Finite Group Databasebeta

← C2×#13072182 #14922027

Subgroups — C2×#13072182 #14922027

Order 2816.

3678 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66928C211yesmaximal, characteristic, complemented
2C22×#66977C211yesmaximal, characteristic, complemented
3C22×#66983C211yesmaximal, characteristic
4C22×C4×Dic22C211yesmaximal, characteristic, complemented
5#13072182C2116yesmaximal, direct product factor
6C2×#10622289C212yesmaximal, complemented
7C2×#10622572C212yesmaximal, complemented
8C2×#10622552C211yesmaximal, characteristic, complemented
9C2×#10622570C212yesmaximal, complemented
10C2×#10622541C211yesmaximal, characteristic, complemented
11C2×#10622282C212yesmaximal, complemented
12C2×C4×C11×C22⋊C4C211yesmaximal, characteristic, complemented
13C24×C4×C11C2211yescharacteristic
14C23×C22⋊D11C2212yes—
15C23×D44C2211yescharacteristic
16C23×Dic22C2211yescharacteristic
17C22×C11×C22⋊C4C2211yescharacteristic
18C22×C11×C22⋊C4C2211yescharacteristic
19C22×C42×C11C2211yescharacteristic
20C22×C42×C11C2211yescharacteristic
21C22×C22⋊Dic11C2212yes—
22C2×#66983—28no—
23C2×#66983C2218yes—
24C22×(C2×D11)⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2212yes—
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2212yes—
28C22×(C2×D11)⋊C4C2212yes—
29C22×C4⋊Dic11C2211yescharacteristic
30C22×C4⋊Dic11C2211yescharacteristic
31C2×#66977—28no—
32C2×#66977C2218yescomplemented
33C22×Dic11⋊C4C2212yes—
34C22×Dic11⋊C4C2212yes—
35C22×Dic11⋊C4C2212yes—
36C22×C4×Dic11C2212yes—
37C2×#66928—28no—
38C2×#66928C2218yescomplemented
39C2×C4×Dic22—28no—
40C2×C4×Dic22C2218yescomplemented
41C4×C11×C22⋊C4C2218yescomplemented
42C2×C11×#95C2211yescharacteristic
43C2×C11×#95C2211yescharacteristic
44C2×C11⋊#95C2212yes—
45#10622572C22116yescomplemented
46#10622570C22116yescomplemented
47#10622552C2218yescomplemented
48#10622541C2218yescomplemented
49#10622289C22116yescomplemented
50#10622282C22116yescomplemented
51C25×C11C2311yescharacteristic
52C24×D11C2312yes—
53C23×C4×C11—24no—
54C23×C4×C11—24no—
55C23×C4×C11—24no—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2311yescharacteristic
58C23×C4×C11C2311yescharacteristic
59C23×C4×C11C2314yes—
60C23×C4×C11C2311yescharacteristic
61C23×C4×C11C2311yescharacteristic
62C23×C4×C11C2311yescharacteristic
63C22×C22⋊D11—28no—
64C22×C22⋊D11—28no—
65C22×C22⋊D11—44no—
66C22×C22⋊D11C2318yes—
67C23×Dic11C2312yes—
68C23×Dic11C2312yes—
69C23×Dic11C2312yes—
70C22×D44—24no—
71C22×D44—24no—
72C22×D44—24no—
73C22×D44C2314yes—
74C22×Dic22—24no—
75C22×Dic22—24no—
76C22×Dic22D418yescomplemented
77C22×Dic22C2314yes—
78C2×C11×C22⋊C4—24no—
79C2×C11×C22⋊C4—24no—
80C2×C11×C22⋊C4C2314yes—
81C2×C11×C22⋊C4C2314yes—
82C2×C42×C11—24no—
83C2×C42×C11—24no—
84C2×C42×C11C2314yes—
85C2×C42×C11C2314yes—
86C2×C22⋊Dic11—28no—
87C2×C22⋊Dic11C2318yes—
88#66983—232no—
89C2×(C2×D11)⋊C4—28no—
90C2×(C2×D11)⋊C4—28no—
91C2×(C2×D11)⋊C4—28no—
92C2×(C2×D11)⋊C4—28no—
93C2×(C2×D11)⋊C4—44no—
94C2×(C2×D11)⋊C4C2318yes—
95C2×(C2×D11)⋊C4C2318yes—
96C2×(C2×D11)⋊C4C2318yes—
97C2×(C2×D11)⋊C4C2318yes—
98C2×(C2×D11)⋊C4C2318yes—
99C2×C4⋊Dic11—24no—
100C2×C4⋊Dic11D418yes—
101C2×C4⋊Dic11C2314yes—
102C2×C4⋊Dic11C2314yes—
103#66977—232no—
104C2×Dic11⋊C4—28no—
105C2×Dic11⋊C4—28no—
106C2×Dic11⋊C4—44no—
107C2×Dic11⋊C4C2318yes—
108C2×Dic11⋊C4C2318yes—
109C2×Dic11⋊C4C2318yes—
110C2×C4×Dic11—28no—
111C2×C4×Dic11C2318yes—
112#66928—232no—
113C4×Dic22—232no—
114C11×#95C2314yes—
115C11×#95C2314yes—
116C11⋊#95C2318yes—
117C2×#2426—111nomaximal, Sylow
118C24×C11—22no—
119C24×C11—22no—
120C24×C11—22no—
121C24×C11—22no—
122C24×C11—22no—
123C24×C11—22no—
124C24×C11#4114yes—
125C24×C11C2412yes—
126C24×C11C2411yescharacteristic
127C23×D11—24no—
128C23×D11—24no—
129C23×D11—24no—
130C23×D11—24no—
131C23×D11—24no—
132C23×D11—24no—
133C23×D11—24no—
134C23×D11C2414yes—
135C22×C4×C11—22no—
136C22×C4×C11—22no—
137C22×C4×C11—22no—
138C22×C4×C11—22no—
139C22×C4×C11—22no—
140C22×C4×C11—22no—
141C22×C4×C11—22no—
142C22×C4×C11—22no—
143C22×C4×C11—22no—
144C22×C4×C11—22no—
145C22×C4×C11—22no—
146C22×C4×C11—22no—
147C22×C4×C11—22no—
148C22×C4×C11—22no—
149C22×C4×C11—28no—
150C22×C4×C11—28no—
151C22×C4×C11—28no—
152C22×C4×C11—416no—
153C22×C4×C11C2×D414yes—
154C22×C4×C11C2×D414yes—
155C22×C4×C11#4114yes—
156C22×C4×C11#4114yes—
157C22×C4×C11C2412yes—
158C22×C4×C11C2412yes—
159C22×C4×C11C2412yes—
160C22×C4×C11C2412yes—
161C22×C4×C11C2412yes—
162C22×C4×C11C2412yes—
163C2×C22⋊D11—216no—
164C2×C22⋊D11—216no—
165C2×C22⋊D11—48no—
166C2×C22⋊D11—432no—
167C22×Dic11—24no—
168C22×Dic11—24no—
169C22×Dic11—24no—
170C22×Dic11—24no—
171C22×Dic11—24no—
172C22×Dic11—24no—
173C22×Dic11—24no—
174C22×Dic11C2×D418yes—
175C22×Dic11C2×D418yes—
176C22×Dic11C2414yes—
177C22×Dic11C2414yes—
178C22×Dic11C2414yes—
179C2×D44—28no—
180C2×D44—28no—
181C2×D44—28no—
182C2×D44—416no—
183C2×Dic22—232no—
184C2×Dic22—28no—
185C2×Dic22—28no—
186C2×Dic22C2×D4116yescomplemented
187C11×C22⋊C4—28no—
188C11×C22⋊C4—28no—
189C42×C11—28no—
190C42×C11—28no—
191C22⋊Dic11—216no—
192(C2×D11)⋊C4—216no—
193(C2×D11)⋊C4—216no—
194(C2×D11)⋊C4—216no—
195(C2×D11)⋊C4—216no—
196(C2×D11)⋊C4—48no—
197C4⋊Dic11—28no—
198C4⋊Dic11C2×D4116yes—
199Dic11⋊C4—216no—
200Dic11⋊C4—216no—
201Dic11⋊C4—48no—
202C4×Dic11—216no—
203C22×#124—111no—
204C22×C22⋊Q8—111no—
205C22×#121—111no—
206C22×C4×Q8—111no—
207#2426—1116no—
208C2×#396—112no—
209C2×#394—112no—
210C2×#390—111no—
211C2×#390—112no—
212C2×#389—111no—
213C2×#388—112no—
214C2×C4×C22⋊C4D1111yescharacteristic, complemented
215C23×C11—28no—
216C23×C11—28no—
217C23×C11—28no—
218C23×C11—22no—
219C23×C11—22no—
220C23×C11—22no—
221C23×C11—22no—
222C23×C11—22no—
223C23×C11—22no—
224C23×C11—44no—
225C23×C11—44no—
226C23×C11—44no—
227C23×C11—44no—
228C23×C11C22×D412yes—
229C23×C11C22×D412yes—
230C23×C11C2×#4114yes—
231C23×C11C2×#4112yes—
232C23×C11C2×#4112yes—
233C23×C11C2×#4112yes—
234C23×C11H(5,𝔽2)12yes—
235C23×C112−1+412yes—
236C23×C11C2511yescharacteristic
237C22×D11—24no—
238C22×D11—24no—
239C22×D11—24no—
240C22×D11—24no—
241C22×D11—24no—
242C22×D11—24no—
243C22×D11—24no—
244C22×D11—48no—
245C22×D11—48no—
246C22×D11—48no—
247C22×D11—48no—
248C22×D11—48no—
249C22×D11—48no—
250C22×D11—48no—
251C2×C4×C11—28no—
252C2×C4×C11—22no—
253C2×C4×C11—22no—
254C2×C4×C11—22no—
255C2×C4×C11—22no—
256C2×C4×C11—22no—
257C2×C4×C11—22no—
258C2×C4×C11—22no—
259C2×C4×C11—22no—
260C2×C4×C11—22no—
261C2×C4×C11—22no—
262C2×C4×C11—22no—
263C2×C4×C11—22no—
264C2×C4×C11—22no—
265C2×C4×C11—22no—
266C2×C4×C11—44no—
267C2×C4×C11—44no—
268C2×C4×C11—44no—
269C2×C4×C11—44no—
270C2×C4×C11—416no—
271C2×C4×C11#121116yes—
272C2×C4×C11C22×D414yes—
273C2×C4×C11C22×D414yes—
274C2×C4×C11C2×#4114yes—
275C2×C4×C11C2×#4114yes—
276C22⋊D11—432no—
277C2×Dic11—216no—
278C2×Dic11—216no—
279C2×Dic11—24no—
280C2×Dic11—24no—
281C2×Dic11—24no—
282C2×Dic11—24no—
283C2×Dic11—24no—
284C2×Dic11—24no—
285C2×Dic11—24no—
286C2×Dic11—48no—
287C2×Dic11C22×D418yes—
288C2×Dic11C22×D418yes—
289D44—416no—
290Dic22—232no—
291C23×Q8—111no—
292C23×D4—112no—
293C23×D4—111no—
294C24×C4C2×D1111yescharacteristic
295C2×#124—118no—
296C2×#124—228no—
297C2×C22⋊Q8—118no—
298C2×C22⋊Q8—228no—
299C2×#121—118no—
300C2×#121—228no—
301C2×C4×Q8—118no—
302C2×C4×Q8—228no—
303C22×C4⋊C4—111no—
304C22×C4⋊C4—112no—
305C22×C4⋊C4—112no—
306C22×C4⋊C4—111no—
307C22×C4⋊C4—112no—
308C22×C22⋊C4—112no—
309C22×C22⋊C4—112no—
310C22×C22⋊C4—112no—
311C22×C22⋊C4—112no—
312C22×C22⋊C4—112no—
313C22×C22⋊C4—112no—
314C22×C22⋊C4C2×D1111yescharacteristic
315C22×C22⋊C4C2×D1111yescharacteristic
316C22×C42—112no—
317C22×C42C2×D1111yescharacteristic
318C22×C42C2×D1111yescharacteristic
319#396—1116no—
320#394—1116no—
321#390—1116no—
322#390—118no—
323#389—118no—
324#388—1116no—
325C4×C22⋊C4C2×D1118yescomplemented
326C2×#95—112no—
327C2×#95C2×D1111yescharacteristic
328C2×#95C2×D1111yescharacteristic
329C22×C11—24no—
330C22×C11—24no—
331C22×C11—24no—
332C22×C11—42no—
333C22×C11—416no—
334C22×C11—42no—
335C22×C11—42no—
336C22×C11—42no—
337C22×C11C2×#12114yes—
338C22×C11#52914yes—
339C22×C11#54214yes—
340C22×C11#54614yes—
341C22×C11#54814yes—
342C22×C11#54814yes—
343C22×C11#55014yes—
344C22×C11C23×D411yescharacteristic
345C22×C11C23×D411yescharacteristic
346C22×C11C22×#4111yescharacteristic
347C22×C11C22×#4111yescharacteristic
348C22×C11C22×#4111yescharacteristic
349C22×C11C2×H(5,𝔽2)11yescharacteristic
350C22×C11C2×2−1+411yescharacteristic
351C2×D11—44no—
352C2×D11—44no—
353C2×D11—44no—
354C2×D11—44no—
355C2×D11—44no—
356C2×D11—44no—
357C2×D11—44no—
358C2×D11—816no—
359C4×C11—24no—
360C4×C11—42no—
361C4×C11—42no—
362C4×C11—42no—
363C4×C11—42no—
364C4×C11C2×#12118yes—
365Dic11—28no—
366Dic11—28no—
367Dic11—44no—
368C25—112no—
369C25C22×D1111yescharacteristic
370C22×Q8—118no—
371C22×Q8—114no—
372C22×Q8—224no—
373C22×Q8—224no—
374C22×D4—118no—
375C22×D4—114no—
376C22×D4—228no—
377C22×D4—228no—
378C22×D4—224no—
379C22×D4—224no—
380C22×D4—224no—
381C22×D4—444no—
382C23×C4—24no—
383C23×C4—24no—
384C23×C4—24no—
385C23×C4—112no—
386C23×C4—112no—
387C23×C4—112no—
388C23×C4C22×D1114yes—
389C23×C4C22×D1111yescharacteristic
390C23×C4C22×D1111yescharacteristic
391C23×C4C22×D1111yescharacteristic
392C23×C4C22×D1111yescharacteristic
393C23×C4C22×D1111yescharacteristic
394C23×C4C22×D1111yescharacteristic
395#124—2232no—
396C22⋊Q8—2232no—
397#121—2232no—
398C4×Q8—2232no—
399C2×C4⋊C4—118no—
400C2×C4⋊C4—114no—
401C2×C4⋊C4—118no—
402C2×C4⋊C4—118no—
403C2×C4⋊C4—114no—
404C2×C4⋊C4—118no—
405C2×C4⋊C4—228no—
406C2×C4⋊C4—228no—
407C2×C4⋊C4—224no—
408C2×C4⋊C4—444no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—118no—
412C2×C22⋊C4—118no—
413C2×C22⋊C4—118no—
414C2×C22⋊C4—118no—
415C2×C22⋊C4—118no—
416C2×C22⋊C4—118no—
417C2×C22⋊C4—228no—
418C2×C22⋊C4—228no—
419C2×C22⋊C4—228no—
420C2×C22⋊C4—228no—
421C2×C22⋊C4—228no—
422C2×C22⋊C4—444no—
423C2×C22⋊C4C22×D1114yes—
424C2×C22⋊C4C22×D1114yes—
425C2×C42—24no—
426C2×C42—24no—
427C2×C42—118no—
428C2×C42—228no—
429C2×C42C22×D1114yes—
430C2×C42C22×D1114yes—
431#95—118no—
432#95C22×D1114yes—
433#95C22×D1114yes—
434C2×C11—44no—
435C2×C11#242618yescomplemented
436C2×C11C22×#12111yescharacteristic
437C2×C11C2×#52911yescharacteristic
438C2×C11C2×#54211yescharacteristic
439C2×C11C2×#54611yescharacteristic
440C2×C11C2×#54811yescharacteristic
441C2×C11C2×#54811yescharacteristic
442C2×C11C2×#55011yescharacteristic
443D11—84no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—114no—
451C24—224no—
452C24—224no—
453C24—224no—
454C24—224no—
455C24—224no—
456C24—224no—
457C24—224no—
458C24#433914yes—
459C24C23×D1112yes—
460C24C23×D1111yescharacteristic
461C2×Q8—1116no—
462C2×Q8—2232no—
463C2×Q8—228no—
464C2×Q8—228no—
465C2×D4—2216no—
466C2×D4—2216no—
467C2×D4—228no—
468C2×D4—228no—
469C2×D4—228no—
470C2×D4—448no—
471C2×D4—4432no—
472C2×D4—4416no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—28no—
489C22×C4—28no—
490C22×C4—416no—
491C22×C4—118no—
492C22×C4—118no—
493C22×C4—114no—
494C22×C4—114no—
495C22×C4—114no—
496C22×C4—224no—
497C22×C4—224no—
498C22×C4—224no—
499C22×C4—224no—
500C22×C4—224no—
501C22×C4—224no—
502C22×C4—224no—
503C22×C4#433914yes—
504C22×C4#433914yes—
505C22×C4D4×D1114yes—
506C22×C4D4×D1114yes—
507C22×C4C23×D1112yes—
508C22×C4C23×D1112yes—
509C22×C4C23×D1112yes—
510C22×C4C23×D1112yes—
511C22×C4C23×D1112yes—
512C22×C4C23×D1112yes—
513C4⋊C4—1116no—
514C4⋊C4—2216no—
515C4⋊C4—2216no—
516C4⋊C4—228no—
517C4⋊C4—448no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—2216no—
521C22⋊C4—2216no—
522C22⋊C4—2216no—
523C22⋊C4—2216no—
524C22⋊C4—2216no—
525C22⋊C4—448no—
526C42—28no—
527C42—28no—
528C42—2216no—
529C11C2×#242611yescharacteristic, complemented, Sylow
530C23—28no—
531C23—28no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—224no—
544C23—224no—
545C23—224no—
546C23—224no—
547C23—224no—
548C23—224no—
549C23—224no—
550C23—448no—
551C23—448no—
552C23—448no—
553C23—448no—
554C23—448no—
555C23—448no—
556C23—448no—
557C23C2×#433914yes—
558C23C2×#433912yes—
559C23C2×#433912yes—
560C23C2×#433912yes—
561C23C2×D4×D1112yes—
562C23C2×D4×D1112yes—
563C23#6704212yes—
564C23#6704312yes—
565C23C24×D1111yescharacteristic
566Q8—2232no—
567D4—4432no—
568D4—4416no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—118no—
590C2×C4—118no—
591C2×C4—2216no—
592C2×C4—2216no—
593C2×C4—224no—
594C2×C4—224no—
595C2×C4—224no—
596C2×C4—224no—
597C2×C4—224no—
598C2×C4—224no—
599C2×C4—224no—
600C2×C4—448no—
601C2×C4#66937116yes—
602C2×C4C2×#433914yes—
603C2×C4C2×#433914yes—
604C2×C4C2×D4×D1114yes—
605C2×C4C2×D4×D1114yes—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—416no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—444no—
615C22—444no—
616C22—444no—
617C22—444no—
618C22—444no—
619C22—444no—
620C22—444no—
621C22—8816no—
622C22#1062301014yes—
623C22C2×#6693714yes—
624C22#1062306514yes—
625C22#1062308114yes—
626C22#1062308314yes—
627C22#1062308314yes—
628C22#1062308414yes—
629C22C22×#433911yescharacteristic
630C22C22×#433911yescharacteristic
631C22C22×#433911yescharacteristic
632C22C22×D4×D1111yescharacteristic
633C22C22×D4×D1111yescharacteristic
634C22C2×#6704211yescharacteristic
635C22C2×#6704311yescharacteristic
636C4—24no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—228no—
642C4—228no—
643C4—444no—
644C4C2×#6693718yes—
645C2—44no—
646C2—884no—
647C2#1307218218yesdirect product factor
648C2C22×#6693711yescharacteristic
649C2C2×#1062301011yescharacteristic
650C2C2×#1062306511yescharacteristic
651C2C2×#1062308111yescharacteristic
652C2C2×#1062308311yescharacteristic
653C2C2×#1062308311yescharacteristic
654C2C2×#1062308411yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C4×C11×C22⋊C4C2
socleC24×C11C24
prime radicalC22×#66983C2
hypercenterC2×C4×C22⋊C4D11
hypocenterC11C2×#2426
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#66928C2semidirect
2. C22×#66977C2semidirect
4. C22×C4×Dic22C2semidirect
5. #13072182C2direct (N × H)
6. C2×#10622289C2semidirect
7. C2×#10622572C2semidirect
8. C2×#10622552C2semidirect
9. C2×#10622570C2semidirect
10. C2×#10622541C2semidirect
11. C2×#10622282C2semidirect
12. C2×C4×C11×C22⋊C4C2semidirect
32. C2×#66977C22semidirect
38. C2×#66928C22semidirect
40. C2×C4×Dic22C22semidirect
41. C4×C11×C22⋊C4C22semidirect
45. #10622572C22semidirect
46. #10622570C22semidirect
47. #10622552C22semidirect
48. #10622541C22semidirect
49. #10622289C22semidirect
50. #10622282C22semidirect
76. C22×Dic22D4semidirect
186. C2×Dic22C2×D4semidirect
214. C2×C4×C22⋊C4D11semidirect
325. C4×C22⋊C4C2×D11semidirect
435. C2×C11#2426semidirect
529. C11C2×#2426semidirect
647. C2#13072182direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669282
C22×#669772
C22×#669832
C22×C4×Dic222
#130721822
C2×#106222892
C2×#106225722
C2×#106225522
C2×#106225702
C2×#106225412
C2×#106222822
C2×C4×C11×C22⋊C42
C2×#242611