Finite Group Databasebeta

← C2×#21446251 #29681694

Subgroups — C2×#21446251 #29681694

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×#14519757C211yesmaximal, characteristic
3C22×#14519702C211yesmaximal, characteristic, complemented
4C22×#14519700C211yesmaximal, characteristic, complemented
5C2×#16839619C211yesmaximal, characteristic
6#21446251C218yesmaximal, direct product factor
7C2×#16839524C211yesmaximal, characteristic, complemented
8C2×C61×#340C211yesmaximal, characteristic, complemented
9C23×Dic122C2211yescharacteristic
10C23×C4×D61C2211yescharacteristic
11C22×#11758189—24no—
12C22×#11758189—24no—
13C22×#11758189—24no—
14C22×#11758189C2214yes—
15C23×#92418C2211yescharacteristic
16C2×#14519757—22no—
17C2×#14519757—24no—
18C2×#14519757C418yescomplemented
19C2×#14519757C2214yes—
20C22×C4⋊Dic61—21no—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758198—21no—
23C22×#11758198—21no—
24C2×#14519702—24no—
25C2×#14519702C2214yescomplemented
26C2×#14519700—24no—
27C2×#14519700C2214yescomplemented
28#16839619—22no—
29#16839619C2214yes—
30C2×#14519684—21no—
31#16839524C2214yescomplemented
32C23×C8×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#340C2214yescomplemented
36C24×D61D411yescharacteristic
37C22×Dic122—22no—
38C22×Dic122—22no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—21no—
42C22×Dic122—22no—
43C22×Dic122C2×C414yes—
44C22×Dic122C2312yes—
45C23×Dic61—21no—
46C23×Dic61—21no—
47C23×Dic61D411yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61C2×C414yes—
53C22×C4×D61D412yes—
54C22×C4×D61D414yes—
55C22×C4×D61C2312yes—
56C2×#11758189—232no—
57C2×#11758189—28no—
58C2×#11758189—28no—
59C2×#11758189—28no—
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418D414yes—
63C22×#92418C2312yes—
64#14519757—216no—
65#14519757—28no—
66#14519757C2×C4116yescomplemented
67C2×C4⋊Dic61—28no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—28no—
74C2×#11758198—24no—
75C2×#11758198—24no—
76C2×#11758198—44no—
77#14519702—28no—
78#14519700—28no—
79#14519684—24no—
80C23×C4×C61—21no—
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C22×C8×C61—22no—
84C22×C8×C61—22no—
85C22×C8×C61D414yes—
86C22×C8×C61C2312yes—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2×C414yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2312yes—
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—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C22×C412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—24no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61C22⋊C414yes—
131C22×Dic61C2×D412yes—
132C2×C4×D61—28no—
133C2×C4×D61—28no—
134C2×C4×D61—28no—
135C2×C4×D61—24no—
136C2×C4×D61—24no—
137C2×C4×D61—28no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61C22⋊C418yes—
143C2×C4×D61C22×C414yes—
144C2×C4×D61C2×D414yes—
145C2×Dic122—28no—
146C2×Dic122—28no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—24no—
152C2×Dic122—22no—
153C2×Dic122—44no—
154C2×Dic122—42no—
155C2×Dic122—42no—
156C2×Dic122—44no—
157C2×Dic122C22×C414yes—
158C2×C8×C61—28no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61C2×D414yes—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61C22×C414yes—
168#11758198—216no—
169#11758198—48no—
170C4⋊Dic61—216no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—42no—
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—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—48no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—84no—
209C2×C4×C61C2×C22⋊C412yes—
210C2×C4×C61C4×D414yes—
211C2×C4×C61C4×D414yes—
212C2×C4×C61#12014yes—
213C2×C4×C61#12112yes—
214C2×C4×C61#12112yes—
215C2×C4×C61#12314yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—28no—
225C2×Dic61—24no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—48no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—84no—
233C2×Dic61C2×C22⋊C414yes—
234C4×D61—24no—
235C4×D61—24no—
236C4×D61—24no—
237C4×D61—24no—
238C4×D61—216no—
239Dic122—24no—
240Dic122—24no—
241Dic122—42no—
242Dic122—42no—
243Dic122—48no—
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×#103—611no—
283C2×#380—611no—
284#1894—618no—
285C2×#340—611no—
286C2×#340D6111yescharacteristic, complemented
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×Q8—611no—
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—1221no—
310C22×C4⋊C4C2×D6111yescharacteristic
311C22×C4⋊C4C2×D6111yescharacteristic
312C22×C22⋊C4—1221no—
313C22×C22⋊C4—1221no—
314C2×#103—614no—
315C2×#103—614no—
316C2×#103—1224no—
317C2×#103—1224no—
318#380—614no—
319#380—1222no—
320C2×#95—1221no—
321#340—614no—
322#340C2×D6114yescomplemented
323C61C2×#189411yescharacteristic, complemented, Sylow
324C25—611no—
325C22×Q8—614no—
326C22×Q8—612no—
327C22×Q8—1221no—
328C22×Q8—1222no—
329C22×Q8—1222no—
330C22×Q8—1222no—
331C22×Q8—1222no—
332C22×Q8—1222no—
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×C8D24414yes—
358C22×C8C22×D6112yes—
359C22⋊Q8—6116no—
360C22⋊Q8—1228no—
361C22⋊Q8—12216no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1224no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C4×D6114yes—
374C2×C4⋊C4C22×D6112yes—
375C2×C4⋊C4C22×D6112yes—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1228no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—2444no—
380#103—1228no—
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—614no—
393C2×Q8—1222no—
394C2×Q8—1224no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—1222no—
399C2×Q8—1228no—
400C2×Q8—1228no—
401C2×Q8—2444no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×Q8—2444no—
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×C4C2×D24412yes—
440C22×C4#1175822212yes—
441C22×C4D4×D6112yes—
442C22×C4D4×D6112yes—
443C22×C4D4×D6111yescharacteristic
444C22×C4#1175822612yes—
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×D24414yes—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—12216no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—2442no—
464C4⋊C4C2×C4×D6114yes—
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#1451975012yes—
483C23#1451975212yes—
484C23#1451980112yes—
485C23#1451980612yes—
486C23#1451980612yes—
487C23#1451980912yes—
488C23C2×D4×D6111yescharacteristic
489Q8—1224no—
490Q8—1224no—
491Q8—2442no—
492Q8—2442no—
493Q8—2448no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
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—2442no—
517C2×C4—2442no—
518C2×C4—2448no—
519C2×C4C22⋊C4×D6112yes—
520C2×C4#1451973914yes—
521C2×C4C4×D24414yes—
522C2×C4#1451974714yes—
523C2×C4#1451975012yes—
524C2×C4#1451975212yes—
525C2×C4#1451976214yes—
526C2×C4C22×C4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C22×D24411yescharacteristic
530C2×C4C2×#1175822211yescharacteristic
531C2×C4C2×#1175822611yescharacteristic
532C8—24no—
533C8—1224no—
534C22—1224no—
535C22—1224no—
536C22—1224no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—24416no—
542C22#1683964014yes—
543C22#1683976014yes—
544C22#1683976914yes—
545C22#1684000714yes—
546C22#1684000914yes—
547C22#1684001914yes—
548C22#1684002114yes—
549C22C2×C22⋊C4×D6111yescharacteristic
550C22C2×#1451975011yescharacteristic
551C22C2×#1451975211yescharacteristic
552C22C2×#1451980111yescharacteristic
553C22C2×#1451980611yescharacteristic
554C22C2×#1451980611yescharacteristic
555C22C2×#1451980911yescharacteristic
556C4—44no—
557C4—82no—
558C4—1224no—
559C4—4882no—
560C4—2444no—
561C4#1683964014yes—
562C4C2×#1451973911yescharacteristic
563C4C2×C4×D24411yescharacteristic
564C4C2×#1451974711yescharacteristic
565C4C2×#1451976211yescharacteristic
566C2—2444no—
567C2#2144625118yesdirect product factor
568C2C2×#1683964011yescharacteristic
569C2C2×#1683976911yescharacteristic
570C2C2×#1683976011yescharacteristic
571C2C2×#1684000711yescharacteristic
572C2C2×#1684000911yescharacteristic
573C2C2×#1684001911yescharacteristic
574C2C2×#1684002111yescharacteristic
subgroupG/H
centerC24D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC24×C61C2×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#340D61
hypocenterC61C2×#1894
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#14519702C2semidirect
4. C22×#14519700C2semidirect
6. #21446251C2direct (N × H)
7. C2×#16839524C2semidirect
8. C2×C61×#340C2semidirect
18. C2×#14519757C4semidirect
25. C2×#14519702C22semidirect
27. C2×#14519700C22semidirect
31. #16839524C22semidirect
35. C61×#340C22semidirect
66. #14519757C2×C4semidirect
286. C2×#340D61semidirect
287. C2×C61#1894semidirect
322. #340C2×D61semidirect
323. C61C2×#1894semidirect
567. C2#21446251direct (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×#145197572
C22×#145197022
C22×#145197002
C2×#168396192
#214462512
C2×#168395242
C2×C61×#3402
C2×#189461