Finite Group Databasebeta

← C2×#21457143 #30552804

Subgroups — C2×#21457143 #30552804

Order 15616.

2210 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic, complemented
2C2×#16840299C211yesmaximal, characteristic, complemented
3#21457143C2116yesmaximal, direct product factor
4C2×#16840057C211yesmaximal, characteristic, complemented
5C2×#16840064C212yesmaximal, complemented
6C2×#16840053C211yesmaximal, characteristic, complemented
7C2×#16839997C212yesmaximal, complemented
8C2×#16839988C212yesmaximal, complemented
9C2×#16839979C212yesmaximal, complemented
10C2×#16839954C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
12C2×C61×#460C211yesmaximal, characteristic, complemented
13C22×#11758224—21no—
14C22×#11758199C2212yes—
15C22×C8×D61C2212yes—
16#16840310—24no—
17#16840310C2218yescomplemented
18C2×#14519777C2211yescharacteristic
19#16840299—24no—
20#16840299C2218yescomplemented
21C2×#14519772—21no—
22C2×#14519768C2211yescharacteristic
23C2×#14519766C2211yescharacteristic
24C2×#14519764—21no—
25C2×#14519762—21no—
26C2×#14519753C2211yescharacteristic
27C2×#14519752C2211yescharacteristic
28C2×#14519748—21no—
29C2×#14519748C2212yes—
30C2×C4×D244—21no—
31C2×C4×D244C2211yescharacteristic
32C2×D4×Dic61—21no—
33C2×D4×Dic61C2212yes—
34C2×#14519734—21no—
35C2×C42×D61C2211yescharacteristic
36#16840064C22116yescomplemented
37#16840057C2218yescomplemented
38#16840053C2218yescomplemented
39#16839997C22116yescomplemented
40#16839988C22116yescomplemented
41#16839979C22116yescomplemented
42#16839954C2218yescomplemented
43C2×#14519710C2212yes—
44C2×#14519709C2212yes—
45C2×C4⋊#92418C2211yescharacteristic
46C2×#14519706C2212yes—
47C4⋊C8×D61C2218yescomplemented
48C2×#14519701C2212yes—
49C2×#14519699C2212yes—
50C2×#14519698C2212yes—
51C22×SD16×C61C2212yes—
52C2×#127×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C61×#460C2218yescomplemented
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2212yes—
57C2×#11758224—28no—
58C2×#11758224—48no—
59C22×Dic122—21no—
60C22×D244—21no—
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—22no—
63C23×Dic61—22no—
64C22×C4×D61—21no—
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C2×#11758199—44no—
69C2×#11758199C2318yes—
70C22×#92418C2312yes—
71C2×C8×D61—44no—
72C2×C8×D61C2318yes—
73#14519777—22no—
74#14519777C2314yes—
75#14519772—28no—
76#14519768—22no—
77#14519768C2314yes—
78#14519766—22no—
79#14519766C2314yes—
80#14519764—28no—
81#14519762—28no—
82#14519753—22no—
83#14519753C2314yes—
84#14519752—22no—
85#14519752C2314yes—
86#14519748—28no—
87#14519748—24no—
88#14519748C2318yes—
89C4×D244—28no—
90C4×D244—22no—
91C4×D244C2314yes—
92D4×Dic61—28no—
93D4×Dic61—24no—
94D4×Dic61C2318yes—
95#14519734—28no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61C2312yes—
99C2×C4⋊Dic61C2311yescharacteristic
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×C22⋊Dic61—22no—
103C2×C22⋊Dic61—22no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×#11758198—21no—
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C42×D61D414yes—
113C42×D61C2314yes—
114#14519710C2318yes—
115#14519709C2318yes—
116C4⋊#92418C2314yes—
117#14519706C2318yes—
118#14519701C2318yes—
119#14519699C2318yes—
120#14519698C2318yes—
121C22×Q8×C61C2311yescharacteristic
122C22×D4×C61—21no—
123C22×D4×C61—21no—
124C22×D4×C61C2312yes—
125C2×SD16×C61—44no—
126C2×SD16×C61C2318yes—
127C22×C8×C61C2312yes—
128#127×C61—22no—
129#127×C61C2314yes—
130C4×Q8×C61—22no—
131C4×Q8×C61C2314yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2314yes—
137#102×C61C2318yes—
138C24×C61—22no—
139C23×D61—21no—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—22no—
143C2×Q8×C61—22no—
144C2×Q8×C61C2412yes—
145C2×D4×C61—24no—
146C2×D4×C61—22no—
147C2×D4×C61—24no—
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—44no—
151C2×D4×C61—44no—
152C2×D4×C61C2414yes—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—28no—
159C2×C22⋊D61—48no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162C22×Dic61—28no—
163C22×Dic61—48no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167#11758224—432no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—24no—
171C2×D244—21no—
172C2×D244—44no—
173C2×D244C2412yes—
174C2×C4×D61—24no—
175C2×C4×D61—24no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—24no—
179C2×C4×D61—44no—
180C2×C4×D61C2×D414yes—
181C2×C4×D61C2×D414yes—
182C2×C4×D61C2×D412yes—
183C2×C4×D61C2412yes—
184C2×Dic122—24no—
185C2×Dic122—44no—
186SD16×C61—48no—
187C2×C8×C61—24no—
188C2×C8×C61C2414yes—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C2412yes—
192C42×C61—24no—
193C42×C61C2×D412yes—
194C42×C61C2412yes—
195C22⋊Dic61—28no—
196C22⋊Dic61—28no—
197#11758199—48no—
198#11758198—24no—
199#11758198—24no—
200#11758198—24no—
201#11758198—24no—
202#11758198—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—24no—
207C4⋊Dic61C2412yes—
208C4⋊Dic61C2414yes—
209#11758196—24no—
210#11758196—24no—
211C4×Dic61C2×D412yes—
212C4×Dic61C2×D414yes—
213C4×Dic61C2×D414yes—
214C4×Dic61C2412yes—
215C2×#92418—24no—
216C2×#92418C2414yes—
217C8×D61—48no—
218C23×C61—24no—
219C23×C61—44no—
220C23×C61—44no—
221C23×C61—44no—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61AGL(1,ℤ/8)14yes—
230C22×D61C22×D412yes—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—42no—
234D4×C61—24no—
235D4×C61—24no—
236D4×C61—48no—
237D4×C61—44no—
238D4×C61—48no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61#13514yes—
245C2×C4×C61C22×D412yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D412yes—
248C2×C4×C61H(5,𝔽2)12yes—
249C2×C4×C612−1+412yes—
250C2×C4×C61C2511yescharacteristic
251C22⋊D61—416no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—24no—
257C2×Dic61—416no—
258C2×Dic61—42no—
259C2×Dic61—44no—
260C2×Dic61C22×D412yes—
261C2×Dic61C22×D412yes—
262C2×Dic61C22×D412yes—
263D244—22no—
264D244—22no—
265D244—48no—
266D244—42no—
267C4×D61—28no—
268C4×D61—24no—
269C4×D61—28no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—48no—
273Dic122—48no—
274C8×C61—24no—
275#92418—24no—
276C2×#3247—611nomaximal, Sylow
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—88no—
281C22×C61#53712yes—
282C22×C61C2×#13512yes—
283C22×C61C2×AGL(1,ℤ/8)12yes—
284C22×C61C23×D411yescharacteristic
285C2×D61—22no—
286C2×D61—28no—
287C2×D61—22no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—84no—
292C2×D61C2×AGL(1,ℤ/8)14yes—
293C4×C61—22no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61#53712yes—
297C4×C61C2×#13514yes—
298C4×C61C2×H(5,𝔽2)11yescharacteristic
299C4×C61C2×2−1+411yescharacteristic
300Dic61—22no—
301Dic61—22no—
302Dic61—22no—
303Dic61—42no—
304Dic61—44no—
305C2×#532—611no—
306C2×#529—611no—
307#3247—6116no—
308C2×#477—611no—
309C2×#478—612no—
310C2×#476—611no—
311C2×#466—612no—
312C2×#465—612no—
313C2×#460D6111yescharacteristic, complemented
314C2×#463—612no—
315C2×#459—611no—
316C22×C4⋊C8—611no—
317C2×C61—84no—
318C2×C61#324714yescomplemented
319C2×C61C2×#53711yescharacteristic
320C2×C61C22×#13511yescharacteristic
321C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
322D61—24no—
323D61—82no—
324C22×#41—1221no—
325C22×SD16C2×D6112yes—
326C22×D8—612no—
327C23×C8—612no—
328#532—618no—
329#532—1224no—
330C2×C4⋊Q8—611no—
331C2×#127C2×D6111yescharacteristic
332#529—618no—
333#529—1224no—
334C2×#126—1221no—
335C2×#125—611no—
336C2×#124—611no—
337C2×#124—1221no—
338C2×#123—1221no—
339C2×C22⋊Q8—611no—
340C2×#121—612no—
341C2×#121—611no—
342C2×#121—1221no—
343C2×C4×Q8C2×D6111yescharacteristic
344C2×C4×D4—612no—
345C2×C4×D4—611no—
346C2×C4×D4—1221no—
347C2×C4×D4—1221no—
348C2×#117—1221no—
349C22×C42—611no—
350#478—6116no—
351#477—618no—
352#476—618no—
353#466—6116no—
354#465—6116no—
355#463—6116no—
356#460C2×D6118yescomplemented
357#459—618no—
358C2×#107—612no—
359C2×#106—612no—
360C2×C4⋊C8—612no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—618no—
363C2×C4⋊C8C2×D6111yescharacteristic
364C2×#103—612no—
365C2×#102—612no—
366C2×#102—612no—
367C2×#102C2×D6112yes—
368C61C2×#324711yescharacteristic, complemented, Sylow
369C2×#41—1228no—
370C2×#41—2448no—
371C22×Q8—1221no—
372C22×Q8C22×D6111yescharacteristic
373C22×D4—21no—
374C22×D4—21no—
375C22×D4—611no—
376C22×D4—1221no—
377C22×D4—1222no—
378C22×D4C22×D6112yes—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—1222no—
383C23×C4—1221no—
384C2×SD16—44no—
385C2×SD16C22×D6118yes—
386C2×D8—618no—
387C2×D8—2444no—
388C22×C8—612no—
389C22×C8—618no—
390C22×C8—2444no—
391C22×C8C22×D6112yes—
392C4⋊Q8—614no—
393C4⋊Q8—1222no—
394#127—22no—
395#127C22×D6114yes—
396#126—1228no—
397#125—614no—
398#125—1222no—
399#124—614no—
400#124—1228no—
401#124—1222no—
402#123—1228no—
403C22⋊Q8—614no—
404C22⋊Q8—1222no—
405#121—618no—
406#121—614no—
407#121—1222no—
408#121—1224no—
409#121—1228no—
410C4×Q8—22no—
411C4×Q8C22×D6114yes—
412C4×D4—618no—
413C4×D4—614no—
414C4×D4—1228no—
415C4×D4—1224no—
416C4×D4—1228no—
417C4×D4—1222no—
418#117—1228no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—612no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1222no—
432C2×C22⋊C4—1222no—
433C2×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—614no—
439C2×C42—614no—
440C2×C42C22×D6111yescharacteristic
441#107—618no—
442#106—618no—
443C4⋊C8—618no—
444C4⋊C8—614no—
445C4⋊C8C22×D6114yes—
446#103—618no—
447#102—618no—
448#102—618no—
449#102C22×D6118yes—
450C24—22no—
451C24—611no—
452C24—1221no—
453#41—24432no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—1224no—
458C2×Q8—2444no—
459C2×Q8C23×D6112yes—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—24no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—612no—
468C2×D4—1224no—
469C2×D4—1228no—
470C2×D4—1221no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—2444no—
474C2×D4—2448no—
475C2×D4C23×D6114yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—614no—
482C22×C4—612no—
483C22×C4—614no—
484C22×C4—612no—
485C22×C4—1224no—
486C22×C4—1228no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1224no—
493C22×C4—2448no—
494C22×C4—2444no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—48no—
499D8—2448no—
500C2×C8—24no—
501C2×C8—614no—
502C2×C8—1224no—
503C2×C8—2448no—
504C2×C8C23×D6114yes—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—612no—
508C4⋊C4—614no—
509C4⋊C4—1224no—
510C4⋊C4—1224no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1224no—
515C4⋊C4C23×D6112yes—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1228no—
521C22⋊C4—1228no—
522C22⋊C4—1224no—
523C42—24no—
524C42—614no—
525C42—612no—
526C42—614no—
527C42—612no—
528C42D4×D6112yes—
529C42C23×D6112yes—
530C23—24no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—614no—
535C23—612no—
536C23—1222no—
537C23—1222no—
538C23—1222no—
539C23—2442no—
540C23—2442no—
541C23—2442no—
542C23C2×D4×D6111yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—42no—
546Q8—2448no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—48no—
551D4—44no—
552D4—1222no—
553D4—1222no—
554D4—2448no—
555D4—24416no—
556D4—2442no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—1228no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1228no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—24416no—
576C2×C4—2442no—
577C2×C4—2444no—
578C2×C4—2448no—
579C2×C4#1451979814yes—
580C2×C4C2×D4×D6112yes—
581C2×C4C2×D4×D6112yes—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4#1451982212yes—
584C2×C4#1451982412yes—
585C2×C4C24×D6111yescharacteristic
586C8—24no—
587C8—1224no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—88no—
592C22—614no—
593C22—1222no—
594C22—1222no—
595C22—1228no—
596C22—4884no—
597C22—2442no—
598C22—2442no—
599C22—2442no—
600C22#1684034112yes—
601C22C2×#1451979812yes—
602C22AGL(1,ℤ/8)×D6112yes—
603C22C22×D4×D6111yescharacteristic
604C4—22no—
605C4—42no—
606C4—42no—
607C4—1222no—
608C4—1222no—
609C4—1222no—
610C4—2442no—
611C4—2444no—
612C4#1684034112yes—
613C4C2×#1451979814yes—
614C4C2×#1451982211yescharacteristic
615C4C2×#1451982411yescharacteristic
616C2—84no—
617C2—1224no—
618C2—4882no—
619C2#2145714314yesdirect product factor
620C2C2×#1684034111yescharacteristic
621C2C22×#1451979811yescharacteristic
622C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#460C2
socleC23×C61C22×D4
hypercenterC2×#460D61
hypocenterC61C2×#3247
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×#16840310C2semidirect
2. C2×#16840299C2semidirect
3. #21457143C2direct (N × H)
4. C2×#16840057C2semidirect
5. C2×#16840064C2semidirect
6. C2×#16840053C2semidirect
7. C2×#16839997C2semidirect
8. C2×#16839988C2semidirect
9. C2×#16839979C2semidirect
10. C2×#16839954C2semidirect
11. C2×C4⋊C8×D61C2semidirect
12. C2×C61×#460C2semidirect
17. #16840310C22semidirect
20. #16840299C22semidirect
36. #16840064C22semidirect
37. #16840057C22semidirect
38. #16840053C22semidirect
39. #16839997C22semidirect
40. #16839988C22semidirect
41. #16839979C22semidirect
42. #16839954C22semidirect
47. C4⋊C8×D61C22semidirect
54. C61×#460C22semidirect
313. C2×#460D61semidirect
318. C2×C61#3247semidirect
356. #460C2×D61semidirect
368. C61C2×#3247semidirect
619. C2#21457143direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403102
C2×#168402992
#214571432
C2×#168400572
C2×#168400642
C2×#168400532
C2×#168399972
C2×#168399882
C2×#168399792
C2×#168399542
C2×C4⋊C8×D612
C2×C61×#4602
C2×#324761