Finite Group Databasebeta

← C2×D4×#4326 #15629046

Subgroups — C2×D4×#4326 #15629046

Order 2816.

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
1C23×#4326C212yesmaximal, complemented
2C2×D4×Dic22C211yesmaximal, characteristic
3D4×#4326C2116yesmaximal, direct product factor
4C2×#10622705C211yesmaximal, characteristic, complemented
5C2×#10622664C212yesmaximal, complemented
6C2×#10622644C211yesmaximal, characteristic, complemented
7C2×#10622703C211yesmaximal, characteristic, complemented
8C2×#10622841C212yesmaximal, complemented
9C2×#10622663C212yesmaximal, complemented
10C2×C4×#4326C211yesmaximal, characteristic, complemented
11C2×D4×C11⋊C8C211yesmaximal, characteristic
12C2×D4×Q8×C11C211yesmaximal, characteristic
13C23×Q8×C11C2212yes—
14C23×Dic22C2212yes—
15D4×Q8×C11—24no—
16D4×Q8×C11C2218yes—
17C2×C11×C4⋊Q8—21no—
18C2×C11×C4⋊Q8C2211yescharacteristic
19C2×C11×C22⋊Q8—22no—
20C2×C11×C22⋊Q8C2212yes—
21C2×C4×Q8×C11C2211yescharacteristic
22C2×C4×D4×C11—21no—
23C2×C4×D4×C11C2211yescharacteristic
24C22×#4326—216no—
25C22×#4326C2212yes—
26C22×#4326C2212yes—
27C22×#4326C2212yes—
28C22×#4326C2212yes—
29C22×#4326C2211yescharacteristic
30C22×#4326C22116yescomplemented
31C2×D4×Dic11—21no—
32C2×#66977C2212yes—
33C23×C11⋊C8C2212yes—
34D4×Dic22—24no—
35D4×Dic22C2218yes—
36C2×#66941—21no—
37C2×#66931—22no—
38C2×#66922C2211yescharacteristic
39C2×C4×Dic22C2211yescharacteristic
40#10622841C22116yescomplemented
41C2×#66899C2212yes—
42C2×#66899C2211yescharacteristic
43C2×C22⋊(C11⋊C8)C2212yes—
44#10622705C2218yescomplemented
45#10622703C2218yescomplemented
46#10622664C22116yescomplemented
47#10622663C22116yescomplemented
48#10622644C2218yescomplemented
49C4×#4326C2218yescomplemented
50D4×C11⋊C8C2218yes—
51C2×#66874C2212yes—
52C2×#66874C2211yescharacteristic
53C2×#66871C2211yescharacteristic
54C2×C4⋊(C11⋊C8)C2211yescharacteristic
55C2×C4×C11⋊C8C2211yescharacteristic
56C22×Q8×C11—22no—
57C22×Q8×C11—24no—
58C22×Q8×C11—28no—
59C22×Q8×C11—44no—
60C22×Q8×C11C2312yes—
61C22×Q8×C11C2318yes—
62C22×Q8×C11C2311yescharacteristic
63C22×D4×C11D411yescharacteristic
64C23×C4×C11—22no—
65C23×C4×C11C2312yes—
66C23×Dic11—22no—
67C22×Dic22—28no—
68C22×Dic22—22no—
69C22×Dic22—24no—
70C22×Dic22—44no—
71C22×Dic22C2312yes—
72C22×Dic22C2318yes—
73C22×Dic22C2311yescharacteristic
74C11×C4⋊Q8—28no—
75C11×C4⋊Q8—22no—
76C11×C4⋊Q8C2314yes—
77C11×C22⋊Q8—24no—
78C11×C22⋊Q8—216no—
79C11×C22⋊Q8C2318yes—
80C4×Q8×C11—22no—
81C4×Q8×C11C2314yes—
82C4×D4×C11—28no—
83C4×D4×C11D414yes—
84C4×D4×C11C2314yes—
85C2×C11×C4⋊C4—21no—
86C2×C11×C4⋊C4—21no—
87C2×C11×C4⋊C4—21no—
88C2×C11×C4⋊C4—22no—
89C2×C11×C4⋊C4D411yescharacteristic
90C2×C11×C4⋊C4C2311yescharacteristic
91C2×C11×C22⋊C4—22no—
92C2×C11×C22⋊C4D412yes—
93C2×C42×C11—21no—
94C2×C42×C11C2311yescharacteristic
95C2×C22⋊Dic11—22no—
96C2×#4326—28no—
97C2×#4326—28no—
98C2×#4326—264no—
99C2×#4326—28no—
100C2×#4326—44no—
101C2×#4326D418yesdirect product factor
102C2×#4326C2318yes—
103C2×#4326C2318yes—
104C2×#4326C2318yes—
105C2×#4326C2318yes—
106C2×#4326C2314yes—
107D4×Dic11—28no—
108C2×C4⋊Dic11—21no—
109C2×C4⋊Dic11—21no—
110C2×C4⋊Dic11C2311yescharacteristic
111#66977—24no—
112#66977C2318yes—
113C2×Dic11⋊C4—21no—
114C2×Dic11⋊C4—22no—
115C2×C4×Dic11—21no—
116C22×C11⋊C8—28no—
117C22×C11⋊C8C2311yescharacteristic
118C22×C11⋊C8C2312yes—
119C22×C11⋊C8C2318yes—
120C22×C11⋊C8C2311yescharacteristic
121#66941—28no—
122#66931—216no—
123#66922—22no—
124#66922C2314yes—
125C4×Dic22—22no—
126C4×Dic22C2314yes—
127#66899C2314yes—
128#66899C2318yes—
129C22⋊(C11⋊C8)C2318yes—
130#66874C2314yes—
131#66874C2318yes—
132#66871C2314yes—
133C4⋊(C11⋊C8)C2314yes—
134C4×C11⋊C8C2314yes—
135C2×D4×Q16—111nomaximal, Sylow
136C24×C11C2×D412yes—
137C2×Q8×C11—21no—
138C2×Q8×C11—28no—
139C2×Q8×C11—22no—
140C2×Q8×C11—22no—
141C2×Q8×C11—216no—
142C2×Q8×C11—24no—
143C2×Q8×C11—24no—
144C2×Q8×C11—48no—
145C2×Q8×C11—48no—
146C2×Q8×C11—416no—
147C2×Q8×C11C2×D414yes—
148C2×Q8×C11C2414yes—
149C2×Q8×C11C2412yes—
150C2×D4×C11Q1618yescomplemented
151C2×D4×C11C2×D414yes—
152C22×C4×C11—21no—
153C22×C4×C11—24no—
154C22×C4×C11—24no—
155C22×C4×C11—21no—
156C22×C4×C11—22no—
157C22×C4×C11—28no—
158C22×C4×C11—48no—
159C22×C4×C11C2×D411yescharacteristic
160C22×C4×C11C2×D412yes—
161C22×C4×C11C2×D414yes—
162C22×C4×C11C2×D411yescharacteristic
163C22×C4×C11C2414yes—
164C22×C4×C11C2411yescharacteristic
165C22×Dic11—21no—
166C22×Dic11—28no—
167C22×Dic11—21no—
168C22×Dic11—22no—
169C22×Dic11—48no—
170C2×Dic22—216no—
171C2×Dic22—21no—
172C2×Dic22—24no—
173C2×Dic22—22no—
174C2×Dic22—28no—
175C2×Dic22—22no—
176C2×Dic22—24no—
177C2×Dic22—48no—
178C2×Dic22—416no—
179C2×Dic22—48no—
180C2×Dic22C2×D414yes—
181C2×Dic22C2414yes—
182C2×Dic22C2412yes—
183C11×C4⋊C4—24no—
184C11×C4⋊C4—24no—
185C11×C4⋊C4—24no—
186C11×C4⋊C4—28no—
187C11×C4⋊C4—21no—
188C11×C4⋊C4C2×D414yes—
189C11×C4⋊C4C2412yes—
190C11×C22⋊C4—28no—
191C11×C22⋊C4C2×D418yes—
192C42×C11—24no—
193C42×C11C2×D412yes—
194C42×C11C2412yes—
195C22⋊Dic11—28no—
196#4326—216no—
197#4326—216no—
198#4326—216no—
199#4326—48no—
200#4326C2×D4116yesdirect product factor
201C4⋊Dic11—21no—
202C4⋊Dic11—24no—
203C4⋊Dic11—24no—
204C4⋊Dic11C2412yes—
205Dic11⋊C4—24no—
206Dic11⋊C4—28no—
207C4×Dic11—24no—
208C2×C11⋊C8—216no—
209C2×C11⋊C8—24no—
210C2×C11⋊C8—22no—
211C2×C11⋊C8C2×D414yes—
212C2×C11⋊C8C2412yes—
213C2×C11⋊C8C2414yes—
214C2×C11⋊C8C2412yes—
215C23×Q16—112no—
216C2×D4×Q8—111no—
217C2×D4×Q8D1111yescharacteristic
218D4×Q16—1116no—
219C2×#494—111no—
220C2×#467—112no—
221C2×#462—111no—
222C2×#462—111no—
223C2×#451—112no—
224C2×#451—112no—
225C2×C4×Q16—111no—
226C2×C8×D4—111no—
227C23×C11—24no—
228C23×C11—24no—
229C23×C11C2×Q1618yescomplemented
230C23×C11C22×D414yes—
231C23×C11C22×D411yescharacteristic
232Q8×C11—24no—
233Q8×C11—22no—
234Q8×C11—24no—
235Q8×C11—24no—
236Q8×C11—416no—
237Q8×C11—44no—
238Q8×C11C22×D414yes—
239D4×C11C2×Q16116yescomplemented
240C2×C4×C11—24no—
241C2×C4×C11—24no—
242C2×C4×C11—22no—
243C2×C4×C11—22no—
244C2×C4×C11—28no—
245C2×C4×C11—24no—
246C2×C4×C11—24no—
247C2×C4×C11—22no—
248C2×C4×C11—24no—
249C2×C4×C11—416no—
250C2×C4×C11—44no—
251C2×C4×C11—42no—
252C2×C4×C11C2×Q1614yescomplemented
253C2×C4×C11C22×D412yes—
254C2×C4×C11C22×D412yes—
255C2×C4×C11C22×D414yes—
256C2×C4×C11C22×D411yescharacteristic
257C2×C4×C11C22×D412yes—
258C2×C4×C11H(5,𝔽2)12yes—
259C2×C4×C11C2511yescharacteristic
260C2×Dic11—24no—
261C2×Dic11—22no—
262C2×Dic11—22no—
263C2×Dic11—24no—
264C2×Dic11—416no—
265C2×Dic11—44no—
266C2×Dic11—42no—
267Dic22—24no—
268Dic22—22no—
269Dic22—24no—
270Dic22—24no—
271Dic22—416no—
272Dic22—44no—
273Dic22C22×D414yes—
274C11⋊C8—24no—
275C11⋊C8—22no—
276C11⋊C8C22×D414yes—
277C23×Q8—112no—
278C23×Q8C2×D1112yes—
279C22×Q16—111no—
280C22×Q16—112no—
281C22×Q16—112no—
282C22×Q16—112no—
283C22×Q16—1116no—
284C22×Q16—112no—
285C22×Q16—2216no—
286C23×C8—112no—
287D4×Q8—24no—
288D4×Q8—118no—
289D4×Q8—224no—
290D4×Q8C2×D1118yes—
291C2×C4⋊Q8—21no—
292C2×C4⋊Q8—111no—
293C2×C4⋊Q8—221no—
294C2×C4⋊Q8C2×D1111yescharacteristic
295C2×C22⋊Q8—22no—
296C2×C22⋊Q8—112no—
297C2×C22⋊Q8—222no—
298C2×C22⋊Q8C2×D1112yes—
299C2×C4×Q8—111no—
300C2×C4×Q8C2×D1111yescharacteristic
301C2×C4×D4—21no—
302C2×C4×D4—221no—
303C2×C4×D4C2×D1111yescharacteristic
304#494—118no—
305#467—1116no—
306#462—118no—
307#462—118no—
308#451—1116no—
309#451—1116no—
310C4×Q16—118no—
311C8×D4—118no—
312C2×#107—111no—
313C2×C4⋊C8—111no—
314C2×#103—111no—
315C2×#103—112no—
316C2×#103—111no—
317C2×#103—112no—
318C2×C22⋊C8—112no—
319C2×C4×C8—111no—
320C22×C11—24no—
321C22×C11—216no—
322C22×C11—24no—
323C22×C11D4212yes—
324C22×C11C22×Q1612yes—
325C22×C11C22×Q1618yescomplemented
326C22×C11#57812yes—
327C22×C11C23×D411yescharacteristic
328C4×C11—24no—
329C4×C11—24no—
330C4×C11—22no—
331C4×C11—44no—
332C4×C11—42no—
333C4×C11D4212yes—
334C4×C11C22×Q1614yescomplemented
335C4×C11C23×D411yescharacteristic
336C4×C11C2×H(5,𝔽2)11yescharacteristic
337Dic11—24no—
338Dic11—44no—
339Dic11—42no—
340C22×Q8—22no—
341C22×Q8—24no—
342C22×Q8—28no—
343C22×Q8—44no—
344C22×Q8—111no—
345C22×Q8—112no—
346C22×Q8—118no—
347C22×Q8—228no—
348C22×Q8—222no—
349C22×Q8—224no—
350C22×Q8—444no—
351C22×Q8C22×D1112yes—
352C22×Q8C22×D1118yes—
353C22×Q8C22×D1111yescharacteristic
354C22×D4C22⋊D1111yescharacteristic
355C23×C4—22no—
356C23×C4—222no—
357C23×C4C22×D1112yes—
358C2×Q16—118no—
359C2×Q16—114no—
360C2×Q16—118no—
361C2×Q16—118no—
362C2×Q16—118no—
363C2×Q16—118no—
364C2×Q16—228no—
365C2×Q16—2264no—
366C2×Q16—228no—
367C2×Q16—228no—
368C2×Q16—444no—
369C22×C8—111no—
370C22×C8—112no—
371C22×C8—118no—
372C22×C8—111no—
373C22×C8—228no—
374C4⋊Q8—28no—
375C4⋊Q8—22no—
376C4⋊Q8—114no—
377C4⋊Q8—228no—
378C4⋊Q8—222no—
379C4⋊Q8C22×D1114yes—
380C22⋊Q8—24no—
381C22⋊Q8—216no—
382C22⋊Q8—118no—
383C22⋊Q8—224no—
384C22⋊Q8—2216no—
385C22⋊Q8C22×D1118yes—
386C4×Q8—22no—
387C4×Q8—114no—
388C4×Q8—222no—
389C4×Q8C22×D1114yes—
390C4×D4—28no—
391C4×D4—228no—
392C4×D4C22⋊D1114yes—
393C4×D4C22×D1114yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—111no—
399C2×C4⋊C4—221no—
400C2×C4⋊C4—221no—
401C2×C4⋊C4—222no—
402C2×C4⋊C4—221no—
403C2×C4⋊C4C22⋊D1111yescharacteristic
404C2×C4⋊C4C22×D1111yescharacteristic
405C2×C22⋊C4—22no—
406C2×C22⋊C4—222no—
407C2×C22⋊C4C22⋊D1112yes—
408C2×C42—21no—
409C2×C42—221no—
410C2×C42C22×D1111yescharacteristic
411#107—114no—
412C4⋊C8—114no—
413#103—114no—
414#103—118no—
415#103—114no—
416#103—118no—
417C22⋊C8—118no—
418C4×C8—114no—
419C2×C11—28no—
420C2×C11D4×Q1614yescomplemented
421C2×C11C2×D4211yescharacteristic
422C2×C11C23×Q1611yescharacteristic
423C2×C11C2×#57811yescharacteristic
424C24C2×C22⋊D1112yes—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—216no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—416no—
435C2×Q8—114no—
436C2×Q8—112no—
437C2×Q8—114no—
438C2×Q8—2216no—
439C2×Q8—222no—
440C2×Q8—221no—
441C2×Q8—228no—
442C2×Q8—222no—
443C2×Q8—224no—
444C2×Q8—224no—
445C2×Q8—448no—
446C2×Q8—448no—
447C2×Q8—4416no—
448C2×Q8D4×D1114yes—
449C2×Q8C23×D1114yes—
450C2×Q8C23×D1112yes—
451C2×D4#432618yesdirect product factor
452C2×D4C2×C22⋊D1114yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—221no—
461C22×C4—228no—
462C22×C4—221no—
463C22×C4—222no—
464C22×C4—448no—
465C22×C4C2×C22⋊D1114yes—
466C22×C4C2×C22⋊D1111yescharacteristic
467C22×C4C2×C22⋊D1112yes—
468C22×C4C2×C22⋊D1111yescharacteristic
469C22×C4C23×D1114yes—
470C22×C4C23×D1111yescharacteristic
471Q16—1116no—
472Q16—2216no—
473Q16—2216no—
474Q16—2216no—
475Q16—448no—
476C2×C8—114no—
477C2×C8—112no—
478C2×C8—112no—
479C2×C8—114no—
480C2×C8—2216no—
481C2×C8—224no—
482C2×C8—222no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—21no—
488C4⋊C4—112no—
489C4⋊C4—221no—
490C4⋊C4—224no—
491C4⋊C4—224no—
492C4⋊C4—228no—
493C4⋊C4—224no—
494C4⋊C4C2×C22⋊D1114yes—
495C4⋊C4C23×D1112yes—
496C22⋊C4—28no—
497C22⋊C4—228no—
498C22⋊C4C2×C22⋊D1118yes—
499C42—24no—
500C42—224no—
501C42C2×C22⋊D1112yes—
502C42C23×D1112yes—
503C11C2×D4×Q1611yescharacteristic, complemented, Sylow
504C23—24no—
505C23—24no—
506C23C2×#432618yescomplemented
507C23C22×C22⋊D1114yes—
508C23C22×C22⋊D1111yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8—24no—
512Q8—24no—
513Q8—416no—
514Q8—44no—
515Q8—114no—
516Q8—224no—
517Q8—222no—
518Q8—224no—
519Q8—224no—
520Q8—4416no—
521Q8—444no—
522Q8C2×D4×D1114yes—
523D4C2×#4326116yesdirect product factor
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—416no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—224no—
537C2×C4—222no—
538C2×C4—222no—
539C2×C4—224no—
540C2×C4—4416no—
541C2×C4—444no—
542C2×C4—442no—
543C2×C4C2×#432614yescomplemented
544C2×C4C2×D4×D1112yes—
545C2×C4#6703712yes—
546C2×C4C22×C22⋊D1112yes—
547C2×C4C22×C22⋊D1114yes—
548C2×C4C22×C22⋊D1111yescharacteristic
549C2×C4C22×C22⋊D1112yes—
550C2×C4C24×D1111yescharacteristic
551C8—114no—
552C8—224no—
553C8—222no—
554C22—216no—
555C22—24no—
556C22—24no—
557C22D4×C22⋊D1112yes—
558C22C22×#432612yes—
559C22C22×#432618yescomplemented
560C22#1062336912yes—
561C22C23×C22⋊D1111yescharacteristic
562C4—24no—
563C4—24no—
564C4—22no—
565C4—44no—
566C4—42no—
567C4—224no—
568C4—444no—
569C4—442no—
570C4D4×C22⋊D1112yes—
571C4C22×#432614yescomplemented
572C4C22×D4×D1111yescharacteristic
573C4C2×#6703711yescharacteristic
574C2—28no—
575C2D4×#432614yesdirect product factor
576C2C2×D4×C22⋊D1111yescharacteristic
577C2C23×#432611yescharacteristic
578C2C2×#1062336911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingC2×D4×Q8×C11C2
socleC23×C11C22×D4
prime radicalC22×D4×C11D4
hypercenterC2×D4×Q8D11
hypocenterC11C2×D4×Q16
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#4326C2semidirect
3. D4×#4326C2direct (N × H)
4. C2×#10622705C2semidirect
5. C2×#10622664C2semidirect
6. C2×#10622644C2semidirect
7. C2×#10622703C2semidirect
8. C2×#10622841C2semidirect
9. C2×#10622663C2semidirect
10. C2×C4×#4326C2semidirect
30. C22×#4326C22semidirect
40. #10622841C22semidirect
44. #10622705C22semidirect
45. #10622703C22semidirect
46. #10622664C22semidirect
47. #10622663C22semidirect
48. #10622644C22semidirect
49. C4×#4326C22semidirect
101. C2×#4326D4direct (N × H)
150. C2×D4×C11Q16semidirect
200. #4326C2×D4direct (N × H)
229. C23×C11C2×Q16semidirect
239. D4×C11C2×Q16semidirect
252. C2×C4×C11C2×Q16semidirect
325. C22×C11C22×Q16semidirect
334. C4×C11C22×Q16semidirect
420. C2×C11D4×Q16semidirect
451. C2×D4#4326direct (N × H)
503. C11C2×D4×Q16semidirect
506. C23C2×#4326semidirect
523. D4C2×#4326direct (N × H)
543. C2×C4C2×#4326semidirect
559. C22C22×#4326semidirect
571. C4C22×#4326semidirect
575. C2D4×#4326direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43262
C2×D4×Dic222
D4×#43262
C2×#106227052
C2×#106226642
C2×#106226442
C2×#106227032
C2×#106228412
C2×#106226632
C2×C4×#43262
C2×D4×C11⋊C82
C2×D4×Q8×C112
C2×D4×Q1611