Finite Group Databasebeta

← C2×#21460276 #30587785

Subgroups — C2×#21460276 #30587785

Order 15616.

6566 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
1#21460276C2132yesmaximal, direct product factor
2C2×#16840516C212yesmaximal, complemented
3C2×#16840510C211yesmaximal, characteristic, complemented
4C2×#16840468C214yesmaximal, complemented
5C2×#16840465C214yesmaximal, complemented
6C2×#16840450C214yesmaximal, complemented
7C2×#16840371C214yesmaximal, complemented
8C2×#16840360C214yesmaximal, complemented
9C2×#16840360C214yesmaximal, complemented
10C2×#16840297C212yesmaximal, complemented
11C2×#125×D61C211yesmaximal, characteristic, complemented
12C2×C61×#564C211yesmaximal, characteristic, complemented
13#16840516C22132yescomplemented
14#16840510C22116yescomplemented
15#16840468C22164yescomplemented
16#16840465C22164yescomplemented
17#16840450C22164yescomplemented
18#16840371C22164yescomplemented
19#16840360C22164yescomplemented
20#16840360C22164yescomplemented
21C2×#14519776C2214yes—
22#16840297C22132yescomplemented
23C2×#14519772C2214yes—
24C2×#14519772C2214yes—
25C2×#14519772C2214yes—
26C2×#14519772C2214yes—
27C2×#14519771C2214yes—
28C2×#14519769C2211yescharacteristic
29C2×#14519768C2212yes—
30C2×#14519768C2214yes—
31C2×#14519767C2214yes—
32C2×#14519767C2214yes—
33#125×D61C22116yescomplemented
34C2×#14519765C2212yes—
35C2×#14519762C2214yes—
36C2×#14519762C2214yes—
37C2×#14519762C2214yes—
38C2×#14519762C2214yes—
39C2×#14519762C2214yes—
40C2×#14519757C2214yes—
41C2×#14519757C2214yes—
42C2×#14519757C2212yes—
43C2×#14519755C2214yes—
44C2×#14519755C2214yes—
45C2×#14519755C2214yes—
46C2×#14519755C2214yes—
47C2×#14519752C2212yes—
48C2×#14519752C2214yes—
49C2×#14519752C2214yes—
50C2×C4×Dic122C2214yes—
51C2×#14519742C2212yes—
52C2×C4×D244C2214yes—
53C2×C4×D244C2214yes—
54C2×#14519739C2214yes—
55C2×#14519739C2212yes—
56C2×#14519739C2214yes—
57C2×#14519735C2214yes—
58C2×#14519735C2214yes—
59C2×C4⋊C4×D61C2214yes—
60C2×C4⋊C4×D61C2212yes—
61C2×C4⋊C4×D61C2214yes—
62C2×C42×D61C2211yescharacteristic
63C61×#564C22116yescomplemented
64C2×#125×C61C2211yescharacteristic
65C2×#125×C61C2214yes—
66C2×#125×C61C2212yes—
67C2×#125×C61C2214yes—
68C2×#125×C61C2214yes—
69C22×Dic122C2312yes—
70C22×Dic122C2314yes—
71C22×D244C2314yes—
72C22×D244C2312yes—
73C22×D244C2314yes—
74C22×C4×D61C2311yescharacteristic
75C22×C4×D61C2312yes—
76C22×C4×D61C2314yes—
77C22×C4×D61C2314yes—
78C22×C4×D61C2314yes—
79C22×C4×D61C2314yes—
80#14519776C23132yes—
81#14519772C23132yes—
82#14519772C23132yes—
83#14519772C23132yes—
84#14519772C23132yes—
85#14519771C23132yes—
86#14519769C2318yes—
87#14519768C23116yes—
88#14519768C23132yes—
89#14519767C23132yes—
90#14519767C23132yes—
91#14519765C23116yes—
92#14519762C23132yes—
93#14519762C23132yes—
94#14519762C23132yes—
95#14519762C23132yes—
96#14519762C23132yes—
97#14519757C23132yes—
98#14519757C23132yes—
99#14519757C23116yes—
100#14519755C23132yes—
101#14519755C23132yes—
102#14519755C23132yes—
103#14519755C23132yes—
104#14519752C23116yes—
105#14519752C23132yes—
106#14519752C23132yes—
107C4×Dic122C23132yes—
108#14519742C23116yes—
109C4×D244C23132yes—
110C4×D244C23132yes—
111#14519739C23132yes—
112#14519739C23132yes—
113#14519739C23116yes—
114#14519735C23132yes—
115#14519735C23132yes—
116C2×C4⋊Dic61C2314yes—
117C2×C4⋊Dic61C2314yes—
118C2×C4⋊Dic61C2312yes—
119C2×C4⋊Dic61C2314yes—
120C2×#11758196C2314yes—
121C2×#11758196C2314yes—
122C2×#11758196C2314yes—
123C2×#11758196C2314yes—
124C2×#11758196C2314yes—
125C2×#11758196C2314yes—
126C4⋊C4×D61C23132yes—
127C4⋊C4×D61C23132yes—
128C4⋊C4×D61C23116yes—
129C2×#11758198C2314yes—
130C2×#11758198C2314yes—
131C2×#11758198C2314yes—
132C2×#11758198C2314yes—
133C2×#11758198C2314yes—
134C2×#11758198C2314yes—
135C2×#11758198C2314yes—
136C2×#11758198C2314yes—
137C2×#11758198C2314yes—
138C2×C4×Dic61C2311yescharacteristic
139C2×C4×Dic61C2312yes—
140C2×C4×Dic61C2314yes—
141C2×C4×Dic61C2314yes—
142C42×D61C2318yes—
143#125×C61C23132yes—
144#125×C61C23132yes—
145#125×C61C23132yes—
146#125×C61C2318yes—
147#125×C61C23116yes—
148C2×C4⋊C4×C61C2314yes—
149C2×C4⋊C4×C61C2314yes—
150C2×C4⋊C4×C61C2312yes—
151C2×C4⋊C4×C61C2314yes—
152C2×C4⋊C4×C61C2314yes—
153C2×C4⋊C4×C61C2312yes—
154C2×C4⋊C4×C61C2314yes—
155C2×C4⋊C4×C61C2314yes—
156C2×C4⋊C4×C61C2312yes—
157C2×C42×C61C2311yescharacteristic
158C2×C42×C61C2314yes—
159C23×D61C2414yes—
160C23×D61C2411yescharacteristic
161C22×C4×C61C2414yes—
162C22×C4×C61C2414yes—
163C22×C4×C61C2412yes—
164C22×C4×C61C2411yescharacteristic
165C22×C4×C61C2414yes—
166C22×Dic61C2412yes—
167C22×Dic61C2414yes—
168C22×Dic61C2414yes—
169C22×Dic61C2411yescharacteristic
170C2×D244—216no—
171C2×D244—44no—
172C2×D244—48no—
173C2×D244C24116yes—
174C2×D244C2418yes—
175C2×D244C24116yes—
176C2×C4×D61—24no—
177C2×C4×D61—48no—
178C2×C4×D61—48no—
179C2×C4×D61—48no—
180C2×C4×D61—48no—
181C2×C4×D61—44no—
182C2×C4×D61C2414yes—
183C2×C4×D61C2418yes—
184C2×C4×D61C24116yes—
185C2×C4×D61C24116yes—
186C2×C4×D61C24116yes—
187C2×C4×D61C24116yes—
188C2×Dic122—44no—
189C2×Dic122—48no—
190C2×Dic122C2418yes—
191C2×Dic122C24116yes—
192C4⋊C4×C61C24116yes—
193C4⋊C4×C61C24116yes—
194C4⋊C4×C61C2418yes—
195C4⋊C4×C61C24116yes—
196C4⋊C4×C61C24116yes—
197C4⋊C4×C61C2418yes—
198C4⋊C4×C61C24116yes—
199C4⋊C4×C61C24116yes—
200C4⋊C4×C61C2418yes—
201C42×C61C2414yes—
202C42×C61C24116yes—
203#11758198C24116yes—
204#11758198C24116yes—
205#11758198C24116yes—
206#11758198C24116yes—
207#11758198C24116yes—
208#11758198C24116yes—
209#11758198C24116yes—
210#11758198C24116yes—
211#11758198C24116yes—
212C4⋊Dic61C24116yes—
213C4⋊Dic61C24116yes—
214C4⋊Dic61C2418yes—
215C4⋊Dic61C24116yes—
216#11758196C24116yes—
217#11758196C24116yes—
218#11758196C24116yes—
219#11758196C24116yes—
220#11758196C24116yes—
221#11758196C24116yes—
222C4×Dic61C2414yes—
223C4×Dic61C2418yes—
224C4×Dic61C24116yes—
225C4×Dic61C24116yes—
226C23×C61C2511yescharacteristic
227C22×D61—24no—
228C22×D61—28no—
229C22×D61—28no—
230C22×D61—28no—
231C22×D612−1+414yes—
232C22×D61C2518yes—
233C22×D61C2512yes—
234C2×C4×C61—22no—
235C2×C4×C61—28no—
236C2×C4×C61—28no—
237C2×C4×C61—24no—
238C2×C4×C61—28no—
239C2×C4×C61C2518yes—
240C2×C4×C61C2514yes—
241C2×C4×C61C2512yes—
242C2×C4×C61C2518yes—
243C2×C4×C61C2518yes—
244C2×Dic61—22no—
245C2×Dic61—28no—
246C2×Dic61—28no—
247C2×Dic61—24no—
248C2×Dic61C2514yes—
249C2×Dic61C2512yes—
250C2×Dic61C2518yes—
251C2×Dic61C2518yes—
252D244—232no—
253D244—48no—
254D244—416no—
255C4×D61—28no—
256C4×D61—416no—
257C4×D61—416no—
258C4×D61—416no—
259C4×D61—416no—
260C4×D61—48no—
261Dic122—48no—
262Dic122—416no—
263C2×#3553—611nomaximal, Sylow
264C22×C61C2×2−1+412yes—
265C22×C61#58514yes—
266C22×C61C2611yescharacteristic
267C2×D61—24no—
268C2×D61—28no—
269C2×D61—28no—
270C2×D61—28no—
271C2×D61—28no—
272C2×D61—416no—
273C2×D61C2×2−1+414yes—
274C4×C61—22no—
275C4×C61—28no—
276C4×C61—28no—
277C4×C61—24no—
278C4×C61—28no—
279Dic61—22no—
280Dic61—28no—
281Dic61—28no—
282Dic61—24no—
283#3553—6132no—
284C2×#564D6111yescharacteristic, complemented
285C2×#563—612no—
286C2×#562—611no—
287C2×#554—614no—
288C2×#553—614no—
289C2×#552—614no—
290C2×#541—614no—
291C2×#539—614no—
292C2×#539—614no—
293C2×#529—612no—
294C22×#125—611no—
295C2×C61#355314yescomplemented
296C2×C61C22×2−1+411yescharacteristic
297C2×C61C2×#58512yes—
298D61—24no—
299D61—48no—
300#564C2×D61116yescomplemented
301#563—6132no—
302#562—6116no—
303#554—6164no—
304#553—6164no—
305#552—6164no—
306#541—6164no—
307#539—6164no—
308#539—6164no—
309C2×C4⋊Q8—614no—
310#529—6132no—
311C2×#126—614no—
312C2×#126—614no—
313C2×#126—614no—
314C2×#126—614no—
315C2×#126—614no—
316C2×#125—6116no—
317C2×#125—614no—
318C2×#125—614no—
319C2×#125—612no—
320C2×#125—614no—
321C2×#125—611no—
322C2×#125C2×D6111yescharacteristic
323C2×#125C2×D6112yes—
324C2×#125C2×D6114yes—
325C2×#125C2×D6114yes—
326C2×#125C2×D6114yes—
327C2×#124—612no—
328C2×#123—614no—
329C2×#123—614no—
330C2×#123—614no—
331C2×#123—614no—
332C2×#123—614no—
333C2×C22⋊Q8—614no—
334C2×C22⋊Q8—614no—
335C2×C22⋊Q8—614no—
336C2×C22⋊Q8—614no—
337C2×C22⋊Q8—612no—
338C2×C22⋊Q8—614no—
339C2×C22⋊Q8—614no—
340C2×#121—612no—
341C2×#121—614no—
342C2×#121—614no—
343C2×C4×Q8—612no—
344C2×C4×Q8—614no—
345C2×C4×D4—612no—
346C2×C4×D4—614no—
347C2×C4×D4—614no—
348C2×C4×D4—614no—
349C2×C4×D4—614no—
350C2×#117—614no—
351C2×#117—614no—
352C22×C4⋊C4—612no—
353C22×C4⋊C4—614no—
354C22×C4⋊C4—614no—
355C22×C42—611no—
356C61C2×#355311yescharacteristic, complemented, Sylow
357C22×Q8—614no—
358C22×Q8—612no—
359C22×D4—614no—
360C22×D4—614no—
361C22×D4—612no—
362C23×C4—611no—
363C23×C4—614no—
364C23×C4—612no—
365C23×C4—614no—
366C23×C4—614no—
367C23×C4—614no—
368C4⋊Q8—6132no—
369#126—6132no—
370#126—6132no—
371#126—6132no—
372#126—6132no—
373#126—6132no—
374#125—6132no—
375#125—6116no—
376#125—6132no—
377#125—6132no—
378#125—618no—
379#125C22×D6118yes—
380#125C22×D61116yes—
381#125C22×D61132yes—
382#125C22×D61132yes—
383#125C22×D61132yes—
384#124—6116no—
385#123—6132no—
386#123—6132no—
387#123—6132no—
388#123—6132no—
389#123—6132no—
390C22⋊Q8—6116no—
391C22⋊Q8—6132no—
392C22⋊Q8—6132no—
393C22⋊Q8—6132no—
394C22⋊Q8—6132no—
395C22⋊Q8—6132no—
396C22⋊Q8—6132no—
397#121—6116no—
398#121—6132no—
399#121—6132no—
400C4×Q8—6116no—
401C4×Q8—6132no—
402C4×D4—6116no—
403C4×D4—6132no—
404C4×D4—6132no—
405C4×D4—6132no—
406C4×D4—6132no—
407#117—6132no—
408#117—6132no—
409C2×C4⋊C4—614no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—614no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—614no—
417C2×C4⋊C4—614no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4—6116no—
420C2×C4⋊C4—6132no—
421C2×C4⋊C4—6132no—
422C2×C4⋊C4C22×D6114yes—
423C2×C4⋊C4C22×D6114yes—
424C2×C4⋊C4C22×D6112yes—
425C2×C4⋊C4C22×D6114yes—
426C2×C4⋊C4C22×D6114yes—
427C2×C4⋊C4C22×D6112yes—
428C2×C4⋊C4C22×D6114yes—
429C2×C4⋊C4C22×D6114yes—
430C2×C4⋊C4C22×D6112yes—
431C2×C22⋊C4—614no—
432C2×C22⋊C4—614no—
433C2×C22⋊C4—614no—
434C2×C22⋊C4—614no—
435C2×C22⋊C4—614no—
436C2×C22⋊C4—614no—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—614no—
439C2×C22⋊C4—614no—
440C2×C42—611no—
441C2×C42—614no—
442C2×C42—614no—
443C2×C42—612no—
444C2×C42—618no—
445C2×C42C22×D6114yes—
446C2×C42C22×D6111yescharacteristic
447C24—611no—
448C24—614no—
449C2×Q8—6116no—
450C2×Q8—618no—
451C2×Q8—2448no—
452C2×Q8—2444no—
453C2×D4—6116no—
454C2×D4—6116no—
455C2×D4—618no—
456C2×D4—12216no—
457C2×D4—2448no—
458C2×D4—2444no—
459C22×C4—612no—
460C22×C4—611no—
461C22×C4—614no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—618no—
465C22×C4—6116no—
466C22×C4—6116no—
467C22×C4—6116no—
468C22×C4—6116no—
469C22×C4—1224no—
470C22×C4—2448no—
471C22×C4—2448no—
472C22×C4—2448no—
473C22×C4—2448no—
474C22×C4—2444no—
475C22×C4C23×D6114yes—
476C22×C4C23×D6111yescharacteristic
477C22×C4C23×D6112yes—
478C22×C4C23×D6114yes—
479C22×C4C23×D6114yes—
480C4⋊C4—6116no—
481C4⋊C4—6116no—
482C4⋊C4—6116no—
483C4⋊C4—6116no—
484C4⋊C4—6116no—
485C4⋊C4—6116no—
486C4⋊C4—618no—
487C4⋊C4—6116no—
488C4⋊C4—6116no—
489C4⋊C4—6116no—
490C4⋊C4C23×D61116yes—
491C4⋊C4C23×D61116yes—
492C4⋊C4C23×D6118yes—
493C4⋊C4C23×D61116yes—
494C4⋊C4C23×D61116yes—
495C4⋊C4C23×D6118yes—
496C4⋊C4C23×D61116yes—
497C4⋊C4C23×D61116yes—
498C4⋊C4C23×D6118yes—
499C22⋊C4—6116no—
500C22⋊C4—6116no—
501C22⋊C4—6116no—
502C22⋊C4—6116no—
503C22⋊C4—6116no—
504C22⋊C4—6116no—
505C22⋊C4—6116no—
506C22⋊C4—6116no—
507C22⋊C4—6116no—
508C42—614no—
509C42—618no—
510C42—6116no—
511C42—6116no—
512C42C23×D6114yes—
513C42C23×D61116yes—
514C23—614no—
515C23—612no—
516C23—618no—
517C23—1224no—
518C23—1228no—
519C23—1228no—
520C23—1228no—
521C23C24×D6111yescharacteristic
522Q8—24416no—
523Q8—2448no—
524D4—12232no—
525D4—24416no—
526D4—2448no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—614no—
533C2×C4—612no—
534C2×C4—618no—
535C2×C4—618no—
536C2×C4—1228no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—1228no—
540C2×C4—1228no—
541C2×C4—24416no—
542C2×C4—24416no—
543C2×C4—24416no—
544C2×C4—24416no—
545C2×C4—2448no—
546C2×C4C24×D6118yes—
547C2×C4C24×D6112yes—
548C2×C4C24×D6114yes—
549C2×C4C24×D6118yes—
550C2×C4C24×D6118yes—
551C22—614no—
552C22—1228no—
553C22—1224no—
554C22—1228no—
555C22—1228no—
556C22—1228no—
557C22—24416no—
558C222−1+4×D6112yes—
559C22#1684059914yes—
560C22C25×D6111yescharacteristic
561C4—22no—
562C4—28no—
563C4—28no—
564C4—28no—
565C4—24no—
566C4—1222no—
567C4—1224no—
568C4—1228no—
569C4—1228no—
570C2—1224no—
571C2—2448no—
572C2#2146027614yesdirect product factor
573C2C2×2−1+4×D6111yescharacteristic
574C2C2×#1684059912yes—
subgroupG/H
centerC23C24×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×C61×#564C2
socleC23×C61C25
hypercenterC2×#564D61
hypocenterC61C2×#3553
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#564

Complements

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

NH ≅ G/Ntype
1. #21460276C2direct (N × H)
2. C2×#16840516C2semidirect
3. C2×#16840510C2semidirect
4. C2×#16840468C2semidirect
5. C2×#16840465C2semidirect
6. C2×#16840450C2semidirect
7. C2×#16840371C2semidirect
8. C2×#16840360C2semidirect
9. C2×#16840360C2semidirect
10. C2×#16840297C2semidirect
11. C2×#125×D61C2semidirect
12. C2×C61×#564C2semidirect
13. #16840516C22semidirect
14. #16840510C22semidirect
15. #16840468C22semidirect
16. #16840465C22semidirect
17. #16840450C22semidirect
18. #16840371C22semidirect
19. #16840360C22semidirect
20. #16840360C22semidirect
22. #16840297C22semidirect
33. #125×D61C22semidirect
63. C61×#564C22semidirect
284. C2×#564D61semidirect
295. C2×C61#3553semidirect
300. #564C2×D61semidirect
356. C61C2×#3553semidirect
572. C2#21460276direct (N × H)

Maximal subgroups

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

subgroupindex
#214602762
C2×#168405162
C2×#168405102
C2×#168404682
C2×#168404652
C2×#168404502
C2×#168403712
C2×#168403602
C2×#168402972
C2×#125×D612
C2×C61×#5642
C2×#355361