Finite Group Databasebeta

← C2×#13072228 #14922409

Subgroups — C2×#13072228 #14922409

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×#66992C212yesmaximal, complemented
3#13072228C2116yesmaximal, direct product factor
4C2×#10622592C211yesmaximal, characteristic, complemented
5C2×#10622604C211yesmaximal, characteristic, complemented
6C2×#10622560C211yesmaximal, characteristic, complemented
7C2×#10622573C211yesmaximal, characteristic, complemented
8C2×#10622571C212yesmaximal, complemented
9C2×#10622569C212yesmaximal, complemented
10C2×#10622602C212yesmaximal, complemented
11C22×C11⋊#95C211yesmaximal, characteristic, complemented
12C22×C11×#123C211yesmaximal, characteristic, complemented
13C23×D4×C11C2211yescharacteristic
14C24×C4×C11C2211yescharacteristic
15C23×C22⋊D11C2212yes—
16C23×C22⋊D11C2212yes—
17C24×Dic11C2212yes—
18C22×#4339—28no—
19C22×#4339—216no—
20C22×#4339C2218yescomplemented
21C23×D44C2211yescharacteristic
22C23×C4×D11C2212yes—
23C23×Dic22C2211yescharacteristic
24C2×C11×#123—28no—
25C2×C11×#123C2218yescomplemented
26C22×C11×C4⋊C4C2211yescharacteristic
27C22×C11×C4⋊C4C2211yescharacteristic
28C22×C11×C22⋊C4C2211yescharacteristic
29C22×C11×C22⋊C4C2211yescharacteristic
30C22×C11×C22⋊C4C2211yescharacteristic
31C22×C22⋊Dic11C2212yes—
32C2×#66992—216no—
33C2×#66992C22116yescomplemented
34C22×(C2×D11)⋊C4C2212yes—
35C22×(C2×D11)⋊C4C2212yes—
36C22×(C2×D11)⋊C4C2212yes—
37C22×Dic11⋊C4C2212yes—
38C2×C11⋊#95C2212yes—
39C2×C11⋊#95C2211yescharacteristic
40C2×C11⋊#95C2211yescharacteristic
41C2×C11⋊#95C2212yes—
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2218yescomplemented
44#10622604C2218yescomplemented
45#10622602C22116yescomplemented
46#10622592C2218yescomplemented
47#10622573C2218yescomplemented
48#10622571C22116yescomplemented
49#10622569C22116yescomplemented
50#10622560C2218yescomplemented
51C25×C11C2311yescharacteristic
52C25×C11C2311yescharacteristic
53C24×D11C2312yes—
54C22×D4×C11—28no—
55C22×D4×C11—24no—
56C22×D4×C11C2314yes—
57C23×C4×C11—28no—
58C23×C4×C11D418yescomplemented
59C23×C4×C11C2311yescharacteristic
60C23×C4×C11C2311yescharacteristic
61C23×C4×C11C2311yescharacteristic
62C23×C4×C11C2311yescharacteristic
63C23×C4×C11C2311yescharacteristic
64C23×C4×C11C2314yes—
65C22×C22⋊D11—28no—
66C22×C22⋊D11—28no—
67C22×C22⋊D11—28no—
68C22×C22⋊D11—28no—
69C22×C22⋊D11—44no—
70C22×C22⋊D11D4116yescomplemented
71C22×C22⋊D11C2318yes—
72C22×C22⋊D11C2318yes—
73C23×Dic11—28no—
74C23×Dic11—28no—
75C23×Dic11—44no—
76C23×Dic11C2312yes—
77C23×Dic11C2318yes—
78C23×Dic11C2312yes—
79C23×Dic11C2312yes—
80C2×#4339—2128no—
81C2×#4339—232no—
82C2×#4339—264no—
83C22×D44—28no—
84C22×D44—24no—
85C22×D44C2314yes—
86C22×C4×D11—28no—
87C22×C4×D11—28no—
88C22×C4×D11—28no—
89C22×C4×D11C2318yes—
90C22×Dic22—24no—
91C22×Dic22—28no—
92C22×Dic22C2314yes—
93C11×#123—232no—
94C2×C11×C4⋊C4—24no—
95C2×C11×C4⋊C4—24no—
96C2×C11×C4⋊C4C2314yes—
97C2×C11×C4⋊C4C2314yes—
98C2×C11×C22⋊C4—24no—
99C2×C11×C22⋊C4—24no—
100C2×C11×C22⋊C4—24no—
101C2×C11×C22⋊C4C2314yes—
102C2×C11×C22⋊C4C2314yes—
103C2×C11×C22⋊C4C2314yes—
104C2×C22⋊Dic11—28no—
105C2×C22⋊Dic11C2318yes—
106#66992—264no—
107C2×(C2×D11)⋊C4—28no—
108C2×(C2×D11)⋊C4—28no—
109C2×(C2×D11)⋊C4—44no—
110C2×(C2×D11)⋊C4C2318yes—
111C2×(C2×D11)⋊C4C2318yes—
112C2×(C2×D11)⋊C4C2318yes—
113C2×Dic11⋊C4—28no—
114C2×Dic11⋊C4C2318yes—
115C11⋊#95C2314yes—
116C11⋊#95C2318yes—
117C11⋊#95C2318yes—
118C11⋊#95C2314yes—
119C11⋊#95C2318yes—
120C2×#2430—111nomaximal, Sylow
121C24×C11—24no—
122C24×C11—22no—
123C24×C11—22no—
124C24×C11—24no—
125C24×C11—22no—
126C24×C11—24no—
127C24×C11—22no—
128C24×C11—22no—
129C24×C11C2×D418yes—
130C24×C11C2×D414yes—
131C24×C11C2412yes—
132C24×C11C2412yes—
133C24×C11C2411yescharacteristic
134C23×D11—24no—
135C23×D11—24no—
136C23×D11—24no—
137C23×D11—24no—
138C23×D11—24no—
139C23×D11—24no—
140C23×D11C2×D418yes—
141C23×D11C2414yes—
142C2×D4×C11—216no—
143C2×D4×C11—28no—
144C2×D4×C11—416no—
145C22×C4×C11—232no—
146C22×C4×C11—24no—
147C22×C4×C11—24no—
148C22×C4×C11—216no—
149C22×C4×C11—24no—
150C22×C4×C11—22no—
151C22×C4×C11—22no—
152C22×C4×C11—22no—
153C22×C4×C11—24no—
154C22×C4×C11—22no—
155C22×C4×C11—22no—
156C22×C4×C11C2×D414yes—
157C22×C4×C11C2×D414yes—
158C22×C4×C11C2×D4116yescomplemented
159C22×C4×C11C2412yes—
160C22×C4×C11C2412yes—
161C22×C4×C11C2412yes—
162C22×C4×C11C2412yes—
163C22×C4×C11C2412yes—
164C2×C22⋊D11—216no—
165C2×C22⋊D11—264no—
166C2×C22⋊D11—216no—
167C2×C22⋊D11—216no—
168C2×C22⋊D11—216no—
169C2×C22⋊D11—48no—
170C2×C22⋊D11—432no—
171C2×C22⋊D11C2×D4132yescomplemented
172C22×Dic11—216no—
173C22×Dic11—24no—
174C22×Dic11—24no—
175C22×Dic11—24no—
176C22×Dic11—24no—
177C22×Dic11—216no—
178C22×Dic11—24no—
179C22×Dic11—24no—
180C22×Dic11—24no—
181C22×Dic11—48no—
182C22×Dic11—432no—
183C22×Dic11C2×D418yes—
184C22×Dic11#4118yescomplemented
185C22×Dic11C2414yes—
186C22×Dic11C2414yes—
187C22×Dic11C2414yes—
188#4339—2256no—
189C2×D44—232no—
190C2×D44—216no—
191C2×D44—28no—
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—
199C11×C4⋊C4—28no—
200C11×C4⋊C4—28no—
201C11×C22⋊C4—28no—
202C11×C22⋊C4—28no—
203C11×C22⋊C4—28no—
204C22⋊Dic11—216no—
205(C2×D11)⋊C4—216no—
206(C2×D11)⋊C4—216no—
207(C2×D11)⋊C4—48no—
208Dic11⋊C4—216no—
209C23×#41—111no—
210C22×#123D1111yescharacteristic, complemented
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×#95—111no—
221C23×C11—28no—
222C23×C11—28no—
223C23×C11—216no—
224C23×C11—216no—
225C23×C11—24no—
226C23×C11—22no—
227C23×C11—22no—
228C23×C11—24no—
229C23×C11—22no—
230C23×C11—24no—
231C23×C11—22no—
232C23×C11—22no—
233C23×C11—44no—
234C23×C11—44no—
235C23×C11—44no—
236C23×C11—48no—
237C23×C11—48no—
238C23×C11#120116yes—
239C23×C11C22×D418yes—
240C23×C11C22×D414yes—
241C23×C11C22×D414yes—
242C23×C11C22×D412yes—
243C23×C11C2×#4114yes—
244C23×C11H(5,𝔽2)12yes—
245C23×C112−1+412yes—
246C23×C11C2511yescharacteristic
247C22×D11—216no—
248C22×D11—216no—
249C22×D11—216no—
250C22×D11—24no—
251C22×D11—24no—
252C22×D11—24no—
253C22×D11—24no—
254C22×D11—24no—
255C22×D11—24no—
256C22×D11—48no—
257C22×D11—48no—
258C22×D11—48no—
259C22×D11—48no—
260C22×D11C22×D418yes—
261D4×C11—416no—
262C2×C4×C11—28no—
263C2×C4×C11—28no—
264C2×C4×C11—232no—
265C2×C4×C11—24no—
266C2×C4×C11—24no—
267C2×C4×C11—24no—
268C2×C4×C11—22no—
269C2×C4×C11—22no—
270C2×C4×C11—22no—
271C2×C4×C11—24no—
272C2×C4×C11—22no—
273C2×C4×C11—22no—
274C2×C4×C11—44no—
275C2×C4×C11—44no—
276C2×C4×C11—44no—
277C2×C4×C11C22×D414yes—
278C2×C4×C11C22×D414yes—
279C22⋊D11—264no—
280C22⋊D11—432no—
281C2×Dic11—216no—
282C2×Dic11—216no—
283C2×Dic11—24no—
284C2×Dic11—24no—
285C2×Dic11—24no—
286C2×Dic11—24no—
287C2×Dic11—24no—
288C2×Dic11—24no—
289C2×Dic11—24no—
290C2×Dic11—432no—
291C2×Dic11—48no—
292C2×Dic11C22×D418yes—
293C2×Dic11C2×#4118yescomplemented
294D44—232no—
295C4×D11—264no—
296Dic22—232no—
297C22×#41—118no—
298C22×#41—2216no—
299C22×#41—228no—
300C23×Q8—111no—
301C23×D4—112no—
302C23×D4—112no—
303C23×D4—111no—
304C23×D4C2×D1111yescharacteristic
305C24×C4—112no—
306C24×C4—112no—
307C24×C4C2×D1111yescharacteristic
308C2×#123—28no—
309C2×#123C2×D1118yescomplemented
310C2×#121—1116no—
311C2×#121—2216no—
312C22×C4⋊C4—112no—
313C22×C4⋊C4C2×D1111yescharacteristic
314C22×C4⋊C4C2×D1111yescharacteristic
315C22×C22⋊C4—112no—
316C22×C22⋊C4—112no—
317C22×C22⋊C4—112no—
318C22×C22⋊C4—112no—
319C22×C22⋊C4C2×D1111yescharacteristic
320C22×C22⋊C4C2×D1111yescharacteristic
321C22×C22⋊C4C2×D1111yescharacteristic
322#395—118no—
323#394—118no—
324#393—118no—
325#392—118no—
326#386—1116no—
327#386—1116no—
328#385—1116no—
329C2×#95—111no—
330C2×#95—111no—
331C2×#95—112no—
332C2×#95—112no—
333C2×#95—112no—
334C2×#95—118no—
335C22×C11—24no—
336C22×C11—24no—
337C22×C11—232no—
338C22×C11—28no—
339C22×C11—28no—
340C22×C11—42no—
341C22×C11—42no—
342C22×C11—42no—
343C22×C11—44no—
344C22×C11—44no—
345C22×C11—88no—
346C22×C11C2×#12018yes—
347C22×C11C2×#12014yes—
348C22×C11#53214yes—
349C22×C11#53714yes—
350C22×C11#54618yes—
351C22×C11#54718yes—
352C22×C11C23×D412yes—
353C22×C11C23×D411yescharacteristic
354C22×C11C22×#4112yes—
355C22×C11C2×H(5,𝔽2)11yescharacteristic
356C22×C11C2×2−1+411yescharacteristic
357C2×D11—28no—
358C2×D11—28no—
359C2×D11—28no—
360C2×D11—44no—
361C2×D11—44no—
362C2×D11—432no—
363C2×D11—44no—
364C2×D11—44no—
365C4×C11—24no—
366C4×C11—24no—
367C4×C11—42no—
368C4×C11—42no—
369C4×C11—42no—
370Dic11—28no—
371Dic11—28no—
372Dic11—44no—
373C25—112no—
374C25C22×D1111yescharacteristic
375C25C22×D1111yescharacteristic
376C2×#41—2264no—
377C2×#41—22128no—
378C2×#41—2232no—
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—228no—
390C22×D4—228no—
391C22×D4—228no—
392C22×D4—228no—
393C22×D4—224no—
394C22×D4—444no—
395C22×D4C22×D1114yes—
396C23×C4—28no—
397C23×C4—112no—
398C23×C4—112no—
399C23×C4—112no—
400C23×C4—118no—
401C23×C4—118no—
402C23×C4—228no—
403C23×C4—228no—
404C23×C4—228no—
405C23×C4—228no—
406C23×C4—228no—
407C23×C4—444no—
408C23×C4C22⋊D1118yescomplemented
409C23×C4C22×D1111yescharacteristic
410C23×C4C22×D1111yescharacteristic
411C23×C4C22×D1111yescharacteristic
412C23×C4C22×D1111yescharacteristic
413C23×C4C22×D1111yescharacteristic
414C23×C4C22×D1114yes—
415#123—232no—
416#121—2264no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—118no—
420C2×C4⋊C4—228no—
421C2×C4⋊C4C22×D1114yes—
422C2×C4⋊C4C22×D1114yes—
423C2×C22⋊C4—24no—
424C2×C22⋊C4—24no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—118no—
427C2×C22⋊C4—118no—
428C2×C22⋊C4—118no—
429C2×C22⋊C4—118no—
430C2×C22⋊C4—228no—
431C2×C22⋊C4—228no—
432C2×C22⋊C4—228no—
433C2×C22⋊C4—444no—
434C2×C22⋊C4C22×D1114yes—
435C2×C22⋊C4C22×D1114yes—
436C2×C22⋊C4C22×D1114yes—
437#95—118no—
438#95—114no—
439#95—114no—
440#95—118no—
441#95—118no—
442C2×C11—28no—
443C2×C11—82no—
444C2×C11#243018yescomplemented
445C2×C11C22×#12011yescharacteristic
446C2×C11C2×#53211yescharacteristic
447C2×C11C2×#53711yescharacteristic
448C2×C11C2×#54612yes—
449C2×C11C2×#54712yes—
450D11—48no—
451C24—24no—
452C24—22no—
453C24—22no—
454C24—24no—
455C24—24no—
456C24—22no—
457C24—22no—
458C24—22no—
459C24—118no—
460C24—114no—
461C24—224no—
462C24—224no—
463C24—224no—
464C24—224no—
465C24—224no—
466C24—224no—
467C24D4×D1118yes—
468C24C2×C22⋊D1114yes—
469C24C23×D1112yes—
470C24C23×D1112yes—
471C24C23×D1111yescharacteristic
472#41—22256no—
473C2×Q8—2216no—
474C2×Q8—2232no—
475C2×Q8—228no—
476C2×D4—216no—
477C2×D4—28no—
478C2×D4—416no—
479C2×D4—1132no—
480C2×D4—2216no—
481C2×D4—2232no—
482C2×D4—2216no—
483C2×D4—2264no—
484C2×D4—2216no—
485C2×D4—2216no—
486C2×D4—2216no—
487C2×D4—228no—
488C2×D4—448no—
489C2×D4—4432no—
490C22×C4—232no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—216no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—118no—
502C22×C4—118no—
503C22×C4—114no—
504C22×C4—114no—
505C22×C4—114no—
506C22×C4—2216no—
507C22×C4—2216no—
508C22×C4—2264no—
509C22×C4—2216no—
510C22×C4—224no—
511C22×C4—224no—
512C22×C4—224no—
513C22×C4—224no—
514C22×C4—2216no—
515C22×C4—224no—
516C22×C4—2216no—
517C22×C4—224no—
518C22×C4—224no—
519C22×C4—448no—
520C22×C4—4432no—
521C22×C4C2×C22⋊D1114yes—
522C22×C4C2×C22⋊D1114yes—
523C22×C4C2×C22⋊D11116yescomplemented
524C22×C4C23×D1112yes—
525C22×C4C23×D1112yes—
526C22×C4C23×D1112yes—
527C22×C4C23×D1112yes—
528C22×C4C23×D1112yes—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—2216no—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—2216no—
536C22⋊C4—2216no—
537C22⋊C4—2216no—
538C22⋊C4—448no—
539C11C2×#243011yescharacteristic, complemented, Sylow
540C23—28no—
541C23—28no—
542C23—216no—
543C23—216no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—48no—
556C23—48no—
557C23—118no—
558C23—2216no—
559C23—2216no—
560C23—2216no—
561C23—224no—
562C23—224no—
563C23—224no—
564C23—224no—
565C23—224no—
566C23—224no—
567C23—448no—
568C23—448no—
569C23—448no—
570C23—448no—
571C23#66990116yes—
572C23C2×D4×D1118yes—
573C23C2×D4×D1114yes—
574C23#41×D1114yes—
575C23#6704212yes—
576C23#6704312yes—
577C23C22×C22⋊D1114yes—
578C23C22×C22⋊D1112yes—
579C23C24×D1111yescharacteristic
580Q8—2232no—
581D4—416no—
582D4—2232no—
583D4—2264no—
584D4—4432no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—232no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—118no—
601C2×C4—118no—
602C2×C4—2264no—
603C2×C4—2216no—
604C2×C4—2216no—
605C2×C4—224no—
606C2×C4—224no—
607C2×C4—224no—
608C2×C4—224no—
609C2×C4—224no—
610C2×C4—224no—
611C2×C4—224no—
612C2×C4—4432no—
613C2×C4—448no—
614C2×C4C22×C22⋊D1114yes—
615C2×C4C22×C22⋊D1114yes—
616C22—24no—
617C22—24no—
618C22—232no—
619C22—28no—
620C22—28no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—88no—
627C22—228no—
628C22—228no—
629C22—228no—
630C22—444no—
631C22—444no—
632C22—4432no—
633C22—444no—
634C22—444no—
635C22#1062318518yes—
636C22#1062318618yes—
637C22C2×#6699018yes—
638C22C2×#6699014yes—
639C22#1062336314yes—
640C22#1062336414yes—
641C22C22×D4×D1112yes—
642C22C2×#41×D1112yes—
643C22C2×#6704211yescharacteristic
644C22C2×#6704311yescharacteristic
645C22C23×C22⋊D1111yescharacteristic
646C4—24no—
647C4—24no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—228no—
652C4—228no—
653C4—444no—
654C2—28no—
655C2—82no—
656C2—448no—
657C2#1307222818yesdirect product factor
658C2C22×#6699011yescharacteristic
659C2C2×#1062336311yescharacteristic
660C2C2×#1062336411yescharacteristic
661C2C2×#1062318512yes—
662C2C2×#1062318612yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC22×C11×#123C2
socleC24×C11C24
hypercenterC22×#123D11
hypocenterC11C2×#2430
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C23×#4339C2semidirect
2. C22×#66992C2semidirect
3. #13072228C2direct (N × H)
4. C2×#10622592C2semidirect
5. C2×#10622604C2semidirect
6. C2×#10622560C2semidirect
7. C2×#10622573C2semidirect
8. C2×#10622571C2semidirect
9. C2×#10622569C2semidirect
10. C2×#10622602C2semidirect
11. C22×C11⋊#95C2semidirect
12. C22×C11×#123C2semidirect
20. C22×#4339C22semidirect
25. C2×C11×#123C22semidirect
33. C2×#66992C22semidirect
43. C2×C11⋊#95C22semidirect
44. #10622604C22semidirect
45. #10622602C22semidirect
46. #10622592C22semidirect
47. #10622573C22semidirect
48. #10622571C22semidirect
49. #10622569C22semidirect
50. #10622560C22semidirect
58. C23×C4×C11D4semidirect
70. C22×C22⋊D11D4semidirect
158. C22×C4×C11C2×D4semidirect
171. C2×C22⋊D11C2×D4semidirect
184. C22×Dic11#41semidirect
210. C22×#123D11semidirect
293. C2×Dic11C2×#41semidirect
309. C2×#123C2×D11semidirect
408. C23×C4C22⋊D11semidirect
444. C2×C11#2430semidirect
523. C22×C4C2×C22⋊D11semidirect
539. C11C2×#2430semidirect
657. C2#13072228direct (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×#669922
#130722282
C2×#106225922
C2×#106226042
C2×#106225602
C2×#106225732
C2×#106225712
C2×#106225692
C2×#106226022
C22×C11⋊#952
C22×C11×#1232
C2×#243011