Finite Group Databasebeta

← C2×D4×#216 #11670460

Subgroups — C2×D4×#216 #11670460

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C23×#216C212yesmaximal, complemented
3C2×D4×Dic6C211yesmaximal, characteristic
4C2×#5228C212yesmaximal, complemented
5D4×#216C2116yesmaximal, direct product factor
6C2×#5090C211yesmaximal, characteristic, complemented
7C2×#5092C211yesmaximal, characteristic, complemented
8C2×#5050C212yesmaximal, complemented
9C2×#5051C212yesmaximal, complemented
10C2×#5031C211yesmaximal, characteristic, complemented
11C2×C4×#216C211yesmaximal, characteristic, complemented
12C2×D4×C3⋊C8C211yesmaximal, characteristic
13C2×D4×Q16—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C23×Dic6C2212yes—
16C3×D4×Q8—24no—
17C3×D4×Q8C2218yes—
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C22×#216—216no—
26C22×#216C2212yes—
27C22×#216C2212yes—
28C22×#216C2212yes—
29C22×#216C2212yes—
30C22×#216C2211yescharacteristic
31C22×#216C22116yescomplemented
32C2×D4×Dic3—21no—
33C2×#924C2212yes—
34C23×C3⋊C8C2212yes—
35D4×Dic6—24no—
36D4×Dic6C2218yes—
37C2×#888—21no—
38C2×#878—22no—
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41#5228C22116yescomplemented
42C2×#835C2212yes—
43C2×#835C2211yescharacteristic
44C2×#830C2212yes—
45#5092C2218yescomplemented
46#5090C2218yescomplemented
47#5051C22116yescomplemented
48#5050C22116yescomplemented
49#5031C2218yescomplemented
50C4×#216C2218yescomplemented
51D4×C3⋊C8C2218yes—
52C2×#810C2212yes—
53C2×#810C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C23×Q16—32no—
58C2×D4×Q8—31no—
59C2×D4×Q8D311yescharacteristic
60D4×Q16—316no—
61C2×#494—31no—
62C2×#467—32no—
63C2×#462—31no—
64C2×#462—31no—
65C2×#451—32no—
66C2×#451—32no—
67C2×C4×Q16—31no—
68C2×C8×D4—31no—
69C22×C3×Q8—22no—
70C22×C3×Q8—24no—
71C22×C3×Q8—28no—
72C22×C3×Q8—44no—
73C22×C3×Q8C2312yes—
74C22×C3×Q8C2318yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4D411yescharacteristic
77C23×C3×C4—22no—
78C23×C3×C4C2312yes—
79C23×Dic3—22no—
80C22×Dic6—28no—
81C22×Dic6—22no—
82C22×Dic6—24no—
83C22×Dic6—44no—
84C22×Dic6C2312yes—
85C22×Dic6C2318yes—
86C22×Dic6C2311yescharacteristic
87C3×C4⋊Q8—28no—
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8C2314yes—
90C3×C22⋊Q8—24no—
91C3×C22⋊Q8—216no—
92C3×C22⋊Q8C2318yes—
93C3×C4×Q8—22no—
94C3×C4×Q8C2314yes—
95C3×C4×D4—28no—
96C3×C4×D4D414yes—
97C3×C4×D4C2314yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D412yes—
106C2×C3×C42—21no—
107C2×C3×C42C2311yescharacteristic
108C2×#217—22no—
109C2×#216—28no—
110C2×#216—28no—
111C2×#216—264no—
112C2×#216—28no—
113C2×#216—44no—
114C2×#216D418yesdirect product factor
115C2×#216C2318yes—
116C2×#216C2318yes—
117C2×#216C2318yes—
118C2×#216C2318yes—
119C2×#216C2314yes—
120D4×Dic3—28no—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2311yescharacteristic
124#924—24no—
125#924C2318yes—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3—21no—
129C22×C3⋊C8—28no—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2312yes—
132C22×C3⋊C8C2318yes—
133C22×C3⋊C8C2311yescharacteristic
134#888—28no—
135#878—216no—
136#869—22no—
137#869C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140#835C2314yes—
141#835C2318yes—
142#830C2318yes—
143#810C2314yes—
144#810C2318yes—
145#807C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C23×Q8—32no—
149C23×Q8C2×D312yes—
150C22×Q16—31no—
151C22×Q16—32no—
152C22×Q16—32no—
153C22×Q16—32no—
154C22×Q16—316no—
155C22×Q16—32no—
156C22×Q16—616no—
157C23×C8—32no—
158D4×Q8—24no—
159D4×Q8—38no—
160D4×Q8—64no—
161D4×Q8C2×D318yes—
162C2×C4⋊Q8—21no—
163C2×C4⋊Q8—31no—
164C2×C4⋊Q8—61no—
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×C22⋊Q8—22no—
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8—62no—
169C2×C22⋊Q8C2×D312yes—
170C2×C4×Q8—31no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—21no—
173C2×C4×D4—61no—
174C2×C4×D4C2×D311yescharacteristic
175#494—38no—
176#467—316no—
177#462—38no—
178#462—38no—
179#451—316no—
180#451—316no—
181C4×Q16—38no—
182C8×D4—38no—
183C2×#107—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—32no—
187C2×#103—31no—
188C2×#103—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3C2×D412yes—
192C2×C3×Q8—21no—
193C2×C3×Q8—28no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—216no—
197C2×C3×Q8—24no—
198C2×C3×Q8—24no—
199C2×C3×Q8—48no—
200C2×C3×Q8—48no—
201C2×C3×Q8—416no—
202C2×C3×Q8C2×D414yes—
203C2×C3×Q8C2414yes—
204C2×C3×Q8C2412yes—
205C2×C3×D4Q1618yescomplemented
206C2×C3×D4C2×D414yes—
207C22×C3×C4—21no—
208C22×C3×C4—24no—
209C22×C3×C4—24no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—28no—
213C22×C3×C4—48no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D414yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2414yes—
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—28no—
222C22×Dic3—21no—
223C22×Dic3—22no—
224C22×Dic3—48no—
225C2×Dic6—21no—
226C2×Dic6—22no—
227C2×Dic6—24no—
228C2×Dic6—24no—
229C2×Dic6—216no—
230C2×Dic6—22no—
231C2×Dic6—28no—
232C2×Dic6—48no—
233C2×Dic6—416no—
234C2×Dic6—48no—
235C2×Dic6C2×D414yes—
236C2×Dic6C2414yes—
237C2×Dic6C2412yes—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—28no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D414yes—
244C3×C4⋊C4C2412yes—
245C3×C22⋊C4—28no—
246C3×C22⋊C4C2×D418yes—
247C3×C42—24no—
248C3×C42C2×D412yes—
249C3×C42C2412yes—
250#217—28no—
251#216—216no—
252#216—216no—
253#216—216no—
254#216—48no—
255#216C2×D4116yesdirect product factor
256C4⋊Dic3—21no—
257C4⋊Dic3—24no—
258C4⋊Dic3—24no—
259C4⋊Dic3C2412yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—28no—
262C4×Dic3—24no—
263C2×C3⋊C8—24no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—216no—
266C2×C3⋊C8C2×D414yes—
267C2×C3⋊C8C2412yes—
268C2×C3⋊C8C2414yes—
269C2×C3⋊C8C2412yes—
270C22×Q8—22no—
271C22×Q8—24no—
272C22×Q8—28no—
273C22×Q8—31no—
274C22×Q8—32no—
275C22×Q8—38no—
276C22×Q8—44no—
277C22×Q8—68no—
278C22×Q8—62no—
279C22×Q8—64no—
280C22×Q8—124no—
281C22×Q8C22×D312yes—
282C22×Q8C22×D318yes—
283C22×Q8C22×D311yescharacteristic
284C22×D4#6711yescharacteristic
285C23×C4—22no—
286C23×C4—62no—
287C23×C4C22×D312yes—
288C2×Q16—38no—
289C2×Q16—34no—
290C2×Q16—38no—
291C2×Q16—38no—
292C2×Q16—38no—
293C2×Q16—38no—
294C2×Q16—68no—
295C2×Q16—664no—
296C2×Q16—68no—
297C2×Q16—68no—
298C2×Q16—124no—
299C22×C8—31no—
300C22×C8—32no—
301C22×C8—38no—
302C22×C8—31no—
303C22×C8—68no—
304C4⋊Q8—28no—
305C4⋊Q8—22no—
306C4⋊Q8—34no—
307C4⋊Q8—68no—
308C4⋊Q8—62no—
309C4⋊Q8C22×D314yes—
310C22⋊Q8—24no—
311C22⋊Q8—216no—
312C22⋊Q8—38no—
313C22⋊Q8—64no—
314C22⋊Q8—616no—
315C22⋊Q8C22×D318yes—
316C4×Q8—22no—
317C4×Q8—34no—
318C4×Q8—62no—
319C4×Q8C22×D314yes—
320C4×D4—28no—
321C4×D4—68no—
322C4×D4#6714yes—
323C4×D4C22×D314yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—62no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4#6711yescharacteristic
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—62no—
337C2×C22⋊C4#6712yes—
338C2×C42—21no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—34no—
342C4⋊C8—34no—
343#103—34no—
344#103—38no—
345#103—34no—
346#103—38no—
347C22⋊C8—38no—
348C4×C8—34no—
349C23×C3—24no—
350C23×C3—24no—
351C23×C3C2×Q1618yescomplemented
352C23×C3C22×D414yes—
353C23×C3C22×D411yescharacteristic
354C3×Q8—24no—
355C3×Q8—22no—
356C3×Q8—24no—
357C3×Q8—24no—
358C3×Q8—416no—
359C3×Q8—44no—
360C3×Q8C22×D414yes—
361C3×D4C2×Q16116yescomplemented
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—28no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—416no—
372C2×C3×C4—44no—
373C2×C3×C4—42no—
374C2×C3×C4C2×Q1614yescomplemented
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D414yes—
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D412yes—
380C2×C3×C4H(5,𝔽2)12yes—
381C2×C3×C4C2511yescharacteristic
382C2×Dic3—24no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—24no—
386C2×Dic3—416no—
387C2×Dic3—44no—
388C2×Dic3—42no—
389Dic6—24no—
390Dic6—22no—
391Dic6—24no—
392Dic6—24no—
393Dic6—416no—
394Dic6—44no—
395Dic6C22×D414yes—
396C3⋊C8—24no—
397C3⋊C8—22no—
398C3⋊C8C22×D414yes—
399C24C2×#6712yes—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—28no—
403C2×Q8—22no—
404C2×Q8—216no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—34no—
408C2×Q8—32no—
409C2×Q8—34no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—416no—
413C2×Q8—616no—
414C2×Q8—62no—
415C2×Q8—61no—
416C2×Q8—68no—
417C2×Q8—62no—
418C2×Q8—64no—
419C2×Q8—64no—
420C2×Q8—128no—
421C2×Q8—128no—
422C2×Q8—1216no—
423C2×Q8D3×D414yes—
424C2×Q8C23×D314yes—
425C2×Q8C23×D312yes—
426C2×D4#21618yesdirect product factor
427C2×D4C2×#6714yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—48no—
435C22×C4—61no—
436C22×C4—68no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—128no—
440C22×C4C2×#6714yes—
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6712yes—
443C22×C4C2×#6711yescharacteristic
444C22×C4C23×D314yes—
445C22×C4C23×D311yescharacteristic
446Q16—316no—
447Q16—616no—
448Q16—616no—
449Q16—616no—
450Q16—128no—
451C2×C8—34no—
452C2×C8—32no—
453C2×C8—32no—
454C2×C8—34no—
455C2×C8—616no—
456C2×C8—64no—
457C2×C8—62no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—21no—
463C4⋊C4—32no—
464C4⋊C4—61no—
465C4⋊C4—64no—
466C4⋊C4—64no—
467C4⋊C4—68no—
468C4⋊C4—64no—
469C4⋊C4C2×#6714yes—
470C4⋊C4C23×D312yes—
471C22⋊C4—28no—
472C22⋊C4—68no—
473C22⋊C4C2×#6718yes—
474C42—24no—
475C42—64no—
476C42C2×#6712yes—
477C42C23×D312yes—
478C22×C3—24no—
479C22×C3—216no—
480C22×C3—24no—
481C22×C3D4212yes—
482C22×C3C22×Q1612yes—
483C22×C3C22×Q1618yescomplemented
484C22×C3#57812yes—
485C22×C3C23×D411yescharacteristic
486C3×C4—24no—
487C3×C4—24no—
488C3×C4—22no—
489C3×C4—44no—
490C3×C4—42no—
491C3×C4D4212yes—
492C3×C4C22×Q1614yescomplemented
493C3×C4C23×D411yescharacteristic
494C3×C4C2×H(5,𝔽2)11yescharacteristic
495Dic3—24no—
496Dic3—44no—
497Dic3—42no—
498C23—24no—
499C23—24no—
500C23C2×#21618yescomplemented
501C23C22×#6714yes—
502C23C22×#6711yescharacteristic
503Q8—22no—
504Q8—24no—
505Q8—24no—
506Q8—24no—
507Q8—34no—
508Q8—416no—
509Q8—44no—
510Q8—62no—
511Q8—64no—
512Q8—64no—
513Q8—64no—
514Q8—1216no—
515Q8—124no—
516Q8C2×D3×D414yes—
517D4C2×#216116yesdirect product factor
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—416no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—1216no—
535C2×C4—124no—
536C2×C4—122no—
537C2×C4C2×#21614yescomplemented
538C2×C4C2×D3×D412yes—
539C2×C4#100412yes—
540C2×C4C22×#6712yes—
541C2×C4C22×#6714yes—
542C2×C4C22×#6711yescharacteristic
543C2×C4C22×#6712yes—
544C2×C4C24×D311yescharacteristic
545C8—34no—
546C8—64no—
547C8—62no—
548C2×C3—28no—
549C2×C3D4×Q1614yescomplemented
550C2×C3C2×D4211yescharacteristic
551C2×C3C23×Q1611yescharacteristic
552C2×C3C2×#57811yescharacteristic
553C22—216no—
554C22—24no—
555C22—24no—
556C22D4×#6712yes—
557C22C22×#21612yes—
558C22C22×#21618yescomplemented
559C22#583812yes—
560C22C23×#6711yescharacteristic
561C4—24no—
562C4—24no—
563C4—22no—
564C4—44no—
565C4—42no—
566C4—64no—
567C4—124no—
568C4—122no—
569C4D4×#6712yes—
570C4C22×#21614yescomplemented
571C4C22×D3×D411yescharacteristic
572C4C2×#100411yescharacteristic
573C3C2×D4×Q1611yescharacteristic, complemented, Sylow
574C2—28no—
575C2D4×#21614yesdirect product factor
576C2C2×D4×#6711yescharacteristic
577C2C23×#21611yescharacteristic
578C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×D4×Q8D3
hypocenterC3C2×D4×Q16
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C23×#216C2semidirect
4. C2×#5228C2semidirect
5. D4×#216C2direct (N × H)
6. C2×#5090C2semidirect
7. C2×#5092C2semidirect
8. C2×#5050C2semidirect
9. C2×#5051C2semidirect
10. C2×#5031C2semidirect
11. C2×C4×#216C2semidirect
31. C22×#216C22semidirect
41. #5228C22semidirect
45. #5092C22semidirect
46. #5090C22semidirect
47. #5051C22semidirect
48. #5050C22semidirect
49. #5031C22semidirect
50. C4×#216C22semidirect
114. C2×#216D4direct (N × H)
205. C2×C3×D4Q16semidirect
255. #216C2×D4direct (N × H)
351. C23×C3C2×Q16semidirect
361. C3×D4C2×Q16semidirect
374. C2×C3×C4C2×Q16semidirect
426. C2×D4#216direct (N × H)
483. C22×C3C22×Q16semidirect
492. C3×C4C22×Q16semidirect
500. C23C2×#216semidirect
517. D4C2×#216direct (N × H)
537. C2×C4C2×#216semidirect
549. C2×C3D4×Q16semidirect
558. C22C22×#216semidirect
570. C4C22×#216semidirect
573. C3C2×D4×Q16semidirect
575. C2D4×#216direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C23×#2162
C2×D4×Dic62
C2×#52282
D4×#2162
C2×#50902
C2×#50922
C2×#50502
C2×#50512
C2×#50312
C2×C4×#2162
C2×D4×C3⋊C82
C2×D4×Q163