2×D3—24no—
516C2×D3C2×#12118yes—
517C3×C4—24no—
518C3×C4—24no—
519C3×C4—44no—
520C3×C4C2×#12318yescomplemented
521Dic3—24no—
522Dic3—24no—
523Dic3—24no—
524Dic3—44no—
525Dic3—44no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—34no—
534C23—34no—
535C23—34no—
536C23—316no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—68no—
545C23—62no—
546C23—62no—
547C23—62no—
548C23—62no—
549C23—68no—
550C23—68no—
551C23—68no—
552C23—68no—
553C23—68no—
554C23C2×#23512yes—
555C23C2×D3×D412yes—
556C23C2×D3×D412yes—
557C23C2×#23712yes—
558C23C2×#23712yes—
559C23D3×#4112yes—
560C23D3×#4112yes—
561C23C24×D311yescharacteristic
562D4—416no—
563D4—1232no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—34no—
574C2×C4—34no—
575C2×C4—34no—
576C2×C4—48no—
577C2×C4—632no—
578C2×C4—632no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—68no—
586C2×C4—68no—
587C2×C4—68no—
588C2×C4—632no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—128no—
596C2×C4—128no—
597C2×C4#883116yescomplemented
598C2×C4C2×#23514yes—
599C2×C4C2×D3×D414yes—
600C2×C4C2×D3×D414yes—
601C2×C4C2×#23714yes—
602C2×C4C2×#23714yes—
603C2×C3—84no—
604C2×C3#285418yescomplemented
605C2×C3C22×#12111yescharacteristic
606C2×C3C22×#12311yescharacteristic
607C2×C3C2×#52511yescharacteristic
608C2×C3C2×#52911yescharacteristic
609C2×C3C2×#52911yescharacteristic
610C2×C3C2×#53211yescharacteristic
611C2×C3C2×#53211yescharacteristic
612D3—28no—
613C22—38no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—64no—
622C22—64no—
623C22—64no—
624C22—64no—
625C22—64no—
626C22—632no—
627C22—64no—
628C22—816no—
629C22C2×#88314yes—
630C22#554814yes—
631C22#555714yes—
632C22D3×#12114yes—
633C22#560614yes—
634C22#568014yes—
635C22#571614yes—
636C22C22×#23511yescharacteristic
637C22C22×D3×D411yescharacteristic
638C22C22×D3×D411yescharacteristic
639C22C22×#23711yescharacteristic
640C22C22×#23711yescharacteristic
641C22C2×D3×#4111yescharacteristic
642C22C2×D3×#4111yescharacteristic
643C4—24no—
644C4—24no—
645C4—44no—
646C4—64no—
647C4—64no—
648C4—64no—
649C4—124no—
650C4—124no—
651C4C2×#88318yescomplemented
652C3C2×#285411yescharacteristic, complemented, Sylow
653C2—68no—
654C2—84no—
655C2#7647518yesdirect product factor
656C2C22×#88311yescharacteristic
657C2C2×#555711yescharacteristic
658C2C2×#554811yescharacteristic
659C2C2×#560611yescharacteristic
660C2C2×D3×#12111yescharacteristic
661C2C2×#568011yescharacteristic
662C2C2×#571611yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#390C2
socleC24×C3C24
prime radicalC22×#938C2
hypercenterC2×#390D3
hypocenterC3C2×#2854
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#883C2semidirect
3. C22×C42×D3C2semidirect
4. C2×C3×#390C2semidirect
5. C2×#5221C2semidirect
6. C2×#4987C2semidirect
7. C2×#4984C2semidirect
8. C2×#4956C2semidirect
9. C2×#4945C2semidirect
10. C2×#4951C2semidirect
11. C2×#4928C2semidirect
12. #76475C2direct (N × H)
37. C2×#883C22semidirect
39. C2×C42×D3C22semidirect
40. C3×#390C22semidirect
41. #5221C22semidirect
45. #4987C22semidirect
46. #4984C22semidirect
47. #4956C22semidirect
48. #4951C22semidirect
49. #4945C22semidirect
50. #4928C22semidirect
56. C2×#390D3semidirect
90. C22×C4×D3D4semidirect
157. #390C2×D3semidirect
239. C2×C4×D3C2×D4semidirect
369. C2×C3×C4#123semidirect
520. C3×C4C2×#123semidirect
597. C2×C4#883semidirect
604. C2×C3#2854semidirect
651. C4C2×#883semidirect
652. C3C2×#2854semidirect
655. C2#76475direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9382
C22×#8832
C22×C42×D32
C2×C3×#3902
C2×#52212
C2×#49872
C2×#49842
C2×#49562
C2×#49452
C2×#49512
C2×#49282
#764752
C2×#28543