Finite Group Databasebeta

← C2×#77764 #10964237

Subgroups — C2×#77764 #10964237

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#894C212yesmaximal
2C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C22×#893C211yesmaximal, characteristic
4C2×C3×#398C211yesmaximal, characteristic, complemented
5C2×#4992C211yesmaximal, characteristic, complemented
6C2×#4981C211yesmaximal, characteristic, complemented
7C2×#4976C212yesmaximal, complemented
8#77764C2116yesmaximal, direct product factor
9C2×#4664C212yesmaximal, complemented
10C2×#4667C212yesmaximal, complemented
11C2×#4667C211yesmaximal, characteristic, complemented
12C2×#4674C211yesmaximal, characteristic, complemented
13C2×#2486—31nomaximal, Sylow
14C23×D12C2211yescharacteristic
15C23×C4×D3C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×C4×D3C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×(C2×D3)⋊C4C2211yescharacteristic
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×C4×Dic3C2211yescharacteristic
31C2×#894—216no—
32C2×#894C22116yes—
33C2×#893—28no—
34C2×#893C2218yes—
35C2×D3×C4⋊C4—28no—
36C2×D3×C4⋊C4C2218yescomplemented
37C3×#398C2218yescomplemented
38C2×C3×#95C2211yescharacteristic
39C2×C3×#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4992C2218yescomplemented
45#4981C2218yescomplemented
46#4976C22116yescomplemented
47#4674C2218yescomplemented
48#4667C22116yescomplemented
49#4667C2218yescomplemented
50#4664C22116yescomplemented
51C22×#123—32no—
52C22×C4×D4—31no—
53C23×C4⋊C4—31no—
54#2486—316no—
55C2×#398—31no—
56C2×#398D311yescharacteristic, complemented
57C2×#397—31no—
58C2×#397—31no—
59C2×#389—32no—
60C2×#381—32no—
61C2×#380—32no—
62C2×#380—31no—
63C24×D3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2312yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2312yes—
73C23×Dic3C2311yescharacteristic
74C22×D12—24no—
75C22×D12—24no—
76C22×D12D418yes—
77C22×D12C2314yes—
78C22×C4×D3—28no—
79C22×C4×D3—28no—
80C22×C4×D3—28no—
81C22×C4×D3—28no—
82C22×C4×D3—24no—
83C22×C4×D3—28no—
84C22×C4×D3D418yes—
85C22×C4×D3C2318yes—
86C22×C4×D3C2314yes—
87C22×C4×D3C2314yes—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4—28no—
91C2×C3×C4⋊C4C2318yes—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C4⋊C4C2314yes—
94C2×(C2×D3)⋊C4—24no—
95C2×(C2×D3)⋊C4—28no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4—24no—
99C2×(C2×D3)⋊C4C2314yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×(C2×D3)⋊C4C2318yes—
102C2×(C2×D3)⋊C4C2318yes—
103C2×(C2×D3)⋊C4C2314yes—
104C2×C4⋊Dic3—24no—
105C2×C4⋊Dic3C2314yes—
106C2×Dic3⋊C4—28no—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4—28no—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3—24no—
113C2×C4×Dic3C2314yes—
114#894—264no—
115#893—232no—
116D3×C4⋊C4—232no—
117C3×#95C2318yes—
118C3×#95C2314yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C3⋊#95C2314yes—
123C23×D4—31no—
124C24×C4—31no—
125C24×C4—32no—
126C24×C4—31no—
127C2×#123—316no—
128C2×#123—616no—
129C2×C4×D4—38no—
130C2×C4×D4—68no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—38no—
136C22×C4⋊C4—68no—
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D312yes—
140C22×C22⋊C4—31no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—31no—
144C22×C22⋊C4—32no—
145C22×C42—31no—
146#398—38no—
147#398C2×D318yescomplemented
148#397—38no—
149#397—38no—
150#389—316no—
151#381—316no—
152#380—316no—
153#380—38no—
154C2×#95—31no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95C2×D311yescharacteristic
159C2×#95C2×D312yes—
160C24×C3C2411yescharacteristic
161C23×D3—24no—
162C23×D3—24no—
163C23×D3—22no—
164C23×D3—24no—
165C23×D3—22no—
166C23×D3—24no—
167C23×D3C2×D414yes—
168C23×D3C2×D414yes—
169C23×D3C2×D414yes—
170C23×D3#4114yes—
171C23×D3C2412yes—
172C23×D3C2412yes—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
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×C4C2×D414yes—
186C22×C3×C4C2412yes—
187C22×C3×C4C2414yes—
188C22×C3×C4C2414yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2412yes—
191C22×Dic3—22no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3C2×D414yes—
200C22×Dic3#4118yes—
201C22×Dic3C2412yes—
202C22×Dic3C2414yes—
203C22×Dic3C2414yes—
204C22×Dic3C2412yes—
205C2×D12—232no—
206C2×D12—28no—
207C2×D12—28no—
208C2×D12C2×D4116yes—
209C2×C4×D3—232no—
210C2×C4×D3—216no—
211C2×C4×D3—216no—
212C2×C4×D3—216no—
213C2×C4×D3—216no—
214C2×C4×D3—28no—
215C2×C4×D3—216no—
216C2×C4×D3—432no—
217C2×C4×D3—416no—
218C2×C4×D3C2×D4116yes—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—28no—
221C3×C4⋊C4—216no—
222(C2×D3)⋊C4—28no—
223(C2×D3)⋊C4—216no—
224(C2×D3)⋊C4—216no—
225(C2×D3)⋊C4—216no—
226(C2×D3)⋊C4—28no—
227C4⋊Dic3—28no—
228Dic3⋊C4—216no—
229Dic3⋊C4—216no—
230Dic3⋊C4—216no—
231C4×Dic3—28no—
232C25—31no—
233C25—31no—
234C22×D4—38no—
235C22×D4—34no—
236C22×D4—64no—
237C22×D4—64no—
238C23×C4—31no—
239C23×C4—38no—
240C23×C4—34no—
241C23×C4—32no—
242C23×C4—31no—
243C23×C4—32no—
244C23×C4—38no—
245C23×C4—34no—
246C23×C4—68no—
247C23×C4—64no—
248C23×C4—68no—
249C23×C4—68no—
250C23×C4—68no—
251C23×C4—68no—
252C23×C4C22×D312yes—
253C23×C4C22×D312yes—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257#123—664no—
258C4×D4—632no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—38no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—38no—
265C2×C4⋊C4—38no—
266C2×C4⋊C4—632no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—68no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—68no—
271C2×C4⋊C4C22×D314yes—
272C2×C4⋊C4C22×D314yes—
273C2×C4⋊C4C22×D318yes—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—38no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—68no—
281C2×C22⋊C4—68no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—64no—
284C2×C42—34no—
285C2×C42—64no—
286#95—38no—
287#95—34no—
288#95—38no—
289#95—38no—
290#95C22×D318yes—
291#95C22×D314yes—
292C23×C3C22×D412yes—
293C23×C3C22×D412yes—
294C23×C3C2×#4112yes—
295C23×C3C2×#4114yes—
296C23×C32−1+414yes—
297C23×C3C2511yescharacteristic
298C22×D3—28no—
299C22×D3—28no—
300C22×D3—216no—
301C22×D3—216no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—28no—
305C22×D3—216no—
306C22×D3—22no—
307C22×D3—24no—
308C22×D3—22no—
309C22×D3—24no—
310C22×D3—44no—
311C22×D3—44no—
312C22×D3—48no—
313C22×D3#121116yes—
314C22×D3C22×D414yes—
315C22×D3C22×D414yes—
316C22×D3C22×D414yes—
317C22×D3C2×#4114yes—
318C2×C3×C4—28no—
319C2×C3×C4—28no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—48no—
332C2×C3×C4—44no—
333C2×C3×C4—48no—
334C2×C3×C4C22×D414yes—
335C2×C3×C4C22×D414yes—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—48no—
345C2×Dic3—44no—
346C2×Dic3—48no—
347C2×Dic3#123116yescomplemented
348C2×Dic3C22×D414yes—
349C2×Dic3C2×#4118yes—
350D12—232no—
351C4×D3—232no—
352C4×D3—432no—
353C4×D3—416no—
354C24—34no—
355C24—34no—
356C24—34no—
357C24—32no—
358C24—34no—
359C24—32no—
360C24—64no—
361C24—64no—
362C24—62no—
363C24—64no—
364C24—64no—
365C24—62no—
366C24C23×D311yescharacteristic
367C2×D4—316no—
368C2×D4—68no—
369C2×D4—68no—
370C2×D4—632no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—34no—
383C22×C4—38no—
384C22×C4—32no—
385C22×C4—316no—
386C22×C4—32no—
387C22×C4—34no—
388C22×C4—34no—
389C22×C4—632no—
390C22×C4—616no—
391C22×C4—68no—
392C22×C4—616no—
393C22×C4—616no—
394C22×C4—616no—
395C22×C4—616no—
396C22×C4—62no—
397C22×C4—64no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—1216no—
405C22×C4—1232no—
406C22×C4D3×D414yes—
407C22×C4D3×D414yes—
408C22×C4C23×D312yes—
409C22×C4C23×D314yes—
410C22×C4C23×D314yes—
411C22×C4C23×D312yes—
412C22×C4C23×D312yes—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—216no—
416C4⋊C4—616no—
417C4⋊C4—616no—
418C4⋊C4—68no—
419C4⋊C4—616no—
420C22⋊C4—68no—
421C22⋊C4—616no—
422C22⋊C4—616no—
423C22⋊C4—616no—
424C22⋊C4—68no—
425C42—68no—
426C22×C3C2×#12114yes—
427C22×C3C2×#12314yes—
428C22×C3#53614yes—
429C22×C3#54718yes—
430C22×C3#55218yes—
431C22×C3C23×D411yescharacteristic
432C22×C3C23×D411yescharacteristic
433C22×C3C22×#4111yescharacteristic
434C22×C3C22×#4112yes—
435C22×C3C2×2−1+412yes—
436C2×D3—24no—
437C2×D3—24no—
438C2×D3—28no—
439C2×D3—28no—
440C2×D3—232no—
441C2×D3—24no—
442C2×D3—28no—
443C2×D3—42no—
444C2×D3—42no—
445C2×D3—44no—
446C2×D3—416no—
447C2×D3C2×#12118yes—
448C3×C4—24no—
449C3×C4—24no—
450C3×C4—44no—
451C3×C4—42no—
452C3×C4—44no—
453Dic3—44no—
454Dic3—42no—
455Dic3—44no—
456Dic3C2×#12318yescomplemented
457C23—34no—
458C23—34no—
459C23—34no—
460C23—316no—
461C23—34no—
462C23—68no—
463C23—616no—
464C23—64no—
465C23—64no—
466C23—68no—
467C23—616no—
468C23—62no—
469C23—64no—
470C23—64no—
471C23—62no—
472C23—68no—
473C23—616no—
474C23—124no—
475C23—128no—
476C23—124no—
477C23C2×D3×D412yes—
478C23C2×D3×D412yes—
479C23#100714yes—
480C23D3×#4114yes—
481C23D3×#4112yes—
482C23C24×D311yescharacteristic
483D4—632no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—316no—
498C2×C4—34no—
499C2×C4—38no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—632no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—1216no—
513C2×C4—1232no—
514C2×C4—124no—
515C2×C4—128no—
516C2×C4—128no—
517C2×C4C2×D3×D414yes—
518C2×C4C2×D3×D414yes—
519C2×C3#248618yescomplemented
520C2×C3C22×#12111yescharacteristic
521C2×C3C22×#12311yescharacteristic
522C2×C3C2×#53611yescharacteristic
523C2×C3C2×#54712yes—
524C2×C3C2×#55212yes—
525D3—28no—
526D3—44no—
527C22—38no—
528C22—68no—
529C22—64no—
530C22—632no—
531C22—68no—
532C22—64no—
533C22—64no—
534C22—68no—
535C22—124no—
536C22—122no—
537C22—1216no—
538C22—122no—
539C22D3×#12114yes—
540C22#563118yes—
541C22D3×#12314yes—
542C22#569318yes—
543C22#572414yes—
544C22C22×D3×D411yescharacteristic
545C22C22×D3×D411yescharacteristic
546C22C2×#100712yes—
547C22C2×D3×#4111yescharacteristic
548C22C2×D3×#4112yes—
549C4—24no—
550C4—24no—
551C4—38no—
552C4—44no—
553C4—44no—
554C4—42no—
555C4—124no—
556C4—122no—
557C4—124no—
558C3C2×#248611yescharacteristic, complemented, Sylow
559C2—68no—
560C2—124no—
561C2#7776418yesdirect product factor
562C2C2×D3×#12111yescharacteristic
563C2C2×#563112yes—
564C2C2×D3×#12311yescharacteristic
565C2C2×#569312yes—
566C2C2×#572411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#398C2
socleC24×C3C24
prime radicalC23×D12C22
hypercenterC2×#398D3
hypocenterC3C2×#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
2. C22×D3×C4⋊C4C2semidirect
4. C2×C3×#398C2semidirect
5. C2×#4992C2semidirect
6. C2×#4981C2semidirect
7. C2×#4976C2semidirect
8. #77764C2direct (N × H)
9. C2×#4664C2semidirect
10. C2×#4667C2semidirect
11. C2×#4667C2semidirect
12. C2×#4674C2semidirect
36. C2×D3×C4⋊C4C22semidirect
37. C3×#398C22semidirect
44. #4992C22semidirect
45. #4981C22semidirect
46. #4976C22semidirect
47. #4674C22semidirect
48. #4667C22semidirect
49. #4667C22semidirect
50. #4664C22semidirect
56. C2×#398D3semidirect
147. #398C2×D3semidirect
347. C2×Dic3#123semidirect
456. Dic3C2×#123semidirect
519. C2×C3#2486semidirect
558. C3C2×#2486semidirect
561. C2#77764direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8942
C22×D3×C4⋊C42
C22×#8932
C2×C3×#3982
C2×#49922
C2×#49812
C2×#49762
#777642
C2×#46642
C2×#46672
C2×#46742
C2×#24863