Finite Group Databasebeta

← C2×#21459966 #30586914

Subgroups — C2×#21459966 #30586914

Order 15616.

6554 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#21459966C2132yesmaximal, direct product factor
2C2×C61⋊#564C211yesmaximal, characteristic, complemented
3C2×#16840516C212yesmaximal, complemented
4C2×#16840513C214yesmaximal, complemented
5C2×#16840511C214yesmaximal, complemented
6C2×#16840510C211yesmaximal, characteristic, complemented
7C2×#16840508C214yesmaximal, complemented
8C2×#16840316C214yesmaximal, complemented
9C2×#16840299C214yesmaximal, complemented
10C2×#16840296C212yesmaximal, complemented
11C2×#16840290C214yesmaximal, complemented
12C2×C61×#533C211yesmaximal, characteristic, complemented
13C61⋊#564C22116yescomplemented
14#16840516C22132yescomplemented
15#16840513C22164yescomplemented
16#16840511C22164yescomplemented
17#16840510C22116yescomplemented
18#16840508C22164yescomplemented
19#16840316C22164yescomplemented
20C2×#14519776C2214yes—
21C2×#14519775C2214yes—
22#16840299C22164yescomplemented
23#16840296C22132yescomplemented
24#16840290C22164yescomplemented
25C2×#14519772C2214yes—
26C2×#14519772C2214yes—
27C2×#14519772C2214yes—
28C2×#14519771C2214yes—
29C2×#14519770C2214yes—
30C2×#14519769C2212yes—
31C2×#14519768C2214yes—
32C2×#14519767C2214yes—
33C2×#14519767C2214yes—
34C2×#14519767C2214yes—
35C2×#14519766C2214yes—
36C2×#14519765C2212yes—
37C2×#14519763C2214yes—
38C2×#14519762C2214yes—
39C2×#14519762C2214yes—
40C2×#14519762C2214yes—
41C2×#14519760C2214yes—
42C2×#14519759C2214yes—
43C2×#14519757C2214yes—
44C2×#14519756C2212yes—
45C2×#14519755C2214yes—
46C2×#14519755C2214yes—
47C2×#14519755C2214yes—
48C2×#14519753C2214yes—
49C2×#14519753C2214yes—
50C2×#14519752C2214yes—
51C2×#14519751C2212yes—
52C2×#14519750C2214yes—
53C2×C4×Dic122C2214yes—
54C2×C4×Dic122C2212yes—
55C2×C4×D244C2214yes—
56C2×C4×D244C2212yes—
57C2×C4×C22⋊D61C2214yes—
58C2×#14519734C2214yes—
59C2×#14519734C2214yes—
60C2×C42×D61C2214yes—
61C61×#533C22116yescomplemented
62C2×C4⋊Q8×C61C2212yes—
63C2×#125×C61C2211yescharacteristic
64C2×#125×C61C2211yescharacteristic
65C2×C22⋊Q8×C61C2214yes—
66C2×C4×Q8×C61C2214yes—
67C2×#117×C61C2212yes—
68C22×C42×C61C2211yescharacteristic
69C22×Dic122C2312yes—
70C22×Dic122C2314yes—
71C22×D244C2314yes—
72C22×D244C2312yes—
73C22×C22⋊D61C2314yes—
74C22×C4×D61C2314yes—
75C22×C4×D61C2314yes—
76C22×C4×D61C2314yes—
77#14519776C23132yes—
78#14519775C23132yes—
79#14519772C23132yes—
80#14519772C23132yes—
81#14519772C23132yes—
82#14519771C23132yes—
83#14519770C23132yes—
84#14519769C23116yes—
85#14519768C23132yes—
86#14519767C23132yes—
87#14519767C23132yes—
88#14519767C23132yes—
89#14519766C23132yes—
90#14519765C23116yes—
91#14519763C23132yes—
92#14519762C23132yes—
93#14519762C23132yes—
94#14519762C23132yes—
95#14519760C23132yes—
96#14519759C23132yes—
97#14519757C23132yes—
98#14519756C23116yes—
99#14519755C23132yes—
100#14519755C23132yes—
101#14519755C23132yes—
102#14519753C23132yes—
103#14519753C23132yes—
104#14519752C23132yes—
105#14519751C23116yes—
106#14519750C23132yes—
107C4×Dic122C23132yes—
108C4×Dic122C23116yes—
109C4×D244C23132yes—
110C4×D244C23116yes—
111C4×C22⋊D61C23132yes—
112#14519734C23132yes—
113#14519734C23132yes—
114C2×C4⋊Dic61C2314yes—
115C2×C4⋊Dic61C2312yes—
116C2×C4⋊Dic61C2312yes—
117C2×C4⋊Dic61C2314yes—
118C2×#11758196C2314yes—
119C2×#11758196C2314yes—
120C2×#11758196C2314yes—
121C2×#11758196C2314yes—
122C2×#11758196C2314yes—
123C2×#11758196C2314yes—
124C2×#11758196C2314yes—
125C2×#11758196C2314yes—
126C2×C22⋊Dic61C2314yes—
127C2×#11758198C2314yes—
128C2×#11758198C2314yes—
129C2×#11758198C2314yes—
130C2×#11758198C2314yes—
131C2×#11758198C2314yes—
132C2×#11758198C2314yes—
133C2×#11758198C2314yes—
134C2×#11758198C2314yes—
135C2×C4×Dic61C2314yes—
136C2×C4×Dic61C2314yes—
137C2×C4×Dic61C2314yes—
138C42×D61C23132yes—
139C22×Q8×C61C2314yes—
140C23×C4×C61C2311yescharacteristic
141C23×C4×C61C2312yes—
142C4⋊Q8×C61C23116yes—
143#125×C61C2318yes—
144#125×C61C2318yes—
145C22⋊Q8×C61C23132yes—
146C4×Q8×C61C23132yes—
147#117×C61C23116yes—
148C2×C4⋊C4×C61C2312yes—
149C2×C4⋊C4×C61C2314yes—
150C2×C4⋊C4×C61C2312yes—
151C2×C4⋊C4×C61C2314yes—
152C2×C4⋊C4×C61C2314yes—
153C2×C22⋊C4×C61C2314yes—
154C2×C42×C61C2311yescharacteristic
155C2×C42×C61C2318yes—
156C2×C42×C61C2314yes—
157C2×C42×C61C2312yes—
158C2×C42×C61C2311yescharacteristic
159C24×C61C2411yescharacteristic
160C23×D61C2414yes—
161C2×Q8×C61—48no—
162C2×Q8×C61C24116yes—
163C22×C4×C61—24no—
164C22×C4×C61—44no—
165C22×C4×C61C2414yes—
166C22×C4×C61C2414yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2411yescharacteristic
169C22×C4×C61C2412yes—
170C22×C4×C61C2414yes—
171C22×C4×C61C2418yes—
172C22×C4×C61C2411yescharacteristic
173C2×C22⋊D61—48no—
174C2×C22⋊D61C24116yes—
175C22×Dic61C2414yes—
176C22×Dic61C2414yes—
177C22×Dic61C2414yes—
178C2×D244—44no—
179C2×D244—48no—
180C2×D244C24116yes—
181C2×D244C2418yes—
182C2×C4×D61—216no—
183C2×C4×D61—48no—
184C2×C4×D61—48no—
185C2×C4×D61C24116yes—
186C2×C4×D61C24116yes—
187C2×C4×D61C24116yes—
188C2×Dic122—44no—
189C2×Dic122—48no—
190C2×Dic122C2418yes—
191C2×Dic122C24116yes—
192C4⋊C4×C61C2418yes—
193C4⋊C4×C61C2418yes—
194C4⋊C4×C61C24116yes—
195C4⋊C4×C61C24116yes—
196C4⋊C4×C61C24116yes—
197C22⋊C4×C61C24116yes—
198C42×C61C2418yes—
199C42×C61C2414yes—
200C42×C61C2414yes—
201C42×C61C24116yes—
202C22⋊Dic61C24116yes—
203#11758198C24116yes—
204#11758198C24116yes—
205#11758198C24116yes—
206#11758198C24116yes—
207#11758198C24116yes—
208#11758198C24116yes—
209#11758198C24116yes—
210#11758198C24116yes—
211C4⋊Dic61C24116yes—
212C4⋊Dic61C2418yes—
213C4⋊Dic61C2418yes—
214C4⋊Dic61C24116yes—
215#11758196C24116yes—
216#11758196C24116yes—
217#11758196C24116yes—
218#11758196C24116yes—
219#11758196C24116yes—
220#11758196C24116yes—
221#11758196C24116yes—
222#11758196C24116yes—
223C4×Dic61C24116yes—
224C4×Dic61C24116yes—
225C4×Dic61C24116yes—
226C23×C61—22no—
227C23×C61—24no—
228C23×C61C2512yes—
229C23×C61C2511yescharacteristic
230C22×D61—28no—
231C22×D61—28no—
232C22×D61—28no—
233C22×D61C2518yes—
234Q8×C61—416no—
235C2×C4×C61—22no—
236C2×C4×C61—28no—
237C2×C4×C61—24no—
238C2×C4×C61—28no—
239C2×C4×C61—24no—
240C2×C4×C61—28no—
241C2×C4×C61—48no—
242C2×C4×C612−1+414yes—
243C2×C4×C61C2518yes—
244C2×C4×C61C2518yes—
245C2×C4×C61C2512yes—
246C2×C4×C61C2514yes—
247C2×C4×C61C2514yes—
248C2×C4×C61C2512yes—
249C22⋊D61—416no—
250C2×Dic61—28no—
251C2×Dic61—28no—
252C2×Dic61—28no—
253C2×Dic61C2518yes—
254C2×Dic61C2518yes—
255C2×Dic61C2518yes—
256D244—48no—
257D244—416no—
258C4×D61—232no—
259C4×D61—416no—
260C4×D61—416no—
261Dic122—48no—
262Dic122—416no—
263C2×#3531—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—44no—
267C22×C61C2×2−1+412yes—
268C22×C61#58514yes—
269C22×C61C2611yescharacteristic
270C2×D61—28no—
271C2×D61—28no—
272C2×D61—28no—
273C2×D61—416no—
274C4×C61—22no—
275C4×C61—28no—
276C4×C61—24no—
277C4×C61—28no—
278C4×C61—24no—
279C4×C61C2×2−1+414yes—
280Dic61—28no—
281Dic61—28no—
282Dic61—28no—
283#3531—6132no—
284C2×#564—611no—
285C2×#563—614no—
286C2×#563—612no—
287C2×#563—614no—
288C2×#562—611no—
289C2×#562—614no—
290C2×#533—614no—
291C2×#533D6111yescharacteristic, complemented
292C2×#529—612no—
293C2×#529—614no—
294C2×#529—614no—
295C2×C61—42no—
296C2×C61#353114yescomplemented
297C2×C61C22×2−1+411yescharacteristic
298C2×C61C2×#58512yes—
299D61—48no—
300#564—6116no—
301#563—6164no—
302#563—6132no—
303#563—6164no—
304#562—6116no—
305#562—6164no—
306#533—6164no—
307#533C2×D61116yescomplemented
308C2×C4⋊Q8—614no—
309C2×C4⋊Q8—614no—
310C2×C4⋊Q8C2×D6112yes—
311#529—6164no—
312#529—6164no—
313#529—6132no—
314C2×#126—614no—
315C2×#126—614no—
316C2×#126—614no—
317C2×#126—614no—
318C2×#126—614no—
319C2×#125—614no—
320C2×#125—614no—
321C2×#125—614no—
322C2×#125—614no—
323C2×#125—612no—
324C2×#125C2×D6111yescharacteristic
325C2×#125C2×D6111yescharacteristic
326C2×#124—614no—
327C2×#124—614no—
328C2×#124—612no—
329C2×#123—614no—
330C2×#123—614no—
331C2×#123—614no—
332C2×#123—614no—
333C2×#123—614no—
334C2×C22⋊Q8—614no—
335C2×C22⋊Q8—614no—
336C2×C22⋊Q8—612no—
337C2×C22⋊Q8—614no—
338C2×C22⋊Q8—614no—
339C2×C22⋊Q8—614no—
340C2×C22⋊Q8—614no—
341C2×C22⋊Q8C2×D6114yes—
342C2×#121—612no—
343C2×#121—614no—
344C2×#121—614no—
345C2×C4×Q8—612no—
346C2×C4×Q8—614no—
347C2×C4×Q8C2×D6114yes—
348C2×C4×D4—614no—
349C2×C4×D4—612no—
350C2×C4×D4—614no—
351C2×#117—614no—
352C2×#117—614no—
353C2×#117C2×D6112yes—
354C22×C42—614no—
355C22×C42C2×D6111yescharacteristic
356C61C2×#353111yescharacteristic, complemented, Sylow
357C22×Q8—612no—
358C22×Q8—614no—
359C22×Q8C22×D6114yes—
360C22×D4—614no—
361C22×D4—614no—
362C22×D4—612no—
363C23×C4—614no—
364C23×C4—614no—
365C23×C4—614no—
366C23×C4C22×D6111yescharacteristic
367C23×C4C22×D6112yes—
368C4⋊Q8—6132no—
369C4⋊Q8—6132no—
370C4⋊Q8C22×D61116yes—
371#126—6132no—
372#126—6132no—
373#126—6132no—
374#126—6132no—
375#126—6132no—
376#125—6132no—
377#125—6132no—
378#125—6132no—
379#125—6132no—
380#125—6116no—
381#125C22×D6118yes—
382#125C22×D6118yes—
383#124—6132no—
384#124—6132no—
385#124—6116no—
386#123—6132no—
387#123—6132no—
388#123—6132no—
389#123—6132no—
390#123—6132no—
391C22⋊Q8—6132no—
392C22⋊Q8—6132no—
393C22⋊Q8—6116no—
394C22⋊Q8—6132no—
395C22⋊Q8—6132no—
396C22⋊Q8—6132no—
397C22⋊Q8—6132no—
398C22⋊Q8C22×D61132yes—
399#121—6116no—
400#121—6132no—
401#121—6132no—
402C4×Q8—6132no—
403C4×Q8—6116no—
404C4×Q8C22×D61132yes—
405C4×D4—6132no—
406C4×D4—6132no—
407C4×D4—6116no—
408#117—6132no—
409#117—6132no—
410#117C22×D61116yes—
411C2×C4⋊C4—614no—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—614no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—614no—
421C2×C4⋊C4—614no—
422C2×C4⋊C4—614no—
423C2×C4⋊C4C22×D6112yes—
424C2×C4⋊C4C22×D6112yes—
425C2×C4⋊C4C22×D6114yes—
426C2×C4⋊C4C22×D6114yes—
427C2×C4⋊C4C22×D6114yes—
428C2×C22⋊C4—614no—
429C2×C22⋊C4—614no—
430C2×C22⋊C4—614no—
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⋊C4C22×D6114yes—
438C2×C42—614no—
439C2×C42—614no—
440C2×C42—614no—
441C2×C42—6132no—
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6118yes—
445C2×C42C22×D6112yes—
446C2×C42C22×D6114yes—
447C24—614no—
448C24C23×D6111yescharacteristic
449C2×Q8—48no—
450C2×Q8—618no—
451C2×Q8—6116no—
452C2×Q8—2444no—
453C2×Q8—2448no—
454C2×Q8C23×D61116yes—
455C2×D4—6116no—
456C2×D4—6116no—
457C2×D4—618no—
458C2×D4—2444no—
459C2×D4—2448no—
460C2×D4—2448no—
461C22×C4—24no—
462C22×C4—44no—
463C22×C4—614no—
464C22×C4—614no—
465C22×C4—614no—
466C22×C4—6116no—
467C22×C4—6116no—
468C22×C4—6116no—
469C22×C4—12216no—
470C22×C4—2448no—
471C22×C4—2448no—
472C22×C4C23×D6114yes—
473C22×C4C23×D6114yes—
474C22×C4C23×D6111yescharacteristic
475C22×C4C23×D6112yes—
476C22×C4C23×D6112yes—
477C22×C4C23×D6114yes—
478C22×C4C23×D6118yes—
479C22×C4C23×D6111yescharacteristic
480C4⋊C4—6116no—
481C4⋊C4—6116no—
482C4⋊C4—6116no—
483C4⋊C4—6116no—
484C4⋊C4—6116no—
485C4⋊C4—6116no—
486C4⋊C4—6116no—
487C4⋊C4—6116no—
488C4⋊C4—6116no—
489C4⋊C4—618no—
490C4⋊C4—618no—
491C4⋊C4—6116no—
492C4⋊C4C23×D6118yes—
493C4⋊C4C23×D6118yes—
494C4⋊C4C23×D61116yes—
495C4⋊C4C23×D61116yes—
496C4⋊C4C23×D61116yes—
497C22⋊C4—6116no—
498C22⋊C4—6116no—
499C22⋊C4—6116no—
500C22⋊C4—6116no—
501C22⋊C4—6116no—
502C22⋊C4—6116no—
503C22⋊C4—6116no—
504C22⋊C4—6116no—
505C22⋊C4—6116no—
506C22⋊C4C23×D61116yes—
507C42—6116no—
508C42—6116no—
509C42—6116no—
510C42C23×D61116yes—
511C42C23×D6118yes—
512C42C23×D6114yes—
513C42C23×D6114yes—
514C23—22no—
515C23—24no—
516C23—618no—
517C23—1228no—
518C23—1228no—
519C23—1228no—
520C23C24×D6112yes—
521C23C24×D6111yescharacteristic
522Q8—416no—
523Q8—2448no—
524Q8—24416no—
525D4—2448no—
526D4—24416no—
527D4—24416no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—48no—
535C2×C4—618no—
536C2×C4—618no—
537C2×C4—618no—
538C2×C4—1228no—
539C2×C4—1228no—
540C2×C4—1228no—
541C2×C4—12232no—
542C2×C4—24416no—
543C2×C4—24416no—
544C2×C4#1451982514yes—
545C2×C4C24×D6118yes—
546C2×C4C24×D6118yes—
547C2×C4C24×D6112yes—
548C2×C4C24×D6114yes—
549C2×C4C24×D6114yes—
550C2×C4C24×D6112yes—
551C22—22no—
552C22—24no—
553C22—44no—
554C22—1228no—
555C22—1228no—
556C22—1228no—
557C22—24416no—
558C22C2×#1451982512yes—
559C22#1684059814yes—
560C22C25×D6111yescharacteristic
561C4—22no—
562C4—24no—
563C4—24no—
564C4—28no—
565C4—28no—
566C4—1228no—
567C4—1228no—
568C4—1228no—
569C4C2×#1451982514yes—
570C2—42no—
571C2—2448no—
572C2#2145996614yesdirect product factor
573C2C22×#1451982511yescharacteristic
574C2C2×#1684059812yes—
subgroupG/H
centerC22×C4C23×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×C61×#533C2
socleC23×C61C25
hypercenterC2×#533D61
hypocenterC61C2×#3531
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. #21459966C2direct (N × H)
2. C2×C61⋊#564C2semidirect
3. C2×#16840516C2semidirect
4. C2×#16840513C2semidirect
5. C2×#16840511C2semidirect
6. C2×#16840510C2semidirect
7. C2×#16840508C2semidirect
8. C2×#16840316C2semidirect
9. C2×#16840299C2semidirect
10. C2×#16840296C2semidirect
11. C2×#16840290C2semidirect
12. C2×C61×#533C2semidirect
13. C61⋊#564C22semidirect
14. #16840516C22semidirect
15. #16840513C22semidirect
16. #16840511C22semidirect
17. #16840510C22semidirect
18. #16840508C22semidirect
19. #16840316C22semidirect
22. #16840299C22semidirect
23. #16840296C22semidirect
24. #16840290C22semidirect
61. C61×#533C22semidirect
291. C2×#533D61semidirect
296. C2×C61#3531semidirect
307. #533C2×D61semidirect
356. C61C2×#3531semidirect
572. C2#21459966direct (N × H)

Maximal subgroups

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

subgroupindex
#214599662
C2×C61⋊#5642
C2×#168405162
C2×#168405132
C2×#168405112
C2×#168405102
C2×#168405082
C2×#168403162
C2×#168402992
C2×#168402962
C2×#168402902
C2×C61×#5332
C2×#353161