Finite Group Databasebeta

← C2×#21456992 #30552048

Subgroups — C2×#21456992 #30552048

Order 15616.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840298C211yesmaximal, characteristic
2C22×#14519768C211yesmaximal, characteristic, complemented
3#21456992C2116yesmaximal, direct product factor
4C2×#16840163C212yesmaximal, complemented
5C2×#16840118C212yesmaximal, complemented
6C2×#16840080C211yesmaximal, characteristic
7C2×#16840090C211yesmaximal, characteristic, complemented
8C2×#16840076C211yesmaximal, characteristic
9C2×#16840071C212yesmaximal, complemented
10C2×#16840065C212yesmaximal, complemented
11C2×#16839812C211yesmaximal, characteristic, complemented
12C2×C61×#484C211yesmaximal, characteristic, complemented
13#16840298—24no—
14#16840298C2218yes—
15C2×#14519772—21no—
16C2×#14519768—21no—
17C2×#14519768—21no—
18C2×#14519768—24no—
19C2×#14519768C2212yes—
20C2×#14519768C2212yes—
21C2×#14519768C2211yescharacteristic
22C2×#14519768C2218yescomplemented
23C2×#14519766C2211yescharacteristic
24C2×#14519761—21no—
25C2×#14519751C2211yescharacteristic
26C2×Q8×Dic61C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28C2×#14519737—21no—
29C2×#14519735—21no—
30C22×C4⋊Dic61—21no—
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196—21no—
33C22×C4×Dic61C2211yescharacteristic
34#16840163C22116yescomplemented
35#16840118C22116yescomplemented
36#16840090C2218yescomplemented
37#16840080C2218yes—
38#16840076C2218yes—
39#16840071C22116yescomplemented
40#16840065C22116yescomplemented
41#16839812C2218yescomplemented
42C2×#14519711C2212yes—
43C2×#14519708C2212yes—
44C2×#14519708C2212yes—
45C2×#14519706C2212yes—
46C2×#14519701C2212yes—
47C2×#14519699C2212yes—
48C2×#14519697C2212yes—
49C2×C22⋊#92418C2211yescharacteristic
50C2×C8×Dic61C2212yes—
51C2×C22⋊Q8×C61C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#484C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#103×C61C2212yes—
56C2×C22⋊C8×C61C2211yescharacteristic
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C23×Dic61—21no—
60C23×Dic61D411yescharacteristic
61C23×Dic61D411yescharacteristic
62C22×C4×D61—21no—
63C22×#92418C2312yes—
64#14519772—28no—
65#14519768—24no—
66#14519768—24no—
67#14519768—28no—
68#14519768—28no—
69#14519768—22no—
70#14519768C2318yes—
71#14519768C2318yes—
72#14519768C2314yes—
73#14519766—22no—
74#14519766C2314yes—
75#14519761—28no—
76#14519751—22no—
77#14519751C2314yes—
78Q8×Dic61—22no—
79Q8×Dic61C2314yes—
80#14519739—22no—
81#14519739C2314yes—
82#14519737—28no—
83#14519735—28no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—22no—
86C2×C4⋊Dic61—22no—
87C2×C4⋊Dic61—22no—
88C2×C4⋊Dic61—28no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2314yes—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—22no—
95C2×#11758196—22no—
96C2×#11758196—28no—
97C2×#11758196—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D414yes—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2314yes—
109#14519711C2318yes—
110#14519708C2318yes—
111#14519708C2318yes—
112#14519706C2318yes—
113#14519701C2318yes—
114#14519699C2318yes—
115#14519697C2318yes—
116C22⋊#92418C2314yes—
117C8×Dic61C2318yes—
118C22×Q8×C61C2311yescharacteristic
119C23×C4×C61—21no—
120C23×C4×C61C2311yescharacteristic
121C22×C8×C61C2312yes—
122C22⋊Q8×C61—22no—
123C22⋊Q8×C61C2314yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2312yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C22⋊C4×C61—21no—
131#107×C61C2318yes—
132#103×C61C2318yes—
133C22⋊C8×C61C2314yes—
134C24×C61C2×D411yescharacteristic
135C23×D61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—24no—
138C2×Q8×C61C2412yes—
139C22×C4×C61—24no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—22no—
144C22×C4×C61—44no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2412yes—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C22×Dic61—24no—
152C22×Dic61—24no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—44no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D414yes—
162C22×Dic61C2×D414yes—
163C2×D244—24no—
164C2×D244—21no—
165C2×D244C2412yes—
166C2×C4×D61—24no—
167C2×C4×D61—44no—
168C2×C8×C61—24no—
169C2×C8×C61C2414yes—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2414yes—
175C4⋊C4×C61C2412yes—
176C22⋊C4×C61—24no—
177#11758198—24no—
178#11758198—24no—
179#11758198—24no—
180#11758198—24no—
181C4⋊Dic61—24no—
182C4⋊Dic61—22no—
183C4⋊Dic61—21no—
184C4⋊Dic61—28no—
185C4⋊Dic61—28no—
186C4⋊Dic61—24no—
187C4⋊Dic61—24no—
188C4⋊Dic61C2414yes—
189C4⋊Dic61C2412yes—
190#11758196—28no—
191#11758196—28no—
192#11758196—24no—
193C4×Dic61—24no—
194C4×Dic61C2×D414yes—
195C4×Dic61C2×D412yes—
196C4×Dic61C2×D412yes—
197C4×Dic61C2×D414yes—
198C4×Dic61C2412yes—
199C4×Dic61C2412yes—
200C2×#92418—24no—
201C2×#92418C2414yes—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61#13514yes—
205C23×C61C22×D412yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211Q8×C61—24no—
212Q8×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—22no—
219C2×C4×C61—48no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61C22×D412yes—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C612−1+414yes—
226C2×C4×C61C2511yescharacteristic
227C22⋊D61—48no—
228C2×Dic61—28no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—44no—
236C2×Dic61—48no—
237C2×Dic61—44no—
238C2×Dic61—42no—
239C2×Dic61C22×D412yes—
240C2×Dic61C22×D412yes—
241C2×Dic61C22×D412yes—
242C2×Dic61C22×D412yes—
243D244—24no—
244D244—42no—
245C4×D61—48no—
246C8×C61—24no—
247#92418—24no—
248C2×#3232—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—28no—
251C22×C61—22no—
252C22×C61#53612yes—
253C22×C61C2×#13514yes—
254C22×C61C2×#13512yes—
255C22×C61#57512yes—
256C22×C61C23×D411yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—84no—
261C4×C61—22no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—44no—
265C4×C61#53612yes—
266C4×C61C2×2−1+412yes—
267Dic61—22no—
268Dic61—22no—
269Dic61—22no—
270Dic61—22no—
271Dic61—44no—
272Dic61—42no—
273C2×#529—611no—
274C22×#125—611no—
275#3232—6116no—
276C2×#499—612no—
277C2×#488—612no—
278C2×#484D6111yescharacteristic, complemented
279C2×#481—611no—
280C2×#483—611no—
281C2×#480—611no—
282C2×#479—612no—
283C2×#478—612no—
284C2×#431—611no—
285C2×C61—24no—
286C2×C61#323214yescomplemented
287C2×C61C2×#53611yescharacteristic
288C2×C61C22×#13511yescharacteristic
289C2×C61C2×#57511yescharacteristic
290D61—82no—
291#529—618no—
292#529—1224no—
293C2×#126—1221no—
294C2×#125—612no—
295C2×#125—612no—
296C2×#125—611no—
297C2×#125—618no—
298C2×#125—1221no—
299C2×#125—1221no—
300C2×#125—1224no—
301C2×#124—611no—
302C2×#123—1221no—
303C2×C22⋊Q8C2×D6111yescharacteristic
304C2×#121—611no—
305C2×C4×Q8—611no—
306C2×C4×D4—611no—
307C2×C4×D4—1221no—
308C2×#117—1221no—
309C22×C4⋊C4—611no—
310C22×C4⋊C4—1221no—
311C22×C4⋊C4—1221no—
312C22×C4⋊C4C2×D6111yescharacteristic
313C22×C42—611no—
314#499—6116no—
315#488—6116no—
316#484C2×D6118yescomplemented
317#483—618no—
318#481—618no—
319#480—618no—
320#479—6116no—
321#478—6116no—
322#431—618no—
323C2×#107—612no—
324C2×#107C2×D6112yes—
325C2×#106—612no—
326C2×#106—612no—
327C2×C4⋊C8—612no—
328C2×#103—612no—
329C2×#103C2×D6112yes—
330C2×#102—612no—
331C2×#102—612no—
332C2×C22⋊C8—611no—
333C2×C22⋊C8C2×D6111yescharacteristic
334C2×C4×C8—612no—
335C61C2×#323211yescharacteristic, complemented, Sylow
336C22×Q8C22×D6111yescharacteristic
337C22×D4—611no—
338C22×D4—1221no—
339C23×C4—21no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—1221no—
343C23×C4—1221no—
344C23×C4C22×D6111yescharacteristic
345C22×C8—612no—
346C22×C8C22×D6112yes—
347#126—1228no—
348#125—618no—
349#125—618no—
350#125—614no—
351#125—1228no—
352#125—1224no—
353#125—1224no—
354#125—1228no—
355#125—1222no—
356#124—614no—
357#124—1222no—
358#123—1228no—
359C22⋊Q8—22no—
360C22⋊Q8C22×D6114yes—
361#121—614no—
362#121—1222no—
363C4×Q8—614no—
364C4×Q8—1222no—
365C4×D4—614no—
366C4×D4—1222no—
367C4×D4—1228no—
368#117—1228no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1228no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1228no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C22×D6112yes—
387C2×C4⋊C4C22×D6114yes—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—614no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—614no—
400C2×C42—1221no—
401#107—618no—
402#107C22×D6118yes—
403#106—618no—
404#106—618no—
405C4⋊C8—618no—
406#103—618no—
407#103C22×D6118yes—
408#102—618no—
409#102—618no—
410C22⋊C8—614no—
411C22⋊C8C22×D6114yes—
412C4×C8—618no—
413C24—1221no—
414C24D4×D6111yescharacteristic
415C2×Q8—21no—
416C2×Q8—24no—
417C2×Q8C23×D6112yes—
418C2×D4—612no—
419C2×D4—1224no—
420C2×D4—1221no—
421C2×D4—1224no—
422C2×D4—2444no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—44no—
429C22×C4—614no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—614no—
435C22×C4—1224no—
436C22×C4—1224no—
437C22×C4—1224no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—2444no—
441C22×C4—2442no—
442C22×C4—2444no—
443C22×C4D4×D6112yes—
444C22×C4D4×D6111yescharacteristic
445C22×C4C23×D6112yes—
446C22×C4C23×D6111yescharacteristic
447C2×C8—24no—
448C2×C8—614no—
449C2×C8—1224no—
450C2×C8C23×D6114yes—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—614no—
456C4⋊C4—612no—
457C4⋊C4—1228no—
458C4⋊C4—1228no—
459C4⋊C4—1228no—
460C4⋊C4—1228no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1222no—
466C4⋊C4—1221no—
467C4⋊C4C23×D6114yes—
468C4⋊C4C23×D6112yes—
469C22⋊C4—24no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C42—614no—
475C42—612no—
476C42—612no—
477C42—614no—
478C42—612no—
479C42—612no—
480C42—1224no—
481C23—22no—
482C23—22no—
483C23—1222no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23#1451980014yes—
488C23C2×D4×D6112yes—
489C23C2×D4×D6111yescharacteristic
490Q8—24no—
491Q8—42no—
492D4—1224no—
493D4—2448no—
494D4—2442no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—1228no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—2448no—
517C2×C4—2444no—
518C2×C4—2448no—
519C2×C4—2444no—
520C2×C4—2442no—
521C2×C4C2×D4×D6112yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4#1451982414yes—
524C2×C4C24×D6111yescharacteristic
525C8—24no—
526C8—1224no—
527C22—22no—
528C22—28no—
529C22—22no—
530C22—4884no—
531C22—2442no—
532C22—2442no—
533C22—2442no—
534C22#1684033412yes—
535C22C2×#1451980014yes—
536C22C2×#1451980012yes—
537C22#1684055812yes—
538C22C22×D4×D6111yescharacteristic
539C4—22no—
540C4—44no—
541C4—42no—
542C4—42no—
543C4—1222no—
544C4—1222no—
545C4—1222no—
546C4—1222no—
547C4—2444no—
548C4—2442no—
549C4#1684033412yes—
550C4C2×#1451982412yes—
551C2—24no—
552C2—4882no—
553C2#2145699214yesdirect product factor
554C2C2×#1684033411yescharacteristic
555C2C22×#1451980011yescharacteristic
556C2C2×#1684055811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#484C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#484D61
hypocenterC61C2×#3232
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#14519768C2semidirect
3. #21456992C2direct (N × H)
4. C2×#16840163C2semidirect
5. C2×#16840118C2semidirect
7. C2×#16840090C2semidirect
9. C2×#16840071C2semidirect
10. C2×#16840065C2semidirect
11. C2×#16839812C2semidirect
12. C2×C61×#484C2semidirect
22. C2×#14519768C22semidirect
34. #16840163C22semidirect
35. #16840118C22semidirect
36. #16840090C22semidirect
39. #16840071C22semidirect
40. #16840065C22semidirect
41. #16839812C22semidirect
53. C61×#484C22semidirect
278. C2×#484D61semidirect
286. C2×C61#3232semidirect
316. #484C2×D61semidirect
335. C61C2×#3232semidirect
553. C2#21456992direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402982
C22×#145197682
#214569922
C2×#168401632
C2×#168401182
C2×#168400802
C2×#168400902
C2×#168400762
C2×#168400712
C2×#168400652
C2×#168398122
C2×C61×#4842
C2×#323261