Finite Group Databasebeta

← C2×#21456424 #30549789

Subgroups — C2×#21456424 #30549789

Order 15616.

1986 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
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840297C211yesmaximal, characteristic
3#21456424C2116yesmaximal, direct product factor
4C2×#16840172C211yesmaximal, characteristic, complemented
5C2×#16840154C211yesmaximal, characteristic, complemented
6C2×#16840122C211yesmaximal, characteristic, complemented
7C2×#16840019C212yesmaximal, complemented
8C2×#16840011C212yesmaximal
9C2×#16839854C212yesmaximal, complemented
10C2×#16839847C212yesmaximal, complemented
11C2×#16839714C211yesmaximal, characteristic, complemented
12C2×C61×#491C211yesmaximal, characteristic, complemented
13C22×#11758202C2212yes—
14C22×#11758201C2212yes—
15C22×#11758189C2212yes—
16#16840317—24no—
17#16840317C2218yescomplemented
18C2×#14519777C2211yescharacteristic
19#16840297—24no—
20#16840297C2218yes—
21C2×#14519772—21no—
22C2×#14519768—21no—
23C2×#14519765C2211yescharacteristic
24C2×#14519762—21no—
25C2×#14519757—21no—
26C2×#14519757C2211yescharacteristic
27C2×#14519753C2212yes—
28C2×#14519752C2211yescharacteristic
29C2×Q8×Dic61C2212yes—
30C2×#14519742—21no—
31C2×#14519742C2211yescharacteristic
32C2×#14519739—21no—
33C2×#14519739C2211yescharacteristic
34C2×#14519735—21no—
35C2×#14519735—21no—
36C2×C42×D61C2211yescharacteristic
37#16840172C2218yescomplemented
38#16840154C2218yescomplemented
39#16840122C2218yescomplemented
40#16840019C22116yescomplemented
41#16840011C22116yes—
42#16839854C22116yescomplemented
43#16839847C22116yescomplemented
44C2×#14519710C2212yes—
45C2×#14519708C2212yes—
46C2×#14519702C2212yes—
47C2×#14519699C2212yes—
48#16839714C2218yescomplemented
49C2×#14519688C2211yescharacteristic
50C2×C8×Dic61C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#491C2218yescomplemented
54C2×#103×C61C2212yes—
55C2×#103×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C22×Dic122—21no—
58C22×Dic122C2311yescharacteristic
59C22×D244—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758202—28no—
66C2×#11758202C2318yes—
67C2×#11758201—28no—
68C2×#11758201C2318yes—
69C2×#11758189—28no—
70C2×#11758189C2318yes—
71C22×#92418C2312yes—
72#14519777—22no—
73#14519777C2314yes—
74#14519772—28no—
75#14519768—28no—
76#14519765—22no—
77#14519765C2314yes—
78#14519762—28no—
79#14519757—28no—
80#14519757—22no—
81#14519757C2314yes—
82#14519753—24no—
83#14519753C2318yes—
84#14519752—22no—
85#14519752C2314yes—
86Q8×Dic61—24no—
87Q8×Dic61C2318yes—
88#14519742—28no—
89#14519742—22no—
90#14519742C2314yes—
91#14519739—28no—
92#14519739—22no—
93#14519739C2314yes—
94#14519735—28no—
95#14519735—28no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61—22no—
99C2×C4⋊Dic61C2312yes—
100C2×#11758196—22no—
101C2×#11758196—21no—
102C2×#11758198—22no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61D414yes—
114C42×D61C2314yes—
115#14519710C2318yes—
116#14519708C2318yes—
117#14519702C2318yes—
118#14519699C2318yes—
119#14519688C2314yes—
120C8×Dic61C2318yes—
121C22×Q8×C61C2312yes—
122C22×C8×C61C2312yes—
123C4⋊Q8×C61—22no—
124C4⋊Q8×C61C2314yes—
125#125×C61—22no—
126#125×C61C2314yes—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134#103×C61C2318yes—
135#103×C61C2318yes—
136#97×C61C2314yes—
137C23×D61—21no—
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—22no—
140C2×Q8×C61—24no—
141C2×Q8×C61—24no—
142C2×Q8×C61C2414yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×D244—24no—
155C2×D244—24no—
156C2×D244—21no—
157C2×D244—44no—
158C2×D244C2412yes—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61C2×D414yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D414yes—
167C2×C4×D61C2412yes—
168C2×Dic122—24no—
169C2×Dic122—24no—
170C2×Dic122—21no—
171C2×Dic122—44no—
172C2×Dic122C2412yes—
173C2×C8×C61—24no—
174C2×C8×C61C2414yes—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2412yes—
182C4⋊C4×C61C2412yes—
183C42×C61C2×D412yes—
184C42×C61C2412yes—
185#11758202—216no—
186#11758201—216no—
187#11758198—28no—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193C4⋊Dic61—24no—
194C4⋊Dic61—22no—
195C4⋊Dic61—24no—
196C4⋊Dic61—28no—
197C4⋊Dic61C2414yes—
198#11758196—28no—
199#11758196—24no—
200C4×Dic61—24no—
201C4×Dic61—28no—
202C4×Dic61C2×D412yes—
203C4×Dic61C2×D414yes—
204C4×Dic61C2×D414yes—
205C4×Dic61C2412yes—
206C2×#92418#4118yes—
207C2×#92418C2414yes—
208#11758189—216no—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61C22×D412yes—
218Q8×C61—24no—
219Q8×C61—24no—
220Q8×C61—44no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—44no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C22×D412yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4114yes—
233C2×C4×C61C2511yescharacteristic
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—44no—
240C2×Dic61#13714yes—
241C2×Dic61C22×D412yes—
242C2×Dic61C22×D412yes—
243C2×Dic61C22×D412yes—
244D244—24no—
245D244—48no—
246D244—42no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—28no—
250C4×D61—44no—
251C4×D61—48no—
252Dic122—24no—
253Dic122—48no—
254Dic122—42no—
255C8×C61—24no—
256#92418C2×#4118yes—
257C2×#3172—611nomaximal, Sylow
258C22×C61#53212yes—
259C22×C61C2×#13712yes—
260C22×C61#57512yes—
261C22×C61C23×D411yescharacteristic
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—44no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—84no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—44no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61#53212yes—
276C4×C61C22×#4112yes—
277Dic61—22no—
278Dic61—22no—
279Dic61—42no—
280Dic61—44no—
281Dic61C2×#13714yes—
282C2×#533—611no—
283C2×#529—611no—
284#3172—6116no—
285C2×#501—611no—
286C2×#497—611no—
287C2×#491D6111yescharacteristic, complemented
288C2×#489—611no—
289C2×#470—612no—
290C2×#468—612no—
291C2×C4×Q16—612no—
292C2×C4×SD16—612no—
293C2×C4×M16—611no—
294C2×C61#317214yescomplemented
295C2×C61C2×#53211yescharacteristic
296C2×C61C22×#13711yescharacteristic
297C2×C61C2×#57511yescharacteristic
298D61—42no—
299D61—82no—
300C22×Q16—612no—
301C22×SD16—612no—
302C22×M16—612no—
303#533—618no—
304#533—1224no—
305C2×C4⋊Q8—611no—
306C2×C4⋊Q8C2×D6111yescharacteristic
307#529—618no—
308#529—1224no—
309C2×#126—1221no—
310C2×#125—1221no—
311C2×#125C2×D6111yescharacteristic
312C2×#124—611no—
313C2×#123—1221no—
314C2×C22⋊Q8—612no—
315C2×C22⋊Q8—611no—
316C2×C22⋊Q8—1221no—
317C2×#121—611no—
318C2×C4×Q8—612no—
319C2×C4×Q8—611no—
320C2×C4×Q8—1221no—
321C2×C4×D4—611no—
322C2×C4×D4—1221no—
323C2×#117—1221no—
324C2×#117—1221no—
325C22×C42—611no—
326#501—618no—
327#497—618no—
328#491C2×D6118yescomplemented
329#489—618no—
330#470—6116no—
331#468—6116no—
332C4×Q16—6116no—
333C4×SD16—6116no—
334C2×#107—612no—
335C2×#106—612no—
336C2×#103—612no—
337C2×#103C2×D6112yes—
338C2×#103C2×D6112yes—
339C2×#102—612no—
340C4×M16—618no—
341C2×#97—611no—
342C2×#97C2×D6111yescharacteristic
343C2×C4×C8—612no—
344C61C2×#317211yescharacteristic, complemented, Sylow
345C22×Q8—611no—
346C22×Q8—1221no—
347C22×Q8C22×D6112yes—
348C22×D4—611no—
349C22×D4—1221no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—1221no—
354C2×Q16—618no—
355C2×Q16—1228no—
356C2×SD16—618no—
357C2×SD16—1228no—
358C2×M16—618no—
359C2×M16—1228no—
360C22×C8—612no—
361C22×C8C22×D6112yes—
362C4⋊Q8—22no—
363C4⋊Q8—614no—
364C4⋊Q8—1222no—
365C4⋊Q8C22×D6114yes—
366#126—1228no—
367#125—22no—
368#125—1228no—
369#125C22×D6114yes—
370#124—614no—
371#124—1222no—
372#123—1228no—
373C22⋊Q8—618no—
374C22⋊Q8—614no—
375C22⋊Q8—1222no—
376C22⋊Q8—1228no—
377C22⋊Q8—1224no—
378#121—614no—
379#121—1222no—
380C4×Q8—618no—
381C4×Q8—614no—
382C4×Q8—1222no—
383C4×Q8—1228no—
384C4×Q8—1224no—
385C4×D4—614no—
386C4×D4—1222no—
387C4×D4—1228no—
388#117—1228no—
389#117—1228no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—614no—
412C2×C42—614no—
413C2×C42—1221no—
414C2×C42—1222no—
415C2×C42C22×D6111yescharacteristic
416#107—618no—
417#106—618no—
418#103—618no—
419#103C22×D6118yes—
420#103C22×D6118yes—
421#102—618no—
422#97—614no—
423#97C22×D6114yes—
424C4×C8—618no—
425C24—611no—
426C24—1221no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—612no—
431C2×Q8—1221no—
432C2×Q8—1224no—
433C2×Q8—1224no—
434C2×Q8—2444no—
435C2×Q8C23×D6114yes—
436C2×D4—612no—
437C2×D4—1224no—
438C2×D4—1221no—
439C2×D4—1224no—
440C2×D4—2444no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—614no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—614no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1224no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4—2442no—
457C22×C4—2444no—
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4C23×D6111yescharacteristic
461Q16—12216no—
462SD16—12216no—
463M16—12216no—
464C2×C8—24no—
465C2×C8—618no—
466C2×C8—614no—
467C2×C8C23×D6114yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—614no—
475C4⋊C4—1224no—
476C4⋊C4—1228no—
477C4⋊C4—1224no—
478C4⋊C4—1228no—
479C4⋊C4—1222no—
480C4⋊C4—1224no—
481C4⋊C4C23×D6112yes—
482C4⋊C4C23×D6112yes—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—1228no—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—1224no—
489C42—614no—
490C42—612no—
491C42—614no—
492C42—612no—
493C42—1224no—
494C42—1228no—
495C42D4×D6112yes—
496C42C23×D6112yes—
497C23—612no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23C2×D4×D6111yescharacteristic
506Q8—24no—
507Q8—24no—
508Q8—44no—
509Q8—1224no—
510Q8—2448no—
511Q8—2442no—
512D4—1224no—
513D4—2448no—
514D4—2442no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—614no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—1228no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1222no—
533C2×C4—1224no—
534C2×C4—2444no—
535C2×C4—2448no—
536C2×C4—2444no—
537C2×C4—2442no—
538C2×C4C2×D4×D6112yes—
539C2×C4C2×D4×D6112yes—
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4#41×D6114yes—
542C2×C4C24×D6111yescharacteristic
543C8—24no—
544C8—618no—
545C22—1222no—
546C22—1222no—
547C22—1222no—
548C22—4884no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22—2444no—
553C22#1684031112yes—
554C22#137×D6112yes—
555C22#1684056112yes—
556C22C22×D4×D6111yescharacteristic
557C4—22no—
558C4—22no—
559C4—42no—
560C4—42no—
561C4—44no—
562C4—614no—
563C4—1222no—
564C4—1222no—
565C4—2444no—
566C4—2442no—
567C4#1684031112yes—
568C4C2×#41×D6112yes—
569C2—4882no—
570C2—2442no—
571C2#2145642414yesdirect product factor
572C2C2×#1684031111yescharacteristic
573C2C2×#137×D6111yescharacteristic
574C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#491C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenterC2×#491D61
hypocenterC61C2×#3172
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
3. #21456424C2direct (N × H)
4. C2×#16840172C2semidirect
5. C2×#16840154C2semidirect
6. C2×#16840122C2semidirect
7. C2×#16840019C2semidirect
9. C2×#16839854C2semidirect
10. C2×#16839847C2semidirect
11. C2×#16839714C2semidirect
12. C2×C61×#491C2semidirect
17. #16840317C22semidirect
37. #16840172C22semidirect
38. #16840154C22semidirect
39. #16840122C22semidirect
40. #16840019C22semidirect
42. #16839854C22semidirect
43. #16839847C22semidirect
48. #16839714C22semidirect
53. C61×#491C22semidirect
287. C2×#491D61semidirect
294. C2×C61#3172semidirect
328. #491C2×D61semidirect
344. C61C2×#3172semidirect
571. C2#21456424direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168402972
#214564242
C2×#168401722
C2×#168401542
C2×#168401222
C2×#168400192
C2×#168400112
C2×#168398542
C2×#168398472
C2×#168397142
C2×C61×#4912
C2×#317261