Finite Group Databasebeta

← C22×#10623239 #15660764

Subgroups — C22×#10623239 #15660764

Order 2816.

7934 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623239C2148yesmaximal, direct product factor
2C22×#66932C211yesmaximal, characteristic, complemented
3C22×#66943C211yesmaximal, characteristic, complemented
4C22×#66942C211yesmaximal, characteristic, complemented
5C22×#66942C211yesmaximal, characteristic, complemented
6C22×#66936C211yesmaximal, characteristic
7C22×#66936C211yesmaximal, characteristic
8C22×#66949C211yesmaximal, characteristic, complemented
9C22×#66931C211yesmaximal, characteristic, complemented
10C22×C4×Dic22C211yesmaximal, characteristic, complemented
11C22×(C22⋊D11)⋊C4C211yesmaximal, characteristic
12C22×#66945C211yesmaximal, characteristic, complemented
13C22×#66925C211yesmaximal, characteristic, complemented
14C22×#66930C211yesmaximal, characteristic, complemented
15C22×C4⋊C4×D11C211yesmaximal, characteristic, complemented
16C22×C11×#126C211yesmaximal, characteristic, complemented
17C23×C22⋊D11C2211yescharacteristic
18C24×Dic11C2211yescharacteristic
19C23×C4×D11C2211yescharacteristic
20C23×C4×D11C2211yescharacteristic
21C23×C4×D11C2211yescharacteristic
22C23×Dic22C2211yescharacteristic
23C2×C11×#126C22124yescomplemented
24C22×C11×C4⋊C4C2211yescharacteristic
25C22×C11×C4⋊C4C2211yescharacteristic
26C22×C11×C4⋊C4C2211yescharacteristic
27C22×C11×C22⋊C4C2211yescharacteristic
28C22×C11×C22⋊C4C2211yescharacteristic
29C22×C11×C22⋊C4C2211yescharacteristic
30C22×C42×C11C2211yescharacteristic
31C22×C22⋊Dic11C2211yescharacteristic
32C22×C22⋊Dic11C2211yescharacteristic
33C22×(C2×D11)⋊C4C2211yescharacteristic
34C22×(C2×D11)⋊C4C2211yescharacteristic
35C22×(C2×D11)⋊C4C2211yescharacteristic
36C22×C4⋊Dic11C2211yescharacteristic
37C22×C4⋊Dic11C2211yescharacteristic
38C22×C4⋊Dic11C2211yescharacteristic
39C22×C4⋊Dic11C2211yescharacteristic
40C22×Dic11⋊C4C2211yescharacteristic
41C22×Dic11⋊C4C2211yescharacteristic
42C22×Dic11⋊C4C2211yescharacteristic
43C22×Dic11⋊C4C2211yescharacteristic
44C22×Dic11⋊C4C2211yescharacteristic
45C22×Dic11⋊C4C2211yescharacteristic
46C22×Dic11⋊C4C2211yescharacteristic
47C22×Dic11⋊C4C2211yescharacteristic
48C22×Dic11⋊C4C2211yescharacteristic
49C22×C4×Dic11C2211yescharacteristic
50C22×C4×Dic11C2211yescharacteristic
51C22×C4×Dic11C2211yescharacteristic
52C22×C4×Dic11C2211yescharacteristic
53#10623239C221256yesdirect product factor
54C2×#66949C22124yescomplemented
55C2×#66945C22124yescomplemented
56C2×C4⋊C4×D11C22124yescomplemented
57C2×#66943C22124yescomplemented
58C2×#66942C22124yescomplemented
59C2×#66942C22124yescomplemented
60C2×#66936C22124yes—
61C2×#66936C22124yes—
62C2×(C22⋊D11)⋊C4C22124yes—
63C2×#66932C22124yescomplemented
64C2×#66931C22124yescomplemented
65C2×#66930C22124yescomplemented
66C2×#66925C22124yescomplemented
67C2×C4×Dic22C22124yescomplemented
68C25×C11C2311yescharacteristic
69C24×D11C2311yescharacteristic
70C23×C4×C11C2311yescharacteristic
71C23×C4×C11C2311yescharacteristic
72C23×C4×C11C2311yescharacteristic
73C23×C4×C11C2311yescharacteristic
74C23×C4×C11C2311yescharacteristic
75C23×C4×C11C2311yescharacteristic
76C22×C22⋊D11—28no—
77C22×C22⋊D11C23112yes—
78C23×Dic11—28no—
79C23×Dic11C23112yes—
80C23×Dic11C2311yescharacteristic
81C23×Dic11C2311yescharacteristic
82C23×Dic11C2311yescharacteristic
83C23×Dic11C2311yescharacteristic
84C23×Dic11C2311yescharacteristic
85C23×Dic11C2311yescharacteristic
86C23×Dic11C2311yescharacteristic
87C22×C4×D11—28no—
88C22×C4×D11—28no—
89C22×C4×D11—28no—
90C22×C4×D11C23112yes—
91C22×C4×D11C23112yes—
92C22×C4×D11C23112yes—
93C22×Dic22—28no—
94C22×Dic22C23112yes—
95C11×#126C23164yescomplemented
96C2×C11×C4⋊C4C23112yes—
97C2×C11×C4⋊C4C23112yes—
98C2×C11×C4⋊C4C23112yes—
99C2×C11×C22⋊C4C23112yes—
100C2×C11×C22⋊C4C23112yes—
101C2×C11×C22⋊C4C23112yes—
102C2×C42×C11C23112yes—
103C2×C22⋊Dic11C23112yes—
104C2×C22⋊Dic11C23112yes—
105C2×(C2×D11)⋊C4C23112yes—
106C2×(C2×D11)⋊C4C23112yes—
107C2×(C2×D11)⋊C4C23112yes—
108C2×C4⋊Dic11C23112yes—
109C2×C4⋊Dic11C23112yes—
110C2×C4⋊Dic11C23112yes—
111C2×C4⋊Dic11C23112yes—
112C2×Dic11⋊C4C23112yes—
113C2×Dic11⋊C4C23112yes—
114C2×Dic11⋊C4C23112yes—
115C2×Dic11⋊C4C23112yes—
116C2×Dic11⋊C4C23112yes—
117C2×Dic11⋊C4C23112yes—
118C2×Dic11⋊C4C23112yes—
119C2×Dic11⋊C4C23112yes—
120C2×Dic11⋊C4C23112yes—
121C2×C4×Dic11C23112yes—
122C2×C4×Dic11C23112yes—
123C2×C4×Dic11C23112yes—
124C2×C4×Dic11C23112yes—
125#66949C23164yescomplemented
126#66945C23164yescomplemented
127C4⋊C4×D11C23164yescomplemented
128#66943C23164yescomplemented
129#66942C23164yescomplemented
130#66942C23164yescomplemented
131#66936C23164yes—
132#66936C23164yes—
133(C22⋊D11)⋊C4C23164yes—
134#66932C23164yescomplemented
135#66931C23164yescomplemented
136#66930C23164yescomplemented
137#66925C23164yescomplemented
138C4×Dic22C23164yescomplemented
139C22×#552—111nomaximal, Sylow
140C24×C11—24no—
141C24×C11—24no—
142C24×C11—24no—
143C24×C11C2416yes—
144C24×C11C2411yescharacteristic
145C23×D11—24no—
146C23×D11—24no—
147C23×D11#4118yes—
148C23×D11C2416yes—
149C22×C4×C11—24no—
150C22×C4×C11—24no—
151C22×C4×C11—24no—
152C22×C4×C11—24no—
153C22×C4×C11—24no—
154C22×C4×C11—24no—
155C22×C4×C11C2416yes—
156C22×C4×C11C2416yes—
157C22×C4×C11C2416yes—
158C22×C4×C11C2416yes—
159C22×C4×C11C2416yes—
160C22×C4×C11C2416yes—
161C2×C22⋊D11—248no—
162C2×C22⋊D11C24116yes—
163C22×Dic11—248no—
164C22×Dic11—24no—
165C22×Dic11—24no—
166C22×Dic11—24no—
167C22×Dic11—24no—
168C22×Dic11—24no—
169C22×Dic11—24no—
170C22×Dic11#4118yes—
171C22×Dic11C24116yes—
172C22×Dic11C2416yes—
173C22×Dic11C2416yes—
174C22×Dic11C2416yes—
175C22×Dic11C2416yes—
176C22×Dic11C2416yes—
177C22×Dic11C2416yes—
178C22×Dic11C2416yes—
179C2×C4×D11—248no—
180C2×C4×D11—248no—
181C2×C4×D11—248no—
182C2×C4×D11C24116yes—
183C2×C4×D11C24116yes—
184C2×C4×D11C24116yes—
185C2×Dic22—248no—
186C2×Dic22C24116yes—
187C11×C4⋊C4C24116yes—
188C11×C4⋊C4C24116yes—
189C11×C4⋊C4C24116yes—
190C11×C22⋊C4C24116yes—
191C11×C22⋊C4C24116yes—
192C11×C22⋊C4C24116yes—
193C42×C11C24116yes—
194C22⋊Dic11C24116yes—
195C22⋊Dic11C24116yes—
196(C2×D11)⋊C4C24116yes—
197(C2×D11)⋊C4C24116yes—
198(C2×D11)⋊C4C24116yes—
199C4⋊Dic11C24116yes—
200C4⋊Dic11C24116yes—
201C4⋊Dic11C24116yes—
202C4⋊Dic11C24116yes—
203Dic11⋊C4C24116yes—
204Dic11⋊C4C24116yes—
205Dic11⋊C4C24116yes—
206Dic11⋊C4C24116yes—
207Dic11⋊C4C24116yes—
208Dic11⋊C4C24116yes—
209Dic11⋊C4C24116yes—
210Dic11⋊C4C24116yes—
211Dic11⋊C4C24116yes—
212C4×Dic11C24116yes—
213C4×Dic11C24116yes—
214C4×Dic11C24116yes—
215C4×Dic11C24116yes—
216C2×#552—1148no—
217C22×#126—111no—
218C22×#126D1111yescharacteristic, complemented
219C22×#125—111no—
220C22×#125—111no—
221C22×#125—111no—
222C22×#123—111no—
223C22×#123—111no—
224C22×C22⋊Q8—111no—
225C22×C22⋊Q8—111no—
226C22×C4×Q8—111no—
227C22×C4×D4—111no—
228C22×#117—111no—
229C22×#117—111no—
230C22×#117—111no—
231C23×C4⋊C4—111no—
232C23×C11—212no—
233C23×C11—212no—
234C23×C11—212no—
235C23×C11—416no—
236C23×C11C2×#4114yes—
237C23×C11C2×#4114yes—
238C23×C112−1+414yes—
239C23×C11C2514yes—
240C23×C11C2513yes—
241C22×D11—232no—
242C22×D11—212no—
243C22×D11—212no—
244C22×D11C2×#41124yes—
245C22×D11C2514yes—
246C2×C4×C11—212no—
247C2×C4×C11—212no—
248C2×C4×C11—212no—
249C2×C4×C11—212no—
250C2×C4×C11—212no—
251C2×C4×C11—212no—
252C2×C4×C11C2514yes—
253C2×C4×C11C2514yes—
254C2×C4×C11C2514yes—
255C2×C4×C11C2514yes—
256C2×C4×C11C2514yes—
257C2×C4×C11C2514yes—
258C22⋊D11—232no—
259C2×Dic11—212no—
260C2×Dic11—212no—
261C2×Dic11—212no—
262C2×Dic11—212no—
263C2×Dic11—212no—
264C2×Dic11—212no—
265C2×Dic11—232no—
266C2×Dic11C2×#41124yes—
267C2×Dic11C2514yes—
268C2×Dic11C2514yes—
269C2×Dic11C2514yes—
270C2×Dic11C2514yes—
271C2×Dic11C2514yes—
272C2×Dic11C2514yes—
273C2×Dic11C2514yes—
274C4×D11—232no—
275C4×D11—232no—
276C4×D11—232no—
277Dic22—232no—
278C23×Q8—111no—
279C23×D4—111no—
280C24×C4—111no—
281C24×C4—111no—
282C24×C4—111no—
283C24×C4—111no—
284#552—11256no—
285C2×#126—1124no—
286C2×#126C2×D11124yescomplemented
287C2×#125—1124no—
288C2×#125—1124no—
289C2×#125—1124no—
290C2×#123—1124no—
291C2×#123—1124no—
292C2×C22⋊Q8—1124no—
293C2×C22⋊Q8—1124no—
294C2×C4×Q8—1124no—
295C2×C4×D4—1124no—
296C2×#117—1124no—
297C2×#117—1124no—
298C2×#117—1124no—
299C22×C4⋊C4—111no—
300C22×C4⋊C4—111no—
301C22×C4⋊C4—111no—
302C22×C4⋊C4—111no—
303C22×C4⋊C4—111no—
304C22×C4⋊C4—111no—
305C22×C4⋊C4—111no—
306C22×C4⋊C4—111no—
307C22×C4⋊C4—111no—
308C22×C4⋊C4—111no—
309C22×C4⋊C4—111no—
310C22×C4⋊C4—111no—
311C22×C4⋊C4—111no—
312C22×C4⋊C4—1124no—
313C22×C4⋊C4C2×D1111yescharacteristic
314C22×C4⋊C4C2×D1111yescharacteristic
315C22×C4⋊C4C2×D1111yescharacteristic
316C22×C22⋊C4—111no—
317C22×C22⋊C4—111no—
318C22×C22⋊C4—111no—
319C22×C22⋊C4—111no—
320C22×C22⋊C4—111no—
321C22×C22⋊C4C2×D1111yescharacteristic
322C22×C22⋊C4C2×D1111yescharacteristic
323C22×C22⋊C4C2×D1111yescharacteristic
324C22×C42—111no—
325C22×C42—111no—
326C22×C42—111no—
327C22×C42—111no—
328C22×C42C2×D1111yescharacteristic
329C22×C11—24no—
330C22×C11—24no—
331C22×C11—24no—
332C22×C11—424no—
333C22×C11#552116yescomplemented
334C22×C11C22×#4116yes—
335C22×C11C22×#4116yes—
336C22×C11C2×2−1+416yes—
337C22×C11C2611yescharacteristic
338C2×D11—24no—
339C2×D11—24no—
340C2×D11—248no—
341C2×D11C22×#4118yes—
342C4×C11—24no—
343C4×C11—24no—
344C4×C11—24no—
345C4×C11—24no—
346C4×C11—24no—
347C4×C11—24no—
348Dic11—24no—
349Dic11—24no—
350Dic11—24no—
351Dic11—24no—
352Dic11—24no—
353Dic11—24no—
354Dic11C22×#4118yes—
355C25—111no—
356C25C22×D1111yescharacteristic
357C22×Q8—1112no—
358C22×Q8—228no—
359C22×D4—1112no—
360C22×D4—228no—
361C23×C4—111no—
362C23×C4—1112no—
363C23×C4—111no—
364C23×C4—111no—
365C23×C4—111no—
366C23×C4—111no—
367C23×C4—111no—
368C23×C4—111no—
369C23×C4—1112no—
370C23×C4—1112no—
371C23×C4—1112no—
372C23×C4—228no—
373C23×C4—228no—
374C23×C4—228no—
375C23×C4—228no—
376C23×C4C22×D1111yescharacteristic
377C23×C4C22×D1111yescharacteristic
378C23×C4C22×D1111yescharacteristic
379C23×C4C22×D1111yescharacteristic
380C23×C4C22×D1111yescharacteristic
381C23×C4C22×D1111yescharacteristic
382#126—1164no—
383#126C22×D11164yescomplemented
384#125—1164no—
385#125—1164no—
386#125—1164no—
387#123—1164no—
388#123—1164no—
389C22⋊Q8—1164no—
390C22⋊Q8—1164no—
391C4×Q8—1164no—
392C4×D4—1164no—
393#117—1164no—
394#117—1164no—
395#117—1164no—
396C2×C4⋊C4—1112no—
397C2×C4⋊C4—1112no—
398C2×C4⋊C4—1112no—
399C2×C4⋊C4—1112no—
400C2×C4⋊C4—1112no—
401C2×C4⋊C4—1112no—
402C2×C4⋊C4—1112no—
403C2×C4⋊C4—1112no—
404C2×C4⋊C4—1112no—
405C2×C4⋊C4—1112no—
406C2×C4⋊C4—1112no—
407C2×C4⋊C4—1112no—
408C2×C4⋊C4—1112no—
409C2×C4⋊C4—1164no—
410C2×C4⋊C4C22×D11112yes—
411C2×C4⋊C4C22×D11112yes—
412C2×C4⋊C4C22×D11112yes—
413C2×C22⋊C4—1112no—
414C2×C22⋊C4—1112no—
415C2×C22⋊C4—1112no—
416C2×C22⋊C4—1112no—
417C2×C22⋊C4—1112no—
418C2×C22⋊C4C22×D11112yes—
419C2×C22⋊C4C22×D11112yes—
420C2×C22⋊C4C22×D11112yes—
421C2×C42—1112no—
422C2×C42—1112no—
423C2×C42—1112no—
424C2×C42—1112no—
425C2×C42C22×D11112yes—
426C2×C11—44no—
427C2×C11C2×#552112yescomplemented
428C2×C11C23×#4111yescharacteristic
429C2×C11C23×#4111yescharacteristic
430C2×C11C22×2−1+411yescharacteristic
431D11—28no—
432C24—24no—
433C24—24no—
434C24—24no—
435C24—118no—
436C24—116no—
437C24—224no—
438C24—224no—
439C24C23×D1116yes—
440C24C23×D1111yescharacteristic
441C2×Q8—1116no—
442C2×Q8—2248no—
443C2×D4—1116no—
444C2×D4—2248no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—118no—
452C22×C4—116no—
453C22×C4—116no—
454C22×C4—116no—
455C22×C4—116no—
456C22×C4—116no—
457C22×C4—116no—
458C22×C4—116no—
459C22×C4—1116no—
460C22×C4—1116no—
461C22×C4—1116no—
462C22×C4—1116no—
463C22×C4—2248no—
464C22×C4—2248no—
465C22×C4—2248no—
466C22×C4—2248no—
467C22×C4—224no—
468C22×C4—224no—
469C22×C4—224no—
470C22×C4—224no—
471C22×C4—224no—
472C22×C4—224no—
473C22×C4C23×D1116yes—
474C22×C4C23×D1116yes—
475C22×C4C23×D1116yes—
476C22×C4C23×D1116yes—
477C22×C4C23×D1116yes—
478C22×C4C23×D1116yes—
479C4⋊C4—1116no—
480C4⋊C4—1116no—
481C4⋊C4—1116no—
482C4⋊C4—1116no—
483C4⋊C4—1116no—
484C4⋊C4—1116no—
485C4⋊C4—1116no—
486C4⋊C4—1116no—
487C4⋊C4—1116no—
488C4⋊C4—1116no—
489C4⋊C4—1116no—
490C4⋊C4—1116no—
491C4⋊C4—1116no—
492C4⋊C4C23×D11116yes—
493C4⋊C4C23×D11116yes—
494C4⋊C4C23×D11116yes—
495C22⋊C4—1116no—
496C22⋊C4—1116no—
497C22⋊C4—1116no—
498C22⋊C4—1116no—
499C22⋊C4—1116no—
500C22⋊C4C23×D11116yes—
501C22⋊C4C23×D11116yes—
502C22⋊C4C23×D11116yes—
503C42—1116no—
504C42—1116no—
505C42—1116no—
506C42—1116no—
507C42C23×D11116yes—
508C11C22×#55211yescharacteristic, complemented, Sylow
509C23—212no—
510C23—212no—
511C23—212no—
512C23—416no—
513C23—1124no—
514C23—114no—
515C23—2232no—
516C23—2212no—
517C23—2212no—
518C23#41×D1114yes—
519C23#41×D1114yes—
520C23#6704314yes—
521C23C24×D1114yes—
522C23C24×D1113yes—
523Q8—2232no—
524D4—2232no—
525C2×C4—212no—
526C2×C4—212no—
527C2×C4—212no—
528C2×C4—212no—
529C2×C4—212no—
530C2×C4—212no—
531C2×C4—1124no—
532C2×C4—114no—
533C2×C4—114no—
534C2×C4—114no—
535C2×C4—114no—
536C2×C4—114no—
537C2×C4—114no—
538C2×C4—114no—
539C2×C4—2232no—
540C2×C4—2232no—
541C2×C4—2232no—
542C2×C4—2232no—
543C2×C4—2212no—
544C2×C4—2212no—
545C2×C4—2212no—
546C2×C4—2212no—
547C2×C4—2212no—
548C2×C4—2212no—
549C2×C4C24×D1114yes—
550C2×C4C24×D1114yes—
551C2×C4C24×D1114yes—
552C2×C4C24×D1114yes—
553C2×C4C24×D1114yes—
554C2×C4C24×D1114yes—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—424no—
559C22—118no—
560C22—2248no—
561C22—224no—
562C22—224no—
563C22#10623239116yesdirect product factor
564C22C2×#41×D1116yes—
565C22C2×#41×D1116yes—
566C22C2×#6704316yes—
567C22C25×D1111yescharacteristic
568C4—24no—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—118no—
575C4—224no—
576C4—224no—
577C4—224no—
578C4—224no—
579C4—224no—
580C4—224no—
581C2—44no—
582C2—228no—
583C2C2×#10623239112yesdirect product factor
584C2C22×#41×D1111yescharacteristic
585C2C22×#41×D1111yescharacteristic
586C2C22×#6704311yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC22×C11×#126C2
socleC24×C11C24
prime radicalC23×C22⋊D11C22
hypercenterC22×#126D11
hypocenterC11C22×#552
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C2×#10623239C2direct (N × H)
2. C22×#66932C2semidirect
3. C22×#66943C2semidirect
4. C22×#66942C2semidirect
5. C22×#66942C2semidirect
8. C22×#66949C2semidirect
9. C22×#66931C2semidirect
10. C22×C4×Dic22C2semidirect
12. C22×#66945C2semidirect
13. C22×#66925C2semidirect
14. C22×#66930C2semidirect
15. C22×C4⋊C4×D11C2semidirect
16. C22×C11×#126C2semidirect
23. C2×C11×#126C22semidirect
53. #10623239C22direct (N × H)
54. C2×#66949C22semidirect
55. C2×#66945C22semidirect
56. C2×C4⋊C4×D11C22semidirect
57. C2×#66943C22semidirect
58. C2×#66942C22semidirect
59. C2×#66942C22semidirect
63. C2×#66932C22semidirect
64. C2×#66931C22semidirect
65. C2×#66930C22semidirect
66. C2×#66925C22semidirect
67. C2×C4×Dic22C22semidirect
95. C11×#126C23semidirect
125. #66949C23semidirect
126. #66945C23semidirect
127. C4⋊C4×D11C23semidirect
128. #66943C23semidirect
129. #66942C23semidirect
130. #66942C23semidirect
134. #66932C23semidirect
135. #66931C23semidirect
136. #66930C23semidirect
137. #66925C23semidirect
138. C4×Dic22C23semidirect
218. C22×#126D11semidirect
286. C2×#126C2×D11semidirect
333. C22×C11#552semidirect
383. #126C22×D11semidirect
427. C2×C11C2×#552semidirect
508. C11C22×#552semidirect
563. C22#10623239direct (N × H)
583. C2C2×#10623239direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106232392
C22×#669322
C22×#669432
C22×#669422
C22×#669362
C22×#669492
C22×#669312
C22×C4×Dic222
C22×(C22⋊D11)⋊C42
C22×#669452
C22×#669252
C22×#669302
C22×C4⋊C4×D112
C22×C11×#1262
C22×#55211