Finite Group Databasebeta

← C2×#76884 #11001341

Subgroups — C2×#76884 #11001341

Order 768.

2674 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic, complemented
2C22×#883C211yesmaximal, characteristic
3C2×C3×#393C211yesmaximal, characteristic, complemented
4C2×#5230C211yesmaximal, characteristic, complemented
5C2×#4959C212yesmaximal
6C2×#4953C211yesmaximal, characteristic, complemented
7C2×#4948C212yesmaximal, complemented
8#76884C2116yesmaximal, direct product factor
9C2×#4667C211yesmaximal, characteristic, complemented
10C2×#4661C211yesmaximal, characteristic, complemented
11C2×#4657C212yesmaximal, complemented
12C2×#4674C212yesmaximal, complemented
13C2×#2814—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×#217C2211yescharacteristic
20C22×#217C2212yes—
21C2×#946—28no—
22C2×#946C2218yescomplemented
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×C4⋊Dic3C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2212yes—
29C22×Dic3⋊C4C2211yescharacteristic
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C2×#883—28no—
34C2×#883C2218yes—
35C3×#393C2218yescomplemented
36C2×C3×#95C2212yes—
37C2×C3×#95C2211yescharacteristic
38#5230C2218yescomplemented
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4959C22116yes—
44#4953C2218yescomplemented
45#4948C22116yescomplemented
46#4674C22116yescomplemented
47#4667C2218yescomplemented
48#4661C2218yescomplemented
49#4657C22116yescomplemented
50C22×#123—31no—
51C22×C22⋊Q8—31no—
52#2814—316no—
53C2×#398—32no—
54C2×#398—31no—
55C2×#397—32no—
56C2×#397—31no—
57C2×#396—32no—
58C2×#395—31no—
59C2×#394—32no—
60C2×#393D311yescharacteristic, complemented
61C2×#380—31no—
62C25×C3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C22×C3×Q8—24no—
65C22×C3×Q8—44no—
66C22×C3×Q8C2314yes—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C22×#67—24no—
72C22×#67—44no—
73C22×#67C2314yes—
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2312yes—
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2311yescharacteristic
79C22×C4×D3—44no—
80C22×C4×D3D418yes—
81C22×C4×D3C2314yes—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—44no—
84C2×C3×C22⋊C4C2314yes—
85C2×C3×C22⋊C4C2318yes—
86C2×#217—24no—
87C2×#217—44no—
88C2×#217C2314yes—
89C2×#217C2318yes—
90#946—232no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—28no—
93C2×(C2×D3)⋊C4—44no—
94C2×(C2×D3)⋊C4C2314yes—
95C2×(C2×D3)⋊C4C2318yes—
96C2×(C2×D3)⋊C4C2318yes—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—44no—
100C2×C4⋊Dic3C2314yes—
101C2×C4⋊Dic3C2314yes—
102C2×C4⋊Dic3C2318yes—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—44no—
106C2×Dic3⋊C4—44no—
107C2×Dic3⋊C4C2318yes—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111#883—232no—
112C3×#95C2318yes—
113C3×#95C2314yes—
114C3⋊#95C2318yes—
115C3⋊#95C2318yes—
116C3⋊#95C2314yes—
117C3⋊#95C2318yes—
118C23×Q8C2×D311yescharacteristic
119C23×D4—31no—
120C24×C4—31no—
121C2×#123—38no—
122C2×#123—68no—
123C2×C22⋊Q8—38no—
124C2×C22⋊Q8—68no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C22⋊C4—31no—
133C22×C22⋊C4—32no—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4C2×D311yescharacteristic
138C22×C22⋊C4C2×D312yes—
139#398—316no—
140#398—38no—
141#397—38no—
142#397—316no—
143#396—316no—
144#395—38no—
145#394—316no—
146#393C2×D318yescomplemented
147#380—38no—
148C2×#95—32no—
149C2×#95—32no—
150C2×#95—31no—
151C2×#95—32no—
152C2×#95C2×D312yes—
153C2×#95C2×D311yescharacteristic
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—24no—
157C24×C3—22no—
158C24×C3—24no—
159C24×C3C2412yes—
160C24×C3C2411yescharacteristic
161C23×D3—22no—
162C23×D3—24no—
163C23×D3—22no—
164C23×D3—24no—
165C23×D3C2×D414yes—
166C23×D3C2412yes—
167C2×C3×Q8—28no—
168C2×C3×Q8—48no—
169C2×C3×Q8—416no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2412yes—
181C22×C3×C4C2414yes—
182C22×C3×C4C2414yes—
183C22×C3×C4C2412yes—
184C2×#67—28no—
185C2×#67—48no—
186C2×#67—416no—
187C22×Dic3—24no—
188C22×Dic3—24no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—22no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—22no—
198C22×Dic3—24no—
199C22×Dic3C2×D414yes—
200C22×Dic3C2414yes—
201C22×Dic3C2412yes—
202C22×Dic3C2414yes—
203C22×Dic3C2412yes—
204C22×Dic3C2412yes—
205C2×C4×D3—416no—
206C2×C4×D3—48no—
207C2×C4×D3C2×D4116yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—48no—
210#217—28no—
211#217—48no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—216no—
214(C2×D3)⋊C4—48no—
215C4⋊Dic3—28no—
216C4⋊Dic3—28no—
217C4⋊Dic3—48no—
218Dic3⋊C4—28no—
219Dic3⋊C4—216no—
220Dic3⋊C4—48no—
221Dic3⋊C4—48no—
222C25—31no—
223C25C22×D311yescharacteristic
224C22×Q8—24no—
225C22×Q8—44no—
226C22×Q8C22×D314yes—
227C22×D4—34no—
228C22×D4—64no—
229C22×D4—124no—
230C23×C4—31no—
231C23×C4—38no—
232C23×C4—34no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—124no—
238C23×C4C22×D312yes—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242#123—632no—
243C22⋊Q8—632no—
244C2×C4⋊C4—38no—
245C2×C4⋊C4—34no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—38no—
249C2×C4⋊C4—38no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—68no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—124no—
256C2×C4⋊C4—124no—
257C2×C4⋊C4—124no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—34no—
260C2×C22⋊C4—38no—
261C2×C22⋊C4—34no—
262C2×C22⋊C4—38no—
263C2×C22⋊C4—38no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—68no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—124no—
269C2×C22⋊C4—124no—
270C2×C22⋊C4C22×D314yes—
271C2×C22⋊C4C22×D318yes—
272#95—34no—
273#95—38no—
274#95—38no—
275#95—38no—
276#95C22×D314yes—
277#95C22×D318yes—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—22no—
282C23×C3—24no—
283C23×C3—44no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3—48no—
287C23×C3—48no—
288C23×C3C22×D412yes—
289C23×C3H(5,𝔽2)14yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C32−1+412yes—
292C23×C32−1+414yes—
293C23×C3C2511yescharacteristic
294C22×D3—28no—
295C22×D3—28no—
296C22×D3—28no—
297C22×D3—22no—
298C22×D3—24no—
299C22×D3—22no—
300C22×D3—24no—
301C22×D3—48no—
302C22×D3—48no—
303C22×D3C22×D414yes—
304C3×Q8—416no—
305C2×C3×C4—28no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—48no—
316C2×C3×C4—44no—
317C2×C3×C4—48no—
318C2×C3×C4C22×D414yes—
319#67—416no—
320C2×Dic3—28no—
321C2×Dic3—24no—
322C2×Dic3—24no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—44no—
334C2×Dic3—48no—
335C2×Dic3—44no—
336C2×Dic3—48no—
337C2×Dic3C22×D414yes—
338C4×D3—416no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—34no—
345C24—32no—
346C24—64no—
347C24—62no—
348C24—62no—
349C24—64no—
350C24C23×D312yes—
351C24C23×D311yescharacteristic
352C2×Q8—28no—
353C2×Q8—48no—
354C2×Q8—416no—
355C2×D4—68no—
356C2×D4—128no—
357C2×D4—1216no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—34no—
368C22×C4—32no—
369C22×C4—316no—
370C22×C4—34no—
371C22×C4—32no—
372C22×C4—34no—
373C22×C4—32no—
374C22×C4—64no—
375C22×C4—62no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4—62no—
381C22×C4—64no—
382C22×C4—64no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—128no—
387C22×C4—1216no—
388C22×C4D3×D414yes—
389C22×C4C23×D312yes—
390C22×C4C23×D314yes—
391C22×C4C23×D314yes—
392C22×C4C23×D312yes—
393C4⋊C4—616no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—128no—
398C4⋊C4—128no—
399C4⋊C4—128no—
400C22⋊C4—28no—
401C22⋊C4—48no—
402C22⋊C4—616no—
403C22⋊C4—68no—
404C22⋊C4—68no—
405C22⋊C4—128no—
406C22⋊C4—128no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—88no—
413C22×C3#53414yes—
414C22×C3#53614yes—
415C22×C3#53714yes—
416C22×C3#56218yes—
417C22×C3#56318yes—
418C22×C3C23×D411yescharacteristic
419C22×C3C2×H(5,𝔽2)12yes—
420C22×C3C2×H(5,𝔽2)11yescharacteristic
421C22×C3C2×2−1+411yescharacteristic
422C22×C3C2×2−1+412yes—
423C2×D3—24no—
424C2×D3—24no—
425C2×D3—24no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—416no—
429C3×C4—24no—
430C3×C4—44no—
431C3×C4—42no—
432C3×C4—44no—
433Dic3—24no—
434Dic3—42no—
435Dic3—44no—
436Dic3—42no—
437Dic3—44no—
438C23—22no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—34no—
444C23—48no—
445C23—44no—
446C23—48no—
447C23—44no—
448C23—44no—
449C23—68no—
450C23—64no—
451C23—62no—
452C23—62no—
453C23—64no—
454C23—68no—
455C23—68no—
456C23—128no—
457C23—128no—
458C23C2×D3×D412yes—
459C23#100414yes—
460C23#100712yes—
461C23#100912yes—
462C23#101014yes—
463C23C24×D311yescharacteristic
464Q8—416no—
465D4—1216no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—34no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—64no—
481C2×C4—68no—
482C2×C4—62no—
483C2×C4—64no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—62no—
492C2×C4—64no—
493C2×C4—1216no—
494C2×C4—128no—
495C2×C4—124no—
496C2×C4—124no—
497C2×C4—128no—
498C2×C4C2×D3×D414yes—
499C2×C3—82no—
500C2×C3#281418yescomplemented
501C2×C3C2×#53411yescharacteristic
502C2×C3C2×#53611yescharacteristic
503C2×C3C2×#53711yescharacteristic
504C2×C3C2×#56212yes—
505C2×C3C2×#56312yes—
506D3—44no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—42no—
512C22—64no—
513C22—64no—
514C22—64no—
515C22—88no—
516C22—124no—
517C22—1216no—
518C22—124no—
519C22#558914yes—
520C22#562914yes—
521C22#564618yes—
522C22#567514yes—
523C22#568418yes—
524C22C22×D3×D411yescharacteristic
525C22C2×#100412yes—
526C22C2×#100711yescharacteristic
527C22C2×#100911yescharacteristic
528C22C2×#101012yes—
529C4—24no—
530C4—44no—
531C4—42no—
532C4—44no—
533C4—64no—
534C4—124no—
535C4—122no—
536C4—122no—
537C4—124no—
538C3C2×#281411yescharacteristic, complemented, Sylow
539C2—82no—
540C2—124no—
541C2#7688418yesdirect product factor
542C2C2×#558911yescharacteristic
543C2C2×#564612yes—
544C2C2×#562911yescharacteristic
545C2C2×#568412yes—
546C2C2×#567511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#393C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#393D3
hypocenterC3C2×#2814
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×#946C2semidirect
3. C2×C3×#393C2semidirect
4. C2×#5230C2semidirect
6. C2×#4953C2semidirect
7. C2×#4948C2semidirect
8. #76884C2direct (N × H)
9. C2×#4667C2semidirect
10. C2×#4661C2semidirect
11. C2×#4657C2semidirect
12. C2×#4674C2semidirect
22. C2×#946C22semidirect
35. C3×#393C22semidirect
38. #5230C22semidirect
44. #4953C22semidirect
45. #4948C22semidirect
46. #4674C22semidirect
47. #4667C22semidirect
48. #4661C22semidirect
49. #4657C22semidirect
60. C2×#393D3semidirect
146. #393C2×D3semidirect
500. C2×C3#2814semidirect
538. C3C2×#2814semidirect
541. C2#76884direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9462
C22×#8832
C2×C3×#3932
C2×#52302
C2×#49592
C2×#49532
C2×#49482
#768842
C2×#46672
C2×#46612
C2×#46572
C2×#46742
C2×#28143