Finite Group Databasebeta

← C2×#77682 #10965212

Subgroups — C2×#77682 #10965212

Order 768.

4326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#895C211yesmaximal, characteristic, complemented
2C22×#884C212yesmaximal, complemented
3C22×#67⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×#397C211yesmaximal, characteristic, complemented
5C2×#4983C211yesmaximal, characteristic, complemented
6C2×#4960C211yesmaximal, characteristic, complemented
7C2×#4958C212yesmaximal, complemented
8#77682C2116yesmaximal, direct product factor
9C2×#4665C212yesmaximal, complemented
10C2×D3×#95C211yesmaximal, characteristic, complemented
11C2×#4672C211yesmaximal, characteristic, complemented
12C2×#4669C212yesmaximal, complemented
13C2×#2543—31nomaximal, Sylow
14C23×#67C2212yes—
15C23×#67C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C23×D12C2212yes—
18C23×D12C2211yescharacteristic
19C23×C4×D3C2212yes—
20C23×C4×D3C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C22⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2211yescharacteristic
28C22×(C2×D3)⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2211yescharacteristic
31C22×C4×Dic3C2211yescharacteristic
32C2×#895—28no—
33C2×#895C2218yescomplemented
34C2×#884—216no—
35C2×#884C22116yescomplemented
36C2×#67⋊C4—28no—
37C2×#67⋊C4C2218yescomplemented
38C3×#397C2218yescomplemented
39C2×C3×#95C2212yes—
40C2×C3×#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2211yescharacteristic
43C2×C3⋊#95C2212yes—
44#4983C2218yescomplemented
45#4960C2218yescomplemented
46#4958C22116yescomplemented
47#4672C2218yescomplemented
48#4669C22116yescomplemented
49#4665C22116yescomplemented
50D3×#95C2218yescomplemented
51C22×#121—31no—
52C22×#121—32no—
53C22×C4×D4—31no—
54#2543—316no—
55C2×#398—31no—
56C2×#397D311yescharacteristic, complemented
57C2×#394—31no—
58C2×#392—31no—
59C2×#388—32no—
60C2×#386—32no—
61C2×#386—32no—
62C22×#95—31no—
63C25×C3C2311yescharacteristic
64C24×D3C2312yes—
65C24×D3C2311yescharacteristic
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×#67—24no—
71C22×#67—24no—
72C22×#67—28no—
73C22×#67—28no—
74C22×#67—44no—
75C22×#67D418yescomplemented
76C22×#67C2318yes—
77C22×#67C2314yes—
78C23×Dic3—24no—
79C23×Dic3—44no—
80C23×Dic3C2311yescharacteristic
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2312yes—
84C23×Dic3C2314yes—
85C22×D12—28no—
86C22×D12—28no—
87C22×D12—28no—
88C22×D12—24no—
89C22×D12—44no—
90C22×D12C2318yes—
91C22×D12C2314yes—
92C22×C4×D3—28no—
93C22×C4×D3—28no—
94C22×C4×D3—44no—
95C22×C4×D3—44no—
96C22×C4×D3D418yescomplemented
97C22×C4×D3C2318yes—
98C22×C4×D3C2314yes—
99C2×C3×C4⋊C4—24no—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C22⋊C4—28no—
102C2×C3×C22⋊C4—24no—
103C2×C3×C22⋊C4C2318yes—
104C2×C3×C22⋊C4C2314yes—
105C2×(C2×D3)⋊C4—24no—
106C2×(C2×D3)⋊C4—28no—
107C2×(C2×D3)⋊C4—28no—
108C2×(C2×D3)⋊C4—28no—
109C2×(C2×D3)⋊C4—44no—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2318yes—
112C2×(C2×D3)⋊C4C2318yes—
113C2×(C2×D3)⋊C4C2318yes—
114C2×(C2×D3)⋊C4C2318yes—
115C2×Dic3⋊C4—24no—
116C2×Dic3⋊C4—28no—
117C2×Dic3⋊C4C2318yes—
118C2×Dic3⋊C4C2314yes—
119C2×C4×Dic3—24no—
120C2×C4×Dic3C2314yes—
121#895—232no—
122#884—264no—
123#67⋊C4—232no—
124C3×#95C2314yes—
125C3×#95C2318yes—
126C3⋊#95C2318yes—
127C3⋊#95C2314yes—
128C3⋊#95C2318yes—
129C23×D4—31no—
130C23×D4—32no—
131C23×D4—32no—
132C23×D4—31no—
133C24×C4—31no—
134C24×C4—31no—
135C24×C4—32no—
136C2×#121—38no—
137C2×#121—316no—
138C2×#121—616no—
139C2×#121—68no—
140C2×C4×D4—38no—
141C2×C4×D4—68no—
142C22×C4⋊C4—31no—
143C22×C4⋊C4—32no—
144C22×C4⋊C4C2×D311yescharacteristic
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4—31no—
148C22×C22⋊C4—32no—
149C22×C22⋊C4—32no—
150C22×C22⋊C4C2×D311yescharacteristic
151C22×C22⋊C4C2×D312yes—
152C22×C42—31no—
153#398—38no—
154#397C2×D318yescomplemented
155#394—38no—
156#392—38no—
157#388—316no—
158#386—316no—
159#386—316no—
160C2×#95—31no—
161C2×#95—32no—
162C2×#95—32no—
163C2×#95—38no—
164C2×#95C2×D311yescharacteristic
165C2×#95C2×D312yes—
166C24×C3—22no—
167C24×C3—22no—
168C24×C3—24no—
169C24×C3—24no—
170C24×C3C2×D414yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—24no—
176C23×D3—24no—
177C23×D3—24no—
178C23×D3—24no—
179C23×D3—24no—
180C23×D3—24no—
181C23×D3—24no—
182C23×D3C2×D414yes—
183C23×D3C2×D414yes—
184C23×D3#4114yes—
185C23×D3C2414yes—
186C23×D3C2412yes—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4C2×D414yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2414yes—
199C22×C3×C4C2412yes—
200C22×C3×C4C2412yes—
201C2×#67—232no—
202C2×#67—28no—
203C2×#67—28no—
204C2×#67—216no—
205C2×#67—216no—
206C2×#67—48no—
207C2×#67—432no—
208C2×#67C2×D4116yescomplemented
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—22no—
215C22×Dic3—24no—
216C22×Dic3—28no—
217C22×Dic3—48no—
218C22×Dic3—416no—
219C22×Dic3C2×D414yes—
220C22×Dic3C2×D414yes—
221C22×Dic3#4118yes—
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C22×Dic3C2412yes—
225C22×Dic3C2414yes—
226C2×D12—216no—
227C2×D12—216no—
228C2×D12—216no—
229C2×D12—28no—
230C2×D12—48no—
231C2×D12—416no—
232C2×D12—432no—
233C2×C4×D3—216no—
234C2×C4×D3—216no—
235C2×C4×D3—416no—
236C2×C4×D3—48no—
237C2×C4×D3—48no—
238C2×C4×D3—432no—
239C2×C4×D3C2×D4116yescomplemented
240C3×C4⋊C4—28no—
241C3×C22⋊C4—216no—
242C3×C22⋊C4—28no—
243(C2×D3)⋊C4—28no—
244(C2×D3)⋊C4—216no—
245(C2×D3)⋊C4—216no—
246(C2×D3)⋊C4—216no—
247(C2×D3)⋊C4—48no—
248Dic3⋊C4—28no—
249Dic3⋊C4—216no—
250C4×Dic3—28no—
251C25—31no—
252C25—32no—
253C25C22×D311yescharacteristic
254C22×D4—38no—
255C22×D4—34no—
256C22×D4—38no—
257C22×D4—38no—
258C22×D4—34no—
259C22×D4—68no—
260C22×D4—68no—
261C22×D4—68no—
262C22×D4—68no—
263C22×D4—64no—
264C22×D4—64no—
265C22×D4—68no—
266C22×D4—64no—
267C22×D4—124no—
268C22×D4—124no—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—32no—
272C23×C4—38no—
273C23×C4—34no—
274C23×C4—31no—
275C23×C4—34no—
276C23×C4—38no—
277C23×C4—68no—
278C23×C4—68no—
279C23×C4—64no—
280C23×C4—124no—
281C23×C4—124no—
282C23×C4—124no—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D312yes—
285C23×C4C22×D312yes—
286C23×C4C22×D311yescharacteristic
287#121—664no—
288#121—632no—
289C4×D4—632no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—34no—
292C2×C4⋊C4—38no—
293C2×C4⋊C4—68no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4C22×D314yes—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—38no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—68no—
304C2×C22⋊C4—68no—
305C2×C22⋊C4—68no—
306C2×C22⋊C4—64no—
307C2×C22⋊C4—124no—
308C2×C22⋊C4C22×D314yes—
309C2×C22⋊C4C22×D318yes—
310C2×C42—34no—
311C2×C42—64no—
312#95—38no—
313#95—34no—
314#95—38no—
315#95C22×D314yes—
316#95C22×D318yes—
317C23×C3—28no—
318C23×C3—216no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—44no—
324C23×C3—44no—
325C23×C3—48no—
326C23×C3C22×D414yes—
327C23×C3C22×D412yes—
328C23×C3C22×D412yes—
329C23×C3C2×#4112yes—
330C23×C3C2×#4114yes—
331C23×C3H(5,𝔽2)14yes—
332C23×C3C2511yescharacteristic
333C22×D3—28no—
334C22×D3—28no—
335C22×D3—216no—
336C22×D3—216no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—24no—
343C22×D3—24no—
344C22×D3—24no—
345C22×D3—24no—
346C22×D3—48no—
347C22×D3—48no—
348C22×D3—48no—
349C22×D3—48no—
350C22×D3—48no—
351C22×D3—48no—
352C22×D3—48no—
353C22×D3#121116yes—
354C22×D3C22×D414yes—
355C22×D3C22×D414yes—
356C22×D3C2×#4114yes—
357C2×C3×C4—28no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—48no—
368C2×C3×C4—44no—
369C2×C3×C4—48no—
370C2×C3×C4C22×D414yes—
371#67—232no—
372#67—432no—
373C2×Dic3—28no—
374C2×Dic3—28no—
375C2×Dic3—216no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—22no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—22no—
382C2×Dic3—24no—
383C2×Dic3—416no—
384C2×Dic3—44no—
385C2×Dic3C22×D414yes—
386C2×Dic3C22×D414yes—
387C2×Dic3C2×#4118yes—
388D12—416no—
389D12—432no—
390C4×D3—416no—
391C4×D3—432no—
392C24—22no—
393C24—24no—
394C24—22no—
395C24—24no—
396C24—34no—
397C24—34no—
398C24—34no—
399C24—32no—
400C24—34no—
401C24—64no—
402C24—64no—
403C24—64no—
404C24—64no—
405C24—64no—
406C24—64no—
407C24—64no—
408C24—64no—
409C24—64no—
410C24D3×D414yes—
411C24C23×D312yes—
412C24C23×D311yescharacteristic
413C2×D4—316no—
414C2×D4—616no—
415C2×D4—616no—
416C2×D4—616no—
417C2×D4—616no—
418C2×D4—68no—
419C2×D4—68no—
420C2×D4—632no—
421C2×D4—616no—
422C2×D4—68no—
423C2×D4—128no—
424C2×D4—1216no—
425C2×D4—1232no—
426C2×D4—128no—
427C2×D4—1232no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—34no—
438C22×C4—34no—
439C22×C4—38no—
440C22×C4—34no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—316no—
444C22×C4—32no—
445C22×C4—616no—
446C22×C4—616no—
447C22×C4—68no—
448C22×C4—62no—
449C22×C4—64no—
450C22×C4—62no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—62no—
454C22×C4—64no—
455C22×C4—1216no—
456C22×C4—128no—
457C22×C4—128no—
458C22×C4—1232no—
459C22×C4—128no—
460C22×C4—1216no—
461C22×C4D3×D414yes—
462C22×C4C23×D314yes—
463C22×C4C23×D314yes—
464C22×C4C23×D312yes—
465C22×C4C23×D312yes—
466C4⋊C4—28no—
467C4⋊C4—616no—
468C4⋊C4—68no—
469C22⋊C4—216no—
470C22⋊C4—28no—
471C22⋊C4—616no—
472C22⋊C4—616no—
473C22⋊C4—616no—
474C22⋊C4—68no—
475C22⋊C4—128no—
476C42—68no—
477C22×C3—24no—
478C22×C3—28no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—44no—
482C22×C3—416no—
483C22×C3C2×#12114yes—
484C22×C3#53214yes—
485C22×C3#53414yes—
486C22×C3#54618yes—
487C22×C3#55318yes—
488C22×C3C23×D411yescharacteristic
489C22×C3C23×D411yescharacteristic
490C22×C3C22×#4111yescharacteristic
491C22×C3C22×#4112yes—
492C22×C3C2×H(5,𝔽2)12yes—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—28no—
496C2×D3—28no—
497C2×D3—232no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—44no—
503C2×D3—44no—
504C2×D3—44no—
505C2×D3—816no—
506C2×D3C2×#12118yes—
507C3×C4—24no—
508C3×C4—44no—
509C3×C4—42no—
510C3×C4—44no—
511Dic3—24no—
512Dic3—24no—
513Dic3—28no—
514Dic3—42no—
515C23—216no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—34no—
522C23—34no—
523C23—34no—
524C23—316no—
525C23—44no—
526C23—44no—
527C23—48no—
528C23—68no—
529C23—616no—
530C23—68no—
531C23—64no—
532C23—64no—
533C23—616no—
534C23—64no—
535C23—64no—
536C23—64no—
537C23—64no—
538C23—64no—
539C23—64no—
540C23—64no—
541C23—128no—
542C23—128no—
543C23—128no—
544C23—128no—
545C23—128no—
546C23—128no—
547C23—128no—
548C23C2×D3×D414yes—
549C23C2×D3×D412yes—
550C23C2×D3×D412yes—
551C23D3×#4114yes—
552C23D3×#4112yes—
553C23#100914yes—
554C23C24×D311yescharacteristic
555D4—632no—
556D4—1216no—
557D4—1232no—
558D4—1232no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—34no—
570C2×C4—34no—
571C2×C4—38no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—68no—
576C2×C4—616no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—68no—
583C2×C4—62no—
584C2×C4—64no—
585C2×C4—1216no—
586C2×C4—1232no—
587C2×C4—1216no—
588C2×C4—124no—
589C2×C4C2×D3×D414yes—
590C2×C3—44no—
591C2×C3#254318yescomplemented
592C2×C3C22×#12111yescharacteristic
593C2×C3C2×#53211yescharacteristic
594C2×C3C2×#53411yescharacteristic
595C2×C3C2×#54612yes—
596C2×C3C2×#55312yes—
597D3—28no—
598D3—84no—
599C22—24no—
600C22—28no—
601C22—38no—
602C22—42no—
603C22—44no—
604C22—416no—
605C22—42no—
606C22—64no—
607C22—68no—
608C22—632no—
609C22—64no—
610C22—68no—
611C22—124no—
612C22—124no—
613C22—124no—
614C22—124no—
615C22—124no—
616C22—124no—
617C22—124no—
618C22—2416no—
619C22D3×#12114yes—
620C22#565314yes—
621C22#565418yes—
622C22#566814yes—
623C22#570918yes—
624C22C22×D3×D411yescharacteristic
625C22C22×D3×D411yescharacteristic
626C22C2×D3×#4112yes—
627C22C2×D3×#4111yescharacteristic
628C22C2×#100912yes—
629C4—24no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—64no—
634C4—64no—
635C4—68no—
636C4—122no—
637C3C2×#254311yescharacteristic, complemented, Sylow
638C2—44no—
639C2—68no—
640C2—244no—
641C2#7768218yesdirect product factor
642C2C2×D3×#12111yescharacteristic
643C2C2×#565311yescharacteristic
644C2C2×#565412yes—
645C2C2×#566811yescharacteristic
646C2C2×#570912yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#397C2
socleC24×C3C24
hypercenterC2×#397D3
hypocenterC3C2×#2543
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C22×#895C2semidirect
2. C22×#884C2semidirect
3. C22×#67⋊C4C2semidirect
4. C2×C3×#397C2semidirect
5. C2×#4983C2semidirect
6. C2×#4960C2semidirect
7. C2×#4958C2semidirect
8. #77682C2direct (N × H)
9. C2×#4665C2semidirect
10. C2×D3×#95C2semidirect
11. C2×#4672C2semidirect
12. C2×#4669C2semidirect
33. C2×#895C22semidirect
35. C2×#884C22semidirect
37. C2×#67⋊C4C22semidirect
38. C3×#397C22semidirect
44. #4983C22semidirect
45. #4960C22semidirect
46. #4958C22semidirect
47. #4672C22semidirect
48. #4669C22semidirect
49. #4665C22semidirect
50. D3×#95C22semidirect
56. C2×#397D3semidirect
75. C22×#67D4semidirect
96. C22×C4×D3D4semidirect
154. #397C2×D3semidirect
208. C2×#67C2×D4semidirect
239. C2×C4×D3C2×D4semidirect
591. C2×C3#2543semidirect
637. C3C2×#2543semidirect
641. C2#77682direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8952
C22×#8842
C22×#67⋊C42
C2×C3×#3972
C2×#49832
C2×#49602
C2×#49582
#776822
C2×#46652
C2×D3×#952
C2×#46722
C2×#46692
C2×#25433