Finite Group Databasebeta

← C2×#21449249 #29848429

Subgroups — C2×#21449249 #29848429

Order 15616.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519753C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊D61C211yesmaximal, characteristic
3C22×#14519736C211yesmaximal, characteristic, complemented
4#21449249C2116yesmaximal, direct product factor
5C2×#16839690C212yesmaximal, complemented
6C2×#16839677C212yesmaximal
7C2×#16839669C211yesmaximal, characteristic, complemented
8C2×#16839658C211yesmaximal, characteristic, complemented
9C2×#16839657C212yesmaximal, complemented
10C2×#16839652C212yesmaximal, complemented
11C2×#16839619C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
13C23×C22⋊D61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C2×#14519753—28no—
16C2×#14519753C2218yescomplemented
17C2×C4×C22⋊D61—28no—
18C2×C4×C22⋊D61C2218yes—
19C2×#14519736—28no—
20C2×#14519736C2218yescomplemented
21C22×C4⋊Dic61C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23C22×#11758196C2211yescharacteristic
24C22×#11758196C2212yes—
25C22×#11758196C2212yes—
26C22×C22⋊Dic61C2212yes—
27C22×C22⋊Dic61C2211yescharacteristic
28C22×#11758198C2211yescharacteristic
29C22×#11758198C2212yes—
30C22×#11758198C2212yes—
31C22×C4×Dic61C2212yes—
32C22×C4×Dic61C2211yescharacteristic
33#16839690C22116yescomplemented
34#16839677C22116yes—
35#16839669C2218yescomplemented
36#16839658C2218yescomplemented
37#16839657C22116yescomplemented
38#16839652C22116yescomplemented
39#16839619C2218yescomplemented
40C2×#14519684C2212yes—
41C2×#14519684C2212yes—
42C2×#14519684C2212yes—
43C2×#14519684C2212yes—
44C23×Q8×C61C2211yescharacteristic
45C24×C4×C61C2211yescharacteristic
46C2×C22⋊Q8×C61—28no—
47C2×C22⋊Q8×C61C2218yescomplemented
48C22×C4⋊C4×C61C2212yes—
49C22×C4⋊C4×C61C2211yescharacteristic
50C22×C22⋊C4×C61C2212yes—
51C24×D61C2311yescharacteristic
52C22×C22⋊D61—24no—
53C22×C22⋊D61—24no—
54C22×C22⋊D61—24no—
55C22×C22⋊D61C2314yes—
56C23×Dic61C2312yes—
57C23×Dic61C2311yescharacteristic
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2312yes—
60C23×Dic61C2311yescharacteristic
61C22×C4×D61—24no—
62C22×C4×D61—24no—
63C22×C4×D61—24no—
64C22×C4×D61C2314yes—
65#14519753—232no—
66C4×C22⋊D61—232no—
67#14519736—232no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61D418yescomplemented
70C2×C4⋊Dic61C2314yes—
71C2×C4⋊Dic61C2314yes—
72C2×#11758196—24no—
73C2×#11758196—28no—
74C2×#11758196—28no—
75C2×#11758196C2318yes—
76C2×#11758196C2314yes—
77C2×#11758196C2318yes—
78C2×C22⋊Dic61—24no—
79C2×C22⋊Dic61—28no—
80C2×C22⋊Dic61C2314yes—
81C2×C22⋊Dic61C2318yes—
82C2×#11758198—28no—
83C2×#11758198—24no—
84C2×#11758198—28no—
85C2×#11758198C2314yes—
86C2×#11758198C2318yes—
87C2×#11758198C2318yes—
88C2×C4×Dic61—24no—
89C2×C4×Dic61—28no—
90C2×C4×Dic61C2318yes—
91C2×C4×Dic61C2314yes—
92#14519684C2318yes—
93#14519684C2318yes—
94#14519684C2318yes—
95#14519684C2318yes—
96C25×C61C2311yescharacteristic
97C22×Q8×C61—24no—
98C22×Q8×C61—24no—
99C22×Q8×C61—24no—
100C22×Q8×C61C2314yes—
101C23×C4×C61—24no—
102C23×C4×C61—24no—
103C23×C4×C61D418yes—
104C23×C4×C61C2312yes—
105C23×C4×C61C2312yes—
106C23×C4×C61C2311yescharacteristic
107C23×C4×C61C2314yes—
108C23×C4×C61C2311yescharacteristic
109C22⋊Q8×C61—232no—
110C2×C4⋊C4×C61—24no—
111C2×C4⋊C4×C61—28no—
112C2×C4⋊C4×C61C2314yes—
113C2×C4⋊C4×C61C2318yes—
114C2×C22⋊C4×C61—28no—
115C2×C22⋊C4×C61C2318yes—
116C24×C61—22no—
117C24×C61—22no—
118C24×C61—22no—
119C24×C61—22no—
120C24×C61—22no—
121C24×C61—22no—
122C24×C61C2×D414yes—
123C24×C61C2412yes—
124C24×C61C2411yescharacteristic
125C23×D61—22no—
126C23×D61—22no—
127C23×D61—22no—
128C23×D61—22no—
129C23×D61—22no—
130C23×D61—22no—
131C23×D61#4114yes—
132C23×D61C2412yes—
133C2×Q8×C61—28no—
134C2×Q8×C61—28no—
135C2×Q8×C61—28no—
136C2×Q8×C61—416no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—232no—
140C22×C4×C61—28no—
141C22×C4×C61—28no—
142C22×C4×C61—24no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—24no—
146C22×C4×C61—24no—
147C22×C4×C61—24no—
148C22×C4×C61C2×D414yes—
149C22×C4×C61C2×D4116yes—
150C22×C4×C61C2×D414yes—
151C22×C4×C61C2×D414yes—
152C22×C4×C61#4114yes—
153C22×C4×C61C2414yes—
154C22×C4×C61C2414yes—
155C22×C4×C61C2412yes—
156C22×C4×C61C2412yes—
157C2×C22⋊D61—232no—
158C2×C22⋊D61—28no—
159C2×C22⋊D61—28no—
160C2×C22⋊D61—28no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—24no—
164C22×Dic61—24no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—24no—
169C22×Dic61—22no—
170C22×Dic61—24no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—24no—
174C22×Dic61C2×D418yes—
175C22×Dic61#4114yes—
176C22×Dic61C2412yes—
177C22×Dic61C2412yes—
178C22×Dic61C2414yes—
179C22×Dic61C2414yes—
180C22×Dic61C2412yes—
181C2×C4×D61—28no—
182C2×C4×D61—28no—
183C2×C4×D61—28no—
184C2×C4×D61—416no—
185C4⋊C4×C61—28no—
186C4⋊C4×C61—216no—
187C22⋊C4×C61—216no—
188C22⋊Dic61—28no—
189C22⋊Dic61—216no—
190#11758198—216no—
191#11758198—28no—
192#11758198—216no—
193C4⋊Dic61—28no—
194C4⋊Dic61C2×D4116yescomplemented
195#11758196—28no—
196#11758196—216no—
197#11758196—216no—
198C4×Dic61—28no—
199C4×Dic61—216no—
200C23×C61—28no—
201C23×C61—28no—
202C23×C61—28no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—44no—
210C23×C61—44no—
211C23×C61—44no—
212C23×C61—44no—
213C23×C61C22×D414yes—
214C23×C61C22×D412yes—
215C23×C61C22×D412yes—
216C23×C61C2×#4112yes—
217C23×C61C2×#4112yes—
218C23×C61C2×#4112yes—
219C23×C612−1+412yes—
220C23×C612−1+412yes—
221C23×C61C2511yescharacteristic
222C22×D61—28no—
223C22×D61—28no—
224C22×D61—28no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61—44no—
234C22×D61—44no—
235C22×D61C2×#4114yes—
236Q8×C61—416no—
237C2×C4×C61—28no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—232no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—48no—
248C2×C4×C61—48no—
249C2×C4×C61#121116yes—
250C2×C4×C61C22×D414yes—
251C2×C4×C61C22×D414yes—
252C2×C4×C61C22×D414yes—
253C2×C4×C61C2×#4114yes—
254C22⋊D61—232no—
255C2×Dic61—28no—
256C2×Dic61—216no—
257C2×Dic61—22no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—24no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—44no—
271C2×Dic61—44no—
272C2×Dic61—48no—
273C2×Dic61C22×D418yes—
274C2×Dic61C2×#4114yes—
275C4×D61—416no—
276C2×#2415—611nomaximal, Sylow
277C22×C61—24no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—416no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61C2×#12114yes—
286C22×C61#52514yes—
287C22×C61#53614yes—
288C22×C61#54714yes—
289C22×C61#54814yes—
290C22×C61#55214yes—
291C22×C61#55614yes—
292C22×C61C23×D411yescharacteristic
293C22×C61C23×D411yescharacteristic
294C22×C61C22×#4111yescharacteristic
295C22×C61C22×#4111yescharacteristic
296C22×C61C22×#4111yescharacteristic
297C22×C61C2×2−1+411yescharacteristic
298C22×C61C2×2−1+411yescharacteristic
299C2×D61—24no—
300C2×D61—24no—
301C2×D61—24no—
302C2×D61—42no—
303C2×D61—416no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C4×C61—24no—
308C4×C61—44no—
309C4×C61—44no—
310C4×C61C2×#12118yes—
311Dic61—24no—
312Dic61—28no—
313Dic61—42no—
314Dic61—42no—
315Dic61—44no—
316C22×C22⋊Q8—611no—
317C22×C22⋊Q8D6111yescharacteristic, complemented
318C22×C4×D4—611no—
319C22×#117—611no—
320#2415—6116no—
321C2×#397—612no—
322C2×#394—612no—
323C2×#391—611no—
324C2×#389—612no—
325C2×#389—611no—
326C2×#388—612no—
327C2×#380—611no—
328C2×C61—44no—
329C2×C61#241518yescomplemented
330C2×C61C22×#12111yescharacteristic
331C2×C61C2×#52511yescharacteristic
332C2×C61C2×#53611yescharacteristic
333C2×C61C2×#54711yescharacteristic
334C2×C61C2×#54811yescharacteristic
335C2×C61C2×#55211yescharacteristic
336C2×C61C2×#55611yescharacteristic
337D61—44no—
338C23×Q8C2×D6111yescharacteristic
339C23×D4—611no—
340C24×C4—611no—
341C24×C4C2×D6111yescharacteristic
342C2×C22⋊Q8—28no—
343C2×C22⋊Q8—618no—
344C2×C22⋊Q8—1228no—
345C2×C22⋊Q8C2×D6118yescomplemented
346C2×C4×D4—618no—
347C2×C4×D4—1228no—
348C2×#117—618no—
349C2×#117—1228no—
350C22×C4⋊C4—611no—
351C22×C4⋊C4—612no—
352C22×C4⋊C4—612no—
353C22×C4⋊C4—611no—
354C22×C4⋊C4—611no—
355C22×C4⋊C4C2×D6111yescharacteristic
356C22×C4⋊C4C2×D6112yes—
357C22×C22⋊C4—612no—
358C22×C22⋊C4—611no—
359C22×C22⋊C4—612no—
360C22×C22⋊C4—612no—
361C22×C22⋊C4—611no—
362C22×C22⋊C4C2×D6112yes—
363C22×C42—612no—
364C22×C42—611no—
365#397—6116no—
366#394—6116no—
367#391—618no—
368#389—6116no—
369#389—618no—
370#388—6116no—
371#380—618no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—612no—
375C2×#95—612no—
376C61C2×#241511yescharacteristic, complemented, Sylow
377C25—611no—
378C25C22×D6111yescharacteristic
379C22×Q8—24no—
380C22×Q8—24no—
381C22×Q8—24no—
382C22×Q8C22×D6114yes—
383C22×D4—614no—
384C22×D4—1224no—
385C22×D4—1224no—
386C22×D4—1224no—
387C23×C4—24no—
388C23×C4—24no—
389C23×C4—612no—
390C23×C4—611no—
391C23×C4—612no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—614no—
395C23×C4—1224no—
396C23×C4—1224no—
397C23×C4—1224no—
398C23×C4C22⋊D6118yes—
399C23×C4C22×D6112yes—
400C23×C4C22×D6112yes—
401C23×C4C22×D6114yes—
402C23×C4C22×D6111yescharacteristic
403C23×C4C22×D6111yescharacteristic
404C22⋊Q8—232no—
405C22⋊Q8—12232no—
406C4×D4—12232no—
407#117—12232no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—28no—
410C2×C4⋊C4—618no—
411C2×C4⋊C4—614no—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—618no—
415C2×C4⋊C4—618no—
416C2×C4⋊C4—1228no—
417C2×C4⋊C4—1228no—
418C2×C4⋊C4—1224no—
419C2×C4⋊C4—1224no—
420C2×C4⋊C4C22×D6114yes—
421C2×C4⋊C4C22×D6118yes—
422C2×C22⋊C4—28no—
423C2×C22⋊C4—618no—
424C2×C22⋊C4—614no—
425C2×C22⋊C4—614no—
426C2×C22⋊C4—618no—
427C2×C22⋊C4—618no—
428C2×C22⋊C4—1228no—
429C2×C22⋊C4—1228no—
430C2×C22⋊C4—1228no—
431C2×C22⋊C4—1224no—
432C2×C22⋊C4—1224no—
433C2×C22⋊C4C22×D6118yes—
434C2×C42—618no—
435C2×C42—614no—
436C2×C42—1228no—
437C2×C42—1224no—
438#95—618no—
439#95—618no—
440#95—618no—
441#95—618no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—614no—
449C24—612no—
450C24—1222no—
451C24—1222no—
452C24—1222no—
453C24—1222no—
454C24—1222no—
455C24—1222no—
456C24C2×C22⋊D6114yes—
457C24C23×D6112yes—
458C24C23×D6111yescharacteristic
459C2×Q8—28no—
460C2×Q8—28no—
461C2×Q8—28no—
462C2×Q8—416no—
463C2×D4—1228no—
464C2×D4—1228no—
465C2×D4—12232no—
466C2×D4—1228no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—232no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—618no—
479C22×C4—614no—
480C22×C4—612no—
481C22×C4—614no—
482C22×C4—612no—
483C22×C4—614no—
484C22×C4—612no—
485C22×C4—1222no—
486C22×C4—1224no—
487C22×C4—1222no—
488C22×C4—1224no—
489C22×C4—1224no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1224no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1224no—
498C22×C4—1228no—
499C22×C4—1228no—
500C22×C4—1228no—
501C22×C4—24416no—
502C22×C4D4×D6114yes—
503C22×C4#1175822414yes—
504C22×C4C2×C22⋊D6114yes—
505C22×C4C2×C22⋊D61116yes—
506C22×C4C2×C22⋊D6114yes—
507C22×C4C23×D6114yes—
508C22×C4C23×D6114yes—
509C22×C4C23×D6112yes—
510C22×C4C23×D6112yes—
511C4⋊C4—28no—
512C4⋊C4—216no—
513C4⋊C4—6116no—
514C4⋊C4—12216no—
515C4⋊C4—12216no—
516C4⋊C4—1228no—
517C4⋊C4—1228no—
518C22⋊C4—216no—
519C22⋊C4—12216no—
520C22⋊C4—12216no—
521C22⋊C4—12216no—
522C22⋊C4—1228no—
523C22⋊C4—1228no—
524C42—12216no—
525C42—1228no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—614no—
540C23—1228no—
541C23—1222no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23—1222no—
546C23—1222no—
547C23—1228no—
548C23—1228no—
549C23—2444no—
550C23—2444no—
551C23—2444no—
552C23—2444no—
553C23C2×D4×D6112yes—
554C23C22×C22⋊D6114yes—
555C23C22×C22⋊D6112yes—
556C23#41×D6112yes—
557C23#41×D6112yes—
558C23C2×#1175822412yes—
559C23#1451982512yes—
560C23#1451982512yes—
561C23C24×D6111yescharacteristic
562Q8—416no—
563D4—12232no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—232no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—618no—
577C2×C4—614no—
578C2×C4—1228no—
579C2×C4—1222no—
580C2×C4—1224no—
581C2×C4—1222no—
582C2×C4—1224no—
583C2×C4—12216no—
584C2×C4—1224no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1224no—
589C2×C4—1222no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—2444no—
594C2×C4—2444no—
595C2×C4—2448no—
596C2×C4—24416no—
597C2×C4#14519749116yes—
598C2×C4C2×D4×D6114yes—
599C2×C4C22×C22⋊D6114yes—
600C2×C4C22×C22⋊D6114yes—
601C2×C4C2×#1175822414yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—416no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—1224no—
611C22—1224no—
612C22—1224no—
613C22—24416no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22C2×#1451974914yes—
619C22#1684026914yes—
620C22#1684033614yes—
621C22#1684041414yes—
622C22#1684042214yes—
623C22#1684044614yes—
624C22#1684048314yes—
625C22C22×D4×D6111yescharacteristic
626C22C23×C22⋊D6111yescharacteristic
627C22C2×#41×D6111yescharacteristic
628C22C2×#41×D6111yescharacteristic
629C22C22×#1175822411yescharacteristic
630C22C2×#1451982511yescharacteristic
631C22C2×#1451982511yescharacteristic
632C4—24no—
633C4—44no—
634C4—44no—
635C4—1228no—
636C4—1224no—
637C4—2444no—
638C4—2442no—
639C4—2442no—
640C4C2×#1451974918yes—
641C2—44no—
642C2—2444no—
643C2#2144924918yesdirect product factor
644C2C22×#1451974911yescharacteristic
645C2C2×#1684026911yescharacteristic
646C2C2×#1684033611yescharacteristic
647C2C2×#1684041411yescharacteristic
648C2C2×#1684042211yescharacteristic
649C2C2×#1684044611yescharacteristic
650C2C2×#1684048311yescharacteristic
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC22×C22⋊Q8×C61C2
socleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenterC22×C22⋊Q8D61
hypocenterC61C2×#2415
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14519753C2semidirect
3. C22×#14519736C2semidirect
4. #21449249C2direct (N × H)
5. C2×#16839690C2semidirect
7. C2×#16839669C2semidirect
8. C2×#16839658C2semidirect
9. C2×#16839657C2semidirect
10. C2×#16839652C2semidirect
11. C2×#16839619C2semidirect
12. C22×C22⋊Q8×C61C2semidirect
16. C2×#14519753C22semidirect
20. C2×#14519736C22semidirect
33. #16839690C22semidirect
35. #16839669C22semidirect
36. #16839658C22semidirect
37. #16839657C22semidirect
38. #16839652C22semidirect
39. #16839619C22semidirect
47. C2×C22⋊Q8×C61C22semidirect
69. C2×C4⋊Dic61D4semidirect
194. C4⋊Dic61C2×D4semidirect
317. C22×C22⋊Q8D61semidirect
329. C2×C61#2415semidirect
345. C2×C22⋊Q8C2×D61semidirect
376. C61C2×#2415semidirect
643. C2#21449249direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197532
C22×C4×C22⋊D612
C22×#145197362
#214492492
C2×#168396902
C2×#168396772
C2×#168396692
C2×#168396582
C2×#168396572
C2×#168396522
C2×#168396192
C22×C22⋊Q8×C612
C2×#241561