Finite Group Databasebeta

← C2×#21450125 #29849065

Subgroups — C2×#21450125 #29849065

Order 15616.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519762C212yesmaximal
2C22×C4×D244C211yesmaximal, characteristic
3C22×C4⋊C4×D61C211yesmaximal, characteristic, complemented
4#21450125C2116yesmaximal, direct product factor
5C2×#16839695C212yesmaximal, complemented
6C2×#16839691C212yesmaximal, complemented
7C2×#16839660C211yesmaximal, characteristic, complemented
8C2×#16839624C212yesmaximal, complemented
9C2×#16839619C211yesmaximal, characteristic, complemented
10C2×#16839619C211yesmaximal, characteristic, complemented
11C2×#16839619C211yesmaximal, characteristic, complemented
12C2×C61×#389C211yesmaximal, characteristic, complemented
13C23×D244C2211yescharacteristic
14C23×C4×D61C2212yes—
15C23×C4×D61C2211yescharacteristic
16C23×C4×D61C2211yescharacteristic
17C2×#14519762—216no—
18C2×#14519762C22116yes—
19C2×C4×D244—28no—
20C2×C4×D244C2218yes—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23C22×C4⋊Dic61C2211yescharacteristic
24C22×#11758196C2212yes—
25C22×#11758196C2212yes—
26C2×C4⋊C4×D61—28no—
27C2×C4⋊C4×D61C2218yescomplemented
28C22×#11758198C2211yescharacteristic
29C22×#11758198C2212yes—
30C22×#11758198C2211yescharacteristic
31C22×#11758198C2212yes—
32C22×#11758198C2212yes—
33#16839695C22116yescomplemented
34#16839691C22116yescomplemented
35#16839660C2218yescomplemented
36#16839624C22116yescomplemented
37#16839619C2218yescomplemented
38#16839619C2218yescomplemented
39#16839619C2218yescomplemented
40C2×#14519684C2212yes—
41C2×#14519684C2212yes—
42C2×#14519684C2212yes—
43C2×#14519684C2212yes—
44C22×C4⋊C4×C61C2211yescharacteristic
45C22×C4⋊C4×C61C2212yes—
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×C42×C61C2211yescharacteristic
48C61×#389C2218yescomplemented
49C2×#95×C61C2212yes—
50C24×D61C2311yescharacteristic
51C24×D61C2311yescharacteristic
52C22×D244—24no—
53C22×D244—24no—
54C22×D244D418yes—
55C22×D244C2314yes—
56C23×Dic61C2312yes—
57C23×Dic61C2312yes—
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C22×C4×D61—24no—
61C22×C4×D61—24no—
62C22×C4×D61—28no—
63C22×C4×D61—28no—
64C22×C4×D61—28no—
65C22×C4×D61—24no—
66C22×C4×D61—24no—
67C22×C4×D61—24no—
68C22×C4×D61D418yes—
69C22×C4×D61C2318yes—
70C22×C4×D61C2314yes—
71C22×C4×D61C2314yes—
72#14519762—264no—
73C4×D244—232no—
74C2×C4⋊Dic61—24no—
75C2×C4⋊Dic61—24no—
76C2×C4⋊Dic61—24no—
77C2×C4⋊Dic61C2314yes—
78C2×C4⋊Dic61C2314yes—
79C2×C4⋊Dic61C2314yes—
80C2×#11758196—28no—
81C2×#11758196—28no—
82C2×#11758196C2318yes—
83C2×#11758196C2318yes—
84C4⋊C4×D61—232no—
85C2×#11758198—24no—
86C2×#11758198—28no—
87C2×#11758198—28no—
88C2×#11758198—28no—
89C2×#11758198—24no—
90C2×#11758198C2314yes—
91C2×#11758198C2318yes—
92C2×#11758198C2318yes—
93C2×#11758198C2314yes—
94C2×#11758198C2318yes—
95#14519684C2318yes—
96#14519684C2318yes—
97#14519684C2318yes—
98#14519684C2318yes—
99C23×C4×C61C2311yescharacteristic
100C23×C4×C61C2312yes—
101C23×C4×C61C2312yes—
102C23×C4×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61—24no—
105C2×C4⋊C4×C61—28no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61C2318yes—
108C2×C4⋊C4×C61C2314yes—
109C2×C4⋊C4×C61C2314yes—
110C2×C42×C61—24no—
111C2×C42×C61C2314yes—
112#95×C61C2318yes—
113C24×C61C2411yescharacteristic
114C23×D61—22no—
115C23×D61—22no—
116C23×D61—22no—
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—22no—
122C23×D61—22no—
123C23×D61—22no—
124C23×D61C2×D414yes—
125C23×D61C2×D414yes—
126C23×D61C2×D414yes—
127C23×D61#4114yes—
128C23×D61C2412yes—
129C23×D61C2412yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—24no—
141C22×C4×C61C2×D414yes—
142C22×C4×C61C2×D414yes—
143C22×C4×C61#4114yes—
144C22×C4×C61#4114yes—
145C22×C4×C61C2414yes—
146C22×C4×C61C2412yes—
147C22×C4×C61C2414yes—
148C22×C4×C61C2412yes—
149C22×C4×C61C2412yes—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—24no—
153C22×Dic61—22no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61C2×D414yes—
162C22×Dic61C2412yes—
163C22×Dic61C2414yes—
164C22×Dic61C2414yes—
165C22×Dic61C2412yes—
166C2×D244—232no—
167C2×D244—28no—
168C2×D244—28no—
169C2×D244C2×D4116yes—
170C2×C4×D61—232no—
171C2×C4×D61—28no—
172C2×C4×D61—28no—
173C2×C4×D61—216no—
174C2×C4×D61—216no—
175C2×C4×D61—216no—
176C2×C4×D61—28no—
177C2×C4×D61—28no—
178C2×C4×D61—28no—
179C2×C4×D61—432no—
180C2×C4×D61—416no—
181C2×C4×D61C2×D4116yes—
182C4⋊C4×C61—28no—
183C4⋊C4×C61—216no—
184C4⋊C4×C61—28no—
185C42×C61—28no—
186#11758198—28no—
187#11758198—216no—
188#11758198—216no—
189#11758198—216no—
190#11758198—28no—
191C4⋊Dic61—28no—
192C4⋊Dic61—28no—
193C4⋊Dic61—28no—
194#11758196—216no—
195#11758196—216no—
196C23×C61C22×D412yes—
197C23×C61C22×D412yes—
198C23×C61C2×#4112yes—
199C23×C61C2×#4112yes—
200C23×C61C2×#4112yes—
201C23×C612−1+412yes—
202C23×C612−1+412yes—
203C23×C61C2511yescharacteristic
204C22×D61—28no—
205C22×D61—28no—
206C22×D61—28no—
207C22×D61—28no—
208C22×D61—28no—
209C22×D61—28no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—28no—
215C22×D61—28no—
216C22×D61—28no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—44no—
224C22×D61—44no—
225C22×D61—44no—
226C22×D61—44no—
227C22×D61#121116yes—
228C22×D61C22×D414yes—
229C22×D61C22×D414yes—
230C22×D61C22×D414yes—
231C22×D61C2×#4114yes—
232C2×C4×C61—28no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—48no—
245C2×C4×C61—44no—
246C2×C4×C61—48no—
247C2×C4×C61#123116yescomplemented
248C2×C4×C61C22×D414yes—
249C2×C4×C61C22×D414yes—
250C2×C4×C61C2×#4114yes—
251C2×C4×C61C2×#4114yes—
252C2×Dic61—28no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—48no—
265C2×Dic61—44no—
266C2×Dic61—48no—
267C2×Dic61C22×D414yes—
268D244—232no—
269C4×D61—232no—
270C4×D61—432no—
271C4×D61—416no—
272C2×#2486—611nomaximal, Sylow
273C22×C61C2×#12114yes—
274C22×C61C2×#12314yes—
275C22×C61#53614yes—
276C22×C61#54714yes—
277C22×C61#54714yes—
278C22×C61#55214yes—
279C22×C61#55214yes—
280C22×C61C23×D411yescharacteristic
281C22×C61C23×D411yescharacteristic
282C22×C61C22×#4111yescharacteristic
283C22×C61C22×#4111yescharacteristic
284C22×C61C22×#4111yescharacteristic
285C22×C61C2×2−1+411yescharacteristic
286C22×C61C2×2−1+411yescharacteristic
287C2×D61—24no—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—232no—
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—416no—
302C2×D61C2×#12118yes—
303C4×C61—24no—
304C4×C61—44no—
305C4×C61—42no—
306C4×C61—44no—
307C4×C61C2×#12318yescomplemented
308Dic61—24no—
309Dic61—44no—
310Dic61—42no—
311Dic61—44no—
312C22×#123—612no—
313C22×C4×D4—611no—
314C23×C4⋊C4—611no—
315#2486—6116no—
316C2×#398—612no—
317C2×#397—612no—
318C2×#389—611no—
319C2×#389D6111yescharacteristic, complemented
320C2×#381—612no—
321C2×#380—611no—
322C2×#380—611no—
323C2×#380—611no—
324C2×C61#248618yescomplemented
325C2×C61C22×#12111yescharacteristic
326C2×C61C22×#12311yescharacteristic
327C2×C61C2×#53611yescharacteristic
328C2×C61C2×#54711yescharacteristic
329C2×C61C2×#54711yescharacteristic
330C2×C61C2×#55211yescharacteristic
331C2×C61C2×#55211yescharacteristic
332D61—28no—
333D61—44no—
334C23×D4—611no—
335C24×C4—611no—
336C24×C4—612no—
337C24×C4—611no—
338C2×#123—6116no—
339C2×#123—12216no—
340C2×C4×D4—618no—
341C2×C4×D4—1228no—
342C22×C4⋊C4—612no—
343C22×C4⋊C4—612no—
344C22×C4⋊C4—611no—
345C22×C4⋊C4—611no—
346C22×C4⋊C4—611no—
347C22×C4⋊C4—618no—
348C22×C4⋊C4—1228no—
349C22×C4⋊C4C2×D6112yes—
350C22×C4⋊C4C2×D6111yescharacteristic
351C22×C4⋊C4C2×D6111yescharacteristic
352C22×C22⋊C4—611no—
353C22×C22⋊C4—612no—
354C22×C22⋊C4—612no—
355C22×C22⋊C4—611no—
356C22×C22⋊C4—612no—
357C22×C42C2×D6111yescharacteristic
358#398—6116no—
359#397—6116no—
360#389—618no—
361#389C2×D6118yescomplemented
362#381—6116no—
363#380—618no—
364#380—618no—
365#380—618no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95C2×D6112yes—
371C61C2×#248611yescharacteristic, complemented, Sylow
372C25—611no—
373C25—611no—
374C22×D4—618no—
375C22×D4—614no—
376C22×D4—1224no—
377C22×D4—1224no—
378C23×C4—611no—
379C23×C4—618no—
380C23×C4—614no—
381C23×C4—612no—
382C23×C4—612no—
383C23×C4—611no—
384C23×C4—618no—
385C23×C4—614no—
386C23×C4—1228no—
387C23×C4—1228no—
388C23×C4—1228no—
389C23×C4—1224no—
390C23×C4—1224no—
391C23×C4—1224no—
392C23×C4—1224no—
393C23×C4—1224no—
394C23×C4C22×D6112yes—
395C23×C4C22×D6111yescharacteristic
396C23×C4C22×D6112yes—
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399#123—12264no—
400C4×D4—12232no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—28no—
404C2×C4⋊C4—614no—
405C2×C4⋊C4—618no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—618no—
408C2×C4⋊C4—614no—
409C2×C4⋊C4—12232no—
410C2×C4⋊C4—1228no—
411C2×C4⋊C4—1224no—
412C2×C4⋊C4—1224no—
413C2×C4⋊C4—1228no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4C22×D6114yes—
416C2×C4⋊C4C22×D6118yes—
417C2×C4⋊C4C22×D6114yes—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—614no—
420C2×C22⋊C4—618no—
421C2×C22⋊C4—618no—
422C2×C22⋊C4—618no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1228no—
426C2×C22⋊C4—1228no—
427C2×C22⋊C4—1228no—
428C2×C42—24no—
429C2×C42C22×D6114yes—
430#95—618no—
431#95—618no—
432#95—618no—
433#95—618no—
434#95C22×D6118yes—
435C24—614no—
436C24—614no—
437C24—614no—
438C24—612no—
439C24—614no—
440C24—612no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24—1222no—
445C24—1222no—
446C24—1222no—
447C24—1222no—
448C24—1222no—
449C24—1222no—
450C24—1222no—
451C24C23×D6111yescharacteristic
452C2×D4—6116no—
453C2×D4—1228no—
454C2×D4—12232no—
455C2×D4—1228no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—614no—
468C22×C4—612no—
469C22×C4—6116no—
470C22×C4—612no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—12216no—
474C22×C4—12216no—
475C22×C4—12216no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1228no—
479C22×C4—1228no—
480C22×C4—12232no—
481C22×C4—1224no—
482C22×C4—1224no—
483C22×C4—1222no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1222no—
487C22×C4—1224no—
488C22×C4—1222no—
489C22×C4—1224no—
490C22×C4—1228no—
491C22×C4—1228no—
492C22×C4—1228no—
493C22×C4—24432no—
494C22×C4—24416no—
495C22×C4#1175822214yes—
496C22×C4D4×D6114yes—
497C22×C4D4×D6114yes—
498C22×C4#1175822614yes—
499C22×C4C23×D6112yes—
500C22×C4C23×D6114yes—
501C22×C4C23×D6114yes—
502C22×C4C23×D6112yes—
503C22×C4C23×D6112yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—12216no—
508C4⋊C4—1228no—
509C4⋊C4—1228no—
510C4⋊C4—12216no—
511C4⋊C4—1228no—
512C22⋊C4—1228no—
513C22⋊C4—1228no—
514C22⋊C4—12216no—
515C22⋊C4—12216no—
516C22⋊C4—12216no—
517C42—28no—
518C23—6116no—
519C23—614no—
520C23—614no—
521C23—614no—
522C23—614no—
523C23—1228no—
524C23—1228no—
525C23—1228no—
526C23—1222no—
527C23—1222no—
528C23—1228no—
529C23—1222no—
530C23—1222no—
531C23—1228no—
532C23—1228no—
533C23—1228no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—1222no—
538C23—1222no—
539C23—1222no—
540C23—1228no—
541C23—1228no—
542C23—2444no—
543C23—2444no—
544C23—2444no—
545C23—2444no—
546C23C2×D4×D6112yes—
547C23C2×D4×D6112yes—
548C23#41×D6112yes—
549C23C2×#1175822212yes—
550C23C2×#1175822612yes—
551C23#1451982412yes—
552C23#1451982512yes—
553C23C24×D6111yescharacteristic
554D4—12232no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—614no—
571C2×C4—1228no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—12232no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1222no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1222no—
581C2×C4—1224no—
582C2×C4—1222no—
583C2×C4—1224no—
584C2×C4—24432no—
585C2×C4—2448no—
586C2×C4—2444no—
587C2×C4—2448no—
588C2×C4—24416no—
589C2×C4#14519762116yescomplemented
590C2×C4C2×D4×D6114yes—
591C2×C4C2×D4×D6114yes—
592C2×C4C2×#1175822214yes—
593C2×C4C2×#1175822614yes—
594C22—618no—
595C22—1224no—
596C22—1224no—
597C22—1224no—
598C22—1224no—
599C22—1224no—
600C22—12232no—
601C22—1224no—
602C22—1224no—
603C22—1224no—
604C22—1224no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22—24416no—
609C22—2442no—
610C22#121×D6114yes—
611C22C2×#1451976214yes—
612C22#1684033514yes—
613C22#1684041514yes—
614C22#1684041714yes—
615C22#1684045014yes—
616C22#1684045314yes—
617C22C22×D4×D6111yescharacteristic
618C22C22×D4×D6111yescharacteristic
619C22C2×#41×D6111yescharacteristic
620C22C22×#1175822211yescharacteristic
621C22C22×#1175822611yescharacteristic
622C22C2×#1451982411yescharacteristic
623C22C2×#1451982511yescharacteristic
624C4—24no—
625C4—44no—
626C4—44no—
627C4—42no—
628C4—1224no—
629C4—2444no—
630C4—2444no—
631C4—2442no—
632C4C2×#1451976218yescomplemented
633C2—1228no—
634C2—2444no—
635C2#2145012518yesdirect product factor
636C2C2×#121×D6111yescharacteristic
637C2C22×#1451976211yescharacteristic
638C2C2×#1684033511yescharacteristic
639C2C2×#1684041511yescharacteristic
640C2C2×#1684041711yescharacteristic
641C2C2×#1684045011yescharacteristic
642C2C2×#1684045311yescharacteristic
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC2×C61×#389C2
socleC24×C61C24
prime radicalC23×D244C22
hypercenterC2×#389D61
hypocenterC61C2×#2486
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D61C2semidirect
4. #21450125C2direct (N × H)
5. C2×#16839695C2semidirect
6. C2×#16839691C2semidirect
7. C2×#16839660C2semidirect
8. C2×#16839624C2semidirect
9. C2×#16839619C2semidirect
10. C2×#16839619C2semidirect
11. C2×#16839619C2semidirect
12. C2×C61×#389C2semidirect
27. C2×C4⋊C4×D61C22semidirect
33. #16839695C22semidirect
34. #16839691C22semidirect
35. #16839660C22semidirect
36. #16839624C22semidirect
37. #16839619C22semidirect
38. #16839619C22semidirect
39. #16839619C22semidirect
48. C61×#389C22semidirect
247. C2×C4×C61#123semidirect
307. C4×C61C2×#123semidirect
319. C2×#389D61semidirect
324. C2×C61#2486semidirect
361. #389C2×D61semidirect
371. C61C2×#2486semidirect
589. C2×C4#14519762semidirect
632. C4C2×#14519762semidirect
635. C2#21450125direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197622
C22×C4×D2442
C22×C4⋊C4×D612
#214501252
C2×#168396952
C2×#168396912
C2×#168396602
C2×#168396242
C2×#168396192
C2×C61×#3892
C2×#248661