Finite Group Databasebeta

← C2×#76944 #10966923

Subgroups — C2×#76944 #10966923

Order 768.

4782 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#939C212yesmaximal, complemented
2C22×#884C212yesmaximal, complemented
3C22×#67⋊C4C212yesmaximal, complemented
4C2×C3×#394C211yesmaximal, characteristic, complemented
5C2×#4991C211yesmaximal, characteristic, complemented
6C2×#4975C211yesmaximal, characteristic, complemented
7C2×#4961C212yesmaximal, complemented
8C2×#4942C212yesmaximal, complemented
9#76944C2116yesmaximal, direct product factor
10C2×#4667C212yesmaximal, complemented
11C2×#2537—31nomaximal, Sylow
12C23×C3×D4C2211yescharacteristic
13C23×#67C2212yes—
14C23×#67C2212yes—
15C24×Dic3C2212yes—
16C23×D12C2211yescharacteristic
17C23×C4×D3C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2212yes—
22C2×#939—216no—
23C2×#939C22116yescomplemented
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×Dic3⋊C4C2212yes—
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×C4×Dic3C2211yescharacteristic
32C2×#884—216no—
33C2×#884C22116yescomplemented
34C2×#67⋊C4—216no—
35C2×#67⋊C4C22116yescomplemented
36C3×#394C2218yescomplemented
37C2×C3×#95C2211yescharacteristic
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2212yes—
40#4991C2218yescomplemented
41#4975C2218yescomplemented
42#4961C22116yescomplemented
43#4942C22116yescomplemented
44#4667C22116yescomplemented
45C22×#121—32no—
46C22×#121—32no—
47C22×C4×D4—32no—
48#2537—316no—
49C2×#394—31no—
50C2×#394—32no—
51C2×#394D311yescharacteristic, complemented
52C2×#384—31no—
53C2×#380—32no—
54C2×#380—32no—
55C25×C3C2312yes—
56C24×D3C2312yes—
57C22×C3×D4—28no—
58C22×C3×D4—42no—
59C22×C3×D4C2314yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—28no—
65C22×#67—28no—
66C22×#67—28no—
67C22×#67—28no—
68C22×#67—44no—
69C22×#67D4116yescomplemented
70C22×#67C2318yes—
71C22×#67C2318yes—
72C23×Dic3—28no—
73C23×Dic3—28no—
74C23×Dic3—28no—
75C23×Dic3C2318yes—
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2312yes—
79C22×D12—28no—
80C22×D12—42no—
81C22×D12C2314yes—
82C22×C4×D3—28no—
83C22×C4×D3—28no—
84C22×C4×D3—28no—
85C22×C4×D3C2318yes—
86C2×C3×C4⋊C4—42no—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C22⋊C4—28no—
89C2×C3×C22⋊C4—28no—
90C2×C3×C22⋊C4C2318yes—
91C2×C3×C22⋊C4C2318yes—
92C2×#217—28no—
93C2×#217C2318yes—
94#939—264no—
95C2×(C2×D3)⋊C4—28no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4C2318yes—
99C2×(C2×D3)⋊C4C2318yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×C4⋊Dic3—42no—
102C2×C4⋊Dic3C2314yes—
103C2×Dic3⋊C4—28no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—28no—
106C2×Dic3⋊C4C2318yes—
107C2×Dic3⋊C4C2318yes—
108C2×Dic3⋊C4C2318yes—
109C2×C4×Dic3D418yescomplemented
110C2×C4×Dic3C2314yes—
111#884—264no—
112#67⋊C4—264no—
113C3×#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2314yes—
116C23×D4—32no—
117C23×D4—32no—
118C23×D4—31no—
119C23×D4C2×D311yescharacteristic
120C24×C4—32no—
121C24×C4—32no—
122C2×#121—316no—
123C2×#121—316no—
124C2×#121—616no—
125C2×#121—616no—
126C2×C4×D4—316no—
127C2×C4×D4—616no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C22⋊C4—32no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4C2×D312yes—
138C22×C22⋊C4C2×D312yes—
139C22×C42—31no—
140#394—38no—
141#394—316no—
142#394C2×D318yescomplemented
143#384—38no—
144#380—316no—
145#380—316no—
146C2×#95—31no—
147C2×#95—32no—
148C2×#95C2×D311yescharacteristic
149C24×C3—24no—
150C24×C3—24no—
151C24×C3—24no—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3C2×D418yes—
156C24×C3C2414yes—
157C24×C3C2411yescharacteristic
158C23×D3—24no—
159C23×D3—24no—
160C23×D3—24no—
161C23×D3—24no—
162C23×D3—24no—
163C23×D3—24no—
164C23×D3C2×D418yes—
165C23×D3C2414yes—
166C2×C3×D4—216no—
167C2×C3×D4—44no—
168C2×C3×D4—416no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4C2×D414yes—
179C22×C3×C4C2414yes—
180C22×C3×C4C2412yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2412yes—
183C2×#67—216no—
184C2×#67—216no—
185C2×#67—264no—
186C2×#67—216no—
187C2×#67—216no—
188C2×#67—48no—
189C2×#67—432no—
190C2×#67C2×D4132yescomplemented
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—24no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—216no—
198C22×Dic3—216no—
199C22×Dic3—216no—
200C22×Dic3—432no—
201C22×Dic3C2×D418yes—
202C22×Dic3C2×D418yes—
203C22×Dic3#4118yes—
204C22×Dic3C2414yes—
205C22×Dic3C2414yes—
206C22×Dic3C2414yes—
207C2×D12—216no—
208C2×D12—44no—
209C2×D12—416no—
210C2×C4×D3—216no—
211C2×C4×D3—216no—
212C2×C4×D3—216no—
213C2×C4×D3—432no—
214C3×C4⋊C4—44no—
215C3×C22⋊C4—216no—
216C3×C22⋊C4—216no—
217#217—216no—
218(C2×D3)⋊C4—216no—
219(C2×D3)⋊C4—216no—
220(C2×D3)⋊C4—216no—
221C4⋊Dic3—44no—
222Dic3⋊C4—216no—
223Dic3⋊C4—216no—
224Dic3⋊C4—216no—
225C4×Dic3C2×D4116yescomplemented
226C25—32no—
227C25C22×D312yes—
228C22×D4—28no—
229C22×D4—316no—
230C22×D4—38no—
231C22×D4—38no—
232C22×D4—34no—
233C22×D4—42no—
234C22×D4—68no—
235C22×D4—68no—
236C22×D4—68no—
237C22×D4—68no—
238C22×D4—68no—
239C22×D4—122no—
240C22×D4—124no—
241C22×D4C22×D314yes—
242C23×C4—32no—
243C23×C4—38no—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—38no—
247C23×C4—68no—
248C23×C4—68no—
249C23×C4—68no—
250C23×C4—68no—
251C23×C4—68no—
252C23×C4—68no—
253C23×C4C22×D311yescharacteristic
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256C23×C4C22×D311yescharacteristic
257#121—664no—
258#121—664no—
259C4×D4—664no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—38no—
262C2×C4⋊C4—38no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—68no—
266C2×C4⋊C4—68no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—122no—
269C2×C4⋊C4C22×D314yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—38no—
273C2×C22⋊C4—38no—
274C2×C22⋊C4—38no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—68no—
277C2×C22⋊C4—68no—
278C2×C22⋊C4—68no—
279C2×C22⋊C4—68no—
280C2×C22⋊C4C22×D318yes—
281C2×C22⋊C4C22×D318yes—
282C2×C42—38no—
283C2×C42—34no—
284#95—34no—
285#95—38no—
286#95C22×D314yes—
287C23×C3—216no—
288C23×C3—216no—
289C23×C3—216no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—48no—
297C23×C3—48no—
298C23×C3—48no—
299C23×C3—48no—
300C23×C3C22×D418yes—
301C23×C3C22×D412yes—
302C23×C3C22×D414yes—
303C23×C3C2×#4114yes—
304C23×C3H(5,𝔽2)12yes—
305C23×C32−1+412yes—
306C23×C3C2511yescharacteristic
307C22×D3—216no—
308C22×D3—216no—
309C22×D3—216no—
310C22×D3—24no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—24no—
316C22×D3—48no—
317C22×D3—48no—
318C22×D3—48no—
319C22×D3—48no—
320C22×D3C22×D418yes—
321C3×D4—416no—
322C2×C3×C4—28no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—48no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4C22×D414yes—
336#67—264no—
337#67—432no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—48no—
345C2×Dic3—48no—
346C2×Dic3—432no—
347C2×Dic3#121132yescomplemented
348C2×Dic3C22×D418yes—
349C2×Dic3C22×D418yes—
350C2×Dic3C2×#4118yes—
351D12—416no—
352C4×D3—432no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—38no—
360C24—34no—
361C24—64no—
362C24—64no—
363C24—64no—
364C24—64no—
365C24—64no—
366C24—64no—
367C24D3×D418yes—
368C24C23×D314yes—
369C24C23×D311yescharacteristic
370C2×D4—216no—
371C2×D4—332no—
372C2×D4—44no—
373C2×D4—416no—
374C2×D4—616no—
375C2×D4—616no—
376C2×D4—616no—
377C2×D4—664no—
378C2×D4—616no—
379C2×D4—616no—
380C2×D4—124no—
381C2×D4—1216no—
382C2×D4—128no—
383C2×D4—1232no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—38no—
394C22×C4—38no—
395C22×C4—38no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—616no—
400C22×C4—616no—
401C22×C4—616no—
402C22×C4—616no—
403C22×C4—616no—
404C22×C4—616no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—1232no—
412C22×C4—1232no—
413C22×C4D3×D414yes—
414C22×C4C23×D314yes—
415C22×C4C23×D312yes—
416C22×C4C23×D312yes—
417C22×C4C23×D312yes—
418C4⋊C4—44no—
419C4⋊C4—616no—
420C4⋊C4—616no—
421C4⋊C4—616no—
422C4⋊C4—124no—
423C22⋊C4—216no—
424C22⋊C4—216no—
425C22⋊C4—616no—
426C22⋊C4—616no—
427C22⋊C4—616no—
428C22⋊C4—616no—
429C42—316no—
430C22×C3—28no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—432no—
438C22×C3C2×#12118yes—
439C22×C3#53714yes—
440C22×C3D4214yes—
441C22×C3#54718yes—
442C22×C3#55514yes—
443C22×C3C23×D411yescharacteristic
444C22×C3C23×D412yes—
445C22×C3C22×#4112yes—
446C22×C3C2×H(5,𝔽2)11yescharacteristic
447C22×C3C2×2−1+411yescharacteristic
448C2×D3—28no—
449C2×D3—28no—
450C2×D3—28no—
451C2×D3—44no—
452C2×D3—44no—
453C2×D3—44no—
454C2×D3—44no—
455C2×D3—432no—
456C3×C4—24no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—42no—
460Dic3—44no—
461Dic3—44no—
462Dic3C2×#121116yescomplemented
463C23—216no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—216no—
471C23—216no—
472C23—38no—
473C23—48no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—616no—
478C23—64no—
479C23—64no—
480C23—64no—
481C23—64no—
482C23—64no—
483C23—64no—
484C23—616no—
485C23—616no—
486C23—128no—
487C23—128no—
488C23—128no—
489C23—128no—
490C23C2×D3×D418yes—
491C23C2×D3×D414yes—
492C23C2×D3×D412yes—
493C23#100412yes—
494C23D3×#4114yes—
495C23#101012yes—
496C23C24×D311yescharacteristic
497D4—416no—
498D4—664no—
499D4—1216no—
500D4—1232no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—38no—
512C2×C4—38no—
513C2×C4—38no—
514C2×C4—332no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—1232no—
525C2×C4—1232no—
526C2×C4—128no—
527C2×C4—128no—
528C2×C4C2×D3×D414yes—
529C2×C3—48no—
530C2×C3#253718yescomplemented
531C2×C3C22×#12112yes—
532C2×C3C2×#53711yescharacteristic
533C2×C3C2×D4211yescharacteristic
534C2×C3C2×#54712yes—
535C2×C3C2×#55511yescharacteristic
536D3—48no—
537C22—28no—
538C22—28no—
539C22—28no—
540C22—44no—
541C22—432no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—68no—
546C22—68no—
547C22—68no—
548C22—124no—
549C22—1232no—
550C22—124no—
551C22—124no—
552C22—124no—
553C22#559014yes—
554C22#559714yes—
555C22D3×#12118yes—
556C22#565518yes—
557C22#567014yes—
558C22C22×D3×D412yes—
559C22C22×D3×D411yescharacteristic
560C22C2×#100411yescharacteristic
561C22C2×D3×#4112yes—
562C22C2×#101011yescharacteristic
563C4—24no—
564C4—316no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—124no—
569C4—124no—
570C3C2×#253711yescharacteristic, complemented, Sylow
571C2—48no—
572C2—128no—
573C2#7694418yesdirect product factor
574C2C2×#559011yescharacteristic
575C2C2×#559711yescharacteristic
576C2C2×D3×#12112yes—
577C2C2×#565512yes—
578C2C2×#567011yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#394C2
socleC24×C3C24
hypercenterC2×#394D3
hypocenterC3C2×#2537
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C22×#939C2semidirect
2. C22×#884C2semidirect
3. C22×#67⋊C4C2semidirect
4. C2×C3×#394C2semidirect
5. C2×#4991C2semidirect
6. C2×#4975C2semidirect
7. C2×#4961C2semidirect
8. C2×#4942C2semidirect
9. #76944C2direct (N × H)
10. C2×#4667C2semidirect
23. C2×#939C22semidirect
33. C2×#884C22semidirect
35. C2×#67⋊C4C22semidirect
36. C3×#394C22semidirect
40. #4991C22semidirect
41. #4975C22semidirect
42. #4961C22semidirect
43. #4942C22semidirect
44. #4667C22semidirect
51. C2×#394D3semidirect
69. C22×#67D4semidirect
109. C2×C4×Dic3D4semidirect
142. #394C2×D3semidirect
190. C2×#67C2×D4semidirect
225. C4×Dic3C2×D4semidirect
347. C2×Dic3#121semidirect
462. Dic3C2×#121semidirect
530. C2×C3#2537semidirect
570. C3C2×#2537semidirect
573. C2#76944direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9392
C22×#8842
C22×#67⋊C42
C2×C3×#3942
C2×#49912
C2×#49752
C2×#49612
C2×#49422
#769442
C2×#46672
C2×#25373