Finite Group Databasebeta

← C2×#21456984 #30552040

Subgroups — C2×#21456984 #30552040

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×#16840293C211yesmaximal, characteristic
2C22×#14519768C211yesmaximal, characteristic, complemented
3#21456984C2116yesmaximal, direct product factor
4C2×#16840163C212yesmaximal, complemented
5C2×#16840115C212yesmaximal, complemented
6C2×#16840098C211yesmaximal, characteristic, complemented
7C2×#16840078C211yesmaximal, characteristic
8C2×#16840092C211yesmaximal, characteristic, complemented
9C2×#16840068C212yesmaximal, complemented
10C2×#16840060C212yesmaximal, complemented
11C2×#16839812C211yesmaximal, characteristic, complemented
12C2×C61×#480C211yesmaximal, characteristic
13#16840293—24no—
14#16840293C2218yes—
15C2×#14519770—21no—
16C2×#14519768—21no—
17C2×#14519768—21no—
18C2×#14519768—24no—
19C2×#14519768C2212yes—
20C2×#14519768C2212yes—
21C2×#14519768C2211yescharacteristic
22C2×#14519768C2218yescomplemented
23C2×#14519764C2211yescharacteristic
24C2×#14519758—21no—
25C2×#14519756C2211yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×D4×Dic61—21no—
28C2×D4×Dic61C2211yescharacteristic
29C2×#14519733—21no—
30C22×C4⋊Dic61—21no—
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196—21no—
33C22×C4×Dic61C2211yescharacteristic
34#16840163C22116yescomplemented
35#16840115C22116yescomplemented
36#16840098C2218yescomplemented
37#16840092C2218yescomplemented
38#16840078C2218yes—
39#16840068C22116yescomplemented
40#16840060C22116yescomplemented
41#16839812C2218yescomplemented
42C2×#14519711C2212yes—
43C2×#14519708C2212yes—
44C2×#14519708C2212yes—
45C2×#14519706C2212yes—
46C2×#14519702C2212yes—
47C2×#14519700C2212yes—
48C2×#14519698C2212yes—
49C2×C22⋊#92418C2211yescharacteristic
50C2×C8×Dic61C2212yes—
51C2×#121×C61C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#480C2218yes—
54C2×#107×C61C2212yes—
55C2×#102×C61C2212yes—
56C2×C22⋊C8×C61C2211yescharacteristic
57C22×Dic122C2311yescharacteristic
58C23×Dic61—21no—
59C23×Dic61—21no—
60C23×Dic61D411yescharacteristic
61C23×Dic61D411yescharacteristic
62C22×#92418C2312yes—
63#14519770—28no—
64#14519768—24no—
65#14519768—24no—
66#14519768—28no—
67#14519768—28no—
68#14519768—22no—
69#14519768C2318yes—
70#14519768C2318yes—
71#14519768C2314yes—
72#14519764—22no—
73#14519764C2314yes—
74#14519758—28no—
75#14519756—22no—
76#14519756C2314yes—
77#14519742—22no—
78#14519742C2314yes—
79D4×Dic61—28no—
80D4×Dic61—22no—
81D4×Dic61C2314yes—
82#14519733—28no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—22no—
85C2×C4⋊Dic61—22no—
86C2×C4⋊Dic61—22no—
87C2×C4⋊Dic61—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2312yes—
90C2×C4⋊Dic61C2314yes—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—22no—
94C2×#11758196—22no—
95C2×#11758196—28no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—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#14519702C2318yes—
114#14519700C2318yes—
115#14519698C2318yes—
116C22⋊#92418C2314yes—
117C8×Dic61C2318yes—
118C22×D4×C61—21no—
119C22×D4×C61C2311yescharacteristic
120C23×C4×C61—21no—
121C23×C4×C61C2311yescharacteristic
122C22×C8×C61C2312yes—
123#121×C61—22no—
124#121×C61C2314yes—
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#102×C61C2318yes—
133C22⋊C8×C61C2314yes—
134C24×C61—21no—
135C24×C61C2×D411yescharacteristic
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—24no—
139C2×D4×C61—44no—
140C2×D4×C61C2412yes—
141C22×C4×C61—24no—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—22no—
145C22×C4×C61—44no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2412yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—24no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—24no—
156C22×Dic61—42no—
157C22×Dic61—44no—
158C22×Dic61—44no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D414yes—
164C22×Dic61C2×D414yes—
165C2×Dic122—24no—
166C2×Dic122—21no—
167C2×Dic122C2412yes—
168C2×C8×C61—24no—
169C2×C8×C61C2414yes—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2414yes—
174C4⋊C4×C61C2412yes—
175C22⋊C4×C61—24no—
176C22⋊Dic61—24no—
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180C4⋊Dic61—24no—
181C4⋊Dic61—22no—
182C4⋊Dic61—21no—
183C4⋊Dic61—28no—
184C4⋊Dic61—28no—
185C4⋊Dic61—24no—
186C4⋊Dic61C2414yes—
187C4⋊Dic61C2412yes—
188#11758196—24no—
189#11758196—28no—
190#11758196—28no—
191#11758196—24no—
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—22no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61#13514yes—
209C23×C61C22×D412yes—
210C23×C61C22×D411yescharacteristic
211D4×C61—24no—
212D4×C61—48no—
213D4×C61—42no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—48no—
220C2×C4×C61—44no—
221C2×C4×C61—42no—
222C2×C4×C61C22×D412yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C612−1+414yes—
225C2×C4×C61C2511yescharacteristic
226C2×Dic61—28no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—22no—
234C2×Dic61—44no—
235C2×Dic61—48no—
236C2×Dic61—48no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—44no—
240C2×Dic61C22×D412yes—
241C2×Dic61C22×D412yes—
242C2×Dic61C22×D412yes—
243C2×Dic61C22×D412yes—
244Dic122—24no—
245Dic122—42no—
246C8×C61—24no—
247#92418—24no—
248C2×#3232—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—28no—
251C22×C61—22no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—84no—
256C22×C61#53612yes—
257C22×C61C2×#13514yes—
258C22×C61C2×#13512yes—
259C22×C61#57512yes—
260C22×C61C23×D411yescharacteristic
261C4×C61—22no—
262C4×C61—44no—
263C4×C61—42no—
264C4×C61#53612yes—
265C4×C61C2×2−1+412yes—
266Dic61—22no—
267Dic61—22no—
268Dic61—22no—
269Dic61—22no—
270Dic61—42no—
271Dic61—44no—
272Dic61—42no—
273C2×#529—611no—
274C22×#125—611no—
275#3232—6116no—
276C2×#499—612no—
277C2×#488—612no—
278C2×#484—611no—
279C2×#481—611no—
280C2×#483—611no—
281C2×#480D6111yescharacteristic
282C2×#479—612no—
283C2×#478—612no—
284C2×#431—611no—
285C2×C61—24no—
286C2×C61—82no—
287C2×C61#323214yescomplemented
288C2×C61C2×#53611yescharacteristic
289C2×C61C22×#13511yescharacteristic
290C2×C61C2×#57511yescharacteristic
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⋊Q8—611no—
304C2×#121C2×D6111yescharacteristic
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#484—618no—
317#483—618no—
318#481—618no—
319#480C2×D6118yes—
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×#103—612no—
330C2×#102—612no—
331C2×#102C2×D6112yes—
332C2×C22⋊C8—611no—
333C2×C22⋊C8C2×D6111yescharacteristic
334C2×C4×C8—612no—
335C61C2×#323211yescharacteristic, complemented, Sylow
336C22×Q8—611no—
337C22×D4—21no—
338C22×D4C22×D6111yescharacteristic
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—614no—
360C22⋊Q8—1222no—
361#121—22no—
362#121C22×D6114yes—
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—612no—
372C2×C4⋊C4—614no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1228no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1228no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
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#103—618no—
408#102—618no—
409#102C22×D6118yes—
410C22⋊C8—614no—
411C22⋊C8C22×D6114yes—
412C4×C8—618no—
413C24—21no—
414C24D4×D6111yescharacteristic
415C2×Q8—612no—
416C2×Q8—1221no—
417C2×Q8—1224no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—24no—
421C2×D4—44no—
422C2×D4C23×D6112yes—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—614no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—614no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—1224no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—2442no—
441C22×C4—2444no—
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—21no—
454C4⋊C4—614no—
455C4⋊C4—612no—
456C4⋊C4—1228no—
457C4⋊C4—1228no—
458C4⋊C4—1228no—
459C4⋊C4—1228no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4—1222no—
465C4⋊C4—1221no—
466C4⋊C4—1224no—
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—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23#1451979614yes—
488C23C2×D4×D6112yes—
489C23C2×D4×D6111yescharacteristic
490Q8—1224no—
491Q8—2442no—
492D4—24no—
493D4—48no—
494D4—42no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—1228no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1224no—
515C2×C4—2444no—
516C2×C4—2448no—
517C2×C4—2448no—
518C2×C4—2442no—
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—42no—
531C22—42no—
532C22—42no—
533C22—84no—
534C22#1684033412yes—
535C22C2×#1451979614yes—
536C22C2×#1451979612yes—
537C22#1684056112yes—
538C22C22×D4×D6111yescharacteristic
539C4—22no—
540C4—44no—
541C4—42no—
542C4—1222no—
543C4—1222no—
544C4—1222no—
545C4—1222no—
546C4—2442no—
547C4—2444no—
548C4—2442no—
549C4#1684033412yes—
550C4C2×#1451982412yes—
551C2—24no—
552C2—82no—
553C2#2145698414yesdirect product factor
554C2C2×#1684033411yescharacteristic
555C2C22×#1451979611yescharacteristic
556C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#480C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenterC2×#480D61
hypocenterC61C2×#3232
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C22×#14519768C2semidirect
3. #21456984C2direct (N × H)
4. C2×#16840163C2semidirect
5. C2×#16840115C2semidirect
6. C2×#16840098C2semidirect
8. C2×#16840092C2semidirect
9. C2×#16840068C2semidirect
10. C2×#16840060C2semidirect
11. C2×#16839812C2semidirect
22. C2×#14519768C22semidirect
34. #16840163C22semidirect
35. #16840115C22semidirect
36. #16840098C22semidirect
37. #16840092C22semidirect
39. #16840068C22semidirect
40. #16840060C22semidirect
41. #16839812C22semidirect
287. C2×C61#3232semidirect
335. C61C2×#3232semidirect
553. C2#21456984direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402932
C22×#145197682
#214569842
C2×#168401632
C2×#168401152
C2×#168400982
C2×#168400782
C2×#168400922
C2×#168400682
C2×#168400602
C2×#168398122
C2×C61×#4802
C2×#323261