Finite Group Databasebeta

← C2×#13072226 #14922407

Subgroups — C2×#13072226 #14922407

Order 2816.

6042 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4339C211yesmaximal, characteristic, complemented
2C22×#66982C211yesmaximal, characteristic, complemented
3C22×#66983C212yesmaximal, complemented
4#13072226C2116yesmaximal, direct product factor
5C2×#10622269C211yesmaximal, characteristic, complemented
6C2×#10622285C211yesmaximal, characteristic, complemented
7C2×#10622286C211yesmaximal, characteristic, complemented
8C2×#10622284C211yesmaximal, characteristic, complemented
9C2×#10622826C212yesmaximal, complemented
10C2×#10622278C212yesmaximal, complemented
11C2×#10622281C212yesmaximal, complemented
12C22×C11×#95C211yesmaximal, characteristic, complemented
13C24×C4×C11C2212yes—
14C24×C4×C11C2211yescharacteristic
15C23×C22⋊D11C2211yescharacteristic
16C23×C22⋊D11C2212yes—
17C22×#4339—28no—
18C22×#4339—216no—
19C22×#4339C2218yescomplemented
20C23×D44C2211yescharacteristic
21C23×D44C2212yes—
22C23×C4×D11C2212yes—
23C23×Dic22C2211yescharacteristic
24C22×C22⋊Dic11C2211yescharacteristic
25C2×#66983—216no—
26C2×#66983C22116yescomplemented
27C2×#66982—28no—
28C2×#66982C2218yescomplemented
29C22×(C2×D11)⋊C4C2212yes—
30C22×(C2×D11)⋊C4C2212yes—
31C22×(C2×D11)⋊C4C2211yescharacteristic
32C22×(C2×D11)⋊C4C2212yes—
33C22×(C2×D11)⋊C4C2211yescharacteristic
34C22×(C2×D11)⋊C4C2212yes—
35C22×C4⋊Dic11C2212yes—
36C22×Dic11⋊C4C2211yescharacteristic
37C22×Dic11⋊C4C2211yescharacteristic
38C2×C11×#95C2212yes—
39C2×C11×#95C2211yescharacteristic
40C2×C11×#95C2211yescharacteristic
41C2×C11×#95C2218yescomplemented
42#10622826C22116yescomplemented
43C2×C11⋊#95C2212yes—
44C2×C11⋊#95C2212yes—
45#10622286C2218yescomplemented
46#10622285C2218yescomplemented
47#10622284C2218yescomplemented
48#10622281C22116yescomplemented
49#10622278C22116yescomplemented
50#10622269C2218yescomplemented
51C25×C11C2311yescharacteristic
52C24×D11C2311yescharacteristic
53C24×D11C2312yes—
54C23×C4×C11—28no—
55C23×C4×C11—28no—
56C23×C4×C11—28no—
57C23×C4×C11—44no—
58C23×C4×C11D418yescomplemented
59C23×C4×C11C2312yes—
60C23×C4×C11C2318yes—
61C23×C4×C11C2311yescharacteristic
62C23×C4×C11C2311yescharacteristic
63C23×C4×C11C2314yes—
64C23×C4×C11C2312yes—
65C22×C22⋊D11—28no—
66C22×C22⋊D11—28no—
67C22×C22⋊D11—24no—
68C22×C22⋊D11—28no—
69C22×C22⋊D11D4116yescomplemented
70C22×C22⋊D11C2314yes—
71C22×C22⋊D11C2318yes—
72C23×Dic11C2311yescharacteristic
73C23×Dic11C2311yescharacteristic
74C23×Dic11C2311yescharacteristic
75C23×Dic11C2312yes—
76C2×#4339—2128no—
77C2×#4339—232no—
78C2×#4339—264no—
79C22×D44—28no—
80C22×D44—24no—
81C22×D44—28no—
82C22×D44—28no—
83C22×D44—44no—
84C22×D44C2314yes—
85C22×D44C2318yes—
86C22×C4×D11—28no—
87C22×C4×D11—28no—
88C22×C4×D11—28no—
89C22×C4×D11C2318yes—
90C22×Dic22—24no—
91C22×Dic22—28no—
92C22×Dic22C2314yes—
93C2×C22⋊Dic11—24no—
94C2×C22⋊Dic11C2314yes—
95#66983—264no—
96#66982—232no—
97C2×(C2×D11)⋊C4—28no—
98C2×(C2×D11)⋊C4—24no—
99C2×(C2×D11)⋊C4—24no—
100C2×(C2×D11)⋊C4—28no—
101C2×(C2×D11)⋊C4—28no—
102C2×(C2×D11)⋊C4—44no—
103C2×(C2×D11)⋊C4C2314yes—
104C2×(C2×D11)⋊C4C2318yes—
105C2×(C2×D11)⋊C4C2318yes—
106C2×(C2×D11)⋊C4C2318yes—
107C2×(C2×D11)⋊C4C2314yes—
108C2×(C2×D11)⋊C4C2318yes—
109C2×C4⋊Dic11—28no—
110C2×C4⋊Dic11C2318yes—
111C2×Dic11⋊C4—24no—
112C2×Dic11⋊C4—24no—
113C2×Dic11⋊C4C2314yes—
114C2×Dic11⋊C4C2314yes—
115C11×#95C2314yes—
116C11×#95C2318yes—
117C11×#95C2314yes—
118C11⋊#95C2318yes—
119C11⋊#95C2318yes—
120C2×#2430—111nomaximal, Sylow
121C24×C11—24no—
122C24×C11—22no—
123C24×C11—22no—
124C24×C11C2×D414yes—
125C24×C11C2×D418yes—
126C24×C11C2412yes—
127C24×C11C2411yescharacteristic
128C23×D11—24no—
129C23×D11—24no—
130C23×D11—24no—
131C23×D11—24no—
132C23×D11—24no—
133C23×D11—24no—
134C23×D11—22no—
135C23×D11—24no—
136C23×D11—24no—
137C23×D11—22no—
138C23×D11—22no—
139C23×D11C2×D418yes—
140C23×D11C2412yes—
141C23×D11C2414yes—
142C22×C4×C11—232no—
143C22×C4×C11—24no—
144C22×C4×C11—24no—
145C22×C4×C11—24no—
146C22×C4×C11—216no—
147C22×C4×C11—24no—
148C22×C4×C11—24no—
149C22×C4×C11—216no—
150C22×C4×C11—24no—
151C22×C4×C11—24no—
152C22×C4×C11—216no—
153C22×C4×C11—48no—
154C22×C4×C11—432no—
155C22×C4×C11C2×D414yes—
156C22×C4×C11C2×D414yes—
157C22×C4×C11C2×D4116yescomplemented
158C22×C4×C11#4118yescomplemented
159C22×C4×C11C2414yes—
160C22×C4×C11C2412yes—
161C22×C4×C11C2412yes—
162C22×C4×C11C2414yes—
163C2×C22⋊D11—264no—
164C2×C22⋊D11—216no—
165C2×C22⋊D11—216no—
166C2×C22⋊D11—28no—
167C2×C22⋊D11—216no—
168C2×C22⋊D11—416no—
169C2×C22⋊D11C2×D4132yescomplemented
170C22×Dic11—24no—
171C22×Dic11—24no—
172C22×Dic11—22no—
173C22×Dic11—22no—
174C22×Dic11—22no—
175C22×Dic11—24no—
176C22×Dic11—24no—
177C22×Dic11—22no—
178C22×Dic11—22no—
179C22×Dic11C2×D418yes—
180C22×Dic11C2412yes—
181C22×Dic11C2412yes—
182C22×Dic11C2412yes—
183C22×Dic11C2414yes—
184#4339—2256no—
185C2×D44—232no—
186C2×D44—216no—
187C2×D44—216no—
188C2×D44—28no—
189C2×D44—216no—
190C2×D44—48no—
191C2×D44—432no—
192C2×C4×D11—264no—
193C2×C4×D11—216no—
194C2×C4×D11—216no—
195C2×C4×D11—216no—
196C2×Dic22—232no—
197C2×Dic22—216no—
198C2×Dic22—28no—
199C22⋊Dic11—28no—
200(C2×D11)⋊C4—216no—
201(C2×D11)⋊C4—216no—
202(C2×D11)⋊C4—28no—
203(C2×D11)⋊C4—28no—
204(C2×D11)⋊C4—216no—
205(C2×D11)⋊C4—48no—
206C4⋊Dic11—216no—
207Dic11⋊C4—28no—
208Dic11⋊C4—28no—
209C23×#41—111no—
210C22×#123—111no—
211C22×#121—112no—
212#2430—1116no—
213C2×#395—111no—
214C2×#394—111no—
215C2×#393—111no—
216C2×#392—111no—
217C2×#386—112no—
218C2×#386—112no—
219C2×#385—112no—
220C22×#95D1111yescharacteristic, complemented
221C23×C11—28no—
222C23×C11—28no—
223C23×C11—216no—
224C23×C11—216no—
225C23×C11—24no—
226C23×C11—22no—
227C23×C11—22no—
228C23×C11#120116yes—
229C23×C11C22×D414yes—
230C23×C11C22×D412yes—
231C23×C11C22×D418yes—
232C23×C11C22×D414yes—
233C23×C11C2×#4114yes—
234C23×C11H(5,𝔽2)12yes—
235C23×C112−1+412yes—
236C23×C11C2511yescharacteristic
237C22×D11—216no—
238C22×D11—216no—
239C22×D11—216no—
240C22×D11—24no—
241C22×D11—24no—
242C22×D11—24no—
243C22×D11—24no—
244C22×D11—24no—
245C22×D11—24no—
246C22×D11—22no—
247C22×D11—24no—
248C22×D11—24no—
249C22×D11—22no—
250C22×D11—22no—
251C22×D11—48no—
252C22×D11—48no—
253C22×D11—48no—
254C22×D11—48no—
255C22×D11—44no—
256C22×D11—44no—
257C22×D11—44no—
258C22×D11—48no—
259C22×D11—48no—
260C22×D11C22×D418yes—
261C2×C4×C11—28no—
262C2×C4×C11—28no—
263C2×C4×C11—232no—
264C2×C4×C11—216no—
265C2×C4×C11—24no—
266C2×C4×C11—24no—
267C2×C4×C11—24no—
268C2×C4×C11—24no—
269C2×C4×C11—24no—
270C2×C4×C11—24no—
271C2×C4×C11—24no—
272C2×C4×C11—432no—
273C2×C4×C11—48no—
274C2×C4×C11C22×D414yes—
275C2×C4×C11C22×D414yes—
276C2×C4×C11C2×#4118yescomplemented
277C22⋊D11—264no—
278C22⋊D11—416no—
279C2×Dic11—216no—
280C2×Dic11—24no—
281C2×Dic11—24no—
282C2×Dic11—22no—
283C2×Dic11—22no—
284C2×Dic11—22no—
285C2×Dic11—24no—
286C2×Dic11—24no—
287C2×Dic11—22no—
288C2×Dic11—22no—
289C2×Dic11—44no—
290C2×Dic11—44no—
291C2×Dic11—44no—
292C2×Dic11C22×D418yes—
293D44—232no—
294D44—432no—
295C4×D11—264no—
296Dic22—232no—
297C22×#41—118no—
298C22×#41—2216no—
299C22×#41—228no—
300C23×Q8—111no—
301C23×D4—112no—
302C23×D4—111no—
303C23×D4—111no—
304C23×D4—112no—
305C24×C4—112no—
306C24×C4C2×D1112yes—
307C24×C4C2×D1111yescharacteristic
308C2×#123—118no—
309C2×#123—228no—
310C2×#121—1116no—
311C2×#121—2216no—
312C22×C4⋊C4—111no—
313C22×C4⋊C4—111no—
314C22×C4⋊C4—112no—
315C22×C22⋊C4—111no—
316C22×C22⋊C4—111no—
317C22×C22⋊C4—112no—
318C22×C22⋊C4—112no—
319C22×C22⋊C4—112no—
320C22×C22⋊C4—111no—
321C22×C22⋊C4—112no—
322#395—118no—
323#394—118no—
324#393—118no—
325#392—118no—
326#386—1116no—
327#386—1116no—
328#385—1116no—
329C2×#95—112no—
330C2×#95—112no—
331C2×#95C2×D1111yescharacteristic
332C2×#95C2×D1111yescharacteristic
333C2×#95C2×D1112yes—
334C2×#95C2×D1118yescomplemented
335C22×C11—24no—
336C22×C11—24no—
337C22×C11—232no—
338C22×C11—28no—
339C22×C11—28no—
340C22×C11C2×#12018yes—
341C22×C11C2×#12014yes—
342C22×C11#53214yes—
343C22×C11#53714yes—
344C22×C11#54618yes—
345C22×C11#54718yes—
346C22×C11C23×D411yescharacteristic
347C22×C11C23×D412yes—
348C22×C11C22×#4112yes—
349C22×C11C2×H(5,𝔽2)11yescharacteristic
350C22×C11C2×2−1+411yescharacteristic
351C2×D11—28no—
352C2×D11—28no—
353C2×D11—28no—
354C2×D11—44no—
355C2×D11—44no—
356C2×D11—432no—
357C2×D11—44no—
358C2×D11—44no—
359C2×D11—42no—
360C2×D11—42no—
361C2×D11—42no—
362C2×D11—44no—
363C2×D11—44no—
364C2×D11—88no—
365C4×C11—24no—
366C4×C11—24no—
367C4×C11—28no—
368C4×C11—44no—
369Dic11—28no—
370Dic11—42no—
371Dic11—42no—
372Dic11—42no—
373C25—112no—
374C25—111no—
375C25C22×D1111yescharacteristic
376C2×#41—2264no—
377C2×#41—22128no—
378C2×#41—2232no—
379C22×Q8—114no—
380C22×Q8—228no—
381C22×Q8—224no—
382C22×D4—1116no—
383C22×D4—118no—
384C22×D4—114no—
385C22×D4—114no—
386C22×D4—118no—
387C22×D4—228no—
388C22×D4—228no—
389C22×D4—228no—
390C22×D4—228no—
391C22×D4—224no—
392C22×D4—228no—
393C22×D4—224no—
394C22×D4—228no—
395C22×D4—444no—
396C23×C4—28no—
397C23×C4—28no—
398C23×C4—28no—
399C23×C4—44no—
400C23×C4—112no—
401C23×C4—111no—
402C23×C4—111no—
403C23×C4—111no—
404C23×C4—118no—
405C23×C4—228no—
406C23×C4—228no—
407C23×C4—228no—
408C23×C4D4418yescomplemented
409C23×C4C22×D1112yes—
410C23×C4C22×D1118yes—
411C23×C4C22×D1111yescharacteristic
412C23×C4C22×D1111yescharacteristic
413C23×C4C22×D1114yes—
414C23×C4C22×D1112yes—
415#123—2232no—
416#121—2264no—
417C2×C4⋊C4—118no—
418C2×C4⋊C4—114no—
419C2×C4⋊C4—114no—
420C2×C4⋊C4—228no—
421C2×C4⋊C4—224no—
422C2×C4⋊C4—224no—
423C2×C22⋊C4—114no—
424C2×C22⋊C4—118no—
425C2×C22⋊C4—118no—
426C2×C22⋊C4—114no—
427C2×C22⋊C4—114no—
428C2×C22⋊C4—118no—
429C2×C22⋊C4—118no—
430C2×C22⋊C4—228no—
431C2×C22⋊C4—228no—
432C2×C22⋊C4—224no—
433C2×C22⋊C4—224no—
434C2×C22⋊C4—224no—
435C2×C22⋊C4—228no—
436C2×C22⋊C4—444no—
437#95—118no—
438#95—118no—
439#95C22×D1118yes—
440#95C22×D1114yes—
441#95C22×D1114yes—
442C2×C11—28no—
443C2×C11#243018yescomplemented
444C2×C11C22×#12011yescharacteristic
445C2×C11C2×#53211yescharacteristic
446C2×C11C2×#53711yescharacteristic
447C2×C11C2×#54612yes—
448C2×C11C2×#54712yes—
449D11—48no—
450D11—82no—
451C24—24no—
452C24—22no—
453C24—22no—
454C24—118no—
455C24—114no—
456C24—112no—
457C24—224no—
458C24—224no—
459C24—224no—
460C24—224no—
461C24—224no—
462C24—224no—
463C24—222no—
464C24—224no—
465C24—224no—
466C24—222no—
467C24—222no—
468C24C2×D4414yes—
469C24D4×D1118yes—
470C24C23×D1112yes—
471C24C23×D1111yescharacteristic
472#41—22256no—
473C2×Q8—2216no—
474C2×Q8—2232no—
475C2×Q8—228no—
476C2×D4—1132no—
477C2×D4—2216no—
478C2×D4—2232no—
479C2×D4—2216no—
480C2×D4—2264no—
481C2×D4—2216no—
482C2×D4—2216no—
483C2×D4—228no—
484C2×D4—2216no—
485C2×D4—228no—
486C2×D4—2216no—
487C2×D4—448no—
488C2×D4—4432no—
489C2×D4—4416no—
490C22×C4—232no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—216no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—216no—
499C22×C4—24no—
500C22×C4—216no—
501C22×C4—48no—
502C22×C4—432no—
503C22×C4—118no—
504C22×C4—114no—
505C22×C4—112no—
506C22×C4—112no—
507C22×C4—112no—
508C22×C4—2216no—
509C22×C4—2216no—
510C22×C4—2264no—
511C22×C4—224no—
512C22×C4—224no—
513C22×C4—2216no—
514C22×C4—222no—
515C22×C4—222no—
516C22×C4—222no—
517C22×C4—224no—
518C22×C4—224no—
519C22×C4—222no—
520C22×C4—222no—
521C22×C4C2×D4414yes—
522C22×C4C2×D4414yes—
523C22×C4C2×D44116yescomplemented
524C22×C4#433918yescomplemented
525C22×C4C23×D1114yes—
526C22×C4C23×D1112yes—
527C22×C4C23×D1112yes—
528C22×C4C23×D1114yes—
529C4⋊C4—2216no—
530C4⋊C4—228no—
531C4⋊C4—228no—
532C22⋊C4—2216no—
533C22⋊C4—2216no—
534C22⋊C4—228no—
535C22⋊C4—228no—
536C22⋊C4—228no—
537C22⋊C4—2216no—
538C22⋊C4—448no—
539C11C2×#243011yescharacteristic, complemented, Sylow
540C23—28no—
541C23—216no—
542C23—216no—
543C23—28no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—118no—
548C23—2216no—
549C23—2216no—
550C23—2216no—
551C23—224no—
552C23—224no—
553C23—224no—
554C23—224no—
555C23—224no—
556C23—224no—
557C23—222no—
558C23—224no—
559C23—224no—
560C23—222no—
561C23—222no—
562C23—448no—
563C23—448no—
564C23—448no—
565C23—448no—
566C23—444no—
567C23—444no—
568C23—448no—
569C23—448no—
570C23—444no—
571C23#66935116yes—
572C23C22×D4414yes—
573C23C22×D4412yes—
574C23C2×#433914yes—
575C23C2×D4×D1118yes—
576C23C2×D4×D1114yes—
577C23#6703712yes—
578C23#6704012yes—
579C23C24×D1111yescharacteristic
580Q8—2232no—
581D4—2232no—
582D4—2264no—
583D4—4432no—
584D4—4416no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—232no—
588C2×C4—216no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—432no—
597C2×C4—48no—
598C2×C4—118no—
599C2×C4—2264no—
600C2×C4—2216no—
601C2×C4—224no—
602C2×C4—224no—
603C2×C4—222no—
604C2×C4—222no—
605C2×C4—222no—
606C2×C4—224no—
607C2×C4—224no—
608C2×C4—222no—
609C2×C4—222no—
610C2×C4—444no—
611C2×C4—444no—
612C2×C4—444no—
613C2×C4C22×D4414yes—
614C2×C4C22×D4414yes—
615C2×C4C2×#433918yescomplemented
616C22—24no—
617C22—24no—
618C22—232no—
619C22—28no—
620C22—28no—
621C22—228no—
622C22—228no—
623C22—228no—
624C22—444no—
625C22—444no—
626C22—4432no—
627C22—444no—
628C22—444no—
629C22—442no—
630C22—442no—
631C22—444no—
632C22—444no—
633C22—442no—
634C22—888no—
635C22#1062300814yes—
636C22C2×#6693518yes—
637C22C2×#6693514yes—
638C22#1062302118yes—
639C22#1062303818yes—
640C22#1062304114yes—
641C22C23×D4411yescharacteristic
642C22C22×#433912yes—
643C22C22×D4×D1112yes—
644C22C2×#6703711yescharacteristic
645C22C2×#6704011yescharacteristic
646C4—28no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—228no—
651C4—442no—
652C4—442no—
653C4—442no—
654C2—28no—
655C2—448no—
656C2—882no—
657C2#1307222618yesdirect product factor
658C2C22×#6693511yescharacteristic
659C2C2×#1062300811yescharacteristic
660C2C2×#1062304111yescharacteristic
661C2C2×#1062302112yes—
662C2C2×#1062303812yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC22×C11×#95C2
socleC24×C11C24
hypercenterC22×#95D11
hypocenterC11C2×#2430
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C23×#4339C2semidirect
2. C22×#66982C2semidirect
3. C22×#66983C2semidirect
4. #13072226C2direct (N × H)
5. C2×#10622269C2semidirect
6. C2×#10622285C2semidirect
7. C2×#10622286C2semidirect
8. C2×#10622284C2semidirect
9. C2×#10622826C2semidirect
10. C2×#10622278C2semidirect
11. C2×#10622281C2semidirect
12. C22×C11×#95C2semidirect
19. C22×#4339C22semidirect
26. C2×#66983C22semidirect
28. C2×#66982C22semidirect
41. C2×C11×#95C22semidirect
42. #10622826C22semidirect
45. #10622286C22semidirect
46. #10622285C22semidirect
47. #10622284C22semidirect
48. #10622281C22semidirect
49. #10622278C22semidirect
50. #10622269C22semidirect
58. C23×C4×C11D4semidirect
69. C22×C22⋊D11D4semidirect
157. C22×C4×C11C2×D4semidirect
158. C22×C4×C11#41semidirect
169. C2×C22⋊D11C2×D4semidirect
220. C22×#95D11semidirect
276. C2×C4×C11C2×#41semidirect
334. C2×#95C2×D11semidirect
408. C23×C4D44semidirect
443. C2×C11#2430semidirect
523. C22×C4C2×D44semidirect
524. C22×C4#4339semidirect
539. C11C2×#2430semidirect
615. C2×C4C2×#4339semidirect
657. C2#13072226direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43392
C22×#669822
C22×#669832
#130722262
C2×#106222692
C2×#106222852
C2×#106222862
C2×#106222842
C2×#106228262
C2×#106222782
C2×#106222812
C22×C11×#952
C2×#243011