Finite Group Databasebeta

← C2×#21456996 #30552089

Subgroups — C2×#21456996 #30552089

Order 15616.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840294C211yesmaximal, characteristic
3#21456996C2116yesmaximal, direct product factor
4C2×#16840163C212yesmaximal, complemented
5C2×#16840115C212yesmaximal, complemented
6C2×#16840105C212yesmaximal, complemented
7C2×#16840084C212yesmaximal
8C2×#16840058C211yesmaximal, characteristic, complemented
9C2×#16840044C211yesmaximal, characteristic, complemented
10C2×#16840052C211yesmaximal, characteristic, complemented
11C2×#16839814C211yesmaximal, characteristic, complemented
12C2×C61×#474C211yesmaximal, characteristic, complemented
13#16840317—24no—
14#16840317C2218yescomplemented
15C2×#14519777C2211yescharacteristic
16#16840294—24no—
17#16840294C2218yes—
18C2×#14519770—21no—
19C2×#14519768C2212yes—
20C2×#14519768C2211yescharacteristic
21C2×#14519764C2211yescharacteristic
22C2×#14519759—21no—
23C2×#14519757—21no—
24C2×#14519757C2211yescharacteristic
25C2×#14519753—21no—
26C2×#14519748C2211yescharacteristic
27C2×Q8×Dic61—21no—
28C2×C4×Dic122C2211yescharacteristic
29C2×#14519742—21no—
30C2×C4×C22⋊D61—21no—
31C2×#14519736—21no—
32C2×#14519735C2212yes—
33C2×C42×D61C2211yescharacteristic
34#16840163C22116yescomplemented
35#16840115C22116yescomplemented
36#16840105C22116yescomplemented
37#16840084C22116yes—
38#16840058C2218yescomplemented
39#16840052C2218yescomplemented
40#16840044C2218yescomplemented
41#16839814C2218yescomplemented
42C2×#14519711C2212yes—
43C2×#14519708C2212yes—
44C2×#14519708C2212yes—
45C2×C4⋊#92418C2211yescharacteristic
46C2×#14519702C2212yes—
47C2×#14519700C2212yes—
48C2×#14519698C2212yes—
49C2×#14519689C2212yes—
50C2×C8×Dic61C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#474C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2212yes—
57C22×Dic122—21no—
58C22×Dic122C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519777—22no—
65#14519777C2314yes—
66#14519770—28no—
67#14519768—22no—
68#14519768—24no—
69#14519768C2314yes—
70#14519768C2318yes—
71#14519764—22no—
72#14519764C2314yes—
73#14519759—28no—
74#14519757—28no—
75#14519757—22no—
76#14519757C2314yes—
77#14519753—28no—
78#14519748—22no—
79#14519748C2314yes—
80Q8×Dic61—28no—
81C4×Dic122—22no—
82C4×Dic122C2314yes—
83#14519742—28no—
84C4×C22⋊D61—28no—
85#14519736—28no—
86#14519735—24no—
87#14519735C2318yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—22no—
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—22no—
97C2×C22⋊Dic61—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×#11758198—22no—
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C42×D61D414yes—
108C42×D61C2314yes—
109#14519711C2318yes—
110#14519708C2318yes—
111#14519708C2318yes—
112C4⋊#92418C2314yes—
113#14519702C2318yes—
114#14519700C2318yes—
115#14519698C2318yes—
116#14519689C2318yes—
117C8×Dic61C2318yes—
118C22×Q8×C61—21no—
119C22×D4×C61C2311yescharacteristic
120C23×C4×C61—21no—
121C22×C8×C61C2312yes—
122C4⋊Q8×C61—22no—
123C4⋊Q8×C61C2314yes—
124C4×D4×C61—22no—
125C4×D4×C61C2314yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C22⋊C4×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#107×C61C2318yes—
132C4⋊C8×C61C2314yes—
133#102×C61C2318yes—
134C24×C61—21no—
135C23×D61C2×D411yescharacteristic
136C2×Q8×C61—24no—
137C2×Q8×C61—44no—
138C2×D4×C61—21no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61C2412yes—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61—24no—
145C22×C4×C61—44no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×C4×D61—24no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—42no—
161C2×C4×D61C2×D414yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D414yes—
164C2×C4×D61C2412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—24no—
168C2×Dic122—21no—
169C2×Dic122—44no—
170C2×Dic122C2412yes—
171C2×C8×C61—24no—
172C2×C8×C61C2414yes—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C2414yes—
177C4⋊C4×C61C2412yes—
178C22⋊C4×C61—24no—
179C42×C61C2×D412yes—
180C42×C61C2412yes—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—24no—
184#11758198—28no—
185#11758198—24no—
186C4⋊Dic61—24no—
187C4⋊Dic61—22no—
188C4⋊Dic61—21no—
189C4⋊Dic61—24no—
190C4⋊Dic61—24no—
191C4⋊Dic61—28no—
192C4⋊Dic61C2414yes—
193C4⋊Dic61C2412yes—
194#11758196—24no—
195#11758196—24no—
196#11758196—28no—
197C4×Dic61—24no—
198C4×Dic61—28no—
199C4×Dic61C2×D412yes—
200C4×Dic61C2×D414yes—
201C4×Dic61C2×D414yes—
202C4×Dic61C2412yes—
203C2×#92418—24no—
204C2×#92418C2414yes—
205C23×C61—22no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61C22×D412yes—
214Q8×C61—48no—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—42no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—48no—
224C2×C4×C61#13514yes—
225C2×C4×C61C22×D412yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D412yes—
228C2×C4×C612−1+412yes—
229C2×C4×C612−1+412yes—
230C2×C4×C61C2511yescharacteristic
231C22⋊D61—48no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—44no—
240C2×Dic61—42no—
241C2×Dic61C22×D412yes—
242C2×Dic61C22×D412yes—
243C2×Dic61C22×D412yes—
244C4×D61—24no—
245C4×D61—28no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—44no—
249Dic122—22no—
250Dic122—22no—
251Dic122—48no—
252Dic122—42no—
253C8×C61—24no—
254#92418—24no—
255C2×#3233—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—84no—
260C22×C61#53612yes—
261C22×C61C2×#13512yes—
262C22×C61#57512yes—
263C22×C61C23×D411yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—44no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—44no—
271C4×C61#53612yes—
272C4×C61C2×#13514yes—
273C4×C61C2×2−1+411yescharacteristic
274C4×C61C2×2−1+411yescharacteristic
275Dic61—22no—
276Dic61—22no—
277Dic61—22no—
278Dic61—42no—
279Dic61—44no—
280Dic61—42no—
281C2×#533—611no—
282C2×#529—611no—
283#3233—6116no—
284C2×#499—612no—
285C2×#488—612no—
286C2×#485—612no—
287C2×#482—612no—
288C2×#477—611no—
289C2×#475—611no—
290C2×#476—611no—
291C2×#474D6111yescharacteristic, complemented
292C2×#431—611no—
293C2×C61—82no—
294C2×C61#323314yescomplemented
295C2×C61C2×#53611yescharacteristic
296C2×C61C22×#13511yescharacteristic
297C2×C61C2×#57511yescharacteristic
298D61—42no—
299#533—618no—
300#533—1224no—
301C2×C4⋊Q8—611no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303#529—618no—
304#529—1224no—
305C2×#126—1221no—
306C2×#125—612no—
307C2×#125—611no—
308C2×#124—611no—
309C2×#123—1221no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—1221no—
312C2×C22⋊Q8—1221no—
313C2×#121—611no—
314C2×C4×Q8—611no—
315C2×C4×Q8—1221no—
316C2×C4×Q8—1221no—
317C2×C4×D4—1221no—
318C2×C4×D4C2×D6111yescharacteristic
319C2×#117—612no—
320C2×#117—1221no—
321C22×C42—611no—
322#499—6116no—
323#488—6116no—
324#485—6116no—
325#482—6116no—
326#477—618no—
327#476—618no—
328#475—618no—
329#474C2×D6118yescomplemented
330#431—618no—
331C2×#107—612no—
332C2×#107C2×D6112yes—
333C2×#106—612no—
334C2×#106—612no—
335C2×C4⋊C8—611no—
336C2×C4⋊C8C2×D6111yescharacteristic
337C2×#103—612no—
338C2×#103—612no—
339C2×#102—612no—
340C2×#102C2×D6112yes—
341C2×C22⋊C8—612no—
342C2×C4×C8—612no—
343C61C2×#323311yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C22×Q8—611no—
346C22×Q8—1221no—
347C22×D4—1221no—
348C22×D4C22×D6111yescharacteristic
349C23×C4—21no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C22×C8—612no—
354C22×C8C22×D6112yes—
355C4⋊Q8—22no—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358C4⋊Q8C22×D6114yes—
359#126—1228no—
360#125—614no—
361#125—618no—
362#125—1222no—
363#125—1224no—
364#124—614no—
365#124—1222no—
366#123—1228no—
367C22⋊Q8—614no—
368C22⋊Q8—1222no—
369C22⋊Q8—1228no—
370C22⋊Q8—1228no—
371#121—614no—
372#121—1222no—
373C4×Q8—614no—
374C4×Q8—1228no—
375C4×Q8—1228no—
376C4×Q8—1222no—
377C4×D4—22no—
378C4×D4—1228no—
379C4×D4C22×D6114yes—
380#117—618no—
381#117—1228no—
382#117—1224no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—614no—
405C2×C42—614no—
406C2×C42—1222no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#107—618no—
410#107C22×D6118yes—
411#106—618no—
412#106—618no—
413C4⋊C8—614no—
414C4⋊C8C22×D6114yes—
415#103—618no—
416#103—618no—
417#102—618no—
418#102C22×D6118yes—
419C22⋊C8—618no—
420C4×C8—618no—
421C24—21no—
422C24—611no—
423C2×Q8—24no—
424C2×Q8—44no—
425C2×Q8—612no—
426C2×Q8—1221no—
427C2×Q8—1224no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—2444no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—1224no—
435C2×D4—2444no—
436C2×D4C23×D6112yes—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—614no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4—2442no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458C2×C8—24no—
459C2×C8—614no—
460C2×C8—1224no—
461C2×C8C23×D6114yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—614no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1228no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4—1228no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4C23×D6114yes—
477C4⋊C4C23×D6112yes—
478C22⋊C4—24no—
479C22⋊C4—1228no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C42—614no—
485C42—614no—
486C42—612no—
487C42—612no—
488C42—1228no—
489C42—1224no—
490C42D4×D6112yes—
491C42C23×D6112yes—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—612no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23C2×D4×D6111yescharacteristic
501Q8—48no—
502Q8—1222no—
503Q8—1222no—
504Q8—2448no—
505Q8—2442no—
506D4—22no—
507D4—22no—
508D4—42no—
509D4—2448no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1228no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—2444no—
530C2×C4—2442no—
531C2×C4—2444no—
532C2×C4—2442no—
533C2×C4#1451979614yes—
534C2×C4C2×D4×D6112yes—
535C2×C4C2×D4×D6112yes—
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4#1451982412yes—
538C2×C4#1451982512yes—
539C2×C4C24×D6111yescharacteristic
540C8—24no—
541C8—1224no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—84no—
546C22—1222no—
547C22—1222no—
548C22—1222no—
549C22—2444no—
550C22#1684033512yes—
551C22C2×#1451979612yes—
552C22#1684056112yes—
553C22C22×D4×D6111yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—1222no—
558C4—1222no—
559C4—1222no—
560C4—2442no—
561C4—2444no—
562C4—2442no—
563C4#1684033512yes—
564C4C2×#1451979614yes—
565C4C2×#1451982411yescharacteristic
566C4C2×#1451982511yescharacteristic
567C2—82no—
568C2—2442no—
569C2#2145699614yesdirect product factor
570C2C2×#1684033511yescharacteristic
571C2C22×#1451979611yescharacteristic
572C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC2×#474D61
hypocenterC61C2×#3233
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
3. #21456996C2direct (N × H)
4. C2×#16840163C2semidirect
5. C2×#16840115C2semidirect
6. C2×#16840105C2semidirect
8. C2×#16840058C2semidirect
9. C2×#16840044C2semidirect
10. C2×#16840052C2semidirect
11. C2×#16839814C2semidirect
12. C2×C61×#474C2semidirect
14. #16840317C22semidirect
34. #16840163C22semidirect
35. #16840115C22semidirect
36. #16840105C22semidirect
38. #16840058C22semidirect
39. #16840052C22semidirect
40. #16840044C22semidirect
41. #16839814C22semidirect
53. C61×#474C22semidirect
291. C2×#474D61semidirect
294. C2×C61#3233semidirect
329. #474C2×D61semidirect
343. C61C2×#3233semidirect
569. C2#21456996direct (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×#168402942
#214569962
C2×#168401632
C2×#168401152
C2×#168401052
C2×#168400842
C2×#168400582
C2×#168400442
C2×#168400522
C2×#168398142
C2×C61×#4742
C2×#323361