Finite Group Databasebeta

← C2×#13073614 #14925616

Subgroups — C2×#13073614 #14925616

Order 2816.

3958 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66993C211yesmaximal, characteristic
2C22×#66938C212yesmaximal, complemented
3C22×#66940C211yesmaximal, characteristic, complemented
4#13073614C2116yesmaximal, direct product factor
5C2×#10622285C211yesmaximal, characteristic, complemented
6C2×#10622604C211yesmaximal, characteristic, complemented
7C2×#10622835C211yesmaximal, characteristic, complemented
8C2×#10622570C212yesmaximal, complemented
9C2×#10622570C212yesmaximal, complemented
10C2×#10622570C212yesmaximal, complemented
11C2×#10622262C211yesmaximal, characteristic, complemented
12C2×C11×#394C211yesmaximal, characteristic, complemented
13C23×D4×C11C2211yescharacteristic
14C23×C22⋊D11C2212yes—
15C23×C22⋊D11C2212yes—
16C23×D44C2211yescharacteristic
17C23×Dic22C2211yescharacteristic
18C22×C11×C4⋊C4C2211yescharacteristic
19C22×C11×C22⋊C4C2212yes—
20C22×C11×C22⋊C4C2212yes—
21C22×C22⋊Dic11C2212yes—
22C22×C22⋊Dic11C2212yes—
23C2×#66993—28no—
24C2×#66993C2218yes—
25C22×(C2×D11)⋊C4C2212yes—
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2212yes—
28C22×(C2×D11)⋊C4C2212yes—
29C22×Dic11⋊C4C2212yes—
30C22×Dic11⋊C4C2211yescharacteristic
31C22×Dic11⋊C4C2211yescharacteristic
32C22×C4×Dic11C2212yes—
33C22×C4×Dic11C2211yescharacteristic
34C22×C4×Dic11C2211yescharacteristic
35C22×C4×Dic11C2211yescharacteristic
36C2×#66940—28no—
37C2×#66940C2218yescomplemented
38C2×#66938—216no—
39C2×#66938C22116yescomplemented
40C11×#394C2218yescomplemented
41C2×C11×#95C2211yescharacteristic
42#10622835C2218yescomplemented
43C2×C11⋊#95C2211yescharacteristic
44#10622604C2218yescomplemented
45#10622570C22116yescomplemented
46#10622570C22116yescomplemented
47#10622570C22116yescomplemented
48#10622285C2218yescomplemented
49#10622262C2218yescomplemented
50C25×C11C2312yes—
51C24×D11C2312yes—
52C22×D4×C11—24no—
53C22×D4×C11—44no—
54C22×D4×C11C2314yes—
55C23×C4×C11C2312yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2311yescharacteristic
58C23×C4×C11C2311yescharacteristic
59C22×C22⋊D11—28no—
60C22×C22⋊D11—28no—
61C22×C22⋊D11—28no—
62C22×C22⋊D11—28no—
63C22×C22⋊D11—28no—
64C22×C22⋊D11—28no—
65C22×C22⋊D11C2318yes—
66C22×C22⋊D11C2318yes—
67C23×Dic11C2311yescharacteristic
68C23×Dic11C2312yes—
69C23×Dic11C2311yescharacteristic
70C23×Dic11C2311yescharacteristic
71C23×Dic11C2311yescharacteristic
72C22×D44—24no—
73C22×D44—44no—
74C22×D44C2314yes—
75C22×Dic22—24no—
76C22×Dic22—24no—
77C22×Dic22D418yescomplemented
78C22×Dic22C2314yes—
79C2×C11×C4⋊C4—24no—
80C2×C11×C4⋊C4C2314yes—
81C2×C11×C22⋊C4—28no—
82C2×C11×C22⋊C4—44no—
83C2×C11×C22⋊C4C2318yes—
84C2×C11×C22⋊C4C2318yes—
85C2×C22⋊Dic11—28no—
86C2×C22⋊Dic11—44no—
87C2×C22⋊Dic11C2318yes—
88C2×C22⋊Dic11C2318yes—
89#66993—232no—
90C2×(C2×D11)⋊C4—28no—
91C2×(C2×D11)⋊C4—28no—
92C2×(C2×D11)⋊C4—44no—
93C2×(C2×D11)⋊C4—44no—
94C2×(C2×D11)⋊C4C2318yes—
95C2×(C2×D11)⋊C4C2318yes—
96C2×(C2×D11)⋊C4C2318yes—
97C2×(C2×D11)⋊C4C2318yes—
98C2×Dic11⋊C4—24no—
99C2×Dic11⋊C4—24no—
100C2×Dic11⋊C4—28no—
101C2×Dic11⋊C4C2314yes—
102C2×Dic11⋊C4C2314yes—
103C2×Dic11⋊C4C2318yes—
104C2×C4×Dic11—28no—
105C2×C4×Dic11—24no—
106C2×C4×Dic11—24no—
107C2×C4×Dic11D418yes—
108C2×C4×Dic11C2318yes—
109C2×C4×Dic11C2314yes—
110C2×C4×Dic11C2314yes—
111C2×C4×Dic11C2314yes—
112#66940—232no—
113#66938—264no—
114C11×#95C2314yes—
115C11⋊#95C2314yes—
116C2×#2548—111nomaximal, Sylow
117C24×C11—24no—
118C24×C11—24no—
119C24×C11—24no—
120C24×C11—24no—
121C24×C11—24no—
122C24×C11—24no—
123C24×C11—24no—
124C24×C11C2414yes—
125C24×C11C2411yescharacteristic
126C23×D11—24no—
127C23×D11—24no—
128C23×D11—24no—
129C23×D11—24no—
130C23×D11—24no—
131C23×D11—24no—
132C23×D11—24no—
133C23×D11C2414yes—
134C2×D4×C11—28no—
135C2×D4×C11—48no—
136C2×D4×C11—416no—
137C22×C4×C11—22no—
138C22×C4×C11—22no—
139C22×C4×C11—24no—
140C22×C4×C11—22no—
141C22×C4×C11—24no—
142C22×C4×C11—24no—
143C22×C4×C11—24no—
144C22×C4×C11—24no—
145C22×C4×C11C2×D414yes—
146C22×C4×C11C2×D414yes—
147C22×C4×C11C2414yes—
148C22×C4×C11C2412yes—
149C22×C4×C11C2412yes—
150C22×C4×C11C2412yes—
151C2×C22⋊D11—216no—
152C2×C22⋊D11—216no—
153C2×C22⋊D11—216no—
154C2×C22⋊D11—216no—
155C2×C22⋊D11—216no—
156C2×C22⋊D11—216no—
157C2×C22⋊D11—432no—
158C2×C22⋊D11—432no—
159C22×Dic11—22no—
160C22×Dic11—22no—
161C22×Dic11—24no—
162C22×Dic11—22no—
163C22×Dic11—24no—
164C22×Dic11—24no—
165C22×Dic11—24no—
166C22×Dic11C2×D414yes—
167C22×Dic11C2×D414yes—
168C22×Dic11C2×D414yes—
169C22×Dic11C2×D414yes—
170C22×Dic11#4114yes—
171C22×Dic11#4118yes—
172C22×Dic11C2414yes—
173C22×Dic11C2412yes—
174C22×Dic11C2412yes—
175C22×Dic11C2412yes—
176C22×Dic11C2412yes—
177C2×D44—28no—
178C2×D44—48no—
179C2×D44—416no—
180C2×Dic22—232no—
181C2×Dic22—28no—
182C2×Dic22—28no—
183C2×Dic22C2×D4116yescomplemented
184C11×C4⋊C4—28no—
185C11×C22⋊C4—216no—
186C11×C22⋊C4—48no—
187C22⋊Dic11—216no—
188C22⋊Dic11—48no—
189(C2×D11)⋊C4—216no—
190(C2×D11)⋊C4—216no—
191(C2×D11)⋊C4—48no—
192(C2×D11)⋊C4—48no—
193Dic11⋊C4—28no—
194Dic11⋊C4—216no—
195Dic11⋊C4—28no—
196C4×Dic11—216no—
197C4×Dic11—28no—
198C4×Dic11—28no—
199C4×Dic11C2×D4116yes—
200C22×#127—111no—
201C22×#124—112no—
202C22×C4×Q8—111no—
203#2548—1116no—
204C2×#394—111no—
205C2×#394—111no—
206C2×#394—111no—
207C2×#394D1111yescharacteristic, complemented
208C2×#390—112no—
209C2×#390—112no—
210C2×#390—112no—
211C2×C4×C4⋊C4—111no—
212C23×C11—24no—
213C23×C11—24no—
214C23×C11—24no—
215C23×C11—24no—
216C23×C11—24no—
217C23×C11—24no—
218C23×C11—24no—
219C23×C11—48no—
220C23×C11—48no—
221C23×C11—48no—
222C23×C11—48no—
223C23×C11—48no—
224C23×C11—48no—
225C23×C11—48no—
226C23×C11C22×D412yes—
227C23×C11C22×D412yes—
228C23×C11C2×#4112yes—
229C23×C11C2×#4114yes—
230C23×C11H(5,𝔽2)14yes—
231C23×C11C2511yescharacteristic
232C22×D11—24no—
233C22×D11—24no—
234C22×D11—24no—
235C22×D11—24no—
236C22×D11—24no—
237C22×D11—24no—
238C22×D11—24no—
239C22×D11—48no—
240C22×D11—48no—
241C22×D11—48no—
242C22×D11—48no—
243C22×D11—48no—
244C22×D11—48no—
245C22×D11—48no—
246D4×C11—416no—
247C2×C4×C11—28no—
248C2×C4×C11—28no—
249C2×C4×C11—22no—
250C2×C4×C11—22no—
251C2×C4×C11—24no—
252C2×C4×C11—22no—
253C2×C4×C11—24no—
254C2×C4×C11—24no—
255C2×C4×C11—24no—
256C2×C4×C11—24no—
257C2×C4×C11—44no—
258C2×C4×C11—48no—
259C2×C4×C11C22×D414yes—
260C2×C4×C11C22×D414yes—
261C22⋊D11—432no—
262C22⋊D11—432no—
263C2×Dic11—28no—
264C2×Dic11—22no—
265C2×Dic11—22no—
266C2×Dic11—22no—
267C2×Dic11—24no—
268C2×Dic11—24no—
269C2×Dic11—24no—
270C2×Dic11—24no—
271C2×Dic11—44no—
272C2×Dic11—48no—
273C2×Dic11#121116yes—
274C2×Dic11#124116yescomplemented
275C2×Dic11C22×D414yes—
276C2×Dic11C22×D414yes—
277C2×Dic11C22×D414yes—
278C2×Dic11C22×D414yes—
279C2×Dic11C2×#4114yes—
280C2×Dic11C2×#4118yes—
281D44—416no—
282Dic22—232no—
283C23×Q8—111no—
284C23×D4—112no—
285C23×D4—112no—
286C23×D4—111no—
287C23×D4C2×D1111yescharacteristic
288C2×#127—118no—
289C2×#127—228no—
290C2×#124—1116no—
291C2×#124—2216no—
292C2×C4×Q8—118no—
293C2×C4×Q8—228no—
294C22×C4⋊C4—111no—
295C22×C4⋊C4—111no—
296C22×C4⋊C4—112no—
297C22×C4⋊C4C2×D1111yescharacteristic
298C22×C22⋊C4—112no—
299C22×C22⋊C4—112no—
300C22×C22⋊C4—112no—
301C22×C22⋊C4—112no—
302C22×C22⋊C4—112no—
303C22×C22⋊C4—112no—
304C22×C22⋊C4C2×D1112yes—
305C22×C22⋊C4C2×D1112yes—
306C22×C42—111no—
307C22×C42—111no—
308C22×C42—111no—
309C22×C42—112no—
310#394—118no—
311#394—118no—
312#394—118no—
313#394C2×D1118yescomplemented
314#390—1116no—
315#390—1116no—
316#390—1116no—
317C4×C4⋊C4—118no—
318C2×#95—111no—
319C2×#95C2×D1111yescharacteristic
320C22×C11—44no—
321C22×C11—44no—
322C22×C11—44no—
323C22×C11—44no—
324C22×C11—44no—
325C22×C11—44no—
326C22×C11—44no—
327C22×C11—816no—
328C22×C11C2×#12114yes—
329C22×C11C2×#12414yes—
330C22×C11#53514yes—
331C22×C11#55018yes—
332C22×C11#55918yes—
333C22×C11C23×D411yescharacteristic
334C22×C11C23×D411yescharacteristic
335C22×C11C22×#4111yescharacteristic
336C22×C11C22×#4112yes—
337C22×C11C2×H(5,𝔽2)12yes—
338C2×D11—44no—
339C2×D11—44no—
340C2×D11—44no—
341C2×D11—44no—
342C2×D11—44no—
343C2×D11—44no—
344C2×D11—44no—
345C2×D11—816no—
346C4×C11—24no—
347C4×C11—24no—
348C4×C11—42no—
349C4×C11—44no—
350Dic11—24no—
351Dic11—42no—
352Dic11—44no—
353Dic11C2×#12118yes—
354Dic11C2×#12418yescomplemented
355C25—112no—
356C25C22×D1112yes—
357C22×Q8—118no—
358C22×Q8—114no—
359C22×Q8—224no—
360C22×Q8—224no—
361C22×D4—24no—
362C22×D4—44no—
363C22×D4—118no—
364C22×D4—118no—
365C22×D4—114no—
366C22×D4—228no—
367C22×D4—228no—
368C22×D4—228no—
369C22×D4—228no—
370C22×D4—228no—
371C22×D4—228no—
372C22×D4—224no—
373C22×D4—444no—
374C22×D4C22×D1114yes—
375C23×C4—111no—
376C23×C4—111no—
377C23×C4—111no—
378C23×C4—111no—
379C23×C4—112no—
380C23×C4C22×D1111yescharacteristic
381C23×C4C22×D1112yes—
382C23×C4C22×D1111yescharacteristic
383C23×C4C22×D1111yescharacteristic
384#127—2232no—
385#124—2264no—
386C4×Q8—2232no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—114no—
389C2×C4⋊C4—114no—
390C2×C4⋊C4—118no—
391C2×C4⋊C4—224no—
392C2×C4⋊C4—228no—
393C2×C4⋊C4—224no—
394C2×C4⋊C4C22×D1114yes—
395C2×C22⋊C4—28no—
396C2×C22⋊C4—44no—
397C2×C22⋊C4—118no—
398C2×C22⋊C4—118no—
399C2×C22⋊C4—118no—
400C2×C22⋊C4—118no—
401C2×C22⋊C4—118no—
402C2×C22⋊C4—118no—
403C2×C22⋊C4—228no—
404C2×C22⋊C4—228no—
405C2×C22⋊C4—228no—
406C2×C22⋊C4—444no—
407C2×C22⋊C4—444no—
408C2×C22⋊C4—444no—
409C2×C22⋊C4C22×D1118yes—
410C2×C22⋊C4C22×D1118yes—
411C2×C42—118no—
412C2×C42—114no—
413C2×C42—114no—
414C2×C42—114no—
415C2×C42—118no—
416C2×C42—224no—
417C2×C42—224no—
418C2×C42—228no—
419#95—114no—
420#95C22×D1114yes—
421C2×C11—84no—
422C2×C11#254818yescomplemented
423C2×C11C22×#12111yescharacteristic
424C2×C11C22×#12411yescharacteristic
425C2×C11C2×#53511yescharacteristic
426C2×C11C2×#55012yes—
427C2×C11C2×#55912yes—
428D11—84no—
429C24—24no—
430C24—24no—
431C24—24no—
432C24—24no—
433C24—24no—
434C24—24no—
435C24—24no—
436C24—114no—
437C24—224no—
438C24—224no—
439C24—224no—
440C24—224no—
441C24—224no—
442C24—224no—
443C24—224no—
444C24C23×D1114yes—
445C24C23×D1111yescharacteristic
446C2×Q8—1116no—
447C2×Q8—228no—
448C2×Q8—2232no—
449C2×Q8—228no—
450C2×D4—28no—
451C2×D4—48no—
452C2×D4—416no—
453C2×D4—2216no—
454C2×D4—2216no—
455C2×D4—2216no—
456C2×D4—2216no—
457C2×D4—2216no—
458C2×D4—2216no—
459C2×D4—228no—
460C2×D4—448no—
461C2×D4—4416no—
462C2×D4—4432no—
463C2×D4—4432no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—114no—
473C22×C4—114no—
474C22×C4—114no—
475C22×C4—118no—
476C22×C4—114no—
477C22×C4—112no—
478C22×C4—112no—
479C22×C4—114no—
480C22×C4—112no—
481C22×C4—114no—
482C22×C4—112no—
483C22×C4—222no—
484C22×C4—222no—
485C22×C4—224no—
486C22×C4—222no—
487C22×C4—224no—
488C22×C4—224no—
489C22×C4—224no—
490C22×C4D4×D1114yes—
491C22×C4D4×D1114yes—
492C22×C4C23×D1114yes—
493C22×C4C23×D1112yes—
494C22×C4C23×D1112yes—
495C22×C4C23×D1112yes—
496C4⋊C4—28no—
497C4⋊C4—228no—
498C4⋊C4—2216no—
499C4⋊C4—228no—
500C22⋊C4—216no—
501C22⋊C4—48no—
502C22⋊C4—2216no—
503C22⋊C4—2216no—
504C22⋊C4—2216no—
505C22⋊C4—448no—
506C22⋊C4—448no—
507C22⋊C4—448no—
508C42—1116no—
509C42—228no—
510C42—228no—
511C42—2216no—
512C11C2×#254811yescharacteristic, complemented, Sylow
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—224no—
528C23—224no—
529C23—224no—
530C23—224no—
531C23—224no—
532C23—224no—
533C23—224no—
534C23—448no—
535C23—448no—
536C23—448no—
537C23—448no—
538C23—448no—
539C23—448no—
540C23—448no—
541C23C2×D4×D1112yes—
542C23C2×D4×D1112yes—
543C23#6703714yes—
544C23#41×D1114yes—
545C23#41×D1112yes—
546C23C24×D1111yescharacteristic
547Q8—2232no—
548D4—416no—
549D4—4416no—
550D4—4432no—
551D4—4432no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—1116no—
565C2×C4—114no—
566C2×C4—114no—
567C2×C4—114no—
568C2×C4—118no—
569C2×C4—114no—
570C2×C4—1116no—
571C2×C4—114no—
572C2×C4—228no—
573C2×C4—222no—
574C2×C4—222no—
575C2×C4—224no—
576C2×C4—222no—
577C2×C4—224no—
578C2×C4—224no—
579C2×C4—224no—
580C2×C4—444no—
581C2×C4—448no—
582C2×C4C2×D4×D1114yes—
583C2×C4C2×D4×D1114yes—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—816no—
592C22—444no—
593C22—444no—
594C22—444no—
595C22—444no—
596C22—444no—
597C22—444no—
598C22—444no—
599C22—8816no—
600C22#1062312314yes—
601C22#1062313018yes—
602C22D11×#12114yes—
603C22#1062319518yes—
604C22D11×#12414yes—
605C22C22×D4×D1111yescharacteristic
606C22C22×D4×D1111yescharacteristic
607C22C2×#6703712yes—
608C22C2×#41×D1112yes—
609C22C2×#41×D1111yescharacteristic
610C4—24no—
611C4—24no—
612C4—42no—
613C4—44no—
614C4—118no—
615C4—118no—
616C4—224no—
617C4—442no—
618C4—444no—
619C2—84no—
620C2—884no—
621C2#1307361418yesdirect product factor
622C2C2×D11×#12111yescharacteristic
623C2C2×D11×#12411yescharacteristic
624C2C2×#1062312311yescharacteristic
625C2C2×#1062313012yes—
626C2C2×#1062319512yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#394C2
socleC24×C11C24
prime radicalC22×#66993C2
hypercenterC2×#394D11
hypocenterC11C2×#2548
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. C22×#66938C2semidirect
3. C22×#66940C2semidirect
4. #13073614C2direct (N × H)
5. C2×#10622285C2semidirect
6. C2×#10622604C2semidirect
7. C2×#10622835C2semidirect
8. C2×#10622570C2semidirect
9. C2×#10622570C2semidirect
10. C2×#10622570C2semidirect
11. C2×#10622262C2semidirect
12. C2×C11×#394C2semidirect
37. C2×#66940C22semidirect
39. C2×#66938C22semidirect
40. C11×#394C22semidirect
42. #10622835C22semidirect
44. #10622604C22semidirect
45. #10622570C22semidirect
46. #10622570C22semidirect
47. #10622570C22semidirect
48. #10622285C22semidirect
49. #10622262C22semidirect
77. C22×Dic22D4semidirect
183. C2×Dic22C2×D4semidirect
207. C2×#394D11semidirect
274. C2×Dic11#124semidirect
313. #394C2×D11semidirect
354. Dic11C2×#124semidirect
422. C2×C11#2548semidirect
512. C11C2×#2548semidirect
621. C2#13073614direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669932
C22×#669382
C22×#669402
#130736142
C2×#106222852
C2×#106226042
C2×#106228352
C2×#106225702
C2×#106222622
C2×C11×#3942
C2×#254811