Finite Group Databasebeta

← C2×#74871 #11001071

Subgroups — C2×#74871 #11001071

Order 768.

3354 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#935C211yesmaximal, characteristic
2C23×C4×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#386C211yesmaximal, characteristic
4C2×#5220C211yesmaximal, characteristic
5C2×#5210C211yesmaximal, characteristic, complemented
6C2×#4945C212yesmaximal, complemented
7C2×#4951C212yesmaximal, complemented
8C2×#4950C211yesmaximal, characteristic, complemented
9C2×#4943C211yesmaximal, characteristic, complemented
10#74871C2116yesmaximal, direct product factor
11C2×#4654C212yesmaximal, complemented
12C2×#4652C212yesmaximal, complemented
13C2×#2853—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×#217C2211yescharacteristic
21C22×#217C2211yescharacteristic
22C22×#217C2212yes—
23C22×#217C2211yescharacteristic
24C22×#217C2211yescharacteristic
25C2×#935—28no—
26C2×#935C2218yes—
27C22×C4⋊Dic3C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2211yescharacteristic
31C22×C4×Dic3—28no—
32C22×C4×Dic3C2212yes—
33C22×C4×Dic3C2212yes—
34C22×C4×Dic3C2218yescomplemented
35C3×#386C2218yes—
36C2×C3×#95C2212yes—
37#5220C2218yes—
38#5210C2218yescomplemented
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4951C22116yescomplemented
45#4950C2218yescomplemented
46#4945C22116yescomplemented
47#4943C2218yescomplemented
48#4654C22116yescomplemented
49#4652C22116yescomplemented
50C22×#123—31no—
51C23×C42—31no—
52#2853—316no—
53C2×#388—32no—
54C2×#388—32no—
55C2×#387—32no—
56C2×#387—32no—
57C2×#386—31no—
58C2×#386D311yescharacteristic
59C2×#385—31no—
60C2×#385—31no—
61C2×#381—31no—
62C25×C3C2311yescharacteristic
63C25×C3C2311yescharacteristic
64C22×C3×D4—24no—
65C22×C3×D4—24no—
66C22×C3×D4—24no—
67C22×C3×D4C2314yes—
68C23×C3×C4—24no—
69C23×C3×C4—28no—
70C23×C3×C4C2314yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2312yes—
73C23×C3×C4C2312yes—
74C23×Dic3—24no—
75C23×Dic3—24no—
76C23×Dic3—24no—
77C23×Dic3—28no—
78C23×Dic3D418yes—
79C23×Dic3C2314yes—
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2314yes—
84C23×Dic3C2311yescharacteristic
85C23×Dic3C2311yescharacteristic
86C23×Dic3C2312yes—
87C2×C3×C22⋊C4—24no—
88C2×C3×C22⋊C4—28no—
89C2×C3×C22⋊C4C2314yes—
90C2×C3×C22⋊C4C2318yes—
91C2×#217—24no—
92C2×#217—24no—
93C2×#217—24no—
94C2×#217—24no—
95C2×#217—28no—
96C2×#217C2314yes—
97C2×#217C2314yes—
98C2×#217C2318yes—
99C2×#217C2314yes—
100C2×#217C2314yes—
101#935—232no—
102C2×C4⋊Dic3—28no—
103C2×C4⋊Dic3C2318yes—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—24no—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2314yes—
110C2×C4×Dic3—28no—
111C2×C4×Dic3—28no—
112C2×C4×Dic3—232no—
113C2×C4×Dic3C2318yes—
114C2×C4×Dic3C2318yes—
115C3×#95C2318yes—
116C3⋊#95C2318yes—
117C3⋊#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C23×D4C2×D311yescharacteristic
122C24×C4—31no—
123C24×C4—31no—
124C24×C4C2×D311yescharacteristic
125C2×#123—38no—
126C2×#123—68no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C22⋊C4—31no—
132C22×C22⋊C4—31no—
133C22×C22⋊C4—31no—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4C2×D311yescharacteristic
137C22×C22⋊C4C2×D312yes—
138C22×C42—38no—
139C22×C42—32no—
140C22×C42—32no—
141C22×C42—68no—
142#388—316no—
143#388—316no—
144#387—316no—
145#387—316no—
146#386—38no—
147#386C2×D318yes—
148#385—38no—
149#385—38no—
150#381—38no—
151C2×#95—32no—
152C2×#95—32no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95C2×D312yes—
157C24×C3—24no—
158C24×C3—24no—
159C24×C3—22no—
160C24×C3—24no—
161C24×C3—24no—
162C24×C3—22no—
163C24×C3—22no—
164C24×C3C2×D414yes—
165C24×C3#4114yes—
166C24×C3#4114yes—
167C24×C3C2412yes—
168C24×C3C2412yes—
169C24×C3C2411yescharacteristic
170C2×C3×D4—28no—
171C2×C3×D4—28no—
172C2×C3×D4—28no—
173C2×C3×D4—416no—
174C22×C3×C4—232no—
175C22×C3×C4—28no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—216no—
179C22×C3×C4—22no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4#4118yes—
186C22×C3×C4C2414yes—
187C22×C3×C4C2412yes—
188C22×C3×C4C2414yes—
189C22×Dic3—232no—
190C22×Dic3—232no—
191C22×Dic3—28no—
192C22×Dic3—24no—
193C22×Dic3—28no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—28no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—24no—
200C22×Dic3—216no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×D414yes—
210C22×Dic3C2×D4116yes—
211C22×Dic3#4118yes—
212C22×Dic3C2414yes—
213C22×Dic3C2412yes—
214C22×Dic3C2412yes—
215C22×Dic3C2412yes—
216C22×Dic3C2412yes—
217C22×Dic3C2414yes—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—216no—
220#217—28no—
221#217—28no—
222#217—28no—
223#217—28no—
224#217—216no—
225C4⋊Dic3—216no—
226Dic3⋊C4—216no—
227Dic3⋊C4—28no—
228Dic3⋊C4—28no—
229C4×Dic3—216no—
230C4×Dic3—216no—
231C25C22×D311yescharacteristic
232C25C22×D311yescharacteristic
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4C22×D314yes—
237C23×C4—24no—
238C23×C4—28no—
239C23×C4—38no—
240C23×C4—34no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—32no—
244C23×C4—34no—
245C23×C4—31no—
246C23×C4—31no—
247C23×C4—32no—
248C23×C4—68no—
249C23×C4—64no—
250C23×C4—64no—
251C23×C4—64no—
252C23×C4C22×D314yes—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D312yes—
256#123—632no—
257C2×C4⋊C4—34no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—68no—
264C2×C4⋊C4—68no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—34no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—34no—
270C2×C22⋊C4—34no—
271C2×C22⋊C4—38no—
272C2×C22⋊C4—64no—
273C2×C22⋊C4—64no—
274C2×C22⋊C4—68no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4C22×D314yes—
278C2×C22⋊C4C22×D318yes—
279C2×C42—38no—
280C2×C42—38no—
281C2×C42—68no—
282C2×C42—68no—
283C2×C42—632no—
284#95—38no—
285#95—38no—
286#95—38no—
287#95—38no—
288#95—38no—
289#95C22×D318yes—
290C23×C3—28no—
291C23×C3—28no—
292C23×C3—216no—
293C23×C3—216no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—44no—
302C23×C3—44no—
303C23×C3—44no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3#123116yes—
307C23×C3C22×D414yes—
308C23×C3C22×D412yes—
309C23×C3C2×#4114yes—
310C23×C3C2×#4112yes—
311C23×C3C2×#4114yes—
312C23×C3C2×#4112yes—
313C23×C3C2×#4114yes—
314C23×C3C2×#4114yes—
315C23×C3C2511yescharacteristic
316C3×D4—416no—
317C2×C3×C4—232no—
318C2×C3×C4—216no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—44no—
328C2×C3×C4—48no—
329C2×C3×C4C2×#4118yes—
330C2×Dic3—232no—
331C2×Dic3—232no—
332C2×Dic3—28no—
333C2×Dic3—28no—
334C2×Dic3—216no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—44no—
349C2×Dic3—44no—
350C2×Dic3—48no—
351C2×Dic3C22×D414yes—
352C2×Dic3C22×D414yes—
353C2×Dic3C2×#4118yes—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24#23514yes—
362C24D3×D414yes—
363C24#23714yes—
364C24C23×D312yes—
365C24C23×D312yes—
366C24C23×D311yescharacteristic
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—416no—
371C22×C4—232no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—34no—
383C22×C4—34no—
384C22×C4—316no—
385C22×C4—38no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—34no—
389C22×C4—34no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—616no—
393C22×C4—632no—
394C22×C4—632no—
395C22×C4—68no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—68no—
399C22×C4—64no—
400C22×C4—68no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4#23718yes—
412C22×C4C23×D314yes—
413C22×C4C23×D312yes—
414C22×C4C23×D314yes—
415C4⋊C4—68no—
416C4⋊C4—68no—
417C4⋊C4—616no—
418C4⋊C4—616no—
419C22⋊C4—28no—
420C22⋊C4—216no—
421C22⋊C4—68no—
422C22⋊C4—68no—
423C22⋊C4—616no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C42—616no—
427C42—616no—
428C22×C3—232no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—88no—
439C22×C3C2×#12318yes—
440C22×C3C2×#12314yes—
441C22×C3#52514yes—
442C22×C3#52514yes—
443C22×C3#52918yes—
444C22×C3#52918yes—
445C22×C3C23×D411yescharacteristic
446C22×C3C22×#4111yescharacteristic
447C22×C3C22×#4111yescharacteristic
448C22×C3C22×#4112yes—
449C22×C3C22×#4112yes—
450C3×C4—28no—
451C3×C4—42no—
452C3×C4—44no—
453Dic3—24no—
454Dic3—24no—
455Dic3—28no—
456Dic3—42no—
457Dic3—42no—
458Dic3—44no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—216no—
463C23—28no—
464C23—216no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—44no—
471C23—44no—
472C23—48no—
473C23—44no—
474C23—48no—
475C23#883116yes—
476C23C2×#23514yes—
477C23C2×#23512yes—
478C23C2×D3×D414yes—
479C23C2×D3×D412yes—
480C23C2×#23714yes—
481C23C2×#23712yes—
482C23C2×#23714yes—
483C23D3×#4114yes—
484C23C24×D311yescharacteristic
485D4—416no—
486C2×C4—232no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—34no—
497C2×C4—34no—
498C2×C4—38no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—632no—
502C2×C4—616no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—632no—
510C2×C4—68no—
511C2×C4—68no—
512C2×C4—64no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—124no—
520C2×C4—128no—
521C2×C4—124no—
522C2×C4C2×#23718yes—
523C2×C3—28no—
524C2×C3—82no—
525C2×C3#285318yescomplemented
526C2×C3C22×#12311yescharacteristic
527C2×C3C2×#52511yescharacteristic
528C2×C3C2×#52511yescharacteristic
529C2×C3C2×#52912yes—
530C2×C3C2×#52912yes—
531C22—232no—
532C22—24no—
533C22—28no—
534C22—24no—
535C22—28no—
536C22—42no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—44no—
541C22—88no—
542C22C2×#88318yes—
543C22C2×#88314yes—
544C22#553818yes—
545C22#558614yes—
546C22#560018yes—
547C22#565614yes—
548C22C22×#23511yescharacteristic
549C22C22×D3×D411yescharacteristic
550C22C22×#23712yes—
551C22C22×#23711yescharacteristic
552C22C2×D3×#4112yes—
553C4—28no—
554C4—44no—
555C4—42no—
556C4—64no—
557C4—64no—
558C4—68no—
559C4—122no—
560C4—122no—
561C4—124no—
562C3C2×#285311yescharacteristic, complemented, Sylow
563C2—28no—
564C2—82no—
565C2#7487118yesdirect product factor
566C2C22×#88311yescharacteristic
567C2C2×#553812yes—
568C2C2×#558611yescharacteristic
569C2C2×#560012yes—
570C2C2×#565611yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#386C2
socleC24×C3C24
prime radicalC23×C3×D4C22
hypercenterC2×#386D3
hypocenterC3C2×#2853
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic3C2semidirect
5. C2×#5210C2semidirect
6. C2×#4945C2semidirect
7. C2×#4951C2semidirect
8. C2×#4950C2semidirect
9. C2×#4943C2semidirect
10. #74871C2direct (N × H)
11. C2×#4654C2semidirect
12. C2×#4652C2semidirect
34. C22×C4×Dic3C22semidirect
38. #5210C22semidirect
44. #4951C22semidirect
45. #4950C22semidirect
46. #4945C22semidirect
47. #4943C22semidirect
48. #4654C22semidirect
49. #4652C22semidirect
525. C2×C3#2853semidirect
562. C3C2×#2853semidirect
565. C2#74871direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9352
C23×C4×Dic32
C2×C3×#3862
C2×#52202
C2×#52102
C2×#49452
C2×#49512
C2×#49502
C2×#49432
#748712
C2×#46542
C2×#46522
C2×#28533