Finite Group Databasebeta

← C2×#76701 #10961896

Subgroups — C2×#76701 #10961896

Order 768.

3670 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic, complemented
2C22×#67⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#392C211yesmaximal, characteristic, complemented
4C2×#5223C212yesmaximal, complemented
5C2×#5223C211yesmaximal, characteristic, complemented
6C2×#5220C212yesmaximal, complemented
7C2×#4956C212yesmaximal, complemented
8C2×#4952C212yesmaximal, complemented
9C2×#4947C211yesmaximal, characteristic, complemented
10#76701C2116yesmaximal, direct product factor
11C2×#4673C211yesmaximal, characteristic, complemented
12C2×#4651C211yesmaximal, characteristic, complemented
13C2×#2631—31nomaximal, Sylow
14C23×C3×D4C2212yes—
15C23×C3×D4C2211yescharacteristic
16C23×#67C2211yescharacteristic
17C23×#67C2212yes—
18C24×Dic3C2211yescharacteristic
19C23×C4×D3C2211yescharacteristic
20C23×Dic6C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C2×#936—28no—
28C2×#936C2218yescomplemented
29C22×(C2×D3)⋊C4C2211yescharacteristic
30C22×(C2×D3)⋊C4C2212yes—
31C22×Dic3⋊C4C2211yescharacteristic
32C22×C4×Dic3C2211yescharacteristic
33C2×#67⋊C4—28no—
34C2×#67⋊C4C2218yescomplemented
35C3×#392C2218yescomplemented
36C2×C3×#95C2211yescharacteristic
37#5223C2218yescomplemented
38#5223C22116yescomplemented
39#5220C22116yescomplemented
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2211yescharacteristic
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45#4956C22116yescomplemented
46#4952C22116yescomplemented
47#4947C2218yescomplemented
48#4673C2218yescomplemented
49#4651C2218yescomplemented
50C22×#124—31no—
51C22×C4×D4—31no—
52#2631—316no—
53C2×#397—32no—
54C2×#393—31no—
55C2×#393—31no—
56C2×#392—32no—
57C2×#392—31no—
58C2×#392D311yescharacteristic, complemented
59C2×#386—32no—
60C2×#386—32no—
61C2×#383—31no—
62C25×C3C2312yes—
63C25×C3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C22×C3×D4—28no—
66C22×C3×D4—24no—
67C22×C3×D4—44no—
68C22×C3×D4—44no—
69C22×C3×D4—44no—
70C22×C3×D4C2318yes—
71C22×C3×D4C2314yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2312yes—
74C23×C3×C4C2311yescharacteristic
75C22×#67—24no—
76C22×#67—24no—
77C22×#67—28no—
78C22×#67—24no—
79C22×#67—44no—
80C22×#67—44no—
81C22×#67C2314yes—
82C22×#67C2318yes—
83C23×Dic3—24no—
84C23×Dic3—44no—
85C23×Dic3C2312yes—
86C23×Dic3C2311yescharacteristic
87C23×Dic3C2312yes—
88C23×Dic3C2311yescharacteristic
89C23×Dic3C2311yescharacteristic
90C23×Dic3C2314yes—
91C22×C4×D3—24no—
92C22×C4×D3—44no—
93C22×C4×D3C2314yes—
94C22×Dic6—24no—
95C22×Dic6—44no—
96C22×Dic6C2314yes—
97C2×C3×C22⋊C4—24no—
98C2×C3×C22⋊C4—44no—
99C2×C3×C22⋊C4C2314yes—
100C2×C3×C22⋊C4C2318yes—
101C2×#217—28no—
102C2×#217—28no—
103C2×#217—44no—
104C2×#217—44no—
105C2×#217C2318yes—
106C2×#217C2318yes—
107C2×#217C2318yes—
108C2×#217C2318yes—
109#936—232no—
110C2×(C2×D3)⋊C4—24no—
111C2×(C2×D3)⋊C4—44no—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2318yes—
114C2×Dic3⋊C4—24no—
115C2×Dic3⋊C4C2314yes—
116C2×C4×Dic3D418yescomplemented
117C2×C4×Dic3C2314yes—
118#67⋊C4—232no—
119C3×#95C2314yes—
120C3⋊#95C2314yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C3⋊#95C2318yes—
125C23×Q8—31no—
126C23×D4—31no—
127C23×D4—32no—
128C23×D4C2×D311yescharacteristic
129C23×D4C2×D312yes—
130C24×C4—31no—
131C24×C4—31no—
132C2×#124—38no—
133C2×#124—68no—
134C2×C4×D4—38no—
135C2×C4×D4—68no—
136C22×C4⋊C4—31no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—31no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4C2×D311yescharacteristic
144C22×C22⋊C4C2×D312yes—
145C22×C42—31no—
146#397—316no—
147#393—38no—
148#393—38no—
149#392—38no—
150#392—316no—
151#392C2×D318yescomplemented
152#386—316no—
153#386—316no—
154#383—38no—
155C2×#95—31no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95—32no—
159C2×#95—32no—
160C2×#95C2×D311yescharacteristic
161C24×C3—22no—
162C24×C3—22no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—24no—
169C24×C3—24no—
170C24×C3—24no—
171C24×C3—24no—
172C24×C3#4114yes—
173C24×C3C2414yes—
174C24×C3C2412yes—
175C24×C3C2411yescharacteristic
176C23×D3—22no—
177C23×D3—22no—
178C23×D3—24no—
179C23×D3—24no—
180C23×D3#4114yes—
181C23×D3C2412yes—
182C2×C3×D4—216no—
183C2×C3×D4—28no—
184C2×C3×D4—48no—
185C2×C3×D4—48no—
186C2×C3×D4—48no—
187C2×C3×D4—416no—
188C2×C3×D4—432no—
189C22×C3×C4—24no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2412yes—
199C2×#67—232no—
200C2×#67—28no—
201C2×#67—28no—
202C2×#67—216no—
203C2×#67—28no—
204C2×#67—48no—
205C2×#67—48no—
206C2×#67—432no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—24no—
210C22×Dic3—22no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3—24no—
217C22×Dic3—24no—
218C22×Dic3—28no—
219C22×Dic3—48no—
220C22×Dic3—416no—
221C22×Dic3C2×D414yes—
222C22×Dic3C2×D414yes—
223C22×Dic3C2414yes—
224C22×Dic3C2412yes—
225C22×Dic3C2414yes—
226C22×Dic3C2412yes—
227C22×Dic3C2412yes—
228C2×C4×D3—28no—
229C2×C4×D3—48no—
230C2×C4×D3—416no—
231C2×Dic6—28no—
232C2×Dic6—48no—
233C2×Dic6—416no—
234C3×C22⋊C4—28no—
235C3×C22⋊C4—48no—
236#217—216no—
237#217—216no—
238#217—48no—
239#217—48no—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—48no—
242Dic3⋊C4—28no—
243C4×Dic3C2×D4116yescomplemented
244C25—31no—
245C25C22×D312yes—
246C25C22×D311yescharacteristic
247C22×Q8—34no—
248C22×Q8—64no—
249C22×Q8—124no—
250C22×D4—28no—
251C22×D4—24no—
252C22×D4—34no—
253C22×D4—38no—
254C22×D4—44no—
255C22×D4—44no—
256C22×D4—44no—
257C22×D4—64no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4—68no—
261C22×D4—124no—
262C22×D4—124no—
263C22×D4C22×D314yes—
264C22×D4C22×D318yes—
265C23×C4—31no—
266C23×C4—31no—
267C23×C4—32no—
268C23×C4—31no—
269C23×C4—32no—
270C23×C4—34no—
271C23×C4—34no—
272C23×C4—64no—
273C23×C4—64no—
274C23×C4—124no—
275C23×C4—124no—
276C23×C4C22×D312yes—
277C23×C4C22×D311yescharacteristic
278C23×C4C22×D311yescharacteristic
279#124—632no—
280C4×D4—632no—
281C2×C4⋊C4—34no—
282C2×C4⋊C4—64no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—38no—
285C2×C22⋊C4—38no—
286C2×C22⋊C4—34no—
287C2×C22⋊C4—38no—
288C2×C22⋊C4—38no—
289C2×C22⋊C4—38no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—68no—
292C2×C22⋊C4—68no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—124no—
295C2×C22⋊C4—124no—
296C2×C22⋊C4—124no—
297C2×C22⋊C4C22×D314yes—
298C2×C22⋊C4C22×D318yes—
299C2×C42—38no—
300C2×C42—34no—
301#95—38no—
302#95—34no—
303#95—38no—
304#95—38no—
305#95—38no—
306#95C22×D314yes—
307C23×C3—28no—
308C23×C3—216no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—48no—
327C23×C3—48no—
328C23×C3—48no—
329C23×C3—48no—
330C23×C3C22×D412yes—
331C23×C3C2×#4114yes—
332C23×C3C2×#4112yes—
333C23×C3C2×#4112yes—
334C23×C3H(5,𝔽2)14yes—
335C23×C3H(5,𝔽2)14yes—
336C23×C3C2511yescharacteristic
337C22×D3—28no—
338C22×D3—216no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—24no—
342C22×D3—24no—
343C22×D3—44no—
344C22×D3—44no—
345C22×D3—48no—
346C22×D3C2×#4114yes—
347C3×D4—416no—
348C3×D4—432no—
349C2×C3×C4—28no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—44no—
357C2×C3×C4—48no—
358C2×C3×C4C22×D414yes—
359#67—232no—
360#67—432no—
361C2×Dic3—28no—
362C2×Dic3—28no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—22no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—416no—
375C2×Dic3—48no—
376C2×Dic3—44no—
377C2×Dic3—48no—
378C2×Dic3C22×D414yes—
379C2×Dic3C22×D414yes—
380C4×D3—416no—
381Dic6—416no—
382C24—22no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—34no—
394C24—32no—
395C24—62no—
396C24—62no—
397C24—64no—
398C24—64no—
399C24#23714yes—
400C24C23×D314yes—
401C24C23×D312yes—
402C24C23×D311yescharacteristic
403C2×Q8—68no—
404C2×Q8—128no—
405C2×Q8—1216no—
406C2×D4—216no—
407C2×D4—28no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—416no—
412C2×D4—432no—
413C2×D4—68no—
414C2×D4—68no—
415C2×D4—632no—
416C2×D4—68no—
417C2×D4—616no—
418C2×D4—128no—
419C2×D4—128no—
420C2×D4—1232no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—34no—
433C22×C4—32no—
434C22×C4—68no—
435C22×C4—68no—
436C22×C4—62no—
437C22×C4—64no—
438C22×C4—62no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—128no—
448C22×C4—1216no—
449C22×C4—128no—
450C22×C4—1216no—
451C22×C4D3×D414yes—
452C22×C4C23×D314yes—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C4⋊C4—68no—
456C22⋊C4—28no—
457C22⋊C4—48no—
458C22⋊C4—616no—
459C22⋊C4—616no—
460C22⋊C4—68no—
461C22⋊C4—128no—
462C22⋊C4—128no—
463C22⋊C4—128no—
464C42—316no—
465C22×C3—24no—
466C22×C3—28no—
467C22×C3—416no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—816no—
479C22×C3#52514yes—
480C22×C3#53514yes—
481C22×C3#53514yes—
482C22×C3#53818yes—
483C22×C3#53818yes—
484C22×C3C23×D411yescharacteristic
485C22×C3C22×#4111yescharacteristic
486C22×C3C22×#4111yescharacteristic
487C22×C3C2×H(5,𝔽2)12yes—
488C22×C3C2×H(5,𝔽2)12yes—
489C2×D3—24no—
490C2×D3—28no—
491C2×D3—416no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—44no—
495C3×C4—24no—
496C3×C4—42no—
497C3×C4—44no—
498Dic3—24no—
499Dic3—24no—
500Dic3—44no—
501Dic3—42no—
502Dic3—44no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—216no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—34no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—62no—
528C23—62no—
529C23—64no—
530C23—64no—
531C23—68no—
532C23—616no—
533C23—124no—
534C23—124no—
535C23—128no—
536C23C2×D3×D412yes—
537C23C2×#23714yes—
538C23C2×#23712yes—
539C23#100414yes—
540C23#100414yes—
541C23D3×#4112yes—
542C23C24×D311yescharacteristic
543Q8—1216no—
544D4—416no—
545D4—432no—
546D4—632no—
547D4—1232no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—34no—
556C2×C4—34no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—68no—
563C2×C4—68no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—1216no—
573C2×C4—1216no—
574C2×C4—124no—
575C2×C4—128no—
576C2×C4—128no—
577C2×C4C2×D3×D414yes—
578C2×C3—44no—
579C2×C3—84no—
580C2×C3#263118yescomplemented
581C2×C3C2×#52511yescharacteristic
582C2×C3C2×#53511yescharacteristic
583C2×C3C2×#53511yescharacteristic
584C2×C3C2×#53812yes—
585C2×C3C2×#53812yes—
586D3—44no—
587C22—28no—
588C22—24no—
589C22—42no—
590C22—416no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—64no—
601C22—68no—
602C22—816no—
603C22—122no—
604C22—124no—
605C22—1216no—
606C22—122no—
607C22#559214yes—
608C22#559318yes—
609C22#560614yes—
610C22#561318yes—
611C22#571514yes—
612C22C22×D3×D411yescharacteristic
613C22C22×#23711yescharacteristic
614C22C2×#100412yes—
615C22C2×#100412yes—
616C22C2×D3×#4111yescharacteristic
617C4—24no—
618C4—44no—
619C4—42no—
620C4—64no—
621C4—64no—
622C4—124no—
623C4—122no—
624C4—124no—
625C3C2×#263111yescharacteristic, complemented, Sylow
626C2—44no—
627C2—84no—
628C2—124no—
629C2#7670118yesdirect product factor
630C2C2×#559211yescharacteristic
631C2C2×#559312yes—
632C2C2×#560611yescharacteristic
633C2C2×#561312yes—
634C2C2×#571511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#392C2
socleC24×C3C24
hypercenterC2×#392D3
hypocenterC3C2×#2631
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
1. C22×#936C2semidirect
2. C22×#67⋊C4C2semidirect
3. C2×C3×#392C2semidirect
4. C2×#5223C2semidirect
5. C2×#5223C2semidirect
6. C2×#5220C2semidirect
7. C2×#4956C2semidirect
8. C2×#4952C2semidirect
9. C2×#4947C2semidirect
10. #76701C2direct (N × H)
11. C2×#4673C2semidirect
12. C2×#4651C2semidirect
28. C2×#936C22semidirect
34. C2×#67⋊C4C22semidirect
35. C3×#392C22semidirect
37. #5223C22semidirect
38. #5223C22semidirect
39. #5220C22semidirect
45. #4956C22semidirect
46. #4952C22semidirect
47. #4947C22semidirect
48. #4673C22semidirect
49. #4651C22semidirect
58. C2×#392D3semidirect
116. C2×C4×Dic3D4semidirect
151. #392C2×D3semidirect
243. C4×Dic3C2×D4semidirect
580. C2×C3#2631semidirect
625. C3C2×#2631semidirect
629. C2#76701direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9362
C22×#67⋊C42
C2×C3×#3922
C2×#52232
C2×#52202
C2×#49562
C2×#49522
C2×#49472
#767012
C2×#46732
C2×#46512
C2×#26313