Finite Group Databasebeta

← C2×#13071952 #14922465

Subgroups — C2×#13071952 #14922465

Order 2816.

5718 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4343C211yesmaximal, characteristic
2C22×#66947C212yesmaximal, complemented
3C22×C4⋊C4×D11C211yesmaximal, characteristic, complemented
4#13071952C2116yesmaximal, direct product factor
5C2×#10622842C211yesmaximal, characteristic, complemented
6C2×#10622285C211yesmaximal, characteristic, complemented
7C2×#10622604C211yesmaximal, characteristic, complemented
8C2×#10622278C212yesmaximal, complemented
9C2×#10622281C212yesmaximal, complemented
10C2×#10622602C212yesmaximal, complemented
11C2×#10622277C211yesmaximal, characteristic, complemented
12C2×C11×#395C211yesmaximal, characteristic, complemented
13C23×Q8×C11C2211yescharacteristic
14C22×#4343—216no—
15C22×#4343—28no—
16C22×#4343C2218yes—
17C23×D44C2211yescharacteristic
18C23×D44C2212yes—
19C23×C4×D11C2212yes—
20C23×C4×D11C2211yescharacteristic
21C23×C4×D11C2211yescharacteristic
22C23×C4×D11C2211yescharacteristic
23C22×C11×C4⋊C4C2211yescharacteristic
24C22×C11×C4⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2212yes—
26C22×(C2×D11)⋊C4C2212yes—
27C22×(C2×D11)⋊C4C2212yes—
28C22×(C2×D11)⋊C4C2212yes—
29C22×(C2×D11)⋊C4C2212yes—
30C22×C4⋊Dic11C2211yescharacteristic
31C22×Dic11⋊C4C2211yescharacteristic
32C22×Dic11⋊C4C2211yescharacteristic
33C22×Dic11⋊C4C2212yes—
34C2×#66947—216no—
35C2×#66947C22116yescomplemented
36C2×C4⋊C4×D11—28no—
37C2×C4⋊C4×D11C2218yescomplemented
38C11×#395C2218yescomplemented
39C2×C11×#95C2211yescharacteristic
40C2×C11×#95C2212yes—
41#10622842C2218yescomplemented
42C2×C11⋊#95C2212yes—
43C2×C11⋊#95C2211yescharacteristic
44C2×C11⋊#95C2212yes—
45#10622604C2218yescomplemented
46#10622602C22116yescomplemented
47#10622285C2218yescomplemented
48#10622281C22116yescomplemented
49#10622278C22116yescomplemented
50#10622277C2218yescomplemented
51C24×D11C2312yes—
52C24×D11C2311yescharacteristic
53C22×Q8×C11—28no—
54C22×Q8×C11—24no—
55C22×Q8×C11C2314yes—
56C23×C4×C11C2311yescharacteristic
57C23×C4×C11C2312yes—
58C23×C4×C11C2312yes—
59C23×C4×C11C2311yescharacteristic
60C23×C4×C11C2311yescharacteristic
61C23×Dic11C2311yescharacteristic
62C23×Dic11C2312yes—
63C23×Dic11C2311yescharacteristic
64C23×Dic11C2311yescharacteristic
65C2×#4343—2128no—
66C2×#4343—264no—
67C2×#4343—232no—
68C22×D44—28no—
69C22×D44—28no—
70C22×D44—28no—
71C22×D44—24no—
72C22×D44D4116yes—
73C22×D44C2314yes—
74C22×D44C2318yes—
75C22×C4×D11—24no—
76C22×C4×D11—28no—
77C22×C4×D11—28no—
78C22×C4×D11—28no—
79C22×C4×D11—24no—
80C22×C4×D11—28no—
81C22×C4×D11—24no—
82C22×C4×D11—24no—
83C22×C4×D11—28no—
84C22×C4×D11D418yes—
85C22×C4×D11C2314yes—
86C22×C4×D11C2314yes—
87C22×C4×D11C2318yes—
88C22×C4×D11C2314yes—
89C2×C11×C4⋊C4—24no—
90C2×C11×C4⋊C4—28no—
91C2×C11×C4⋊C4C2314yes—
92C2×C11×C4⋊C4C2318yes—
93C2×(C2×D11)⋊C4—28no—
94C2×(C2×D11)⋊C4—28no—
95C2×(C2×D11)⋊C4—28no—
96C2×(C2×D11)⋊C4—28no—
97C2×(C2×D11)⋊C4—28no—
98C2×(C2×D11)⋊C4C2318yes—
99C2×(C2×D11)⋊C4C2318yes—
100C2×(C2×D11)⋊C4C2318yes—
101C2×(C2×D11)⋊C4C2318yes—
102C2×(C2×D11)⋊C4C2318yes—
103C2×C4⋊Dic11—24no—
104C2×C4⋊Dic11C2314yes—
105C2×Dic11⋊C4—24no—
106C2×Dic11⋊C4—28no—
107C2×Dic11⋊C4—24no—
108C2×Dic11⋊C4C2314yes—
109C2×Dic11⋊C4C2314yes—
110C2×Dic11⋊C4C2318yes—
111#66947—264no—
112C4⋊C4×D11—232no—
113C11×#95C2314yes—
114C11×#95C2318yes—
115C11⋊#95C2318yes—
116C11⋊#95C2314yes—
117C11⋊#95C2318yes—
118C2×#2406—111nomaximal, Sylow
119C24×C11C2411yescharacteristic
120C23×D11—22no—
121C23×D11—22no—
122C23×D11—24no—
123C23×D11—24no—
124C23×D11—24no—
125C23×D11—24no—
126C23×D11—24no—
127C23×D11—24no—
128C23×D11—24no—
129C23×D11C2×D418yes—
130C23×D11C2×D418yes—
131C23×D11C2×D414yes—
132C23×D11C2414yes—
133C23×D11C2412yes—
134C2×Q8×C11—232no—
135C2×Q8×C11—216no—
136C2×Q8×C11—28no—
137C22×C4×C11—22no—
138C22×C4×C11—24no—
139C22×C4×C11—24no—
140C22×C4×C11—24no—
141C22×C4×C11—22no—
142C22×C4×C11—22no—
143C22×C4×C11—24no—
144C22×C4×C11—24no—
145C22×C4×C11—24no—
146C22×C4×C11—24no—
147C22×C4×C11C2×D418yes—
148C22×C4×C11C2×D414yes—
149C22×C4×C11#4114yescomplemented
150C22×C4×C11C2414yes—
151C22×C4×C11C2412yes—
152C22×C4×C11C2414yes—
153C22×C4×C11C2412yes—
154C22×C4×C11C2412yes—
155C22×Dic11—22no—
156C22×Dic11—24no—
157C22×Dic11—22no—
158C22×Dic11—24no—
159C22×Dic11—24no—
160C22×Dic11—22no—
161C22×Dic11—24no—
162C22×Dic11—24no—
163C22×Dic11C2×D414yes—
164C22×Dic11#4114yescomplemented
165C22×Dic11C2414yes—
166C22×Dic11C2412yes—
167C22×Dic11C2412yes—
168C22×Dic11C2412yes—
169#4343—2256no—
170C2×D44—232no—
171C2×D44—264no—
172C2×D44—216no—
173C2×D44—216no—
174C2×D44—216no—
175C2×D44—28no—
176C2×D44C2×D4132yes—
177C2×C4×D11—232no—
178C2×C4×D11—264no—
179C2×C4×D11—216no—
180C2×C4×D11—28no—
181C2×C4×D11—232no—
182C2×C4×D11—28no—
183C2×C4×D11—216no—
184C2×C4×D11—216no—
185C2×C4×D11—28no—
186C2×C4×D11—216no—
187C2×C4×D11—216no—
188C2×C4×D11—28no—
189C2×C4×D11—416no—
190C2×C4×D11C2×D4116yes—
191C11×C4⋊C4—28no—
192C11×C4⋊C4—216no—
193(C2×D11)⋊C4—216no—
194(C2×D11)⋊C4—216no—
195(C2×D11)⋊C4—216no—
196(C2×D11)⋊C4—216no—
197(C2×D11)⋊C4—216no—
198C4⋊Dic11—28no—
199Dic11⋊C4—28no—
200Dic11⋊C4—28no—
201Dic11⋊C4—216no—
202C23×#41—111no—
203C22×#123—112no—
204C23×C4⋊C4—111no—
205#2406—1116no—
206C2×#395—111no—
207C2×#395D1111yescharacteristic, complemented
208C2×#394—111no—
209C2×#394—111no—
210C2×#386—112no—
211C2×#385—112no—
212C2×#385—112no—
213C2×#381—111no—
214C23×C11C22×D412yes—
215C23×C11C22×D414yes—
216C23×C11C2×#4112yes—
217C23×C11C2×#4112yes—
218C23×C112−1+414yes—
219C23×C11C2511yescharacteristic
220C22×D11—28no—
221C22×D11—28no—
222C22×D11—216no—
223C22×D11—216no—
224C22×D11—22no—
225C22×D11—22no—
226C22×D11—24no—
227C22×D11—216no—
228C22×D11—216no—
229C22×D11—216no—
230C22×D11—24no—
231C22×D11—24no—
232C22×D11—24no—
233C22×D11—24no—
234C22×D11—24no—
235C22×D11—24no—
236C22×D11—48no—
237C22×D11—48no—
238C22×D11—48no—
239C22×D11—48no—
240C22×D11#120116yes—
241C22×D11C22×D418yes—
242C22×D11C22×D418yes—
243C22×D11C22×D414yes—
244Q8×C11—232no—
245C2×C4×C11—216no—
246C2×C4×C11—28no—
247C2×C4×C11—28no—
248C2×C4×C11—22no—
249C2×C4×C11—24no—
250C2×C4×C11—24no—
251C2×C4×C11—24no—
252C2×C4×C11—22no—
253C2×C4×C11—22no—
254C2×C4×C11—24no—
255C2×C4×C11—24no—
256C2×C4×C11—24no—
257C2×C4×C11—24no—
258C2×C4×C11—44no—
259C2×C4×C11—48no—
260C2×C4×C11C22×D418yes—
261C2×C4×C11C22×D414yes—
262C2×C4×C11C2×#4114yescomplemented
263C2×Dic11—28no—
264C2×Dic11—28no—
265C2×Dic11—22no—
266C2×Dic11—24no—
267C2×Dic11—22no—
268C2×Dic11—22no—
269C2×Dic11—24no—
270C2×Dic11—24no—
271C2×Dic11—24no—
272C2×Dic11—24no—
273C2×Dic11—44no—
274C2×Dic11—48no—
275C2×Dic11C22×D414yes—
276C2×Dic11C2×#4114yescomplemented
277D44—232no—
278D44—264no—
279C4×D11—232no—
280C4×D11—264no—
281C4×D11—232no—
282C4×D11—416no—
283C22×#41—118no—
284C22×#41—228no—
285C22×#41—2216no—
286C23×Q8C2×D1111yescharacteristic
287C23×D4—112no—
288C23×D4—111no—
289C24×C4—111no—
290C24×C4—112no—
291C24×C4—111no—
292C24×C4—111no—
293C2×#123—1116no—
294C2×#123—2216no—
295C22×C4⋊C4—111no—
296C22×C4⋊C4—112no—
297C22×C4⋊C4—111no—
298C22×C4⋊C4—111no—
299C22×C4⋊C4—118no—
300C22×C4⋊C4—228no—
301C22×C4⋊C4C2×D1111yescharacteristic
302C22×C4⋊C4C2×D1112yes—
303C22×C22⋊C4—112no—
304C22×C22⋊C4—112no—
305C22×C22⋊C4—112no—
306C22×C22⋊C4—112no—
307C22×C22⋊C4—112no—
308#395—118no—
309#395C2×D1118yescomplemented
310#394—118no—
311#394—118no—
312#386—1116no—
313#385—1116no—
314#385—1116no—
315#381—118no—
316C2×#95—111no—
317C2×#95—112no—
318C2×#95—112no—
319C2×#95C2×D1111yescharacteristic
320C2×#95C2×D1112yes—
321C22×C11C2×#12014yes—
322C22×C11#52514yes—
323C22×C11#53614yes—
324C22×C11#54718yes—
325C22×C11#54718yes—
326C22×C11C23×D411yescharacteristic
327C22×C11C23×D412yes—
328C22×C11C22×#4111yescharacteristic
329C22×C11C22×#4111yescharacteristic
330C22×C11C2×2−1+412yes—
331C2×D11—232no—
332C2×D11—24no—
333C2×D11—24no—
334C2×D11—28no—
335C2×D11—28no—
336C2×D11—28no—
337C2×D11—28no—
338C2×D11—28no—
339C2×D11—432no—
340C2×D11—44no—
341C2×D11—44no—
342C2×D11—44no—
343C2×D11—44no—
344C2×D11C2×#12018yes—
345C4×C11—28no—
346C4×C11—24no—
347C4×C11—24no—
348C4×C11—42no—
349C4×C11—44no—
350Dic11—24no—
351Dic11—24no—
352Dic11—42no—
353Dic11—44no—
354C25—111no—
355C25—112no—
356C2×#41—2232no—
357C2×#41—2264no—
358C2×#41—22128no—
359C22×Q8—24no—
360C22×Q8—28no—
361C22×Q8C22×D1114yes—
362C22×D4—1116no—
363C22×D4—118no—
364C22×D4—114no—
365C22×D4—228no—
366C22×D4—228no—
367C22×D4—228no—
368C22×D4—224no—
369C23×C4—111no—
370C23×C4—111no—
371C23×C4—118no—
372C23×C4—114no—
373C23×C4—112no—
374C23×C4—111no—
375C23×C4—114no—
376C23×C4—114no—
377C23×C4—118no—
378C23×C4—228no—
379C23×C4—224no—
380C23×C4—224no—
381C23×C4—228no—
382C23×C4—228no—
383C23×C4—224no—
384C23×C4—228no—
385C23×C4—228no—
386C23×C4—224no—
387C23×C4C22×D1112yes—
388C23×C4C22×D1111yescharacteristic
389C23×C4C22×D1111yescharacteristic
390C23×C4C22×D1112yes—
391C23×C4C22×D1111yescharacteristic
392#123—2264no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—28no—
395C2×C4⋊C4—114no—
396C2×C4⋊C4—114no—
397C2×C4⋊C4—114no—
398C2×C4⋊C4—118no—
399C2×C4⋊C4—2232no—
400C2×C4⋊C4—224no—
401C2×C4⋊C4—224no—
402C2×C4⋊C4—224no—
403C2×C4⋊C4—228no—
404C2×C4⋊C4C22×D1114yes—
405C2×C4⋊C4C22×D1118yes—
406C2×C22⋊C4—118no—
407C2×C22⋊C4—118no—
408C2×C22⋊C4—118no—
409C2×C22⋊C4—118no—
410C2×C22⋊C4—118no—
411C2×C22⋊C4—228no—
412C2×C22⋊C4—228no—
413C2×C22⋊C4—228no—
414C2×C22⋊C4—228no—
415C2×C22⋊C4—228no—
416#95—118no—
417#95—114no—
418#95—118no—
419#95C22×D1114yes—
420#95C22×D1118yes—
421C2×C11#240618yescomplemented
422C2×C11C22×#12011yescharacteristic
423C2×C11C2×#52511yescharacteristic
424C2×C11C2×#53611yescharacteristic
425C2×C11C2×#54712yes—
426C2×C11C2×#54712yes—
427D11—28no—
428D11—48no—
429C24—118no—
430C24—114no—
431C24—112no—
432C24—118no—
433C24—114no—
434C24—222no—
435C24—224no—
436C24—222no—
437C24—224no—
438C24—224no—
439C24—224no—
440C24—224no—
441C24—224no—
442C24—224no—
443C24C23×D1111yescharacteristic
444#41—22256no—
445C2×Q8—232no—
446C2×Q8—28no—
447C2×Q8—216no—
448C2×D4—1132no—
449C2×D4—2264no—
450C2×D4—2216no—
451C2×D4—2216no—
452C2×D4—2216no—
453C2×D4—2232no—
454C2×D4—228no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—114no—
466C22×C4—114no—
467C22×C4—112no—
468C22×C4—112no—
469C22×C4—1116no—
470C22×C4—114no—
471C22×C4—112no—
472C22×C4—2216no—
473C22×C4—228no—
474C22×C4—228no—
475C22×C4—2232no—
476C22×C4—2216no—
477C22×C4—2216no—
478C22×C4—2264no—
479C22×C4—2232no—
480C22×C4—228no—
481C22×C4—2216no—
482C22×C4—222no—
483C22×C4—224no—
484C22×C4—222no—
485C22×C4—2216no—
486C22×C4—228no—
487C22×C4—224no—
488C22×C4—224no—
489C22×C4—222no—
490C22×C4—224no—
491C22×C4—224no—
492C22×C4—4416no—
493C22×C4D4×D1114yes—
494C22×C4D4×D1118yes—
495C22×C4#434314yescomplemented
496C22×C4C23×D1114yes—
497C22×C4C23×D1112yes—
498C22×C4C23×D1112yes—
499C22×C4C23×D1114yes—
500C22×C4C23×D1112yes—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—228no—
504C4⋊C4—228no—
505C4⋊C4—228no—
506C4⋊C4—2216no—
507C22⋊C4—2216no—
508C22⋊C4—2216no—
509C22⋊C4—2216no—
510C22⋊C4—2216no—
511C22⋊C4—2216no—
512C11C2×#240611yescharacteristic, complemented, Sylow
513C23—1116no—
514C23—118no—
515C23—114no—
516C23—118no—
517C23—228no—
518C23—2216no—
519C23—228no—
520C23—2216no—
521C23—222no—
522C23—224no—
523C23—222no—
524C23—2216no—
525C23—2216no—
526C23—2216no—
527C23—224no—
528C23—224no—
529C23—224no—
530C23—224no—
531C23—224no—
532C23—224no—
533C23—448no—
534C23—448no—
535C23—448no—
536C23—448no—
537C23C2×D4×D1112yes—
538C23C2×D4×D1114yes—
539C23C2×#434312yes—
540C23#6704014yes—
541C23#41×D1112yes—
542C23C24×D1111yescharacteristic
543Q8—232no—
544D4—2264no—
545D4—2232no—
546C2×C4—28no—
547C2×C4—216no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—114no—
562C2×C4—114no—
563C2×C4—2232no—
564C2×C4—2264no—
565C2×C4—2232no—
566C2×C4—228no—
567C2×C4—228no—
568C2×C4—222no—
569C2×C4—224no—
570C2×C4—222no—
571C2×C4—224no—
572C2×C4—224no—
573C2×C4—222no—
574C2×C4—224no—
575C2×C4—224no—
576C2×C4—4416no—
577C2×C4—448no—
578C2×C4—444no—
579C2×C4C2×D4×D1114yes—
580C2×C4C2×D4×D1118yes—
581C2×C4C2×#434314yescomplemented
582C22—118no—
583C22—224no—
584C22—224no—
585C22—2232no—
586C22—228no—
587C22—228no—
588C22—228no—
589C22—228no—
590C22—228no—
591C22—444no—
592C22—444no—
593C22—4432no—
594C22—444no—
595C22—444no—
596C22D11×#12014yes—
597C22#1062315814yes—
598C22#1062316218yes—
599C22#1062325514yes—
600C22#1062325718yes—
601C22C22×D4×D1111yescharacteristic
602C22C22×D4×D1112yes—
603C22C22×#434311yescharacteristic
604C22C2×#6704012yes—
605C22C2×#41×D1111yescharacteristic
606C4—24no—
607C4—24no—
608C4—28no—
609C4—44no—
610C4—42no—
611C4—224no—
612C4—224no—
613C4—444no—
614C4—442no—
615C2—228no—
616C2—448no—
617C2#1307195218yesdirect product factor
618C2C2×D11×#12011yescharacteristic
619C2C2×#1062315811yescharacteristic
620C2C2×#1062325511yescharacteristic
621C2C2×#1062316212yes—
622C2C2×#1062325712yes—
subgroupG/H
centerC24C23×D11
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC2×C11×#395C2
socleC24×C11C24
prime radicalC23×#4343C2
hypercenterC2×#395D11
hypocenterC11C2×#2406
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C22×#66947C2semidirect
3. C22×C4⋊C4×D11C2semidirect
4. #13071952C2direct (N × H)
5. C2×#10622842C2semidirect
6. C2×#10622285C2semidirect
7. C2×#10622604C2semidirect
8. C2×#10622278C2semidirect
9. C2×#10622281C2semidirect
10. C2×#10622602C2semidirect
11. C2×#10622277C2semidirect
12. C2×C11×#395C2semidirect
35. C2×#66947C22semidirect
37. C2×C4⋊C4×D11C22semidirect
38. C11×#395C22semidirect
41. #10622842C22semidirect
45. #10622604C22semidirect
46. #10622602C22semidirect
47. #10622285C22semidirect
48. #10622281C22semidirect
49. #10622278C22semidirect
50. #10622277C22semidirect
149. C22×C4×C11#41semidirect
164. C22×Dic11#41semidirect
207. C2×#395D11semidirect
262. C2×C4×C11C2×#41semidirect
276. C2×Dic11C2×#41semidirect
309. #395C2×D11semidirect
421. C2×C11#2406semidirect
495. C22×C4#4343semidirect
512. C11C2×#2406semidirect
581. C2×C4C2×#4343semidirect
617. C2#13071952direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43432
C22×#669472
C22×C4⋊C4×D112
#130719522
C2×#106228422
C2×#106222852
C2×#106226042
C2×#106222782
C2×#106222812
C2×#106226022
C2×#106222772
C2×C11×#3952
C2×#240611