Finite Group Databasebeta

← C2×#82736 #11665708

Subgroups — C2×#82736 #11665708

Order 768.

2698 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#930C211yesmaximal, characteristic
3C2×#5227C212yesmaximal, complemented
4C22×#830C211yesmaximal, characteristic, complemented
5#82736C2116yesmaximal, direct product factor
6C2×#5046C212yesmaximal, complemented
7C2×#5048C214yesmaximal, complemented
8C2×#5043C212yesmaximal, complemented
9C2×#4968C212yesmaximal, complemented
10C2×#3187—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic, complemented
12C24×C3×C4C2211yescharacteristic
13C23×#67—21no—
14C23×D12C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8—24no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C22⋊Q8C2212yescomplemented
20C2×C3×C22⋊Q8C2218yescomplemented
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4C2211yescharacteristic, complemented
23C22×C3×C22⋊C4—21no—
24C22×#215C2214yescomplemented
25C2×#930—21no—
26C2×#930—21no—
27C2×#930—24no—
28C2×#930C2212yes—
29C2×#930C2218yes—
30C2×#930C2212yes—
31C22×(C2×D3)⋊C4—21no—
32C22×C4⋊Dic3C2211yescharacteristic
33C23×C3⋊C8C2212yescomplemented
34#5227C22116yescomplemented
35C2×#835C2214yescomplemented
36C2×#830C2212yes—
37C2×#830C2218yescomplemented
38C2×#830C2212yescomplemented
39#5048C22132yescomplemented
40#5046C22116yescomplemented
41#5043C22116yescomplemented
42#4968C22116yescomplemented
43C2×#809C2214yescomplemented
44C2×#808C2214yescomplemented
45C22×C22⋊Q8D311yescharacteristic, complemented
46C22×#121—31no—
47#3187—316no—
48C2×#481—32no—
49C2×#483—32no—
50C2×#465—34no—
51C2×#448—32no—
52C2×#450—32no—
53C22×C22⋊C8—31no—
54C25×C3D411yescharacteristic
55C24×D3—21no—
56C22×C3×Q8—21no—
57C22×C3×Q8—22no—
58C22×C3×Q8—44no—
59C22×C3×Q8C2312yes—
60C22×C3×Q8C2314yescomplemented
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C23×C3×C4—44no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D414yes—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2314yes—
72C22×#67—22no—
73C22×#67—22no—
74C22×#67—28no—
75C22×#67—48no—
76C23×Dic3—21no—
77C22×D12—21no—
78C22×D12—22no—
79C22×D12—44no—
80C22×D12C2312yes—
81C22×D12C2314yes—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—28no—
84C3×C22⋊Q8—28no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8C2318yes—
87C3×C22⋊Q8C2318yescomplemented
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—28no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2314yescomplemented
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—28no—
97C2×C3×C22⋊C4—22no—
98C2×#215—48no—
99C2×#215C23116yescomplemented
100#930—24no—
101#930—24no—
102#930—28no—
103#930—28no—
104#930C2318yes—
105#930C2318yes—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—22no—
108C2×(C2×D3)⋊C4—28no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—22no—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2314yes—
113C22×C3⋊C8—44no—
114C22×C3⋊C8C2314yes—
115C22×C3⋊C8C2318yescomplemented
116#835C23116yescomplemented
117#830C2318yes—
118#830C2318yescomplemented
119#809C23116yescomplemented
120#808C23116yescomplemented
121C23×Q8C2×D311yescharacteristic, complemented
122C23×D4—31no—
123C23×D4—61no—
124C24×C4C2×D311yescharacteristic
125C22×SD16—34no—
126C23×C8—32no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—24no—
130C2×C22⋊Q8C2×D312yes—
131C2×C22⋊Q8C2×D312yescomplemented
132C2×C22⋊Q8C2×D318yescomplemented
133C2×#121—32no—
134C2×#121—38no—
135C2×#121—32no—
136C2×#121—61no—
137C2×#121—64no—
138C2×#121—61no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—31no—
141C22×C4⋊C4C2×D311yescharacteristic, complemented
142C22×C22⋊C4—21no—
143C22×C22⋊C4—61no—
144#483—316no—
145#481—316no—
146#465—332no—
147#450—316no—
148#448—316no—
149C2×#106—34no—
150C2×#103—34no—
151C2×#102—34no—
152C2×C22⋊C8—32no—
153C2×C22⋊C8—32no—
154C2×C22⋊C8—38no—
155C24×C3—42no—
156C24×C3—42no—
157C24×C3SD1618yescomplemented
158C24×C3C2×D411yescharacteristic
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C24×C3C2×D414yes—
162C23×D3—24no—
163C23×D3—22no—
164C23×D3—44no—
165C23×D3—44no—
166C23×D3—44no—
167C2×C3×Q8—22no—
168C2×C3×Q8—24no—
169C2×C3×Q8—28no—
170C2×C3×Q8—48no—
171C2×C3×Q8—44no—
172C2×C3×Q8—48no—
173C2×C3×Q8C2414yes—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—24no—
181C22×C3×C4—48no—
182C22×C3×C4—48no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4—42no—
186C22×C3×C4—44no—
187C22×C3×C4—44no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D414yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2412yes—
200C22×C3×C4C2411yescharacteristic
201C2×#67—28no—
202C2×#67—28no—
203C2×#67—48no—
204C2×#67—48no—
205C2×#67—432no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—44no—
209C2×D12—28no—
210C2×D12—24no—
211C2×D12—22no—
212C2×D12—48no—
213C2×D12—44no—
214C2×D12—48no—
215C2×D12C2414yes—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4C2414yes—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—28no—
223#215—416no—
224(C2×D3)⋊C4—28no—
225(C2×D3)⋊C4—28no—
226C4⋊Dic3—24no—
227C4⋊Dic3—22no—
228C4⋊Dic3C2414yes—
229C2×C3⋊C8—28no—
230C2×C3⋊C8—48no—
231C2×C3⋊C8C2418yes—
232C25—61no—
233C25#6711yescharacteristic
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—44no—
237C22×Q8C22×D312yes—
238C22×Q8C22×D314yescomplemented
239C22×D4—32no—
240C22×D4—34no—
241C22×D4—68no—
242C22×D4—62no—
243C22×D4—62no—
244C22×D4—62no—
245C22×D4—61no—
246C22×D4—124no—
247C22×D4—128no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—44no—
251C23×C4—61no—
252C23×C4#6711yescharacteristic
253C23×C4#6711yescharacteristic
254C23×C4#6714yes—
255C23×C4#6711yescharacteristic
256C23×C4C22×D311yescharacteristic
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D314yes—
260C2×SD16—316no—
261C2×SD16—128no—
262C22×C8—34no—
263C22×C8—38no—
264C22×C8—124no—
265C22⋊Q8—24no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8C22×D318yes—
270C22⋊Q8C22×D318yescomplemented
271#121—38no—
272#121—38no—
273#121—64no—
274#121—64no—
275#121—68no—
276#121—68no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—34no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D312yes—
287C2×C4⋊C4C22×D314yescomplemented
288C2×C22⋊C4—22no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—68no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—62no—
294#106—316no—
295#103—316no—
296#102—316no—
297C22⋊C8—38no—
298C22⋊C8—38no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—44no—
306C23×C3—416no—
307C23×C3—44no—
308C23×C3C2×SD1614yes—
309C23×C3C2×SD1614yes—
310C23×C3C2×SD1614yes—
311C23×C3C2×SD16116yescomplemented
312C23×C3C22×D412yes—
313C23×C3C22×D412yes—
314C23×C3C22×D412yes—
315C23×C3C22×D411yescharacteristic
316C22×D3—24no—
317C22×D3—48no—
318C22×D3—48no—
319C22×D3—44no—
320C22×D3—48no—
321C22×D3—44no—
322C22×D3—44no—
323C22×D3—816no—
324C3×Q8—28no—
325C3×Q8—48no—
326C3×Q8—44no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—44no—
341C2×C3×C4—48no—
342C2×C3×C4—48no—
343C2×C3×C4—44no—
344C2×C3×C4—44no—
345C2×C3×C4C22×D412yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C22×D412yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4H(5,𝔽2)14yes—
350C2×C3×C4C2511yescharacteristic
351#67—416no—
352#67—416no—
353C2×Dic3—24no—
354C2×Dic3—48no—
355C2×Dic3—44no—
356D12—28no—
357D12—48no—
358D12—44no—
359C3⋊C8—28no—
360C24—42no—
361C24—42no—
362C24—64no—
363C24—62no—
364C24—124no—
365C24—124no—
366C24—124no—
367C24#21518yescomplemented
368C24C2×#6711yescharacteristic
369C24C2×#6711yescharacteristic
370C24C2×#6711yescharacteristic
371C24C2×#6714yes—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—28no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—48no—
378C2×Q8C23×D314yes—
379C2×D4—34no—
380C2×D4—68no—
381C2×D4—68no—
382C2×D4—68no—
383C2×D4—64no—
384C2×D4—62no—
385C2×D4—128no—
386C2×D4—124no—
387C2×D4—128no—
388C2×D4—128no—
389C2×D4—128no—
390C2×D4—1232no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—64no—
406C22×C4—62no—
407C22×C4—124no—
408C22×C4C2×#6712yes—
409C22×C4C2×#6712yes—
410C22×C4C2×#6712yes—
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6714yes—
415C22×C4C2×#6714yes—
416C22×C4C2×#6714yes—
417C22×C4C23×D312yes—
418C22×C4C23×D312yes—
419C22×C4C23×D312yes—
420C22×C4C23×D311yescharacteristic
421SD16—1216no—
422C2×C8—38no—
423C2×C8—68no—
424C2×C8—128no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—34no—
430C4⋊C4—64no—
431C4⋊C4—62no—
432C4⋊C4C23×D314yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—68no—
436C22⋊C4—68no—
437C22×C3—28no—
438C22×C3—28no—
439C22×C3—28no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—416no—
447C22×C3#53412yes—
448C22×C3C22×SD1614yes—
449C22×C3C22×SD1614yes—
450C22×C3C22×SD1614yes—
451C22×C3C22×SD1612yes—
452C22×C3#57512yes—
453C22×C3C23×D411yescharacteristic
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—44no—
457C2×D3—816no—
458C2×D3—88no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—44no—
463C3×C4—44no—
464C3×C4#53412yes—
465C3×C4C2×H(5,𝔽2)12yes—
466Dic3—44no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—44no—
474C23—416no—
475C23—44no—
476C23—64no—
477C23—128no—
478C23—128no—
479C23—124no—
480C23—128no—
481C23—124no—
482C23—124no—
483C23—2416no—
484C23C2×#21514yes—
485C23C2×#21514yes—
486C23C2×#21514yes—
487C23C2×#215116yescomplemented
488C23C22×#6712yes—
489C23C22×#6712yes—
490C23C22×#6712yes—
491C23C22×#6711yescharacteristic
492Q8—28no—
493Q8—48no—
494Q8—44no—
495D4—68no—
496D4—128no—
497D4—1216no—
498D4—1216no—
499D4—124no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—64no—
519C2×C4—128no—
520C2×C4—124no—
521C2×C4#100414yes—
522C2×C4C22×#6712yes—
523C2×C4C22×#6712yes—
524C2×C4C22×#6712yes—
525C2×C4C22×#6711yescharacteristic
526C2×C4C24×D311yescharacteristic
527C8—68no—
528C2×C3—24no—
529C2×C3—24no—
530C2×C3—24no—
531C2×C3#318714yescomplemented
532C2×C3C2×#53411yescharacteristic
533C2×C3C23×SD1611yescharacteristic
534C2×C3C2×#57511yescharacteristic
535D3—84no—
536C22—28no—
537C22—28no—
538C22—28no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—416no—
546C22—124no—
547C22—124no—
548C22—124no—
549C22—2416no—
550C22—248no—
551C22#580412yes—
552C22C22×#21514yes—
553C22C22×#21514yes—
554C22C22×#21514yes—
555C22C22×#21512yes—
556C22#582212yes—
557C22C23×#6711yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—44no—
562C4—44no—
563C4—124no—
564C4#580412yes—
565C4C2×#100412yes—
566C3C2×#318711yescharacteristic, complemented, Sylow
567C2—24no—
568C2—24no—
569C2—24no—
570C2—244no—
571C2#8273614yesdirect product factor
572C2C2×#580411yescharacteristic
573C2C23×#21511yescharacteristic
574C2C2×#582211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC22×#930C2
hypercenterC22×C22⋊Q8D3
hypocenterC3C2×#3187
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊Q8C2semidirect
3. C2×#5227C2semidirect
4. C22×#830C2semidirect
5. #82736C2direct (N × H)
6. C2×#5046C2semidirect
7. C2×#5048C2semidirect
8. C2×#5043C2semidirect
9. C2×#4968C2semidirect
11. C23×C3×Q8C22semidirect
19. C2×C3×C22⋊Q8C22semidirect
20. C2×C3×C22⋊Q8C22semidirect
22. C22×C3×C4⋊C4C22semidirect
24. C22×#215C22semidirect
33. C23×C3⋊C8C22semidirect
34. #5227C22semidirect
35. C2×#835C22semidirect
37. C2×#830C22semidirect
38. C2×#830C22semidirect
39. #5048C22semidirect
40. #5046C22semidirect
41. #5043C22semidirect
42. #4968C22semidirect
43. C2×#809C22semidirect
44. C2×#808C22semidirect
45. C22×C22⋊Q8D3semidirect
60. C22×C3×Q8C23semidirect
87. C3×C22⋊Q8C23semidirect
94. C2×C3×C4⋊C4C23semidirect
99. C2×#215C23semidirect
115. C22×C3⋊C8C23semidirect
116. #835C23semidirect
118. #830C23semidirect
119. #809C23semidirect
120. #808C23semidirect
121. C23×Q8C2×D3semidirect
131. C2×C22⋊Q8C2×D3semidirect
132. C2×C22⋊Q8C2×D3semidirect
141. C22×C4⋊C4C2×D3semidirect
157. C24×C3SD16semidirect
238. C22×Q8C22×D3semidirect
270. C22⋊Q8C22×D3semidirect
287. C2×C4⋊C4C22×D3semidirect
311. C23×C3C2×SD16semidirect
367. C24#215semidirect
487. C23C2×#215semidirect
531. C2×C3#3187semidirect
566. C3C2×#3187semidirect
571. C2#82736direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×#9302
C2×#52272
C22×#8302
#827362
C2×#50462
C2×#50482
C2×#50432
C2×#49682
C2×#31873