Finite Group Databasebeta

← C2×#81836 #11664949

Subgroups — C2×#81836 #11664949

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×#5716C211yesmaximal, characteristic, complemented
2C2×#5680C211yesmaximal, characteristic, complemented
3C2×C3×#496C211yesmaximal, characteristic, complemented
4C2×#5174C212yesmaximal, complemented
5C2×Dic3×SD16C212yesmaximal, complemented
6C2×#5159C212yesmaximal, complemented
7C2×Dic3×D8C212yesmaximal, complemented
8C2×#5071C211yesmaximal, characteristic, complemented
9C2×#5055C211yesmaximal, characteristic, complemented
10#81836C2116yesmaximal, direct product factor
11C2×#4707C211yesmaximal, characteristic, complemented
12C2×#4702C211yesmaximal, characteristic, complemented
13C2×#3171—31nomaximal, Sylow
14C22×#237—21no—
15C22×C3×SD16C2212yes—
16C22×C3×D8C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#938—21no—
22C2×#938C2211yescharacteristic
23C2×#938C2212yes—
24C2×#936—21no—
25C2×D4×Dic3—21no—
26C2×D4×Dic3C2211yescharacteristic
27C2×D4×Dic3C2212yes—
28C22×ΓL(1,𝔽25)C2212yes—
29#5716—24no—
30#5716C2218yescomplemented
31#5680—24no—
32#5680C2218yescomplemented
33C2×#883—21no—
34C2×#67⋊C4—21no—
35C2×#879—21no—
36C2×#877—21no—
37C2×C42×D3C2211yescharacteristic
38C2×#870C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C3×#496C2218yescomplemented
41C2×C3×#97C2211yescharacteristic
42C2×#835C2212yes—
43C2×#832C2212yes—
44C2×#832C2212yes—
45C2×#832C2212yes—
46#5174C22116yescomplemented
47Dic3×SD16C22116yescomplemented
48#5159C22116yescomplemented
49Dic3×D8C22116yescomplemented
50C2×#818C2212yes—
51C2×#817C2212yes—
52C2×C8×Dic3C2212yes—
53#5071C2218yescomplemented
54#5055C2218yescomplemented
55C2×#803C2211yescharacteristic
56#4707C2218yescomplemented
57#4702C2218yescomplemented
58C2×#532—31no—
59C2×#529—31no—
60#3171—316no—
61C2×#501—31no—
62C2×#496D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4—21no—
73C22×C3×D4C2311yescharacteristic
74C22×C3×D4C2312yes—
75C22×#67—22no—
76C22×#67—21no—
77C23×Dic3—22no—
78C23×Dic3—21no—
79C2×#237—28no—
80C2×#237—48no—
81C22×C4×D3D411yescharacteristic
82C22×C4×D3D411yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C2×C3×SD16—28no—
86C2×C3×SD16C2318yes—
87C2×C3×D8—28no—
88C2×C3×D8C2318yes—
89C22×C3×C8C2312yes—
90C3×#127—22no—
91C3×#127C2314yes—
92C3×#124—22no—
93C3×#124C2314yes—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—22no—
100C2×#217—22no—
101#946—22no—
102#946C2314yes—
103Q8×Dic3—22no—
104Q8×Dic3C2314yes—
105#938—22no—
106#938—24no—
107#938—28no—
108#938C2314yes—
109#938C2318yes—
110#936—28no—
111D4×Dic3—22no—
112D4×Dic3—24no—
113D4×Dic3—28no—
114D4×Dic3C2314yes—
115D4×Dic3C2318yes—
116C2×(C2×D3)⋊C4—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3D411yescharacteristic
127C2×C4×Dic3D411yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2312yes—
130C2×ΓL(1,𝔽25)—28no—
131C2×ΓL(1,𝔽25)C2318yes—
132#883—28no—
133#67⋊C4—28no—
134#879—28no—
135#877—28no—
136C42×D3D414yes—
137C42×D3C2314yes—
138#870—22no—
139#870C2314yes—
140#869—22no—
141#869C2314yes—
142C3×#97C2314yes—
143#835C2318yes—
144#832C2318yes—
145#832C2318yes—
146#832C2318yes—
147#818C2318yes—
148#817C2318yes—
149C8×Dic3C2318yes—
150#803C2314yes—
151C22×#41—61no—
152C22×SD16C2×D312yes—
153C22×D8C2×D312yes—
154C22×M16—32no—
155#532—38no—
156#532—64no—
157C2×C4⋊Q8—31no—
158C2×#127C2×D311yescharacteristic
159#529—38no—
160#529—64no—
161C2×#126—61no—
162C2×#125—31no—
163C2×#124—61no—
164C2×#124C2×D311yescharacteristic
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#501—38no—
178#496C2×D318yescomplemented
179#490—38no—
180#489—38no—
181#469—316no—
182#468—316no—
183C4×SD16—316no—
184C4×D8—316no—
185C2×#107—32no—
186C2×#106—32no—
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—
195C24×C3—21no—
196C24×C3—22no—
197C23×D3C2×D411yescharacteristic
198C2×C3×Q8—21no—
199C2×C3×Q8—24no—
200C2×C3×Q8C2412yes—
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—24no—
204C2×C3×D4—24no—
205C2×C3×D4—24no—
206C2×C3×D4—24no—
207C2×C3×D4—24no—
208C2×C3×D4—44no—
209C2×C3×D4—44no—
210C2×C3×D4C2412yes—
211C2×C3×D4C2414yes—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—24no—
217C2×#67—28no—
218C2×#67—48no—
219C2×#67—44no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—22no—
223C22×Dic3—24no—
224C22×Dic3—28no—
225C22×Dic3—48no—
226C22×Dic3—44no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230#237—432no—
231C2×C4×D3—24no—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3C2×D412yes—
235C2×C4×D3C2×D414yes—
236C2×C4×D3C2×D414yes—
237C2×C4×D3C2412yes—
238C2×Dic6—24no—
239C2×Dic6—44no—
240C3×SD16—216no—
241C3×D8—216no—
242C2×C3×C8#4118yescomplemented
243C2×C3×C8C2414yes—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—24no—
246C3×C42C2×D412yes—
247C3×C42C2412yes—
248#217—24no—
249#217—24no—
250#217—28no—
251#217—28no—
252(C2×D3)⋊C4—24no—
253C4⋊Dic3—22no—
254C4⋊Dic3—24no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—24no—
258C4⋊Dic3C2412yes—
259C4⋊Dic3C2414yes—
260C4⋊Dic3C2412yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264C4×Dic3—24no—
265C4×Dic3C2×D412yes—
266C4×Dic3C2×D414yes—
267C4×Dic3C2×D414yes—
268C4×Dic3C2412yes—
269C2×C3⋊C8—24no—
270C2×C3⋊C8C2414yes—
271ΓL(1,𝔽25)—216no—
272C2×#41—68no—
273C2×#41—128no—
274C22×Q8—61no—
275C22×Q8C22×D311yescharacteristic
276C22×D4—21no—
277C22×D4—21no—
278C22×D4—61no—
279C22×D4—62no—
280C22×D4C22×D311yescharacteristic
281C22×D4C22×D312yes—
282C23×C4—31no—
283C23×C4—31no—
284C23×C4—31no—
285C23×C4—61no—
286C23×C4—62no—
287C2×SD16—28no—
288C2×SD16C22×D318yes—
289C2×D8—28no—
290C2×D8C22×D318yes—
291C2×M16—38no—
292C2×M16—68no—
293C22×C8—32no—
294C22×C8C22×D312yes—
295C4⋊Q8—34no—
296C4⋊Q8—62no—
297#127—22no—
298#127C22×D314yes—
299#126—68no—
300#125—34no—
301#125—62no—
302#124—22no—
303#124—68no—
304#124C22×D314yes—
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—32no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—62no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—62no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—34no—
341C2×C42—34no—
342C2×C42—61no—
343C2×C42C22×D311yescharacteristic
344#107—38no—
345#106—38no—
346#103—38no—
347#102—38no—
348#102—38no—
349#102—38no—
350#97—34no—
351#97C22×D314yes—
352C4×C8—38no—
353C23×C3—22no—
354C23×C3—24no—
355C23×C3—42no—
356C23×C3—44no—
357C23×C3—44no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3—44no—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—22no—
364C22×D3—22no—
365C22×D3C22×D412yes—
366C3×Q8—24no—
367C3×Q8—42no—
368C3×D4—24no—
369C3×D4—24no—
370C3×D4—24no—
371C3×D4—48no—
372C3×D4—48no—
373C3×D4—42no—
374C3×D4—44no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—42no—
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C2×#4114yes—
383C2×C3×C4C2511yescharacteristic
384#67—416no—
385#67—48no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—24no—
391C2×Dic3—44no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—416no—
395C2×Dic3—48no—
396C2×Dic3AGL(1,ℤ/8)14yes—
397C2×Dic3C22×D412yes—
398C2×Dic3C22×D412yes—
399C2×Dic3C22×D412yes—
400C4×D3—24no—
401C4×D3—24no—
402C4×D3—28no—
403C4×D3—44no—
404Dic6—48no—
405C3×C8C2×#4118yescomplemented
406C3⋊C8—24no—
407C24—21no—
408C24—22no—
409C24—31no—
410#41—1232no—
411C2×Q8—21no—
412C2×Q8—24no—
413C2×Q8—64no—
414C2×Q8—124no—
415C2×Q8C23×D312yes—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—64no—
426C2×D4—68no—
427C2×D4—128no—
428C2×D4—124no—
429C2×D4C23×D312yes—
430C2×D4C23×D314yes—
431C22×C4—21no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—34no—
436C22×C4—32no—
437C22×C4—32no—
438C22×C4—34no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—68no—
442C22×C4—64no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—62no—
446C22×C4—122no—
447C22×C4—128no—
448C22×C4—124no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452SD16—216no—
453D8—216no—
454M16—616no—
455C2×C8—34no—
456C2×C8—64no—
457C2×C8#23718yescomplemented
458C2×C8C23×D314yes—
459C4⋊C4—32no—
460C4⋊C4—32no—
461C4⋊C4—34no—
462C4⋊C4—64no—
463C4⋊C4—64no—
464C4⋊C4—62no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—64no—
473C22⋊C4—64no—
474C22⋊C4—68no—
475C22⋊C4—64no—
476C22⋊C4—68no—
477C42—34no—
478C42—32no—
479C42—34no—
480C42—32no—
481C42—64no—
482C42D3×D412yes—
483C42C23×D312yes—
484C22×C3—42no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—44no—
490C22×C3—88no—
491C22×C3—84no—
492C22×C3#53212yes—
493C22×C3C2×AGL(1,ℤ/8)12yes—
494C22×C3#57512yes—
495C22×C3C23×D411yescharacteristic
496C2×D3—22no—
497C2×D3—22no—
498C2×D3—22no—
499C2×D3—44no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—42no—
503C3×C4#53212yes—
504C3×C4C22×#4112yes—
505Dic3—22no—
506Dic3—22no—
507Dic3—44no—
508Dic3—42no—
509Dic3—42no—
510Dic3C2×AGL(1,ℤ/8)14yes—
511C23—22no—
512C23—24no—
513C23—32no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23C2×D3×D411yescharacteristic
524Q8—24no—
525Q8—42no—
526Q8—128no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4—48no—
531D4—48no—
532D4—42no—
533D4—44no—
534D4—1216no—
535D4—128no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—32no—
540C2×C4—34no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—42no—
544C2×C4—68no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—64no—
552C2×C4—124no—
553C2×C4—1216no—
554C2×C4—128no—
555C2×C4—124no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4C2×D3×D412yes—
559C2×C4C2×D3×D412yes—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×#23714yes—
562C2×C4C24×D311yescharacteristic
563C8—64no—
564C8C2×#23718yescomplemented
565C2×C3—84no—
566C2×C3—82no—
567C2×C3#317114yescomplemented
568C2×C3C2×#53211yescharacteristic
569C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
570C2×C3C2×#57511yescharacteristic
571D3—42no—
572C22—44no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—62no—
579C22—62no—
580C22—62no—
581C22—88no—
582C22—84no—
583C22—124no—
584C22#571612yes—
585C22D3×AGL(1,ℤ/8)12yes—
586C22#577312yes—
587C22C22×D3×D411yescharacteristic
588C4—22no—
589C4—22no—
590C4—34no—
591C4—42no—
592C4—62no—
593C4—62no—
594C4—124no—
595C4—122no—
596C4—122no—
597C4#571612yes—
598C4C22×#23712yes—
599C3C2×#317111yescharacteristic, complemented, Sylow
600C2—84no—
601C2—82no—
602C2—122no—
603C2#8183614yesdirect product factor
604C2C2×#571611yescharacteristic
605C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
606C2C2×#577311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#496C2
socleC23×C3C22×D4
hypercenterC2×#496D3
hypocenterC3C2×#3171
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
1. C2×#5716C2semidirect
2. C2×#5680C2semidirect
3. C2×C3×#496C2semidirect
4. C2×#5174C2semidirect
5. C2×Dic3×SD16C2semidirect
6. C2×#5159C2semidirect
7. C2×Dic3×D8C2semidirect
8. C2×#5071C2semidirect
9. C2×#5055C2semidirect
10. #81836C2direct (N × H)
11. C2×#4707C2semidirect
12. C2×#4702C2semidirect
30. #5716C22semidirect
32. #5680C22semidirect
40. C3×#496C22semidirect
46. #5174C22semidirect
47. Dic3×SD16C22semidirect
48. #5159C22semidirect
49. Dic3×D8C22semidirect
53. #5071C22semidirect
54. #5055C22semidirect
56. #4707C22semidirect
57. #4702C22semidirect
62. C2×#496D3semidirect
178. #496C2×D3semidirect
242. C2×C3×C8#41semidirect
405. C3×C8C2×#41semidirect
457. C2×C8#237semidirect
564. C8C2×#237semidirect
567. C2×C3#3171semidirect
599. C3C2×#3171semidirect
603. C2#81836direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57162
C2×#56802
C2×C3×#4962
C2×#51742
C2×Dic3×SD162
C2×#51592
C2×Dic3×D82
C2×#50712
C2×#50552
#818362
C2×#47072
C2×#47022
C2×#31713