Finite Group Databasebeta

← C2×C4×#66939 #14912211

Subgroups — C2×C4×#66939 #14912211

Order 2816.

4070 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
1C22×#66939C211yesmaximal, characteristic
2C22×C4×C22⋊D11C211yesmaximal, characteristic
3C23×C4×Dic11C211yesmaximal, characteristic, complemented
4C4×#66939C2116yesmaximal, direct product factor
5C2×#10622283C212yesmaximal, complemented
6C2×#10622266C212yesmaximal, complemented
7C2×#10622571C211yesmaximal, characteristic
8C2×#10622563C211yesmaximal, characteristic, complemented
9C2×#10622556C211yesmaximal, characteristic, complemented
10C2×C4×C4⋊Dic11C212yesmaximal, complemented
11C2×C4×(C2×D11)⋊C4C212yesmaximal, complemented
12C2×C4×C11×C22⋊C4C211yesmaximal, characteristic, complemented
13C24×C4×C11C2211yescharacteristic
14C23×C22⋊D11C2211yescharacteristic
15C24×Dic11C2211yescharacteristic
16C24×Dic11C2211yescharacteristic
17C23×C4×D11C2211yescharacteristic
18C22×C11×C22⋊C4C2211yescharacteristic
19C22×C11×C22⋊C4C2211yescharacteristic
20C22×C42×C11C2212yes—
21C22×C22⋊Dic11C2211yescharacteristic
22C2×C4×C22⋊D11—28no—
23C2×C4×C22⋊D11C2218yes—
24C22×(C2×D11)⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2211yescharacteristic
26C22×(C2×D11)⋊C4C2212yes—
27C22×C4⋊Dic11C2212yes—
28C22×C4⋊Dic11C2212yes—
29C22×Dic11⋊C4C2211yescharacteristic
30C22×C4×Dic11—28no—
31C22×C4×Dic11C2212yes—
32C22×C4×Dic11C2211yescharacteristic
33C22×C4×Dic11C2212yes—
34C22×C4×Dic11C2218yescomplemented
35C2×#66939C4116yesdirect product factor
36C2×#66939C2218yes—
37C4×C11×C22⋊C4C2218yescomplemented
38C2×C11×#95C2212yes—
39C2×C11⋊#95C2212yes—
40C2×C11⋊#95C2212yes—
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43#10622571C2218yes—
44#10622563C2218yescomplemented
45#10622556C2218yescomplemented
46C4×(C2×D11)⋊C4C22116yescomplemented
47C4×C4⋊Dic11C22116yescomplemented
48#10622283C22116yescomplemented
49#10622266C22116yescomplemented
50C25×C11C2311yescharacteristic
51C24×D11C2311yescharacteristic
52C23×C4×C11—28no—
53C23×C4×C11D418yes—
54C23×C4×C11C2312yes—
55C23×C4×C11C2312yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2314yes—
58C23×C4×C11C2311yescharacteristic
59C22×C22⋊D11—24no—
60C22×C22⋊D11—24no—
61C22×C22⋊D11C2×C418yes—
62C22×C22⋊D11C2314yes—
63C23×Dic11—24no—
64C23×Dic11—24no—
65C23×Dic11—24no—
66C23×Dic11—28no—
67C23×Dic11C2×C418yescomplemented
68C23×Dic11C2311yescharacteristic
69C23×Dic11C2314yes—
70C23×Dic11C2311yescharacteristic
71C23×Dic11C2312yes—
72C23×Dic11C2311yescharacteristic
73C23×Dic11C2314yes—
74C23×Dic11C2312yes—
75C22×C4×D11—24no—
76C22×C4×D11—28no—
77C22×C4×D11C2314yes—
78C2×C11×C22⋊C4—24no—
79C2×C11×C22⋊C4C2×C418yescomplemented
80C2×C11×C22⋊C4C2314yes—
81C2×C11×C22⋊C4C2314yes—
82C2×C42×C11—28no—
83C2×C42×C11C2318yes—
84C2×C22⋊Dic11—24no—
85C2×C22⋊Dic11C2314yes—
86C4×C22⋊D11—232no—
87C2×(C2×D11)⋊C4—24no—
88C2×(C2×D11)⋊C4—28no—
89C2×(C2×D11)⋊C4C2×C4116yescomplemented
90C2×(C2×D11)⋊C4C2314yes—
91C2×(C2×D11)⋊C4C2318yes—
92C2×(C2×D11)⋊C4C2318yes—
93C2×C4⋊Dic11—28no—
94C2×C4⋊Dic11C2×C4116yescomplemented
95C2×C4⋊Dic11C2318yes—
96C2×C4⋊Dic11C2318yes—
97C2×Dic11⋊C4—24no—
98C2×Dic11⋊C4C2314yes—
99C2×C4×Dic11—28no—
100C2×C4×Dic11—28no—
101C2×C4×Dic11—232no—
102C2×C4×Dic11—24no—
103C2×C4×Dic11C2318yes—
104C2×C4×Dic11C2318yes—
105C2×C4×Dic11C2314yes—
106#66939C2×C4164yesdirect product factor
107C11×#95C2318yes—
108C11⋊#95C2318yes—
109C11⋊#95C2318yes—
110C11⋊#95C2318yes—
111C11⋊#95C2318yes—
112C2×C4×#123—111nomaximal, Sylow
113C24×C11—24no—
114C24×C11—24no—
115C24×C11C22×C414yes—
116C24×C11C2×D414yes—
117C24×C11#4114yes—
118C24×C11C2412yes—
119C24×C11C2411yescharacteristic
120C23×D11—22no—
121C23×D11—24no—
122C23×D11—24no—
123C23×D11—22no—
124C23×D11C22×C414yes—
125C23×D11C2412yes—
126C22×C4×C11—24no—
127C22×C4×C11—24no—
128C22×C4×C11—24no—
129C22×C4×C11—232no—
130C22×C4×C11—216no—
131C22×C4×C11—24no—
132C22×C4×C11—24no—
133C22×C4×C11—24no—
134C22×C4×C11C22×C418yes—
135C22×C4×C11C2×D414yes—
136C22×C4×C11C2×D4116yes—
137C22×C4×C11C2×D414yes—
138C22×C4×C11#4118yes—
139C22×C4×C11C2414yes—
140C22×C4×C11C2414yes—
141C22×C4×C11C2412yes—
142C22×C4×C11C2412yes—
143C2×C22⋊D11—232no—
144C2×C22⋊D11—28no—
145C2×C22⋊D11—28no—
146C2×C22⋊D11C22×C4116yes—
147C22×Dic11—232no—
148C22×Dic11—216no—
149C22×Dic11—232no—
150C22×Dic11—28no—
151C22×Dic11—28no—
152C22×Dic11—24no—
153C22×Dic11—24no—
154C22×Dic11—22no—
155C22×Dic11—22no—
156C22×Dic11—22no—
157C22×Dic11—28no—
158C22×Dic11—24no—
159C22×Dic11—24no—
160C22×Dic11—22no—
161C22×Dic11—24no—
162C22×Dic11—24no—
163C22×Dic11C22×C414yes—
164C22×Dic11C22×C418yes—
165C22×Dic11C22×C4116yescomplemented
166C22×Dic11#4118yes—
167C22×Dic11C2412yes—
168C22×Dic11C2412yes—
169C22×Dic11C2414yes—
170C22×Dic11C2412yes—
171C22×Dic11C2414yes—
172C2×C4×D11—28no—
173C2×C4×D11—216no—
174C2×C4×D11—416no—
175C11×C22⋊C4—28no—
176C11×C22⋊C4C22×C4116yescomplemented
177C42×C11—216no—
178C22⋊Dic11—28no—
179(C2×D11)⋊C4—216no—
180(C2×D11)⋊C4—28no—
181(C2×D11)⋊C4C22×C4132yescomplemented
182C4⋊Dic11—216no—
183C4⋊Dic11C22×C4132yescomplemented
184Dic11⋊C4—28no—
185C4×Dic11—216no—
186C4×Dic11—216no—
187C4×Dic11—28no—
188C22×#123—111no—
189C22×C4×D4—111no—
190C23×C42—111no—
191C4×#123—1116no—
192C2×#388—112no—
193C2×#387—112no—
194C2×#386—111no—
195C2×#385—111no—
196C2×#381—111no—
197C2×C4×C4⋊C4—112no—
198C2×C4×C22⋊C4—112no—
199C2×C4×C22⋊C4D1111yescharacteristic, complemented
200C23×C11—216no—
201C23×C11—28no—
202C23×C11—28no—
203C23×C11—216no—
204C23×C11—24no—
205C23×C11—24no—
206C23×C11C4×D4116yes—
207C23×C11C23×C414yes—
208C23×C11C23×C412yes—
209C23×C11C22×D414yes—
210C23×C11C22×D412yes—
211C23×C11C2×#4114yes—
212C23×C11C2×#4112yes—
213C23×C11C2×#4114yes—
214C23×C11C2×#4114yes—
215C23×C11C2511yescharacteristic
216C22×D11—28no—
217C22×D11—216no—
218C22×D11—22no—
219C22×D11—24no—
220C22×D11—24no—
221C22×D11—22no—
222C22×D11—44no—
223C22×D11—44no—
224C22×D11—48no—
225C22×D11C23×C414yes—
226C2×C4×C11—216no—
227C2×C4×C11—28no—
228C2×C4×C11—24no—
229C2×C4×C11—24no—
230C2×C4×C11—24no—
231C2×C4×C11—232no—
232C2×C4×C11—24no—
233C2×C4×C11—24no—
234C2×C4×C11—24no—
235C2×C4×C11—48no—
236C2×C4×C11#123116yescomplemented
237C2×C4×C11C23×C418yes—
238C2×C4×C11C22×D414yes—
239C2×C4×C11C22×D414yes—
240C2×C4×C11C2×#4118yes—
241C22⋊D11—232no—
242C2×Dic11—232no—
243C2×Dic11—216no—
244C2×Dic11—232no—
245C2×Dic11—28no—
246C2×Dic11—216no—
247C2×Dic11—24no—
248C2×Dic11—24no—
249C2×Dic11—22no—
250C2×Dic11—24no—
251C2×Dic11—24no—
252C2×Dic11—22no—
253C2×Dic11—22no—
254C2×Dic11—22no—
255C2×Dic11—24no—
256C2×Dic11—24no—
257C2×Dic11—44no—
258C2×Dic11—44no—
259C2×Dic11C23×C414yes—
260C2×Dic11C23×C418yes—
261C2×Dic11C2×#4118yes—
262C4×D11—416no—
263C23×D4—111no—
264C24×C4—111no—
265C24×C4—111no—
266C24×C4—111no—
267C24×C4C2×D1111yescharacteristic
268C2×#123—118no—
269C2×#123—1116no—
270C2×C4×D4—118no—
271C2×C4×D4—228no—
272C22×C4⋊C4—112no—
273C22×C4⋊C4—111no—
274C22×C4⋊C4—112no—
275C22×C22⋊C4—112no—
276C22×C22⋊C4—111no—
277C22×C22⋊C4—111no—
278C22×C22⋊C4—112no—
279C22×C22⋊C4C2×D1111yescharacteristic
280C22×C22⋊C4C2×D1111yescharacteristic
281C22×C42—112no—
282C22×C42—118no—
283C22×C42—112no—
284C22×C42—111no—
285C22×C42—228no—
286C22×C42C2×D1112yes—
287#388—1116no—
288#387—1116no—
289#386—118no—
290#385—118no—
291#381—118no—
292C4×C4⋊C4—1116no—
293C4×C22⋊C4—1116no—
294C4×C22⋊C4C2×D1118yescomplemented
295C2×#95—112no—
296C2×#95—112no—
297C2×#95—112no—
298C2×#95—112no—
299C2×#95C2×D1112yes—
300C22×C11—28no—
301C22×C11—24no—
302C22×C11—24no—
303C22×C11—28no—
304C22×C11—232no—
305C22×C11C2×C4×D418yes—
306C22×C11C2×C4×D414yes—
307C22×C11C4×#4118yes—
308C22×C11C2×#12314yes—
309C22×C11#52514yes—
310C22×C11#52918yes—
311C22×C11C24×C411yescharacteristic
312C22×C11C23×D411yescharacteristic
313C22×C11C22×#4111yescharacteristic
314C22×C11C22×#4112yes—
315C22×C11C22×#4112yes—
316C2×D11—24no—
317C2×D11—28no—
318C2×D11—42no—
319C2×D11—42no—
320C2×D11—44no—
321C2×D11—416no—
322C4×C11—28no—
323C4×C11—24no—
324C4×C11—44no—
325C4×C11C2×#12318yescomplemented
326Dic11—28no—
327Dic11—24no—
328Dic11—28no—
329Dic11—42no—
330Dic11—42no—
331C25—111no—
332C25C22×D1111yescharacteristic
333C22×D4—118no—
334C22×D4—114no—
335C22×D4—224no—
336C22×D4—224no—
337C23×C4—28no—
338C23×C4—118no—
339C23×C4—112no—
340C23×C4—112no—
341C23×C4—111no—
342C23×C4—114no—
343C23×C4—111no—
344C23×C4—111no—
345C23×C4—114no—
346C23×C4—114no—
347C23×C4—228no—
348C23×C4—224no—
349C23×C4—224no—
350C23×C4—224no—
351C23×C4—224no—
352C23×C4—228no—
353C23×C4D4418yes—
354C23×C4C22×D1112yes—
355C23×C4C22×D1112yes—
356C23×C4C22×D1111yescharacteristic
357C23×C4C22×D1114yes—
358C23×C4C22×D1111yescharacteristic
359#123—1164no—
360C4×D4—2232no—
361C2×C4⋊C4—1116no—
362C2×C4⋊C4—118no—
363C2×C4⋊C4—114no—
364C2×C4⋊C4—118no—
365C2×C4⋊C4—228no—
366C2×C4⋊C4—224no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—1116no—
369C2×C22⋊C4—118no—
370C2×C22⋊C4—114no—
371C2×C22⋊C4—114no—
372C2×C22⋊C4—118no—
373C2×C22⋊C4—228no—
374C2×C22⋊C4—224no—
375C2×C22⋊C4—224no—
376C2×C22⋊C4C4×D1118yescomplemented
377C2×C22⋊C4C22×D1114yes—
378C2×C22⋊C4C22×D1114yes—
379C2×C42—28no—
380C2×C42—114no—
381C2×C42—118no—
382C2×C42—118no—
383C2×C42—224no—
384C2×C42—228no—
385C2×C42—2232no—
386C2×C42—228no—
387C2×C42C22×D1118yes—
388#95—118no—
389#95—118no—
390#95—118no—
391#95—118no—
392#95C22×D1118yes—
393C2×C11—28no—
394C2×C11C4×#12318yescomplemented
395C2×C11C22×C4×D411yescharacteristic
396C2×C11C2×C4×#4112yes—
397C2×C11C22×#12311yescharacteristic
398C2×C11C2×#52511yescharacteristic
399C2×C11C2×#52912yes—
400D11—44no—
401C24—24no—
402C24—24no—
403C24—114no—
404C24—112no—
405C24—222no—
406C24—222no—
407C24—224no—
408C24—224no—
409C24C2×C4×D1114yes—
410C24C2×D4414yes—
411C24#433914yes—
412C24C23×D1112yes—
413C24C23×D1111yescharacteristic
414C2×D4—1116no—
415C2×D4—2232no—
416C2×D4—228no—
417C2×D4—228no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—232no—
422C22×C4—216no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—118no—
427C22×C4—114no—
428C22×C4—1116no—
429C22×C4—118no—
430C22×C4—114no—
431C22×C4—114no—
432C22×C4—112no—
433C22×C4—112no—
434C22×C4—112no—
435C22×C4—2216no—
436C22×C4—2232no—
437C22×C4—2232no—
438C22×C4—228no—
439C22×C4—228no—
440C22×C4—224no—
441C22×C4—224no—
442C22×C4—222no—
443C22×C4—228no—
444C22×C4—224no—
445C22×C4—222no—
446C22×C4—224no—
447C22×C4—222no—
448C22×C4—222no—
449C22×C4—224no—
450C22×C4—224no—
451C22×C4—228no—
452C22×C4—2216no—
453C22×C4—4416no—
454C22×C4C2×C4×D1118yes—
455C22×C4C2×D4414yes—
456C22×C4C2×D44116yes—
457C22×C4C2×D4414yes—
458C22×C4#434118yes—
459C22×C4C23×D1114yes—
460C22×C4C23×D1114yes—
461C22×C4C23×D1112yes—
462C22×C4C23×D1112yes—
463C4⋊C4—1132no—
464C4⋊C4—2216no—
465C4⋊C4—228no—
466C22⋊C4—28no—
467C22⋊C4—1132no—
468C22⋊C4—2216no—
469C22⋊C4—228no—
470C22⋊C4—228no—
471C22⋊C4C2×C4×D11116yescomplemented
472C42—216no—
473C42—2216no—
474C42—2216no—
475C42—228no—
476C11C2×C4×#12311yescharacteristic, complemented, Sylow
477C23—28no—
478C23—28no—
479C23—216no—
480C23—216no—
481C23—24no—
482C23—24no—
483C23—114no—
484C23—228no—
485C23—2216no—
486C23—222no—
487C23—222no—
488C23—224no—
489C23—224no—
490C23—444no—
491C23—444no—
492C23—448no—
493C23C4×D44116yes—
494C23C22×C4×D1114yes—
495C23C22×C4×D1112yes—
496C23C22×D4414yes—
497C23C22×D4412yes—
498C23C2×#433914yes—
499C23C2×#433912yes—
500C23C2×#434114yes—
501C23#41×D1114yes—
502C23C24×D1111yescharacteristic
503D4—2232no—
504C2×C4—28no—
505C2×C4—216no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—232no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—118no—
515C2×C4—114no—
516C2×C4—118no—
517C2×C4—2232no—
518C2×C4—2232no—
519C2×C4—2216no—
520C2×C4—228no—
521C2×C4—2216no—
522C2×C4—224no—
523C2×C4—224no—
524C2×C4—222no—
525C2×C4—224no—
526C2×C4—222no—
527C2×C4—224no—
528C2×C4—222no—
529C2×C4—222no—
530C2×C4—224no—
531C2×C4—224no—
532C2×C4—444no—
533C2×C4—444no—
534C2×C4—4416no—
535C2×C4#66939116yesdirect product factor
536C2×C4C22×C4×D1118yes—
537C2×C4C22×D4414yes—
538C2×C4C22×D4414yes—
539C2×C4C2×#434118yes—
540C22—24no—
541C22—28no—
542C22—232no—
543C22—28no—
544C22—24no—
545C22—224no—
546C22—228no—
547C22—442no—
548C22—444no—
549C22—4416no—
550C22—442no—
551C22C2×C4×D4418yes—
552C22C2×C4×D4414yes—
553C22C2×#6693914yes—
554C22#1062305514yes—
555C22C4×#434118yes—
556C22#1062306918yes—
557C22C23×C4×D1111yescharacteristic
558C22C23×D4411yescharacteristic
559C22C22×#433911yescharacteristic
560C22C22×#434112yes—
561C22C2×#41×D1112yes—
562C4—28no—
563C4—24no—
564C4—44no—
565C4—228no—
566C4—228no—
567C4—224no—
568C4—442no—
569C4—442no—
570C4C2×#6693918yesdirect product factor
571C2—28no—
572C2—444no—
573C2C4×#6693918yesdirect product factor
574C2C22×C4×D4411yescharacteristic
575C2C2×C4×#434112yes—
576C2C22×#6693911yescharacteristic
577C2C2×#1062305511yescharacteristic
578C2C2×#1062306912yes—
subgroupG/H
centerC23×C4C22×D11
commutatorC22×C11C24×C4
FrattiniC23C24×D11
FittingC2×C4×C11×C22⋊C4C2
socleC24×C11C24
prime radicalC23×C22⋊D11C22
cosocleC23×C11C25
hypercenterC2×C4×C22⋊C4D11
hypocenterC11C2×C4×#123
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C23×C4×Dic11C2semidirect
4. C4×#66939C2direct (N × H)
5. C2×#10622283C2semidirect
6. C2×#10622266C2semidirect
8. C2×#10622563C2semidirect
9. C2×#10622556C2semidirect
10. C2×C4×C4⋊Dic11C2semidirect
11. C2×C4×(C2×D11)⋊C4C2semidirect
12. C2×C4×C11×C22⋊C4C2semidirect
34. C22×C4×Dic11C22semidirect
35. C2×#66939C4direct (N × H)
37. C4×C11×C22⋊C4C22semidirect
44. #10622563C22semidirect
45. #10622556C22semidirect
46. C4×(C2×D11)⋊C4C22semidirect
47. C4×C4⋊Dic11C22semidirect
48. #10622283C22semidirect
49. #10622266C22semidirect
67. C23×Dic11C2×C4semidirect
79. C2×C11×C22⋊C4C2×C4semidirect
89. C2×(C2×D11)⋊C4C2×C4semidirect
94. C2×C4⋊Dic11C2×C4semidirect
106. #66939C2×C4direct (N × H)
165. C22×Dic11C22×C4semidirect
176. C11×C22⋊C4C22×C4semidirect
181. (C2×D11)⋊C4C22×C4semidirect
183. C4⋊Dic11C22×C4semidirect
199. C2×C4×C22⋊C4D11semidirect
236. C2×C4×C11#123semidirect
294. C4×C22⋊C4C2×D11semidirect
325. C4×C11C2×#123semidirect
376. C2×C22⋊C4C4×D11semidirect
394. C2×C11C4×#123semidirect
471. C22⋊C4C2×C4×D11semidirect
476. C11C2×C4×#123semidirect
535. C2×C4#66939direct (N × H)
570. C4C2×#66939direct (N × H)
573. C2C4×#66939direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669392
C22×C4×C22⋊D112
C23×C4×Dic112
C4×#669392
C2×#106222832
C2×#106222662
C2×#106225712
C2×#106225632
C2×#106225562
C2×C4×C4⋊Dic112
C2×C4×(C2×D11)⋊C42
C2×C4×C11×C22⋊C42
C2×C4×#12311