Finite Group Databasebeta

← C2×#77121 #10965698

Subgroups — C2×#77121 #10965698

Order 768.

3438 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#938C211yesmaximal, characteristic
2C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#394C211yesmaximal, characteristic, complemented
4C2×#5222C211yesmaximal, characteristic, complemented
5C2×#4986C211yesmaximal, characteristic, complemented
6C2×#4956C212yesmaximal, complemented
7C2×#4956C212yesmaximal, complemented
8C2×#4944C212yesmaximal, complemented
9C2×#4951C212yesmaximal, complemented
10#77121C2116yesmaximal, direct product factor
11C2×#4659C211yesmaximal, characteristic, complemented
12C2×#4666C211yesmaximal, characteristic, complemented
13C2×#2591—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×#217C2212yes—
23C22×#217C2212yes—
24C22×#217C2212yes—
25C2×#938—28no—
26C2×#938C2218yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2211yescharacteristic
32C22×C4×Dic3C2212yes—
33C2×D3×C4⋊C4—28no—
34C2×D3×C4⋊C4C2218yescomplemented
35C3×#394C2218yescomplemented
36C2×C3×#95C2211yescharacteristic
37#5222C2218yescomplemented
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4986C2218yescomplemented
44#4956C22116yescomplemented
45#4956C22116yescomplemented
46#4951C22116yescomplemented
47#4944C22116yescomplemented
48#4666C2218yescomplemented
49#4659C2218yescomplemented
50C22×#121—31no—
51C23×C4⋊C4—31no—
52#2591—316no—
53C2×#400—31no—
54C2×#400—31no—
55C2×#394—31no—
56C2×#394D311yescharacteristic, complemented
57C2×#388—32no—
58C2×#388—32no—
59C2×#386—32no—
60C2×#386—32no—
61C2×C4×C22⋊C4—31no—
62C25×C3C2312yes—
63C24×D3C2311yescharacteristic
64C22×C3×D4—24no—
65C22×C3×D4—44no—
66C22×C3×D4C2314yes—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C22×#67—28no—
72C22×#67—28no—
73C22×#67—28no—
74C22×#67C2318yes—
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C22×C4×D3—24no—
81C22×C4×D3—24no—
82C22×C4×D3—24no—
83C22×C4×D3—24no—
84C22×C4×D3—44no—
85C22×C4×D3—44no—
86C22×C4×D3D418yes—
87C22×C4×D3C2314yes—
88C22×C4×D3C2314yes—
89C22×C4×D3C2314yes—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4C2314yes—
92C2×C3×C22⋊C4—28no—
93C2×C3×C22⋊C4—44no—
94C2×C3×C22⋊C4C2318yes—
95C2×C3×C22⋊C4C2318yes—
96C2×#217—28no—
97C2×#217—28no—
98C2×#217—44no—
99C2×#217C2318yes—
100C2×#217C2318yes—
101C2×#217C2318yes—
102#938—232no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4C2318yes—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3—24no—
107C2×C4⋊Dic3C2314yes—
108C2×C4⋊Dic3C2314yes—
109C2×Dic3⋊C4—24no—
110C2×Dic3⋊C4—24no—
111C2×Dic3⋊C4C2314yes—
112C2×Dic3⋊C4C2314yes—
113C2×C4×Dic3—44no—
114C2×C4×Dic3C2318yes—
115D3×C4⋊C4—232no—
116C3×#95C2314yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2314yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C23×D4—32no—
123C23×D4C2×D311yescharacteristic
124C24×C4—31no—
125C24×C4—31no—
126C24×C4—31no—
127C2×#121—38no—
128C2×#121—68no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—38no—
134C22×C4⋊C4—68no—
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4C2×D312yes—
141C22×C22⋊C4C2×D312yes—
142C22×C42—32no—
143#400—38no—
144#400—38no—
145#394—38no—
146#394C2×D318yescomplemented
147#388—316no—
148#388—316no—
149#386—316no—
150#386—316no—
151C4×C22⋊C4—38no—
152C2×#95—31no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95C2×D311yescharacteristic
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3—24no—
162C24×C3—24no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3C2414yes—
166C24×C3C2411yescharacteristic
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—24no—
170C23×D3C2×D414yes—
171C23×D3#4118yes—
172C23×D3C2412yes—
173C2×C3×D4—28no—
174C2×C3×D4—48no—
175C2×C3×D4—416no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4C2×D414yes—
185C22×C3×C4#4114yes—
186C22×C3×C4C2414yes—
187C22×C3×C4C2412yes—
188C22×C3×C4C2412yes—
189C22×C3×C4C2412yes—
190C2×#67—216no—
191C2×#67—216no—
192C2×#67—216no—
193C2×#67—432no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—24no—
197C22×Dic3—22no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3C2×D414yes—
206C22×Dic3#4114yes—
207C22×Dic3C2414yes—
208C22×Dic3C2414yes—
209C22×Dic3C2412yes—
210C22×Dic3C2412yes—
211C22×Dic3C2412yes—
212C2×C4×D3—232no—
213C2×C4×D3—28no—
214C2×C4×D3—28no—
215C2×C4×D3—28no—
216C2×C4×D3—28no—
217C2×C4×D3—416no—
218C2×C4×D3—48no—
219C2×C4×D3—48no—
220C2×C4×D3—416no—
221C2×C4×D3C2×D4116yes—
222C3×C4⋊C4—28no—
223C3×C22⋊C4—216no—
224C3×C22⋊C4—48no—
225#217—216no—
226#217—216no—
227#217—48no—
228(C2×D3)⋊C4—216no—
229C4⋊Dic3—28no—
230C4⋊Dic3—28no—
231Dic3⋊C4—28no—
232Dic3⋊C4—28no—
233C4×Dic3—48no—
234C25—31no—
235C25C22×D312yes—
236C22×D4—24no—
237C22×D4—38no—
238C22×D4—44no—
239C22×D4—68no—
240C22×D4—68no—
241C22×D4—68no—
242C22×D4C22×D314yes—
243C23×C4—31no—
244C23×C4—31no—
245C23×C4—38no—
246C23×C4—34no—
247C23×C4—32no—
248C23×C4—31no—
249C23×C4—32no—
250C23×C4—34no—
251C23×C4—34no—
252C23×C4—64no—
253C23×C4—64no—
254C23×C4—64no—
255C23×C4—64no—
256C23×C4—124no—
257C23×C4—124no—
258C23×C4C22×D312yes—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D311yescharacteristic
262#121—632no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—34no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—632no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4C22×D314yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—68no—
281C2×C22⋊C4—68no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—124no—
284C2×C22⋊C4C22×D318yes—
285C2×C22⋊C4C22×D318yes—
286C2×C42—38no—
287C2×C42—124no—
288#95—34no—
289#95—38no—
290#95—38no—
291#95—38no—
292#95—38no—
293#95C22×D314yes—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3—48no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—48no—
307C23×C3—48no—
308C23×C3C22×D412yes—
309C23×C3C2×#4112yes—
310C23×C3C2×#4112yes—
311C23×C3C2×#4114yes—
312C23×C3H(5,𝔽2)14yes—
313C23×C3C2511yescharacteristic
314C22×D3—28no—
315C22×D3—28no—
316C22×D3—216no—
317C22×D3—216no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—24no—
321C22×D3#124116yes—
322C22×D3C22×D414yes—
323C22×D3C2×#4118yes—
324C3×D4—416no—
325C2×C3×C4—28no—
326C2×C3×C4—28no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—44no—
336C2×C3×C4—48no—
337C2×C3×C4C22×D414yes—
338C2×C3×C4C2×#4114yes—
339#67—432no—
340C2×Dic3—28no—
341C2×Dic3—28no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—22no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
352C2×Dic3—24no—
353C2×Dic3—48no—
354C2×Dic3—44no—
355C2×Dic3—48no—
356C2×Dic3C22×D414yes—
357C2×Dic3C2×#4114yes—
358C4×D3—232no—
359C4×D3—416no—
360C4×D3—416no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—34no—
369C24—38no—
370C24—32no—
371C24—62no—
372C24—64no—
373C24—62no—
374C24C23×D314yes—
375C24C23×D311yescharacteristic
376C2×D4—28no—
377C2×D4—48no—
378C2×D4—416no—
379C2×D4—616no—
380C2×D4—616no—
381C2×D4—616no—
382C2×D4—1232no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—316no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—32no—
399C22×C4—68no—
400C22×C4—68no—
401C22×C4—632no—
402C22×C4—68no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—64no—
406C22×C4—68no—
407C22×C4—64no—
408C22×C4—62no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—64no—
412C22×C4—64no—
413C22×C4—64no—
414C22×C4—64no—
415C22×C4—128no—
416C22×C4—1216no—
417C22×C4—128no—
418C22×C4—1216no—
419C22×C4D3×D414yes—
420C22×C4#23714yes—
421C22×C4C23×D314yes—
422C22×C4C23×D312yes—
423C22×C4C23×D312yes—
424C22×C4C23×D312yes—
425C4⋊C4—28no—
426C4⋊C4—68no—
427C4⋊C4—68no—
428C4⋊C4—68no—
429C4⋊C4—68no—
430C22⋊C4—216no—
431C22⋊C4—48no—
432C22⋊C4—616no—
433C22⋊C4—616no—
434C22⋊C4—616no—
435C22⋊C4—128no—
436C42—128no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3—816no—
445C22×C3#52514yes—
446C22×C3C2×#12414yes—
447C22×C3#53414yes—
448C22×C3#55318yes—
449C22×C3#55318yes—
450C22×C3C23×D411yescharacteristic
451C22×C3C22×#4111yescharacteristic
452C22×C3C22×#4111yescharacteristic
453C22×C3C22×#4112yes—
454C22×C3C2×H(5,𝔽2)12yes—
455C2×D3—232no—
456C2×D3—24no—
457C2×D3—24no—
458C2×D3—28no—
459C2×D3—28no—
460C2×D3C2×#12418yes—
461C3×C4—24no—
462C3×C4—24no—
463C3×C4—42no—
464C3×C4—44no—
465Dic3—24no—
466Dic3—24no—
467Dic3—44no—
468Dic3—42no—
469Dic3—44no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—316no—
478C23—34no—
479C23—38no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—616no—
488C23—62no—
489C23—64no—
490C23—62no—
491C23—68no—
492C23—68no—
493C23—616no—
494C23C2×D3×D412yes—
495C23C2×#23712yes—
496C23#100414yes—
497C23D3×#4112yes—
498C23D3×#4114yes—
499C23C24×D311yescharacteristic
500D4—416no—
501D4—1232no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—34no—
513C2×C4—34no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—632no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—64no—
520C2×C4—68no—
521C2×C4—68no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—1216no—
531C2×C4—1216no—
532C2×C4—128no—
533C2×C4—128no—
534C2×C4—124no—
535C2×C4C2×D3×D414yes—
536C2×C4C2×#23714yes—
537C2×C3—84no—
538C2×C3#259118yescomplemented
539C2×C3C2×#52511yescharacteristic
540C2×C3C22×#12411yescharacteristic
541C2×C3C2×#53411yescharacteristic
542C2×C3C2×#55312yes—
543C2×C3C2×#55312yes—
544D3—28no—
545C22—38no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—632no—
554C22—64no—
555C22—64no—
556C22—68no—
557C22—68no—
558C22—816no—
559C22#558914yes—
560C22#560614yes—
561C22#561418yes—
562C22#566118yes—
563C22D3×#12414yes—
564C22C22×D3×D411yescharacteristic
565C22C22×#23711yescharacteristic
566C22C2×#100412yes—
567C22C2×D3×#4112yes—
568C22C2×D3×#4111yescharacteristic
569C4—24no—
570C4—24no—
571C4—42no—
572C4—44no—
573C4—64no—
574C4—64no—
575C4—124no—
576C4—122no—
577C4—124no—
578C3C2×#259111yescharacteristic, complemented, Sylow
579C2—68no—
580C2—84no—
581C2#7712118yesdirect product factor
582C2C2×#558911yescharacteristic
583C2C2×#560611yescharacteristic
584C2C2×#561412yes—
585C2C2×#566112yes—
586C2C2×D3×#12411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#394C2
socleC24×C3C24
prime radicalC22×#938C2
hypercenterC2×#394D3
hypocenterC3C2×#2591
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×D3×C4⋊C4C2semidirect
3. C2×C3×#394C2semidirect
4. C2×#5222C2semidirect
5. C2×#4986C2semidirect
6. C2×#4956C2semidirect
7. C2×#4956C2semidirect
8. C2×#4944C2semidirect
9. C2×#4951C2semidirect
10. #77121C2direct (N × H)
11. C2×#4659C2semidirect
12. C2×#4666C2semidirect
34. C2×D3×C4⋊C4C22semidirect
35. C3×#394C22semidirect
37. #5222C22semidirect
43. #4986C22semidirect
44. #4956C22semidirect
45. #4956C22semidirect
46. #4951C22semidirect
47. #4944C22semidirect
48. #4666C22semidirect
49. #4659C22semidirect
56. C2×#394D3semidirect
146. #394C2×D3semidirect
538. C2×C3#2591semidirect
578. C3C2×#2591semidirect
581. C2#77121direct (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×D3×C4⋊C42
C2×C3×#3942
C2×#52222
C2×#49862
C2×#49562
C2×#49442
C2×#49512
#771212
C2×#46592
C2×#46662
C2×#25913