Finite Group Databasebeta

← C2×#21446249 #29681692

Subgroups — C2×#21446249 #29681692

Order 15616.

1966 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758189C211yesmaximal, characteristic
2C22×#14519753C211yesmaximal, characteristic
3C22×#14519701C211yesmaximal, characteristic, complemented
4C2×#16839619C211yesmaximal, characteristic
5#21446249C218yesmaximal, direct product factor
6C2×#16839525C211yesmaximal, characteristic, complemented
7C2×#16839524C211yesmaximal, characteristic, complemented
8C22×#103×C61C211yesmaximal, characteristic, complemented
9C23×C4×D61C2211yescharacteristic
10C22×#11758189—24no—
11C22×#11758189—24no—
12C22×#11758189—24no—
13C22×#11758189C2214yes—
14C23×#92418C2211yescharacteristic
15C2×#14519753—22no—
16C2×#14519753—24no—
17C2×#14519753C418yescomplemented
18C2×#14519753C2214yes—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758196—21no—
22C22×#11758198—21no—
23C22×#11758198—21no—
24C2×#14519701—24no—
25C2×#14519701C2214yescomplemented
26#16839619—22no—
27#16839619C2214yes—
28C2×#14519684—21no—
29#16839525C2214yescomplemented
30#16839524C2214yescomplemented
31C23×Q8×C61C2211yescharacteristic
32C23×C8×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C2×#103×C61—24no—
35C2×#103×C61C2214yescomplemented
36C24×D61D411yescharacteristic
37C23×Dic61—21no—
38C23×Dic61—21no—
39C23×Dic61D411yescharacteristic
40C22×C4×D61—22no—
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61C2×C414yes—
45C22×C4×D61D412yes—
46C22×C4×D61D414yes—
47C22×C4×D61C2312yes—
48C2×#11758189—232no—
49C2×#11758189—28no—
50C2×#11758189—28no—
51C2×#11758189—28no—
52C22×#92418—22no—
53C22×#92418—22no—
54C22×#92418D414yes—
55C22×#92418C2312yes—
56#14519753—216no—
57#14519753—28no—
58#14519753C2×C4116yescomplemented
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2×C414yes—
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—28no—
68C2×#11758196—24no—
69C2×#11758198—28no—
70C2×#11758198—24no—
71C2×#11758198—24no—
72C2×#11758198—44no—
73#14519701—28no—
74#14519684—24no—
75C22×Q8×C61—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61—22no—
78C22×Q8×C61—22no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61C2×C414yes—
82C22×Q8×C61C2312yes—
83C23×C4×C61—21no—
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61—22no—
87C22×C8×C61—22no—
88C22×C8×C61D414yes—
89C22×C8×C61C2312yes—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61C2312yes—
94#103×C61—28no—
95C24×C61C2×D411yescharacteristic
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61C22⋊C414yes—
103C23×D61C2×D412yes—
104C2×Q8×C61—24no—
105C2×Q8×C61—22no—
106C2×Q8×C61—28no—
107C2×Q8×C61—28no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—44no—
113C2×Q8×C61—42no—
114C2×Q8×C61—42no—
115C2×Q8×C61—44no—
116C2×Q8×C61C22×C414yes—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—24no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61C22⋊C414yes—
144C22×Dic61C2×D412yes—
145C2×C4×D61—28no—
146C2×C4×D61—28no—
147C2×C4×D61—28no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—28no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61C22⋊C418yes—
156C2×C4×D61C22×C414yes—
157C2×C4×D61C2×D414yes—
158C2×C8×C61—28no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61C2×D414yes—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—42no—
165#11758198—216no—
166#11758198—48no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61C22×C414yes—
173#11758196—216no—
174C2×#92418—28no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418C2×D414yes—
178#11758189—232no—
179C23×C61C2×C22⋊C412yes—
180C23×C61#12112yes—
181C23×C61#12112yes—
182C23×C61AGL(1,ℤ/8)12yes—
183C23×C61#13712yes—
184C23×C61#13712yes—
185C23×C61#13712yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—28no—
194C22×D61—28no—
195C22×D61—28no—
196C22×D61—44no—
197C22×D61—44no—
198C22×D61—44no—
199C22×D61—44no—
200C22×D61C2×C22⋊C414yes—
201Q8×C61—24no—
202Q8×C61—24no—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—48no—
206C2×C4×C61—24no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—48no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—84no—
214C2×C4×C61C2×C22⋊C412yes—
215C2×C4×C61C4×D414yes—
216C2×C4×C61C4×D414yes—
217C2×C4×C61#12014yes—
218C2×C4×C61#12112yes—
219C2×C4×C61#12112yes—
220C2×C4×C61#12314yes—
221C2×C4×C61C23×C411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—28no—
230C2×Dic61—24no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—48no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—84no—
238C2×Dic61C2×C22⋊C414yes—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—216no—
244C8×C61—24no—
245#92418—24no—
246C2×#1894—611nomaximal, Sylow
247C22×C61#38614yes—
248C22×C61#41714yes—
249C22×C61#41914yes—
250C22×C61#46814yes—
251C22×C61#46814yes—
252C22×C61#47014yes—
253C22×C61#47014yes—
254C22×C61C22×C22⋊C411yescharacteristic
255C22×C61C2×#12111yescharacteristic
256C22×C61C2×#12111yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61C2×#13711yescharacteristic
259C22×C61C2×#13711yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C2×D61—24no—
262C2×D61—24no—
263C2×D61—24no—
264C2×D61—416no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C4×C61—44no—
270C4×C61—82no—
271C4×C61#38614yes—
272C4×C61C2×C4×D411yescharacteristic
273C4×C61C2×C4×D411yescharacteristic
274C4×C61C2×#12011yescharacteristic
275C4×C61C2×#12311yescharacteristic
276Dic61—24no—
277Dic61—44no—
278Dic61—82no—
279C23×M16—611no—
280C22×C22⋊Q8—611no—
281C22×#103—611no—
282C22×#103D6111yescharacteristic, complemented
283C2×#380—611no—
284#1894—618no—
285C2×#340—611no—
286C2×#340—611no—
287C2×C61#189418yescomplemented
288C2×C61C2×#38611yescharacteristic
289C2×C61C2×#41911yescharacteristic
290C2×C61C2×#41711yescharacteristic
291C2×C61C2×#46811yescharacteristic
292C2×C61C2×#46811yescharacteristic
293C2×C61C2×#47011yescharacteristic
294C2×C61C2×#47011yescharacteristic
295D61—44no—
296C23×Q8C2×D6111yescharacteristic
297C24×C4—611no—
298C22×M16—614no—
299C22×M16—1224no—
300C22×M16—1224no—
301C22×M16—1224no—
302C23×C8—611no—
303C23×C8C2×D6111yescharacteristic
304C2×C22⋊Q8—618no—
305C2×C22⋊Q8—614no—
306C2×C22⋊Q8—1224no—
307C2×C22⋊Q8—1222no—
308C22×C4⋊C4—611no—
309C22×C4⋊C4—611no—
310C22×C4⋊C4—1221no—
311C22×C4⋊C4C2×D6111yescharacteristic
312C22×C22⋊C4—1221no—
313C22×C22⋊C4—1221no—
314C2×#103—24no—
315C2×#103—614no—
316C2×#103—1224no—
317C2×#103C2×D6114yescomplemented
318#380—614no—
319#380—1222no—
320C2×#95—1221no—
321#340—614no—
322#340—614no—
323C61C2×#189411yescharacteristic, complemented, Sylow
324C25—611no—
325C22×Q8—21no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8C4×D6114yes—
332C22×Q8C22×D6112yes—
333C23×C4—21no—
334C23×C4—21no—
335C23×C4—611no—
336C23×C4—614no—
337C23×C4—614no—
338C23×C4—612no—
339C23×C4—612no—
340C23×C4—1222no—
341C23×C4—1222no—
342C23×C4—1222no—
343C23×C4—1222no—
344C23×C4—1221no—
345C23×C4—1221no—
346C23×C4C22×D6111yescharacteristic
347C2×M16—1228no—
348C2×M16—1228no—
349C2×M16—12232no—
350C2×M16—1228no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—614no—
354C22×C8—612no—
355C22×C8—1222no—
356C22×C8—1222no—
357C22×C8C22⋊D6114yes—
358C22×C8C22×D6112yes—
359C22⋊Q8—6116no—
360C22⋊Q8—1228no—
361C22⋊Q8—12216no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22×D6112yes—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1228no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—2444no—
380#103—28no—
381#103—1228no—
382#95—1224no—
383C24—612no—
384C24—614no—
385C24—1222no—
386C24—1222no—
387C24—1222no—
388C24—1222no—
389C24—1222no—
390C24—1222no—
391C24D4×D6111yescharacteristic
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8C2×C4×D6114yes—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—612no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—618no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1228no—
425C22×C4—1228no—
426C22×C4—1228no—
427C22×C4—1224no—
428C22×C4—1224no—
429C22×C4—1228no—
430C22×C4—1222no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4C2×C4×D6112yes—
439C22×C4#1175822212yes—
440C22×C4D4×D6112yes—
441C22×C4D4×D6112yes—
442C22×C4D4×D6111yescharacteristic
443C22×C4#1175822412yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C23×D6111yescharacteristic
446M16—12232no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—28no—
450C2×C8—614no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1228no—
454C2×C8C2×C22⋊D6114yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—614no—
459C4⋊C4—12216no—
460C4⋊C4—1222no—
461C4⋊C4—1224no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—2442no—
465C22⋊C4—12216no—
466C22⋊C4—2448no—
467C23—614no—
468C23—1222no—
469C23—1222no—
470C23—1228no—
471C23—1228no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—1222no—
476C23—1228no—
477C23—2444no—
478C23—2444no—
479C23—2444no—
480C23—2444no—
481C23C22⋊C4×D6112yes—
482C23#1451974812yes—
483C23#1451975012yes—
484C23#1451980412yes—
485C23#1451980612yes—
486C23#1451980912yes—
487C23#1451980912yes—
488C23C2×D4×D6111yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—42no—
492Q8—42no—
493Q8—48no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—614no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1228no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—12216no—
511C2×C4—1224no—
512C2×C4—4884no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2448no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4C22⋊C4×D6112yes—
520C2×C4#1451973714yes—
521C2×C4C4×C22⋊D6114yes—
522C2×C4#1451974514yes—
523C2×C4#1451974812yes—
524C2×C4#1451975012yes—
525C2×C4#1451975914yes—
526C2×C4C22×C4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C2×#1175822411yescharacteristic
531C2×C4C2×#1175822211yescharacteristic
532C8—24no—
533C8—1224no—
534C22—1224no—
535C22—1224no—
536C22—1224no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—24416no—
542C22#1683963814yes—
543C22#1683976214yes—
544C22#1683976914yes—
545C22#1684000514yes—
546C22#1684001114yes—
547C22#1684001714yes—
548C22#1684001914yes—
549C22C2×C22⋊C4×D6111yescharacteristic
550C22C2×#1451974811yescharacteristic
551C22C2×#1451975011yescharacteristic
552C22C2×#1451980411yescharacteristic
553C22C2×#1451980611yescharacteristic
554C22C2×#1451980911yescharacteristic
555C22C2×#1451980911yescharacteristic
556C4—44no—
557C4—82no—
558C4—1224no—
559C4—4882no—
560C4—2444no—
561C4#1683963814yes—
562C4C2×#1451973711yescharacteristic
563C4C2×C4×C22⋊D6111yescharacteristic
564C4C2×#1451974511yescharacteristic
565C4C2×#1451975911yescharacteristic
566C2—2444no—
567C2#2144624918yesdirect product factor
568C2C2×#1683963811yescharacteristic
569C2C2×#1683976911yescharacteristic
570C2C2×#1683976211yescharacteristic
571C2C2×#1684000511yescharacteristic
572C2C2×#1684001111yescharacteristic
573C2C2×#1684001711yescharacteristic
574C2C2×#1684001911yescharacteristic
subgroupG/H
centerC24D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC24×C61C2×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC22×#103D61
hypocenterC61C2×#1894
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14519701C2semidirect
5. #21446249C2direct (N × H)
6. C2×#16839525C2semidirect
7. C2×#16839524C2semidirect
8. C22×#103×C61C2semidirect
17. C2×#14519753C4semidirect
25. C2×#14519701C22semidirect
29. #16839525C22semidirect
30. #16839524C22semidirect
35. C2×#103×C61C22semidirect
58. #14519753C2×C4semidirect
282. C22×#103D61semidirect
287. C2×C61#1894semidirect
317. C2×#103C2×D61semidirect
323. C61C2×#1894semidirect
567. C2#21446249direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117581892
C22×#145197532
C22×#145197012
C2×#168396192
#214462492
C2×#168395252
C2×#168395242
C22×#103×C612
C2×#189461