Finite Group Databasebeta

← C2×#13072221 #14922404

Subgroups — C2×#13072221 #14922404

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×#4341C211yesmaximal, characteristic, complemented
2C22×#66947C211yesmaximal, characteristic, complemented
3C22×#66937C212yesmaximal, complemented
4#13072221C2116yesmaximal, direct product factor
5C2×#10622592C211yesmaximal, characteristic, complemented
6C2×#10622286C211yesmaximal, characteristic, complemented
7C2×#10622836C211yesmaximal, characteristic, complemented
8C2×#10622571C212yesmaximal, complemented
9C2×#10622569C212yesmaximal, complemented
10C2×#10622559C212yesmaximal, complemented
11C2×D11×#95C211yesmaximal, characteristic, complemented
12C2×C11×#394C211yesmaximal, characteristic, complemented
13C23×D4×C11C2211yescharacteristic
14C23×C22⋊D11C2212yes—
15C23×C22⋊D11C2212yes—
16C24×Dic11C2212yes—
17C22×#4341—216no—
18C22×#4341—28no—
19C22×#4341C2218yescomplemented
20C23×D44C2211yescharacteristic
21C23×C4×D11C2212yes—
22C23×C4×D11C2211yescharacteristic
23C23×Dic22C2211yescharacteristic
24C22×C11×C4⋊C4C2211yescharacteristic
25C22×C11×C22⋊C4C2212yes—
26C22×C11×C22⋊C4C2212yes—
27C22×C22⋊Dic11C2212yes—
28C22×(C2×D11)⋊C4C2211yescharacteristic
29C22×(C2×D11)⋊C4C2211yescharacteristic
30C22×(C2×D11)⋊C4C2212yes—
31C22×(C2×D11)⋊C4C2211yescharacteristic
32C22×Dic11⋊C4C2211yescharacteristic
33C22×Dic11⋊C4C2212yes—
34C2×#66947—28no—
35C2×#66947C2218yescomplemented
36C2×#66937—216no—
37C2×#66937C22116yescomplemented
38C11×#394C2218yescomplemented
39C2×C11×#95C2211yescharacteristic
40#10622836C2218yescomplemented
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2211yescharacteristic
43C2×C11⋊#95C2212yes—
44C2×C11⋊#95C2212yes—
45#10622592C2218yescomplemented
46#10622571C22116yescomplemented
47#10622569C22116yescomplemented
48#10622559C22116yescomplemented
49#10622286C2218yescomplemented
50D11×#95C2218yescomplemented
51C25×C11C2312yes—
52C24×D11C2311yescharacteristic
53C24×D11C2311yescharacteristic
54C22×D4×C11—28no—
55C22×D4×C11—24no—
56C22×D4×C11C2314yes—
57C23×C4×C11C2311yescharacteristic
58C23×C4×C11C2311yescharacteristic
59C23×C4×C11C2312yes—
60C23×C4×C11C2311yescharacteristic
61C22×C22⋊D11—28no—
62C22×C22⋊D11—28no—
63C22×C22⋊D11—28no—
64C22×C22⋊D11—28no—
65C22×C22⋊D11—44no—
66C22×C22⋊D11D4116yescomplemented
67C22×C22⋊D11C2318yes—
68C22×C22⋊D11C2318yes—
69C23×Dic11—28no—
70C23×Dic11—28no—
71C23×Dic11—28no—
72C23×Dic11C2311yescharacteristic
73C23×Dic11C2318yes—
74C23×Dic11C2312yes—
75C23×Dic11C2311yescharacteristic
76C23×Dic11C2312yes—
77C2×#4341—2128no—
78C2×#4341—264no—
79C2×#4341—232no—
80C22×D44—28no—
81C22×D44—24no—
82C22×D44C2314yes—
83C22×C4×D11—28no—
84C22×C4×D11—28no—
85C22×C4×D11—28no—
86C22×C4×D11—44no—
87C22×C4×D11D418yescomplemented
88C22×C4×D11C2318yes—
89C22×C4×D11C2314yes—
90C22×Dic22—24no—
91C22×Dic22—28no—
92C22×Dic22C2314yes—
93C2×C11×C4⋊C4—24no—
94C2×C11×C4⋊C4C2314yes—
95C2×C11×C22⋊C4—28no—
96C2×C11×C22⋊C4—44no—
97C2×C11×C22⋊C4C2318yes—
98C2×C11×C22⋊C4C2318yes—
99C2×C22⋊Dic11—28no—
100C2×C22⋊Dic11C2318yes—
101C2×(C2×D11)⋊C4—24no—
102C2×(C2×D11)⋊C4—24no—
103C2×(C2×D11)⋊C4—28no—
104C2×(C2×D11)⋊C4—24no—
105C2×(C2×D11)⋊C4C2318yes—
106C2×(C2×D11)⋊C4C2314yes—
107C2×(C2×D11)⋊C4C2314yes—
108C2×(C2×D11)⋊C4C2314yes—
109C2×Dic11⋊C4—28no—
110C2×Dic11⋊C4—24no—
111C2×Dic11⋊C4C2318yes—
112C2×Dic11⋊C4C2314yes—
113#66947—232no—
114#66937—264no—
115C11×#95C2314yes—
116C11⋊#95C2318yes—
117C11⋊#95C2314yes—
118C11⋊#95C2318yes—
119C11⋊#95C2318yes—
120C2×#2430—111nomaximal, Sylow
121C24×C11—24no—
122C24×C11—24no—
123C24×C11—24no—
124C24×C11—24no—
125C24×C11—24no—
126C24×C11—24no—
127C24×C11C2×D418yes—
128C24×C11C2414yes—
129C24×C11C2411yescharacteristic
130C23×D11—24no—
131C23×D11—22no—
132C23×D11—22no—
133C23×D11—22no—
134C23×D11—24no—
135C23×D11—24no—
136C23×D11—22no—
137C23×D11—22no—
138C23×D11C2×D414yes—
139C23×D11C2×D418yes—
140C23×D11C2412yes—
141C23×D11C2412yes—
142C2×D4×C11—232no—
143C2×D4×C11—216no—
144C2×D4×C11—28no—
145C22×C4×C11—24no—
146C22×C4×C11—24no—
147C22×C4×C11—22no—
148C22×C4×C11—22no—
149C22×C4×C11—24no—
150C22×C4×C11—24no—
151C22×C4×C11—22no—
152C22×C4×C11—24no—
153C22×C4×C11—22no—
154C22×C4×C11C2×D414yes—
155C22×C4×C11C2414yes—
156C22×C4×C11C2412yes—
157C22×C4×C11C2412yes—
158C22×C4×C11C2412yes—
159C2×C22⋊D11—216no—
160C2×C22⋊D11—216no—
161C2×C22⋊D11—264no—
162C2×C22⋊D11—216no—
163C2×C22⋊D11—216no—
164C2×C22⋊D11—48no—
165C2×C22⋊D11—432no—
166C2×C22⋊D11C2×D4132yescomplemented
167C22×Dic11—264no—
168C22×Dic11—216no—
169C22×Dic11—216no—
170C22×Dic11—24no—
171C22×Dic11—24no—
172C22×Dic11—24no—
173C22×Dic11—24no—
174C22×Dic11—216no—
175C22×Dic11—22no—
176C22×Dic11—24no—
177C22×Dic11—24no—
178C22×Dic11C2×D418yes—
179C22×Dic11C2×D414yes—
180C22×Dic11#4118yescomplemented
181C22×Dic11C2412yes—
182C22×Dic11C2414yes—
183C22×Dic11C2412yes—
184C22×Dic11C2414yes—
185#4341—2256no—
186C2×D44—216no—
187C2×D44—28no—
188C2×D44—416no—
189C2×C4×D11—216no—
190C2×C4×D11—232no—
191C2×C4×D11—216no—
192C2×C4×D11—216no—
193C2×C4×D11—48no—
194C2×C4×D11—432no—
195C2×C4×D11C2×D4116yescomplemented
196C2×Dic22—232no—
197C2×Dic22—216no—
198C2×Dic22—28no—
199C11×C4⋊C4—28no—
200C11×C22⋊C4—216no—
201C11×C22⋊C4—48no—
202C22⋊Dic11—216no—
203(C2×D11)⋊C4—216no—
204(C2×D11)⋊C4—28no—
205(C2×D11)⋊C4—28no—
206(C2×D11)⋊C4—28no—
207Dic11⋊C4—216no—
208Dic11⋊C4—28no—
209C23×#41—111no—
210C22×#123—111no—
211C22×#121—112no—
212#2430—1116no—
213C2×#395—111no—
214C2×#394D1111yescharacteristic, complemented
215C2×#393—111no—
216C2×#392—111no—
217C2×#386—112no—
218C2×#386—112no—
219C2×#385—112no—
220C22×#95—111no—
221C23×C11—216no—
222C23×C11—216no—
223C23×C11—216no—
224C23×C11—24no—
225C23×C11—24no—
226C23×C11—24no—
227C23×C11—24no—
228C23×C11—24no—
229C23×C11—24no—
230C23×C11—48no—
231C23×C11—48no—
232C23×C11—48no—
233C23×C11—48no—
234C23×C11C22×D418yes—
235C23×C11C22×D412yes—
236C23×C11C22×D414yes—
237C23×C11C2×#4114yes—
238C23×C11H(5,𝔽2)12yes—
239C23×C112−1+412yes—
240C23×C11C2511yescharacteristic
241C22×D11—28no—
242C22×D11—28no—
243C22×D11—216no—
244C22×D11—216no—
245C22×D11—24no—
246C22×D11—22no—
247C22×D11—22no—
248C22×D11—22no—
249C22×D11—24no—
250C22×D11—24no—
251C22×D11—22no—
252C22×D11—22no—
253C22×D11—44no—
254C22×D11—44no—
255C22×D11—44no—
256C22×D11—48no—
257C22×D11—48no—
258C22×D11#120116yes—
259C22×D11C22×D414yes—
260C22×D11C22×D418yes—
261D4×C11—232no—
262C2×C4×C11—28no—
263C2×C4×C11—24no—
264C2×C4×C11—24no—
265C2×C4×C11—22no—
266C2×C4×C11—22no—
267C2×C4×C11—24no—
268C2×C4×C11—24no—
269C2×C4×C11—22no—
270C2×C4×C11—24no—
271C2×C4×C11—22no—
272C2×C4×C11—44no—
273C2×C4×C11—44no—
274C2×C4×C11—48no—
275C2×C4×C11C22×D414yes—
276C22⋊D11—264no—
277C22⋊D11—432no—
278C2×Dic11—264no—
279C2×Dic11—216no—
280C2×Dic11—28no—
281C2×Dic11—216no—
282C2×Dic11—24no—
283C2×Dic11—24no—
284C2×Dic11—24no—
285C2×Dic11—24no—
286C2×Dic11—24no—
287C2×Dic11—22no—
288C2×Dic11—24no—
289C2×Dic11—44no—
290C2×Dic11C22×D418yes—
291C2×Dic11C22×D414yes—
292C2×Dic11C2×#4118yescomplemented
293D44—416no—
294C4×D11—232no—
295C4×D11—432no—
296Dic22—232no—
297C22×#41—118no—
298C22×#41—228no—
299C22×#41—2216no—
300C23×Q8—111no—
301C23×D4—112no—
302C23×D4—112no—
303C23×D4—111no—
304C23×D4C2×D1111yescharacteristic
305C24×C4—111no—
306C24×C4—112no—
307C24×C4—112no—
308C2×#123—118no—
309C2×#123—228no—
310C2×#121—1116no—
311C2×#121—2216no—
312C22×C4⋊C4—112no—
313C22×C4⋊C4—111no—
314C22×C4⋊C4C2×D1111yescharacteristic
315C22×C22⋊C4—112no—
316C22×C22⋊C4—111no—
317C22×C22⋊C4—112no—
318C22×C22⋊C4—111no—
319C22×C22⋊C4—111no—
320C22×C22⋊C4C2×D1112yes—
321C22×C22⋊C4C2×D1112yes—
322#395—118no—
323#394C2×D1118yescomplemented
324#393—118no—
325#392—118no—
326#386—1116no—
327#386—1116no—
328#385—1116no—
329C2×#95—111no—
330C2×#95—112no—
331C2×#95—112no—
332C2×#95—112no—
333C2×#95—118no—
334C2×#95C2×D1111yescharacteristic
335C22×C11—28no—
336C22×C11—28no—
337C22×C11—28no—
338C22×C11—44no—
339C22×C11—44no—
340C22×C11—432no—
341C22×C11—44no—
342C22×C11—44no—
343C22×C11C2×#12014yes—
344C22×C11#53214yes—
345C22×C11#53714yes—
346C22×C11#54618yes—
347C22×C11#54718yes—
348C22×C11C23×D411yescharacteristic
349C22×C11C23×D412yes—
350C22×C11C22×#4112yes—
351C22×C11C2×H(5,𝔽2)11yescharacteristic
352C22×C11C2×2−1+411yescharacteristic
353C2×D11—24no—
354C2×D11—24no—
355C2×D11—232no—
356C2×D11—28no—
357C2×D11—28no—
358C2×D11—42no—
359C2×D11—42no—
360C2×D11—42no—
361C2×D11—44no—
362C2×D11—44no—
363C2×D11—88no—
364C2×D11C2×#12018yes—
365C4×C11—24no—
366C4×C11—42no—
367C4×C11—42no—
368C4×C11—44no—
369Dic11—28no—
370Dic11—24no—
371Dic11—28no—
372Dic11—42no—
373C25—111no—
374C25—111no—
375C25C22×D1112yes—
376C2×#41—2232no—
377C2×#41—2264no—
378C2×#41—22128no—
379C22×Q8—114no—
380C22×Q8—228no—
381C22×Q8—224no—
382C22×D4—28no—
383C22×D4—24no—
384C22×D4—1116no—
385C22×D4—118no—
386C22×D4—118no—
387C22×D4—114no—
388C22×D4—228no—
389C22×D4—224no—
390C22×D4—228no—
391C22×D4—228no—
392C22×D4—228no—
393C22×D4—228no—
394C22×D4—444no—
395C22×D4C22×D1114yes—
396C23×C4—111no—
397C23×C4—112no—
398C23×C4—112no—
399C23×C4—118no—
400C23×C4—114no—
401C23×C4—118no—
402C23×C4—111no—
403C23×C4—118no—
404C23×C4—228no—
405C23×C4—228no—
406C23×C4—228no—
407C23×C4—228no—
408C23×C4—228no—
409C23×C4—228no—
410C23×C4—444no—
411C23×C4C22×D1111yescharacteristic
412C23×C4C22×D1112yes—
413C23×C4C22×D1111yescharacteristic
414C23×C4C22×D1111yescharacteristic
415#123—2232no—
416#121—2264no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—118no—
419C2×C4⋊C4—114no—
420C2×C4⋊C4—228no—
421C2×C4⋊C4—224no—
422C2×C4⋊C4C22×D1114yes—
423C2×C22⋊C4—28no—
424C2×C22⋊C4—44no—
425C2×C22⋊C4—118no—
426C2×C22⋊C4—118no—
427C2×C22⋊C4—114no—
428C2×C22⋊C4—114no—
429C2×C22⋊C4—114no—
430C2×C22⋊C4—228no—
431C2×C22⋊C4—224no—
432C2×C22⋊C4—224no—
433C2×C22⋊C4—228no—
434C2×C22⋊C4—224no—
435C2×C22⋊C4C22×D1118yes—
436C2×C22⋊C4C22×D1118yes—
437#95—118no—
438#95—114no—
439#95—118no—
440#95—118no—
441#95C22×D1114yes—
442C2×C11—48no—
443C2×C11#243018yescomplemented
444C2×C11C22×#12011yescharacteristic
445C2×C11C2×#53211yescharacteristic
446C2×C11C2×#53711yescharacteristic
447C2×C11C2×#54612yes—
448C2×C11C2×#54712yes—
449D11—28no—
450D11—82no—
451C24—24no—
452C24—24no—
453C24—24no—
454C24—24no—
455C24—24no—
456C24—24no—
457C24—114no—
458C24—118no—
459C24—112no—
460C24—112no—
461C24—222no—
462C24—224no—
463C24—222no—
464C24—222no—
465C24—224no—
466C24—222no—
467C24—224no—
468C24—222no—
469C24D4×D1118yes—
470C24C23×D1114yes—
471C24C23×D1111yescharacteristic
472#41—22256no—
473C2×Q8—2216no—
474C2×Q8—2232no—
475C2×Q8—228no—
476C2×D4—232no—
477C2×D4—216no—
478C2×D4—28no—
479C2×D4—1132no—
480C2×D4—2216no—
481C2×D4—228no—
482C2×D4—2216no—
483C2×D4—2264no—
484C2×D4—2216no—
485C2×D4—2216no—
486C2×D4—2216no—
487C2×D4—4416no—
488C2×D4—448no—
489C2×D4—4432no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—114no—
500C22×C4—118no—
501C22×C4—118no—
502C22×C4—114no—
503C22×C4—112no—
504C22×C4—114no—
505C22×C4—1116no—
506C22×C4—112no—
507C22×C4—2216no—
508C22×C4—2232no—
509C22×C4—2216no—
510C22×C4—2216no—
511C22×C4—2216no—
512C22×C4—2216no—
513C22×C4—2264no—
514C22×C4—224no—
515C22×C4—224no—
516C22×C4—224no—
517C22×C4—224no—
518C22×C4—2216no—
519C22×C4—224no—
520C22×C4—222no—
521C22×C4—224no—
522C22×C4—448no—
523C22×C4—4432no—
524C22×C4D4×D1114yes—
525C22×C4C23×D1114yes—
526C22×C4C23×D1112yes—
527C22×C4C23×D1112yes—
528C22×C4C23×D1112yes—
529C4⋊C4—28no—
530C4⋊C4—2216no—
531C4⋊C4—228no—
532C22⋊C4—216no—
533C22⋊C4—48no—
534C22⋊C4—2216no—
535C22⋊C4—228no—
536C22⋊C4—228no—
537C22⋊C4—2216no—
538C22⋊C4—228no—
539C11C2×#243011yescharacteristic, complemented, Sylow
540C23—216no—
541C23—216no—
542C23—216no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—1116no—
554C23—114no—
555C23—118no—
556C23—228no—
557C23—2216no—
558C23—2216no—
559C23—222no—
560C23—224no—
561C23—222no—
562C23—228no—
563C23—222no—
564C23—224no—
565C23—222no—
566C23—224no—
567C23—222no—
568C23—444no—
569C23—444no—
570C23—448no—
571C23—448no—
572C23—444no—
573C23C2×D4×D1118yes—
574C23C2×D4×D1112yes—
575C23C2×D4×D1114yes—
576C23#6703712yes—
577C23#41×D1114yes—
578C23#6704312yes—
579C23C24×D1111yescharacteristic
580Q8—2232no—
581D4—232no—
582D4—2264no—
583D4—4416no—
584D4—4432no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—114no—
599C2×C4—118no—
600C2×C4—118no—
601C2×C4—2232no—
602C2×C4—2264no—
603C2×C4—228no—
604C2×C4—2216no—
605C2×C4—2216no—
606C2×C4—224no—
607C2×C4—224no—
608C2×C4—224no—
609C2×C4—224no—
610C2×C4—224no—
611C2×C4—222no—
612C2×C4—224no—
613C2×C4—4432no—
614C2×C4—444no—
615C2×C4C2×D4×D1114yes—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—44no—
620C22—44no—
621C22—432no—
622C22—44no—
623C22—44no—
624C22—118no—
625C22—224no—
626C22—224no—
627C22—2232no—
628C22—228no—
629C22—228no—
630C22—442no—
631C22—442no—
632C22—444no—
633C22—444no—
634C22—442no—
635C22—888no—
636C22D11×#12014yes—
637C22#1062313914yes—
638C22#1062314118yes—
639C22#1062318618yes—
640C22#1062319914yes—
641C22C22×D4×D1111yescharacteristic
642C22C22×D4×D1112yes—
643C22C2×#6703711yescharacteristic
644C22C2×#41×D1112yes—
645C22C2×#6704311yescharacteristic
646C4—24no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—228no—
651C4—224no—
652C4—228no—
653C4—442no—
654C2—48no—
655C2—228no—
656C2—882no—
657C2#1307222118yesdirect product factor
658C2C2×D11×#12011yescharacteristic
659C2C2×#1062319911yescharacteristic
660C2C2×#1062313911yescharacteristic
661C2C2×#1062314112yes—
662C2C2×#1062318612yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#394C2
socleC24×C11C24
hypercenterC2×#394D11
hypocenterC11C2×#2430
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C23×#4341C2semidirect
2. C22×#66947C2semidirect
3. C22×#66937C2semidirect
4. #13072221C2direct (N × H)
5. C2×#10622592C2semidirect
6. C2×#10622286C2semidirect
7. C2×#10622836C2semidirect
8. C2×#10622571C2semidirect
9. C2×#10622569C2semidirect
10. C2×#10622559C2semidirect
11. C2×D11×#95C2semidirect
12. C2×C11×#394C2semidirect
19. C22×#4341C22semidirect
35. C2×#66947C22semidirect
37. C2×#66937C22semidirect
38. C11×#394C22semidirect
40. #10622836C22semidirect
45. #10622592C22semidirect
46. #10622571C22semidirect
47. #10622569C22semidirect
48. #10622559C22semidirect
49. #10622286C22semidirect
50. D11×#95C22semidirect
66. C22×C22⋊D11D4semidirect
87. C22×C4×D11D4semidirect
166. C2×C22⋊D11C2×D4semidirect
180. C22×Dic11#41semidirect
195. C2×C4×D11C2×D4semidirect
214. C2×#394D11semidirect
292. C2×Dic11C2×#41semidirect
323. #394C2×D11semidirect
443. C2×C11#2430semidirect
539. C11C2×#2430semidirect
657. C2#13072221direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43412
C22×#669472
C22×#669372
#130722212
C2×#106225922
C2×#106222862
C2×#106228362
C2×#106225712
C2×#106225692
C2×#106225592
C2×D11×#952
C2×C11×#3942
C2×#243011