Finite Group Databasebeta

← C2×#13074344 #14920819

Subgroups — C2×#13074344 #14920819

Order 2816.

5390 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4341C211yesmaximal, characteristic
2C22×#66949C211yesmaximal, characteristic, complemented
3C22×#66931C212yesmaximal, complemented
4#13074344C2116yesmaximal, direct product factor
5C2×#10622269C211yesmaximal, characteristic, complemented
6C2×#10622592C211yesmaximal, characteristic, complemented
7C2×#10622574C212yesmaximal, complemented
8C2×#10622572C212yesmaximal, complemented
9C2×#10622835C211yesmaximal, characteristic, complemented
10C2×#10622280C211yesmaximal, characteristic, complemented
11C2×#10622555C212yesmaximal, complemented
12C2×C11×#394C211yesmaximal, characteristic, complemented
13C23×D4×C11C2211yescharacteristic
14C23×C22⋊D11C2212yes—
15C24×Dic11C2212yes—
16C22×#4341—216no—
17C22×#4341—28no—
18C22×#4341C2218yes—
19C23×C4×D11C2211yescharacteristic
20C23×Dic22C2211yescharacteristic
21C23×Dic22C2211yescharacteristic
22C22×C11×C4⋊C4C2211yescharacteristic
23C22×C11×C22⋊C4C2212yes—
24C22×C11×C22⋊C4C2212yes—
25C22×C22⋊Dic11C2212yes—
26C22×C22⋊Dic11C2212yes—
27C22×(C2×D11)⋊C4C2211yescharacteristic
28C22×(C2×D11)⋊C4C2211yescharacteristic
29C22×(C2×D11)⋊C4C2212yes—
30C22×C4⋊Dic11C2212yes—
31C22×Dic11⋊C4C2211yescharacteristic
32C22×Dic11⋊C4C2212yes—
33C22×Dic11⋊C4C2211yescharacteristic
34C22×Dic11⋊C4C2212yes—
35C2×#66949—28no—
36C2×#66949C2218yescomplemented
37C2×#66931—216no—
38C2×#66931C22116yescomplemented
39C11×#394C2218yescomplemented
40C2×C11×#95C2211yescharacteristic
41#10622835C2218yescomplemented
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2212yes—
44C2×C11⋊#95C2211yescharacteristic
45#10622592C2218yescomplemented
46#10622574C22116yescomplemented
47#10622572C22116yescomplemented
48#10622555C22116yescomplemented
49#10622280C2218yescomplemented
50#10622269C2218yescomplemented
51C25×C11C2312yes—
52C24×D11C2311yescharacteristic
53C22×D4×C11—28no—
54C22×D4×C11—24no—
55C22×D4×C11C2314yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2312yes—
58C23×C4×C11C2311yescharacteristic
59C23×C4×C11C2311yescharacteristic
60C22×C22⋊D11—28no—
61C22×C22⋊D11—28no—
62C22×C22⋊D11—28no—
63C22×C22⋊D11C2318yes—
64C23×Dic11—28no—
65C23×Dic11—28no—
66C23×Dic11D4116yes—
67C23×Dic11C2311yescharacteristic
68C23×Dic11C2312yes—
69C23×Dic11C2311yescharacteristic
70C23×Dic11C2318yes—
71C23×Dic11C2312yes—
72C23×Dic11C2311yescharacteristic
73C2×#4341—2128no—
74C2×#4341—264no—
75C2×#4341—232no—
76C22×C4×D11—28no—
77C22×C4×D11D418yes—
78C22×C4×D11C2314yes—
79C22×Dic22—24no—
80C22×Dic22—28no—
81C22×Dic22—28no—
82C22×Dic22—24no—
83C22×Dic22C2314yes—
84C22×Dic22C2314yes—
85C2×C11×C4⋊C4—24no—
86C2×C11×C4⋊C4C2314yes—
87C2×C11×C22⋊C4—28no—
88C2×C11×C22⋊C4—44no—
89C2×C11×C22⋊C4C2318yes—
90C2×C11×C22⋊C4C2318yes—
91C2×C22⋊Dic11—28no—
92C2×C22⋊Dic11—44no—
93C2×C22⋊Dic11C2318yes—
94C2×C22⋊Dic11C2318yes—
95C2×(C2×D11)⋊C4—24no—
96C2×(C2×D11)⋊C4—24no—
97C2×(C2×D11)⋊C4—44no—
98C2×(C2×D11)⋊C4C2314yes—
99C2×(C2×D11)⋊C4C2314yes—
100C2×(C2×D11)⋊C4C2318yes—
101C2×C4⋊Dic11—28no—
102C2×C4⋊Dic11C2318yes—
103C2×Dic11⋊C4—24no—
104C2×Dic11⋊C4—28no—
105C2×Dic11⋊C4—28no—
106C2×Dic11⋊C4—24no—
107C2×Dic11⋊C4C2318yes—
108C2×Dic11⋊C4C2318yes—
109C2×Dic11⋊C4C2314yes—
110C2×Dic11⋊C4C2314yes—
111#66949—232no—
112#66931—264no—
113C11×#95C2314yes—
114C11⋊#95C2318yes—
115C11⋊#95C2314yes—
116C11⋊#95C2318yes—
117C2×#2616—111nomaximal, Sylow
118C24×C11—24no—
119C24×C11—24no—
120C24×C11—24no—
121C24×C11—24no—
122C24×C11—24no—
123C24×C11—24no—
124C24×C11C2×D418yes—
125C24×C11C2414yes—
126C24×C11C2411yescharacteristic
127C23×D11—24no—
128C23×D11—22no—
129C23×D11—24no—
130C23×D11—22no—
131C23×D11C2×D414yes—
132C23×D11C2412yes—
133C2×D4×C11—232no—
134C2×D4×C11—216no—
135C2×D4×C11—28no—
136C22×C4×C11—24no—
137C22×C4×C11—24no—
138C22×C4×C11—22no—
139C22×C4×C11—24no—
140C22×C4×C11—22no—
141C22×C4×C11—24no—
142C22×C4×C11—22no—
143C22×C4×C11—24no—
144C22×C4×C11—22no—
145C22×C4×C11C2×D414yes—
146C22×C4×C11C2412yes—
147C22×C4×C11C2414yes—
148C22×C4×C11C2412yes—
149C22×C4×C11C2412yes—
150C2×C22⋊D11—264no—
151C2×C22⋊D11—216no—
152C2×C22⋊D11—216no—
153C2×C22⋊D11—216no—
154C22×Dic11—264no—
155C22×Dic11—216no—
156C22×Dic11—216no—
157C22×Dic11—24no—
158C22×Dic11—24no—
159C22×Dic11—22no—
160C22×Dic11—24no—
161C22×Dic11—22no—
162C22×Dic11—24no—
163C22×Dic11—24no—
164C22×Dic11—22no—
165C22×Dic11—24no—
166C22×Dic11—22no—
167C22×Dic11C2×D4132yes—
168C22×Dic11C2×D418yes—
169C22×Dic11C2×D418yes—
170C22×Dic11C2×D414yes—
171C22×Dic11C2412yes—
172C22×Dic11C2414yes—
173C22×Dic11C2412yes—
174C22×Dic11C2414yes—
175C22×Dic11C2412yes—
176#4341—2256no—
177C2×C4×D11—232no—
178C2×C4×D11—216no—
179C2×C4×D11C2×D4116yes—
180C2×Dic22—232no—
181C2×Dic22—216no—
182C2×Dic22—216no—
183C2×Dic22—28no—
184C2×Dic22—28no—
185C2×Dic22—416no—
186C11×C4⋊C4—28no—
187C11×C22⋊C4—216no—
188C11×C22⋊C4—48no—
189C22⋊Dic11—216no—
190C22⋊Dic11—48no—
191(C2×D11)⋊C4—28no—
192(C2×D11)⋊C4—28no—
193(C2×D11)⋊C4—48no—
194C4⋊Dic11—216no—
195Dic11⋊C4—216no—
196Dic11⋊C4—216no—
197Dic11⋊C4—28no—
198Dic11⋊C4—28no—
199C23×#41—111no—
200C22×C22⋊Q8—111no—
201C22×C22⋊Q8—112no—
202#2616—1116no—
203C2×#395—111no—
204C2×#395—111no—
205C2×#394—111no—
206C2×#394—112no—
207C2×#394—112no—
208C2×#394D1111yescharacteristic, complemented
209C2×#380—111no—
210C2×#380—112no—
211C23×C11—216no—
212C23×C11—216no—
213C23×C11—216no—
214C23×C11—24no—
215C23×C11—24no—
216C23×C11—24no—
217C23×C11—24no—
218C23×C11—24no—
219C23×C11—24no—
220C23×C11—48no—
221C23×C11—48no—
222C23×C11—48no—
223C23×C11—48no—
224C23×C11C22×D418yes—
225C23×C11C22×D412yes—
226C23×C11C22×D414yes—
227C23×C11H(5,𝔽2)12yes—
228C23×C112−1+414yes—
229C23×C112−1+412yes—
230C23×C11C2511yescharacteristic
231C22×D11—28no—
232C22×D11—28no—
233C22×D11—28no—
234C22×D11—24no—
235C22×D11—22no—
236C22×D11—24no—
237C22×D11—22no—
238C22×D11—48no—
239C22×D11—48no—
240C22×D11C22×D414yes—
241D4×C11—232no—
242C2×C4×C11—28no—
243C2×C4×C11—24no—
244C2×C4×C11—24no—
245C2×C4×C11—22no—
246C2×C4×C11—24no—
247C2×C4×C11—22no—
248C2×C4×C11—24no—
249C2×C4×C11—22no—
250C2×C4×C11—22no—
251C2×C4×C11—24no—
252C2×C4×C11—44no—
253C2×C4×C11—48no—
254C2×C4×C11—44no—
255C2×C4×C11C22×D414yes—
256C22⋊D11—264no—
257C2×Dic11—216no—
258C2×Dic11—24no—
259C2×Dic11—24no—
260C2×Dic11—264no—
261C2×Dic11—22no—
262C2×Dic11—24no—
263C2×Dic11—22no—
264C2×Dic11—24no—
265C2×Dic11—24no—
266C2×Dic11—22no—
267C2×Dic11—24no—
268C2×Dic11—22no—
269C2×Dic11—44no—
270C2×Dic11—48no—
271C2×Dic11—44no—
272C2×Dic11#120116yes—
273C2×Dic11C22×D418yes—
274C2×Dic11C22×D418yes—
275C2×Dic11C22×D414yes—
276C4×D11—232no—
277Dic22—232no—
278Dic22—416no—
279C22×#41—118no—
280C22×#41—2216no—
281C22×#41—228no—
282C23×Q8—111no—
283C23×Q8—111no—
284C23×D4—112no—
285C23×D4C2×D1111yescharacteristic
286C24×C4—112no—
287C24×C4—111no—
288C2×C22⋊Q8—118no—
289C2×C22⋊Q8—1116no—
290C2×C22⋊Q8—2216no—
291C2×C22⋊Q8—228no—
292C22×C4⋊C4—112no—
293C22×C4⋊C4—111no—
294C22×C4⋊C4—112no—
295C22×C4⋊C4—112no—
296C22×C4⋊C4—111no—
297C22×C4⋊C4C2×D1111yescharacteristic
298C22×C22⋊C4—112no—
299C22×C22⋊C4—111no—
300C22×C22⋊C4—111no—
301C22×C22⋊C4—112no—
302C22×C22⋊C4—112no—
303C22×C22⋊C4C2×D1112yes—
304C22×C22⋊C4C2×D1112yes—
305#395—118no—
306#395—118no—
307#394—118no—
308#394—1116no—
309#394—1116no—
310#394C2×D1118yescomplemented
311#380—1116no—
312#380—118no—
313C2×#95—112no—
314C2×#95—112no—
315C2×#95—111no—
316C2×#95C2×D1111yescharacteristic
317C22×C11—28no—
318C22×C11—28no—
319C22×C11—28no—
320C22×C11—432no—
321C22×C11—44no—
322C22×C11—44no—
323C22×C11—44no—
324C22×C11—44no—
325C22×C11C2×#12014yes—
326C22×C11#53618yes—
327C22×C11#53614yes—
328C22×C11#53718yes—
329C22×C11#53714yes—
330C22×C11C23×D412yes—
331C22×C11C23×D411yescharacteristic
332C22×C11C2×H(5,𝔽2)11yescharacteristic
333C22×C11C2×2−1+412yes—
334C22×C11C2×2−1+411yescharacteristic
335C2×D11—24no—
336C2×D11—24no—
337C2×D11—24no—
338C2×D11—416no—
339C2×D11—44no—
340C2×D11—44no—
341C4×C11—24no—
342C4×C11—42no—
343C4×C11—44no—
344C4×C11—42no—
345Dic11—28no—
346Dic11—42no—
347Dic11—44no—
348Dic11—42no—
349Dic11C2×#12018yes—
350C25—111no—
351C25C22×D1112yes—
352C2×#41—2264no—
353C2×#41—2232no—
354C2×#41—22128no—
355C22×Q8—114no—
356C22×Q8—114no—
357C22×Q8—228no—
358C22×Q8—228no—
359C22×Q8—224no—
360C22×Q8—224no—
361C22×D4—28no—
362C22×D4—24no—
363C22×D4—118no—
364C22×D4—228no—
365C22×D4—228no—
366C22×D4—228no—
367C22×D4C22×D1114yes—
368C23×C4—111no—
369C23×C4—1116no—
370C23×C4—118no—
371C23×C4—112no—
372C23×C4—118no—
373C23×C4—114no—
374C23×C4—111no—
375C23×C4—112no—
376C23×C4—111no—
377C23×C4—228no—
378C23×C4—228no—
379C23×C4—228no—
380C23×C4C22×D1111yescharacteristic
381C23×C4C22×D1112yes—
382C23×C4C22×D1111yescharacteristic
383C23×C4C22×D1111yescharacteristic
384C22⋊Q8—2264no—
385C22⋊Q8—2232no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—114no—
388C2×C4⋊C4—118no—
389C2×C4⋊C4—118no—
390C2×C4⋊C4—114no—
391C2×C4⋊C4—118no—
392C2×C4⋊C4—228no—
393C2×C4⋊C4—228no—
394C2×C4⋊C4—224no—
395C2×C4⋊C4—228no—
396C2×C4⋊C4—224no—
397C2×C4⋊C4C22×D1114yes—
398C2×C22⋊C4—28no—
399C2×C22⋊C4—44no—
400C2×C22⋊C4—118no—
401C2×C22⋊C4—114no—
402C2×C22⋊C4—114no—
403C2×C22⋊C4—118no—
404C2×C22⋊C4—118no—
405C2×C22⋊C4—224no—
406C2×C22⋊C4—224no—
407C2×C22⋊C4—228no—
408C2×C22⋊C4—444no—
409C2×C22⋊C4—444no—
410C2×C22⋊C4C22×D1118yes—
411C2×C22⋊C4C22×D1118yes—
412#95—118no—
413#95—118no—
414#95—114no—
415#95C22×D1114yes—
416C2×C11—48no—
417C2×C11#261618yescomplemented
418C2×C11C22×#12011yescharacteristic
419C2×C11C2×#53611yescharacteristic
420C2×C11C2×#53612yes—
421C2×C11C2×#53712yes—
422C2×C11C2×#53711yescharacteristic
423D11—44no—
424C24—24no—
425C24—24no—
426C24—24no—
427C24—24no—
428C24—24no—
429C24—24no—
430C24—114no—
431C24—112no—
432C24—224no—
433C24—222no—
434C24—222no—
435C24—224no—
436C24D4×D1118yes—
437C24C23×D1114yes—
438C24C23×D1111yescharacteristic
439#41—22256no—
440C2×Q8—2216no—
441C2×Q8—2216no—
442C2×Q8—228no—
443C2×Q8—2232no—
444C2×Q8—228no—
445C2×Q8—4416no—
446C2×D4—232no—
447C2×D4—216no—
448C2×D4—28no—
449C2×D4—2216no—
450C2×D4—2216no—
451C2×D4—2264no—
452C2×D4—2216no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—118no—
463C22×C4—114no—
464C22×C4—112no—
465C22×C4—118no—
466C22×C4—1132no—
467C22×C4—114no—
468C22×C4—1116no—
469C22×C4—114no—
470C22×C4—112no—
471C22×C4—112no—
472C22×C4—2232no—
473C22×C4—2216no—
474C22×C4—2264no—
475C22×C4—2216no—
476C22×C4—2216no—
477C22×C4—224no—
478C22×C4—224no—
479C22×C4—224no—
480C22×C4—222no—
481C22×C4—224no—
482C22×C4—222no—
483C22×C4—224no—
484C22×C4—222no—
485C22×C4—224no—
486C22×C4—222no—
487C22×C4D4×D1114yes—
488C22×C4C23×D1112yes—
489C22×C4C23×D1114yes—
490C22×C4C23×D1112yes—
491C22×C4C23×D1112yes—
492C4⋊C4—28no—
493C4⋊C4—2216no—
494C4⋊C4—2216no—
495C4⋊C4—228no—
496C4⋊C4—2216no—
497C4⋊C4—228no—
498C22⋊C4—216no—
499C22⋊C4—48no—
500C22⋊C4—228no—
501C22⋊C4—228no—
502C22⋊C4—2216no—
503C22⋊C4—448no—
504C22⋊C4—448no—
505C11C2×#261611yescharacteristic, complemented, Sylow
506C23—216no—
507C23—216no—
508C23—216no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—114no—
520C23—228no—
521C23—228no—
522C23—224no—
523C23—222no—
524C23—224no—
525C23—222no—
526C23—228no—
527C23—448no—
528C23—448no—
529C23C2×D4×D1118yes—
530C23C2×D4×D1114yes—
531C23C2×D4×D1112yes—
532C23#6703712yes—
533C23#6704314yes—
534C23#6704312yes—
535C23C24×D1111yescharacteristic
536Q8—2232no—
537Q8—4416no—
538D4—232no—
539D4—2264no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—1116no—
554C2×C4—114no—
555C2×C4—118no—
556C2×C4—118no—
557C2×C4—2232no—
558C2×C4—2264no—
559C2×C4—2216no—
560C2×C4—224no—
561C2×C4—224no—
562C2×C4—224no—
563C2×C4—222no—
564C2×C4—222no—
565C2×C4—224no—
566C2×C4—224no—
567C2×C4—222no—
568C2×C4—224no—
569C2×C4—222no—
570C2×C4—444no—
571C2×C4—448no—
572C2×C4—444no—
573C2×C4C2×D4×D1114yes—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—432no—
582C22—224no—
583C22—224no—
584C22—224no—
585C22—444no—
586C22—444no—
587C22—4416no—
588C22D11×#12014yes—
589C22#1062312818yes—
590C22#1062313914yes—
591C22#1062317918yes—
592C22#1062320614yes—
593C22C22×D4×D1111yescharacteristic
594C22C22×D4×D1112yes—
595C22C2×#6703711yescharacteristic
596C22C2×#6704312yes—
597C22C2×#6704311yescharacteristic
598C4—24no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—118no—
603C4—228no—
604C4—444no—
605C4—442no—
606C4—442no—
607C2—48no—
608C2—444no—
609C2#1307434418yesdirect product factor
610C2C2×D11×#12011yescharacteristic
611C2C2×#1062320611yescharacteristic
612C2C2×#1062317912yes—
613C2C2×#1062312812yes—
614C2C2×#1062313911yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#394C2
socleC24×C11C24
prime radicalC23×#4341C2
hypercenterC2×#394D11
hypocenterC11C2×#2616
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#66949C2semidirect
3. C22×#66931C2semidirect
4. #13074344C2direct (N × H)
5. C2×#10622269C2semidirect
6. C2×#10622592C2semidirect
7. C2×#10622574C2semidirect
8. C2×#10622572C2semidirect
9. C2×#10622835C2semidirect
10. C2×#10622280C2semidirect
11. C2×#10622555C2semidirect
12. C2×C11×#394C2semidirect
36. C2×#66949C22semidirect
38. C2×#66931C22semidirect
39. C11×#394C22semidirect
41. #10622835C22semidirect
45. #10622592C22semidirect
46. #10622574C22semidirect
47. #10622572C22semidirect
48. #10622555C22semidirect
49. #10622280C22semidirect
50. #10622269C22semidirect
208. C2×#394D11semidirect
310. #394C2×D11semidirect
417. C2×C11#2616semidirect
505. C11C2×#2616semidirect
609. C2#13074344direct (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×#669492
C22×#669312
#130743442
C2×#106222692
C2×#106225922
C2×#106225742
C2×#106225722
C2×#106228352
C2×#106222802
C2×#106225552
C2×C11×#3942
C2×#261611