Finite Group Databasebeta

← C2×#82307 #11662017

Subgroups — C2×#82307 #11662017

Order 768.

3242 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C22×#932C212yesmaximal, complemented
3C2×#5477C211yesmaximal, characteristic, complemented
4C2×#5019C212yesmaximal, complemented
5#82307C2116yesmaximal, direct product factor
6C2×#5016C214yesmaximal, complemented
7C2×#5015C212yesmaximal, complemented
8C2×#4966C212yesmaximal, complemented
9C2×#4924C211yesmaximal, characteristic, complemented
10C2×#3036—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4—21no—
13C22×#235C2212yes—
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4—21no—
16C2×C3×C4×D4—24no—
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2212yes—
19C2×C3×C4×D4C2218yescomplemented
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#214C2214yes—
24C2×#932—216no—
25C2×#932C22116yescomplemented
26C22×#213C2214yes—
27C2×#930—21no—
28C2×#930—21no—
29C22×#208C2212yes—
30#5477—24no—
31#5477C2218yescomplemented
32C2×#875C2212yescomplemented
33C2×#874C2211yescharacteristic, complemented
34C2×C4×D12—21no—
35C2×C4×D12—21no—
36C2×#869C2211yescharacteristic, complemented
37#5019C22116yescomplemented
38#5016C22132yescomplemented
39#5015C22116yescomplemented
40C2×#810C2214yescomplemented
41#4966C22116yescomplemented
42C2×#809C2214yescomplemented
43#4924C2218yescomplemented
44C2×C4⋊(C3⋊C8)C2212yescomplemented
45C2×C4⋊(C3⋊C8)C2212yescomplemented
46C22×AGL(1,ℤ/8)—32no—
47C2×#532—31no—
48C22×C4×D4D311yescharacteristic, complemented
49#3036—316no—
50C2×#463—34no—
51C2×#461—32no—
52C2×#459—32no—
53C2×#423—31no—
54C2×#417—32no—
55C25×C3—21no—
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4—24no—
59C22×C3×D4D418yescomplemented
60C22×C3×D4C2312yes—
61C22×C3×D4C2314yes—
62C23×C3×C4—21no—
63C23×C3×C4—22no—
64C23×C3×C4—22no—
65C23×C3×C4—28no—
66C23×C3×C4—48no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×#67—22no—
71C2×#235—24no—
72C2×#235—28no—
73C2×#235—28no—
74C2×#235C2318yes—
75C22×D12—21no—
76C22×D12—21no—
77C22×D12C2312yes—
78C22×C4×D3—22no—
79C22×Dic6C2312yes—
80C3×C4×D4—24no—
81C3×C4×D4—24no—
82C3×C4×D4—28no—
83C3×C4×D4—28no—
84C3×C4×D4C2318yes—
85C3×C4×D4C2318yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—28no—
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D411yescharacteristic
95C2×C3×C42D414yes—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2314yes—
99C2×#214—216no—
100C2×#214C23116yes—
101#932—264no—
102C2×#213—216no—
103C2×#213C23116yes—
104#930—28no—
105#930—28no—
106C2×(C2×D3)⋊C4—22no—
107C2×(C2×D3)⋊C4—22no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×#208—28no—
111C2×#208C2318yes—
112C22×C3⋊C8C2314yes—
113#875—24no—
114#875C2318yescomplemented
115#874—22no—
116#874C2314yescomplemented
117C4×D12—28no—
118C4×D12—28no—
119#869—22no—
120#869C2314yescomplemented
121#810C23116yescomplemented
122#809C23116yescomplemented
123C4⋊(C3⋊C8)C2318yescomplemented
124C4⋊(C3⋊C8)C2318yescomplemented
125C22×#41—32no—
126C23×D4C2×D311yescharacteristic
127C24×C4—21no—
128C2×AGL(1,ℤ/8)—316no—
129C2×AGL(1,ℤ/8)—616no—
130C22×SD16—34no—
131C22×D8—34no—
132C22×M16—32no—
133#532—38no—
134#532—64no—
135C2×C4⋊Q8—31no—
136C2×#127—31no—
137C2×#124—32no—
138C2×#121—61no—
139C2×#121—61no—
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×C4×D4—24no—
143C2×C4×D4—61no—
144C2×C4×D4—61no—
145C2×C4×D4C2×D312yes—
146C2×C4×D4C2×D312yes—
147C2×C4×D4C2×D318yescomplemented
148C22×C4⋊C4C2×D311yescharacteristic
149C22×C22⋊C4—21no—
150C22×C42C2×D311yescharacteristic
151#463—332no—
152#461—316no—
153#459—316no—
154#423—38no—
155C2×C4⋊C8—32no—
156C2×C4⋊C8—32no—
157#417—316no—
158C2×#103—34no—
159C2×#102—34no—
160C24×C3—28no—
161C24×C3—24no—
162C24×C3—22no—
163C24×C3—44no—
164C24×C3—44no—
165C24×C3C2×D411yescharacteristic
166C23×D3—22no—
167C2×C3×D4—28no—
168C2×C3×D4—216no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—28no—
173C2×C3×D4C2×D418yes—
174C2×C3×D4C2×D4116yescomplemented
175C2×C3×D4C2414yes—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—28no—
180C22×C3×C4—28no—
181C22×C3×C4—42no—
182C22×C3×C4—48no—
183C22×C3×C4—432no—
184C22×C3×C4—44no—
185C22×C3×C4—48no—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D414yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4#4114yes—
196C22×C3×C4C2411yescharacteristic
197C22×C3×C4C2412yes—
198C2×#67—28no—
199C2×#67—48no—
200C22×Dic3—22no—
201#235—216no—
202#235—216no—
203#235—416no—
204C2×D12—24no—
205C2×D12—24no—
206C2×D12—24no—
207C2×D12—24no—
208C2×D12—22no—
209C2×D12—44no—
210C2×D12—44no—
211C2×D12C2414yes—
212C2×C4×D3—28no—
213C2×C4×D3—48no—
214C2×Dic6—24no—
215C2×Dic6—24no—
216C2×Dic6—22no—
217C2×Dic6C2414yes—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4C2414yes—
221C3×C22⋊C4—28no—
222C3×C22⋊C4—28no—
223C3×C42C2×D412yes—
224C3×C42C2×D414yes—
225C3×C42C2×D414yes—
226C3×C42C2×D412yes—
227C3×C42C2412yes—
228C3×C42C2412yes—
229#214—232no—
230#213—232no—
231(C2×D3)⋊C4—28no—
232(C2×D3)⋊C4—28no—
233C4⋊Dic3—24no—
234C4⋊Dic3—24no—
235#208—216no—
236C2×C3⋊C8—28no—
237C2×C3⋊C8C2418yes—
238C25—21no—
239C2×#41—38no—
240C2×#41—64no—
241C2×#41—68no—
242C2×#41—68no—
243C22×Q8—32no—
244C22×D4—24no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—32no—
248C22×D4—62no—
249C22×D4—61no—
250C22×D4—61no—
251C22×D4D1218yescomplemented
252C22×D4C22×D312yes—
253C22×D4C22×D314yes—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—28no—
258C23×C4—48no—
259C23×C4—62no—
260C23×C4#6711yescharacteristic
261C23×C4#6711yescharacteristic
262C23×C4C22×D311yescharacteristic
263AGL(1,ℤ/8)—664no—
264C2×SD16—316no—
265C2×SD16—616no—
266C2×D8—316no—
267C2×D8—616no—
268C2×M16—38no—
269C2×M16—68no—
270C22×C8—34no—
271C4⋊Q8—34no—
272C4⋊Q8—62no—
273#127—34no—
274#127—62no—
275#124—38no—
276#124—64no—
277#121—68no—
278#121—68no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—68no—
284C4×D4—68no—
285C4×D4C22×D318yes—
286C4×D4C22×D318yes—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4C22×D312yes—
292C2×C4⋊C4C22×D314yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—62no—
297C2×C22⋊C4—62no—
298C2×C42#6711yescharacteristic
299C2×C42#6711yescharacteristic
300C2×C42#6714yes—
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303C2×C42C22×D314yes—
304C4⋊C8—38no—
305C4⋊C8—38no—
306#103—316no—
307#102—316no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—28no—
311C23×C3—216no—
312C23×C3—24no—
313C23×C3—48no—
314C23×C3—432no—
315C23×C3—48no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3#12114yes—
319C23×C3C22×D412yes—
320C23×C3C22×D411yescharacteristic
321C22×D3—24no—
322C22×D3—44no—
323C22×D3—44no—
324C22×D3—44no—
325C3×D4—28no—
326C3×D4—216no—
327C3×D4—24no—
328C3×D4C22×D418yes—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—28no—
333C2×C3×C4—24no—
334C2×C3×C4—44no—
335C2×C3×C4—416no—
336C2×C3×C4—48no—
337C2×C3×C4—44no—
338C2×C3×C4—416no—
339C2×C3×C4#12114yes—
340C2×C3×C4#12118yes—
341C2×C3×C4AGL(1,ℤ/8)14yescomplemented
342C2×C3×C4C22×D412yes—
343C2×C3×C4C22×D414yes—
344C2×C3×C4C22×D412yes—
345C2×C3×C4C22×D412yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D412yes—
349C2×C3×C4C22×D412yes—
350C2×C3×C4C2×#4112yes—
351C2×C3×C4C2×#4114yes—
352C2×C3×C4C2511yescharacteristic
353#67—416no—
354C2×Dic3—24no—
355C2×Dic3—44no—
356D12—24no—
357D12—24no—
358D12—48no—
359D12—44no—
360D12—48no—
361C4×D3—416no—
362Dic6—24no—
363Dic6—24no—
364Dic6—44no—
365C3⋊C8—28no—
366C24—28no—
367C24—24no—
368C24—22no—
369C24—44no—
370C24—44no—
371C24—62no—
372C24C2×#6711yescharacteristic
373#41—616no—
374#41—616no—
375#41—1216no—
376C2×Q8—34no—
377C2×Q8—62no—
378C2×Q8—64no—
379C2×Q8—64no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—34no—
387C2×D4—68no—
388C2×D4—64no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—64no—
392C2×D4—62no—
393C2×D4—124no—
394C2×D4—128no—
395C2×D4—124no—
396C2×D4C2×D1218yes—
397C2×D4C2×D12116yescomplemented
398C2×D4C23×D314yes—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—432no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—62no—
410C22×C4—68no—
411C22×C4—128no—
412C22×C4C2×D1214yes—
413C22×C4#23514yes—
414C22×C4C2×#6714yes—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6714yes—
419C22×C4C2×#6712yes—
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C23×D312yes—
423C22×C4C23×D311yescharacteristic
424SD16—632no—
425D8—632no—
426M16—616no—
427C2×C8—38no—
428C2×C8—68no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4C23×D314yes—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—68no—
437C22⋊C4—68no—
438C42C2×#6714yes—
439C42C2×#6714yes—
440C42C2×#6712yes—
441C42C2×#6712yes—
442C42C23×D312yes—
443C42C23×D312yes—
444C22×C3—28no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—28no—
448C22×C3—432no—
449C22×C3—416no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3C2×#12114yes—
453C22×C3C2×#12112yes—
454C22×C3C2×AGL(1,ℤ/8)12yes—
455C22×C3#57512yes—
456C22×C3C23×D411yescharacteristic
457C2×D3—44no—
458C2×D3—44no—
459C2×D3—44no—
460C2×D3—88no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—44no—
465C3×C4C2×#12112yes—
466C3×C4C2×#12114yes—
467C3×C4C2×AGL(1,ℤ/8)14yescomplemented
468C3×C4C23×D411yescharacteristic
469C3×C4C22×#4111yescharacteristic
470Dic3—44no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—28no—
475C23—216no—
476C23—48no—
477C23—432no—
478C23—48no—
479C23—44no—
480C23—44no—
481C23—64no—
482C23—124no—
483C23—124no—
484C23—124no—
485C23#93014yes—
486C23C22×#6712yes—
487C23C22×#6711yescharacteristic
488Q8—64no—
489Q8—64no—
490Q8—124no—
491D4—24no—
492D4—28no—
493D4—216no—
494D4—64no—
495D4—64no—
496D4—128no—
497D4—1216no—
498D4—124no—
499D4—128no—
500D4C22×D1218yes—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—416no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—416no—
511C2×C4—64no—
512C2×C4—124no—
513C2×C4—1216no—
514C2×C4#93014yes—
515C2×C4#93018yes—
516C2×C4#93214yescomplemented
517C2×C4C22×D1212yes—
518C2×C4C22×D1214yes—
519C2×C4C2×#23512yes—
520C2×C4C2×#23514yes—
521C2×C4C22×#6712yes—
522C2×C4C22×#6712yes—
523C2×C4C22×#6712yes—
524C2×C4C22×#6712yes—
525C2×C4C22×#6711yescharacteristic
526C2×C4C22×#6712yes—
527C2×C4C24×D311yescharacteristic
528C8—68no—
529C2×C3—24no—
530C2×C3—48no—
531C2×C3#303614yescomplemented
532C2×C3C22×#12111yescharacteristic
533C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
534C2×C3C2×#57511yescharacteristic
535D3—84no—
536C22—28no—
537C22—22no—
538C22—22no—
539C22—28no—
540C22—432no—
541C22—416no—
542C22—44no—
543C22—44no—
544C22—124no—
545C22—124no—
546C22—124no—
547C22—248no—
548C22C2×#93014yes—
549C22C2×#93012yes—
550C22C2×#93212yes—
551C22#582212yes—
552C22C23×#6711yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—44no—
557C4—124no—
558C4C2×#93014yes—
559C4C2×#93012yes—
560C4C2×#93214yescomplemented
561C4C23×D1211yescharacteristic
562C4C22×#23511yescharacteristic
563C3C2×#303611yescharacteristic, complemented, Sylow
564C2—24no—
565C2—48no—
566C2—244no—
567C2#8230714yesdirect product factor
568C2C22×#93011yescharacteristic
569C2C22×#93211yescharacteristic
570C2C2×#582211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×C4×D4C2
socleC23×C3C22×D4
hypercenterC22×C4×D4D3
hypocenterC3C2×#3036
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
2. C22×#932C2semidirect
3. C2×#5477C2semidirect
4. C2×#5019C2semidirect
5. #82307C2direct (N × H)
6. C2×#5016C2semidirect
7. C2×#5015C2semidirect
8. C2×#4966C2semidirect
9. C2×#4924C2semidirect
19. C2×C3×C4×D4C22semidirect
25. C2×#932C22semidirect
31. #5477C22semidirect
32. C2×#875C22semidirect
33. C2×#874C22semidirect
36. C2×#869C22semidirect
37. #5019C22semidirect
38. #5016C22semidirect
39. #5015C22semidirect
40. C2×#810C22semidirect
41. #4966C22semidirect
42. C2×#809C22semidirect
43. #4924C22semidirect
44. C2×C4⋊(C3⋊C8)C22semidirect
45. C2×C4⋊(C3⋊C8)C22semidirect
48. C22×C4×D4D3semidirect
59. C22×C3×D4D4semidirect
114. #875C23semidirect
116. #874C23semidirect
120. #869C23semidirect
121. #810C23semidirect
122. #809C23semidirect
123. C4⋊(C3⋊C8)C23semidirect
124. C4⋊(C3⋊C8)C23semidirect
147. C2×C4×D4C2×D3semidirect
174. C2×C3×D4C2×D4semidirect
251. C22×D4D12semidirect
341. C2×C3×C4AGL(1,ℤ/8)semidirect
397. C2×D4C2×D12semidirect
467. C3×C4C2×AGL(1,ℤ/8)semidirect
516. C2×C4#932semidirect
531. C2×C3#3036semidirect
560. C4C2×#932semidirect
563. C3C2×#3036semidirect
567. C2#82307direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C22×#9322
C2×#54772
C2×#50192
#823072
C2×#50162
C2×#50152
C2×#49662
C2×#49242
C2×#30363