Finite Group Databasebeta

← C2×#81958 #11664954

Subgroups — C2×#81958 #11664954

Order 768.

2266 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic, complemented
2C2×#5691C211yesmaximal, characteristic, complemented
3C2×C3×#501C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5C2×#5067C211yesmaximal, characteristic, complemented
6#81958C2116yesmaximal, direct product factor
7C2×#4886C212yesmaximal, complemented
8C2×#4872C212yesmaximal, complemented
9C2×#4865C212yesmaximal, complemented
10C2×#4852C212yesmaximal, complemented
11C2×#4712C211yesmaximal, characteristic, complemented
12C2×#4707C211yesmaximal, characteristic, complemented
13C2×#3171—31nomaximal, Sylow
14C22×#239—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2211yescharacteristic
17C2×#947—21no—
18C22×D24C2212yes—
19C22×#204C2212yes—
20C22×ΓL(1,𝔽25)C2212yes—
21#5725—24no—
22#5725C2218yescomplemented
23#5691—24no—
24#5691C2218yescomplemented
25C2×#898—21no—
26C2×#897C2211yescharacteristic
27C2×#895—21no—
28C2×#895C2211yescharacteristic
29C2×#895C2212yes—
30C2×#894—21no—
31C2×#893—21no—
32C2×#893—21no—
33C2×#893C2211yescharacteristic
34C2×#893C2212yes—
35C2×#892—21no—
36C2×#887C2211yescharacteristic
37C2×#875C2211yescharacteristic
38C2×#874C2211yescharacteristic
39C2×C42×D3C2211yescharacteristic
40C3×#501C2218yescomplemented
41C2×C3×#107C2212yes—
42C2×C3×#106C2212yes—
43C2×C3×#97C2211yescharacteristic
44C2×C8×Dic3C2212yes—
45#5089C2218yescomplemented
46#5067C2218yescomplemented
47C2×#810C2212yes—
48C2×#809C2212yes—
49C2×#809C2212yes—
50C2×#809C2212yes—
51C2×#803C2211yescharacteristic
52#4886C22116yescomplemented
53#4872C22116yescomplemented
54#4865C22116yescomplemented
55#4852C22116yescomplemented
56#4712C2218yescomplemented
57#4707C2218yescomplemented
58C2×#532—31no—
59C2×#529—31no—
60#3171—316no—
61C2×#501D311yescharacteristic, complemented
62C2×#496—31no—
63C2×#490—31no—
64C2×#489—31no—
65C2×#469—32no—
66C2×#468—32no—
67C2×C4×SD16—32no—
68C2×C4×D8—32no—
69C2×C4×M16—31no—
70C22×C3×Q8—21no—
71C2×#239—28no—
72C2×#239—48no—
73C22×D12—22no—
74C22×D12—21no—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2312yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3—22no—
80C22×C4×D3—21no—
81C22×C4×D3D411yescharacteristic
82C22×C4×D3D411yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2312yes—
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8C2314yes—
88C3×#125—22no—
89C3×#125C2314yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C42C2311yescharacteristic
97#947—28no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—22no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—22no—
105C2×C4⋊Dic3—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112C2×D24—28no—
113C2×D24C2318yes—
114C2×#204—28no—
115C2×#204C2318yes—
116C2×ΓL(1,𝔽25)—28no—
117C2×ΓL(1,𝔽25)C2318yes—
118#898—28no—
119#897—22no—
120#897C2314yes—
121#895—22no—
122#895—24no—
123#895—28no—
124#895C2314yes—
125#895C2318yes—
126#894—28no—
127#893—28no—
128#893—22no—
129#893—24no—
130#893—28no—
131#893C2314yes—
132#893C2318yes—
133#892—28no—
134#887—22no—
135#887C2314yes—
136#875—22no—
137#875C2314yes—
138#874—22no—
139#874C2314yes—
140C42×D3D414yes—
141C42×D3C2314yes—
142C3×#107C2318yes—
143C3×#106C2318yes—
144C3×#97C2314yes—
145C8×Dic3C2318yes—
146#810C2318yes—
147#809C2318yes—
148#809C2318yes—
149#809C2318yes—
150#803C2314yes—
151C22×#41—61no—
152C22×SD16—32no—
153C22×D8—32no—
154C22×M16—32no—
155#532—38no—
156#532—64no—
157C2×C4⋊Q8C2×D311yescharacteristic
158C2×#127—31no—
159#529—38no—
160#529—64no—
161C2×#126—61no—
162C2×#125C2×D311yescharacteristic
163C2×#124—31no—
164C2×#124—61no—
165C2×#123—61no—
166C2×C22⋊Q8—31no—
167C2×#121—31no—
168C2×#121—32no—
169C2×#121—61no—
170C2×C4×Q8—31no—
171C2×C4×D4—31no—
172C2×C4×D4—32no—
173C2×C4×D4—61no—
174C2×C4×D4—61no—
175C2×#117—61no—
176C22×C42—31no—
177#501C2×D318yescomplemented
178#496—38no—
179#490—38no—
180#489—38no—
181#469—316no—
182#468—316no—
183C4×SD16—316no—
184C4×D8—316no—
185C2×#107C2×D312yes—
186C2×#106C2×D312yes—
187C2×#103—32no—
188C2×#102—32no—
189C2×#102—32no—
190C2×#102—32no—
191C4×M16—38no—
192C2×#97—31no—
193C2×#97C2×D311yescharacteristic
194C2×C4×C8—32no—
195C23×D3—21no—
196C23×D3—22no—
197C23×D3C2×D411yescharacteristic
198C2×C3×Q8—24no—
199C2×C3×Q8—44no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210#239—432no—
211C2×D12—24no—
212C2×D12—24no—
213C2×D12—24no—
214C2×D12—24no—
215C2×D12—28no—
216C2×D12—22no—
217C2×D12—24no—
218C2×D12—24no—
219C2×D12—21no—
220C2×D12—44no—
221C2×D12—44no—
222C2×D12—44no—
223C2×D12—48no—
224C2×D12C2414yes—
225C2×D12C2412yes—
226C2×C4×D3—24no—
227C2×C4×D3—24no—
228C2×C4×D3—24no—
229C2×C4×D3—28no—
230C2×C4×D3—42no—
231C2×C4×D3—44no—
232C2×C4×D3—48no—
233C2×C4×D3C2×D412yes—
234C2×C4×D3C2×D414yes—
235C2×C4×D3C2×D414yes—
236C2×C4×D3C2412yes—
237C2×Dic6—24no—
238C2×Dic6—21no—
239C2×Dic6C2412yes—
240C2×C3×C8#4118yescomplemented
241C2×C3×C8C2414yes—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4C2412yes—
249C3×C4⋊C4C2412yes—
250C3×C4⋊C4C2414yes—
251C3×C42C2×D412yes—
252C3×C42C2412yes—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—28no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—28no—
260C4⋊Dic3—24no—
261Dic3⋊C4—24no—
262C4×Dic3—24no—
263C4×Dic3C2×D412yes—
264C4×Dic3C2×D414yes—
265C4×Dic3C2×D414yes—
266C4×Dic3C2412yes—
267C2×C3⋊C8—24no—
268C2×C3⋊C8C2414yes—
269D24—216no—
270#204—216no—
271ΓL(1,𝔽25)—216no—
272C2×#41—68no—
273C2×#41—128no—
274C22×Q8—21no—
275C22×Q8—31no—
276C22×D4—31no—
277C22×D4—32no—
278C22×D4—61no—
279C22×D4—62no—
280C22×D4—61no—
281C22×D4—61no—
282C23×C4—31no—
283C23×C4—31no—
284C23×C4—31no—
285C23×C4—61no—
286C23×C4—62no—
287C2×SD16—38no—
288C2×SD16—68no—
289C2×D8—38no—
290C2×D8—68no—
291C2×M16—38no—
292C2×M16—68no—
293C22×C8—32no—
294C22×C8C22×D312yes—
295C4⋊Q8—22no—
296C4⋊Q8C22×D314yes—
297#127—34no—
298#127—62no—
299#126—68no—
300#125—22no—
301#125C22×D314yes—
302#124—34no—
303#124—68no—
304#124—62no—
305#123—68no—
306C22⋊Q8—34no—
307C22⋊Q8—62no—
308#121—34no—
309#121—38no—
310#121—62no—
311#121—64no—
312#121—68no—
313C4×Q8—34no—
314C4×Q8—62no—
315C4×D4—34no—
316C4×D4—38no—
317C4×D4—62no—
318C4×D4—64no—
319C4×D4—68no—
320C4×D4—68no—
321#117—68no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—61no—
331C2×C22⋊C4—62no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—62no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—34no—
341C2×C42—34no—
342C2×C42—61no—
343C2×C42C22×D311yescharacteristic
344#107C22×D318yes—
345#106C22×D318yes—
346#103—38no—
347#102—38no—
348#102—38no—
349#102—38no—
350#97—34no—
351#97C22×D314yes—
352C4×C8—38no—
353C23×C3C22×D411yescharacteristic
354C22×D3—22no—
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—22no—
358C22×D3—24no—
359C22×D3—44no—
360C22×D3—42no—
361C22×D3—44no—
362C22×D3—42no—
363C22×D3—44no—
364C22×D3—42no—
365C22×D3C22×D412yes—
366C3×Q8—48no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—44no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4114yes—
379C2×C3×C4C2511yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—42no—
384C2×Dic3AGL(1,ℤ/8)14yes—
385C2×Dic3C22×D412yes—
386C2×Dic3C22×D412yes—
387C2×Dic3C22×D412yes—
388D12—24no—
389D12—24no—
390D12—24no—
391D12—48no—
392D12—48no—
393D12—48no—
394D12—416no—
395D12—44no—
396D12—42no—
397C4×D3—24no—
398C4×D3—24no—
399C4×D3—28no—
400C4×D3—44no—
401C4×D3—48no—
402C4×D3—416no—
403Dic6—24no—
404Dic6—42no—
405C3×C8C2×#4118yescomplemented
406C3⋊C8—24no—
407C24—31no—
408C24—61no—
409C24—62no—
410#41—1232no—
411C2×Q8—24no—
412C2×Q8—32no—
413C2×Q8—44no—
414C2×Q8—61no—
415C2×Q8—64no—
416C2×D4—32no—
417C2×D4—34no—
418C2×D4—64no—
419C2×D4—68no—
420C2×D4—61no—
421C2×D4—64no—
422C2×D4—62no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—64no—
426C2×D4—64no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—128no—
430C2×D4—124no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—34no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—34no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—68no—
444C22×C4—64no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—128no—
448C22×C4—124no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452SD16—616no—
453D8—616no—
454M16—616no—
455C2×C8—34no—
456C2×C8—64no—
457C2×C8#23918yescomplemented
458C2×C8C23×D314yes—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4C23×D312yes—
468C4⋊C4C23×D312yes—
469C4⋊C4C23×D314yes—
470C22⋊C4—64no—
471C22⋊C4—68no—
472C22⋊C4—64no—
473C22⋊C4—68no—
474C22⋊C4—64no—
475C22⋊C4—64no—
476C22⋊C4—64no—
477C42—34no—
478C42—32no—
479C42—34no—
480C42—32no—
481C42—64no—
482C42D3×D412yes—
483C42C23×D312yes—
484C22×C3#53212yes—
485C22×C3C2×AGL(1,ℤ/8)12yes—
486C22×C3#57512yes—
487C22×C3C23×D411yescharacteristic
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—44no—
492C2×D3—44no—
493C2×D3—42no—
494C2×D3—44no—
495C2×D3—42no—
496C2×D3—44no—
497C2×D3—42no—
498C2×D3—88no—
499C2×D3—84no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—44no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4#53212yes—
506C3×C4C22×#4112yes—
507Dic3—22no—
508Dic3—22no—
509Dic3—42no—
510Dic3C2×AGL(1,ℤ/8)14yes—
511C23—32no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23—62no—
516C23—64no—
517C23—122no—
518C23—124no—
519C23—124no—
520C23—122no—
521C23—124no—
522C23—122no—
523C23C2×D3×D411yescharacteristic
524Q8—48no—
525Q8—64no—
526Q8—122no—
527D4—64no—
528D4—64no—
529D4—64no—
530D4—128no—
531D4—128no—
532D4—1216no—
533D4—128no—
534D4—122no—
535D4—124no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—32no—
542C2×C4—34no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—68no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—124no—
555C2×C4—1216no—
556C2×C4—128no—
557C2×C4—122no—
558C2×C4C2×D3×D412yes—
559C2×C4C2×D3×D412yes—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×#23914yes—
562C2×C4C24×D311yescharacteristic
563C8—64no—
564C8C2×#23918yescomplemented
565C2×C3#317114yescomplemented
566C2×C3C2×#53211yescharacteristic
567C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
568C2×C3C2×#57511yescharacteristic
569D3—42no—
570D3—84no—
571D3—82no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—124no—
576C22—124no—
577C22—122no—
578C22—124no—
579C22—122no—
580C22—122no—
581C22—124no—
582C22—248no—
583C22—244no—
584C22#572512yes—
585C22D3×AGL(1,ℤ/8)12yes—
586C22#577712yes—
587C22C22×D3×D411yescharacteristic
588C4—22no—
589C4—22no—
590C4—34no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—62no—
595C4—62no—
596C4—122no—
597C4#572512yes—
598C4C22×#23912yes—
599C3C2×#317111yescharacteristic, complemented, Sylow
600C2—122no—
601C2—244no—
602C2—242no—
603C2#8195814yesdirect product factor
604C2C2×#572511yescharacteristic
605C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
606C2C2×#577711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#501C2
socleC23×C3C22×D4
hypercenterC2×#501D3
hypocenterC3C2×#3171
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×#5725C2semidirect
2. C2×#5691C2semidirect
3. C2×C3×#501C2semidirect
4. C2×#5089C2semidirect
5. C2×#5067C2semidirect
6. #81958C2direct (N × H)
7. C2×#4886C2semidirect
8. C2×#4872C2semidirect
9. C2×#4865C2semidirect
10. C2×#4852C2semidirect
11. C2×#4712C2semidirect
12. C2×#4707C2semidirect
22. #5725C22semidirect
24. #5691C22semidirect
40. C3×#501C22semidirect
45. #5089C22semidirect
46. #5067C22semidirect
52. #4886C22semidirect
53. #4872C22semidirect
54. #4865C22semidirect
55. #4852C22semidirect
56. #4712C22semidirect
57. #4707C22semidirect
61. C2×#501D3semidirect
177. #501C2×D3semidirect
240. C2×C3×C8#41semidirect
405. C3×C8C2×#41semidirect
457. C2×C8#239semidirect
564. C8C2×#239semidirect
565. C2×C3#3171semidirect
599. C3C2×#3171semidirect
603. C2#81958direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57252
C2×#56912
C2×C3×#5012
C2×#50892
C2×#50672
#819582
C2×#48862
C2×#48722
C2×#48652
C2×#48522
C2×#47122
C2×#47072
C2×#31713