Finite Group Databasebeta

← C2×#78532 #11664952

Subgroups — C2×#78532 #11664952

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×#5477C211yesmaximal, characteristic, complemented
2C2×#5479C211yesmaximal, characteristic, complemented
3C2×C3×C4×M16C211yesmaximal, characteristic, complemented
4C2×#5135C212yesmaximal, complemented
5C2×#5134C212yesmaximal, complemented
6#78532C2116yesmaximal, direct product factor
7C2×#4712C211yesmaximal, characteristic, complemented
8C2×#4713C211yesmaximal, characteristic, complemented
9C2×#4714C211yesmaximal, characteristic, complemented
10C2×#4702C211yesmaximal, characteristic, complemented
11C2×C4×#204C212yesmaximal, complemented
12C2×C4×D24C212yesmaximal, complemented
13C2×#3171—31nomaximal, Sylow
14C22×#235—21no—
15C22×C3×M16C2212yes—
16C22×C3×C42C2211yescharacteristic
17C2×#930—21no—
18C2×#930C2212yes—
19C2×#930C2211yescharacteristic
20C2×#929—21no—
21C2×C4×#67—21no—
22C2×#924C2211yescharacteristic
23C22×D24C2212yes—
24C22×#204C2212yes—
25#5479—24no—
26#5479C2218yescomplemented
27C2×#876—21no—
28#5477—24no—
29#5477C2218yescomplemented
30C2×#875—21no—
31C2×#875C2211yescharacteristic
32C2×#874C2211yescharacteristic
33C2×C4×D12—21no—
34C2×C4×D12C2211yescharacteristic
35C2×C4×D12C2212yes—
36C2×#872—21no—
37C2×#870C2211yescharacteristic
38C2×#869C2211yescharacteristic
39C2×C4×Dic6C2211yescharacteristic
40C3×C4×M16C2218yescomplemented
41C2×C3×#97C2211yescharacteristic
42C2×C3×#97C2211yescharacteristic
43C2×C3×C4×C8C2212yes—
44#5135C22116yescomplemented
45#5134C22116yescomplemented
46C2×#821C2212yes—
47C2×#821C2212yes—
48C2×#821C2212yes—
49C2×#818C2212yes—
50C2×#817C2212yes—
51C2×#816C2212yes—
52#4714C2218yescomplemented
53#4713C2218yescomplemented
54#4712C2218yescomplemented
55#4702C2218yescomplemented
56C4×D24C22116yescomplemented
57C4×#204C22116yescomplemented
58C2×#532—31no—
59C2×#529—31no—
60#3171—316no—
61C2×#501—31no—
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×M16D311yescharacteristic, complemented
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D411yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×#67—22no—
74C22×#67—21no—
75C2×#235—28no—
76C2×#235—48no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2312yes—
80C22×D12C2311yescharacteristic
81C22×C4×D3—22no—
82C22×C4×D3—21no—
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C2×C3×M16—28no—
86C2×C3×M16C2318yes—
87C22×C3×C8C2312yes—
88C22×C3×C8C2312yes—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D414yes—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2314yes—
95C2×#217—21no—
96#930—24no—
97#930—28no—
98#930—22no—
99#930C2318yes—
100#930C2314yes—
101#929—28no—
102C4×#67—28no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—22no—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2311yescharacteristic
113#924—22no—
114#924C2314yes—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×D24—28no—
121C2×D24C2318yes—
122C2×#204—28no—
123C2×#204C2318yes—
124#876—28no—
125#875—28no—
126#875—22no—
127#875C2314yes—
128#874—22no—
129#874C2314yes—
130C4×D12—28no—
131C4×D12—24no—
132C4×D12—22no—
133C4×D12C2318yes—
134C4×D12C2314yes—
135#872—28no—
136#870—22no—
137#870C2314yes—
138#869—22no—
139#869C2314yes—
140C4×Dic6—22no—
141C4×Dic6C2314yes—
142C3×#97C2314yes—
143C3×#97C2314yes—
144C3×C4×C8C2318yes—
145#821C2318yes—
146#821C2318yes—
147#821C2318yes—
148#818C2318yes—
149#817C2318yes—
150#816C2318yes—
151C22×#41—61no—
152C22×SD16—32no—
153C22×D8—32no—
154C22×M16C2×D312yes—
155#532—38no—
156#532—64no—
157C2×C4⋊Q8—31no—
158C2×#127—31no—
159#529—38no—
160#529—64no—
161C2×#126—61no—
162C2×#125—31no—
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×C42C2×D311yescharacteristic
177#501—38no—
178#496—38no—
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×M16C2×D318yescomplemented
192C2×#97C2×D311yescharacteristic
193C2×#97C2×D311yescharacteristic
194C2×C4×C8C2×D312yes—
195C24×C3C2×D411yescharacteristic
196C23×D3—21no—
197C23×D3—22no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—42no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D414yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2412yes—
210C22×C3×C4C2411yescharacteristic
211C2×#67—24no—
212C2×#67—28no—
213C2×#67—48no—
214C2×#67—44no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—22no—
219#235—432no—
220C2×D12—24no—
221C2×D12—24no—
222C2×D12—24no—
223C2×D12—22no—
224C2×D12—24no—
225C2×D12—21no—
226C2×D12—24no—
227C2×D12—44no—
228C2×D12—44no—
229C2×D12C2414yes—
230C2×D12C2412yes—
231C2×C4×D3—24no—
232C2×C4×D3—28no—
233C2×C4×D3—44no—
234C2×C4×D3—48no—
235C2×Dic6—24no—
236C2×Dic6—21no—
237C2×Dic6—24no—
238C2×Dic6—44no—
239C2×Dic6C2412yes—
240C3×M16—216no—
241C2×C3×C8—24no—
242C2×C3×C8#4118yescomplemented
243C2×C3×C8C2414yes—
244C2×C3×C8C2414yes—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247C3×C42C2×D414yes—
248C3×C42C2×D414yes—
249C3×C42C2412yes—
250C3×C42C2412yes—
251#217—24no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—28no—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—22no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3C2414yes—
263C4⋊Dic3C2412yes—
264C4⋊Dic3C2412yes—
265Dic3⋊C4—24no—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—24no—
269C4×Dic3—24no—
270D24—216no—
271#204—216no—
272C2×#41—68no—
273C2×#41—128no—
274C22×Q8—31no—
275C22×Q8—61no—
276C22×D4—31no—
277C22×D4—32no—
278C22×D4—61no—
279C22×D4—62no—
280C22×D4—61no—
281C22×D4—61no—
282C23×C4—61no—
283C23×C4—62no—
284C23×C4D1211yescharacteristic
285C23×C4D1211yescharacteristic
286C23×C4C22×D311yescharacteristic
287C2×SD16—38no—
288C2×SD16—68no—
289C2×D8—38no—
290C2×D8—68no—
291C2×M16—28no—
292C2×M16C22×D318yes—
293C22×C8C22×D312yes—
294C22×C8C22×D312yes—
295C4⋊Q8—34no—
296C4⋊Q8—62no—
297#127—34no—
298#127—62no—
299#126—68no—
300#125—34no—
301#125—62no—
302#124—34no—
303#124—62no—
304#124—68no—
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—68no—
318C4×D4—62no—
319C4×D4—64no—
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—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—62no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—62no—
336C2×C22⋊C4—61no—
337C2×C42—61no—
338C2×C42D1211yescharacteristic
339C2×C42D1214yes—
340C2×C42D1211yescharacteristic
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D314yes—
343C2×C42C22×D311yescharacteristic
344#107—38no—
345#106—38no—
346#103—38no—
347#102—38no—
348#102—38no—
349#102—38no—
350#97C22×D314yes—
351#97C22×D314yes—
352C4×C8C22×D318yes—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3C22×D412yes—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—24no—
360C22×D3—44no—
361C22×D3—42no—
362C22×D3—44no—
363C22×D3—42no—
364C22×D3—44no—
365C22×D3—42no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—28no—
372C2×C3×C4—24no—
373C2×C3×C4—44no—
374C2×C3×C4AGL(1,ℤ/8)14yes—
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D412yes—
381C2×C3×C4C2×#4114yes—
382C2×C3×C4C2511yescharacteristic
383#67—416no—
384#67—48no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—24no—
389C2×Dic3—42no—
390C2×Dic3—44no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393D12—24no—
394D12—24no—
395D12—24no—
396D12—48no—
397D12—48no—
398D12—44no—
399D12—42no—
400C4×D3—48no—
401C4×D3—416no—
402Dic6—24no—
403Dic6—48no—
404Dic6—42no—
405C3×C8—24no—
406C3×C8C2×#4118yescomplemented
407C24—61no—
408C24—62no—
409C24C2×D1211yescharacteristic
410#41—1232no—
411C2×Q8—32no—
412C2×Q8—61no—
413C2×Q8—64no—
414C2×Q8—64no—
415C2×Q8—124no—
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—24no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—62no—
438C22×C4—64no—
439C22×C4—68no—
440C22×C4—128no—
441C22×C4—124no—
442C22×C4C2×D1211yescharacteristic
443C22×C4C2×D1212yes—
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1214yes—
448C22×C4C2×D1214yes—
449C22×C4C2×D1211yescharacteristic
450C22×C4C23×D312yes—
451C22×C4C23×D311yescharacteristic
452SD16—616no—
453D8—616no—
454M16—216no—
455C2×C8—24no—
456C2×C8#23518yescomplemented
457C2×C8C23×D314yes—
458C2×C8C23×D314yes—
459C4⋊C4—32no—
460C4⋊C4—32no—
461C4⋊C4—34no—
462C4⋊C4—62no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C22⋊C4—64no—
471C22⋊C4—64no—
472C22⋊C4—64no—
473C22⋊C4—68no—
474C22⋊C4—64no—
475C22⋊C4—68no—
476C22⋊C4—64no—
477C42—64no—
478C42C2×D1214yes—
479C42C2×D1212yes—
480C42C2×D1214yes—
481C42C2×D1212yes—
482C42C23×D312yes—
483C42C23×D312yes—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—44no—
488C22×C3#53212yes—
489C22×C3C2×AGL(1,ℤ/8)12yes—
490C22×C3#57512yes—
491C22×C3C23×D411yescharacteristic
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—22no—
503C3×C4—22no—
504C3×C4#53212yes—
505C3×C4C2×AGL(1,ℤ/8)14yes—
506C3×C4C22×#4112yes—
507Dic3—42no—
508Dic3—44no—
509Dic3—42no—
510Dic3—42no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—62no—
515C23—64no—
516C23—122no—
517C23—124no—
518C23—124no—
519C23—122no—
520C23—124no—
521C23—122no—
522C23C22×D1212yes—
523C23C22×D1211yescharacteristic
524Q8—64no—
525Q8—128no—
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—24no—
542C2×C4—28no—
543C2×C4—44no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—122no—
549C2×C4—124no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—1216no—
553C2×C4—128no—
554C2×C4#90814yes—
555C2×C4C22×D1212yes—
556C2×C4C22×D1212yes—
557C2×C4C22×D1212yes—
558C2×C4C22×D1212yes—
559C2×C4C22×D1211yescharacteristic
560C2×C4C22×D1212yes—
561C2×C4C2×#23514yes—
562C2×C4C24×D311yescharacteristic
563C8—24no—
564C8C2×#23518yescomplemented
565C2×C3—42no—
566C2×C3#317114yescomplemented
567C2×C3C2×#53211yescharacteristic
568C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
569C2×C3C2×#57511yescharacteristic
570D3—84no—
571D3—82no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—44no—
576C22—124no—
577C22—122no—
578C22—124no—
579C22—122no—
580C22—122no—
581C22—124no—
582C22—248no—
583C22—244no—
584C22#547712yes—
585C22C2×#90812yes—
586C22#574812yes—
587C22C23×D1211yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—124no—
593C4—122no—
594C4—122no—
595C4—122no—
596C4#547712yes—
597C4C2×#90814yes—
598C4C22×#23512yes—
599C3C2×#317111yescharacteristic, complemented, Sylow
600C2—42no—
601C2—244no—
602C2—242no—
603C2#7853214yesdirect product factor
604C2C2×#547711yescharacteristic
605C2C22×#90811yescharacteristic
606C2C2×#574811yescharacteristic
subgroupG/H
centerC22×C4C2×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×M16C2
socleC23×C3C22×D4
hypercenterC2×C4×M16D3
hypocenterC3C2×#3171
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C2×#5479C2semidirect
3. C2×C3×C4×M16C2semidirect
4. C2×#5135C2semidirect
5. C2×#5134C2semidirect
6. #78532C2direct (N × H)
7. C2×#4712C2semidirect
8. C2×#4713C2semidirect
9. C2×#4714C2semidirect
10. C2×#4702C2semidirect
11. C2×C4×#204C2semidirect
12. C2×C4×D24C2semidirect
26. #5479C22semidirect
29. #5477C22semidirect
40. C3×C4×M16C22semidirect
44. #5135C22semidirect
45. #5134C22semidirect
52. #4714C22semidirect
53. #4713C22semidirect
54. #4712C22semidirect
55. #4702C22semidirect
56. C4×D24C22semidirect
57. C4×#204C22semidirect
69. C2×C4×M16D3semidirect
191. C4×M16C2×D3semidirect
242. C2×C3×C8#41semidirect
406. C3×C8C2×#41semidirect
456. C2×C8#235semidirect
564. C8C2×#235semidirect
566. C2×C3#3171semidirect
599. C3C2×#3171semidirect
603. C2#78532direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54772
C2×#54792
C2×C3×C4×M162
C2×#51352
C2×#51342
#785322
C2×#47122
C2×#47132
C2×#47142
C2×#47022
C2×C4×#2042
C2×C4×D242
C2×#31713