Finite Group Databasebeta

← C2×#21457159 #30552821

Subgroups — C2×#21457159 #30552821

Order 15616.

1930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840299C211yesmaximal, characteristic
3#21457159C2116yesmaximal, direct product factor
4C2×#16840072C212yesmaximal, complemented
5C2×#16840039C211yesmaximal, characteristic, complemented
6C2×#16840002C212yesmaximal, complemented
7C2×#16839992C212yesmaximal
8C2×#16839987C212yesmaximal, complemented
9C2×#16839972C211yesmaximal, characteristic, complemented
10C2×#16839968C211yesmaximal, characteristic, complemented
11C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
12C2×C61×#475C211yesmaximal, characteristic, complemented
13C22×#11758202C2212yes—
14C22×#11758190C2212yes—
15C22×C8×D61C2212yes—
16#16840317—24no—
17#16840317C2218yescomplemented
18C2×#14519777C2211yescharacteristic
19#16840299—24no—
20#16840299C2218yes—
21C2×#14519772—21no—
22C2×#14519768—21no—
23C2×#14519768C2211yescharacteristic
24C2×#14519766C2211yescharacteristic
25C2×#14519762—21no—
26C2×#14519757C2212yes—
27C2×#14519753—21no—
28C2×#14519753C2211yescharacteristic
29C2×#14519752C2211yescharacteristic
30C2×Q8×Dic61—21no—
31C2×#14519742C2212yes—
32C2×C4×D244—21no—
33C2×C4×D244C2211yescharacteristic
34C2×#14519735—21no—
35C2×#14519734—21no—
36C2×C42×D61C2211yescharacteristic
37#16840072C22116yescomplemented
38#16840039C2218yescomplemented
39#16840002C22116yescomplemented
40#16839992C22116yes—
41#16839987C22116yescomplemented
42#16839972C2218yescomplemented
43#16839968C2218yescomplemented
44C2×#14519710C2212yes—
45C2×C4⋊#92418C2211yescharacteristic
46C2×#14519706C2212yes—
47C4⋊C8×D61C2218yescomplemented
48C2×#14519702C2212yes—
49C2×#14519700C2212yes—
50C2×#14519697C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#475C2218yescomplemented
54C2×#106×C61C2212yes—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2212yes—
57C22×Dic122C2312yes—
58C22×D244—21no—
59C22×D244C2311yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758202—44no—
65C2×#11758202C2318yes—
66C2×#11758190—44no—
67C2×#11758190C2318yes—
68C22×#92418C2312yes—
69C2×C8×D61—44no—
70C2×C8×D61C2318yes—
71#14519777—22no—
72#14519777C2314yes—
73#14519772—28no—
74#14519768—28no—
75#14519768—22no—
76#14519768C2314yes—
77#14519766—22no—
78#14519766C2314yes—
79#14519762—28no—
80#14519757—24no—
81#14519757C2318yes—
82#14519753—28no—
83#14519753—22no—
84#14519753C2314yes—
85#14519752—22no—
86#14519752C2314yes—
87Q8×Dic61—28no—
88#14519742—24no—
89#14519742C2318yes—
90C4×D244—28no—
91C4×D244—22no—
92C4×D244C2314yes—
93#14519735—28no—
94#14519734—28no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61—22no—
99C2×C4⋊Dic61C2311yescharacteristic
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—22no—
103C2×#11758198—22no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×#11758198—21no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61D414yes—
114C42×D61C2314yes—
115#14519710C2318yes—
116C4⋊#92418C2314yes—
117#14519706C2318yes—
118#14519702C2318yes—
119#14519700C2318yes—
120#14519697C2318yes—
121C22×Q8×C61—21no—
122C22×Q8×C61C2311yescharacteristic
123C22×C8×C61C2312yes—
124C4⋊Q8×C61—22no—
125C4⋊Q8×C61C2314yes—
126C4×Q8×C61—22no—
127C4×Q8×C61C2314yes—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61C2312yes—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#106×C61C2318yes—
135C4⋊C8×C61C2314yes—
136#103×C61C2318yes—
137C23×D61—21no—
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—24no—
141C2×Q8×C61—22no—
142C2×Q8×C61—22no—
143C2×Q8×C61—44no—
144C2×Q8×C61C2412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—24no—
159C2×D244—21no—
160C2×D244—44no—
161C2×D244C2412yes—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—24no—
167C2×C4×D61—44no—
168C2×C4×D61C2×D414yes—
169C2×C4×D61C2×D414yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2412yes—
172C2×Dic122—24no—
173C2×Dic122—24no—
174C2×Dic122—22no—
175C2×Dic122C2414yes—
176C2×C8×C61—24no—
177C2×C8×C61C2414yes—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C2414yes—
183C4⋊C4×C61C2412yes—
184C42×C61—24no—
185C42×C61C2×D412yes—
186C42×C61C2412yes—
187#11758202—48no—
188#11758198—28no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—24no—
195C4⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—24no—
198C4⋊Dic61—28no—
199C4⋊Dic61C2412yes—
200#11758196—24no—
201#11758196—24no—
202#11758196—28no—
203C4×Dic61—28no—
204C4×Dic61C2×D412yes—
205C4×Dic61C2×D414yes—
206C4×Dic61C2×D414yes—
207C4×Dic61C2412yes—
208C2×#92418—24no—
209C2×#92418C2414yes—
210#11758190—48no—
211C8×D61—48no—
212C23×C61C22×D411yescharacteristic
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61#13714yes—
220C22×D61C22×D412yes—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—42no—
224Q8×C61—48no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—42no—
230C2×C4×C61—44no—
231C2×C4×C61—42no—
232C2×C4×C61#13514yes—
233C2×C4×C61C22×D412yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D412yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C2×C4×C612−1+412yes—
238C2×C4×C61C2511yescharacteristic
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—44no—
245C2×Dic61—42no—
246C2×Dic61C22×D412yes—
247C2×Dic61C22×D412yes—
248C2×Dic61C22×D412yes—
249D244—22no—
250D244—22no—
251D244—48no—
252D244—42no—
253C4×D61—28no—
254C4×D61—24no—
255C4×D61—28no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—48no—
259Dic122—24no—
260Dic122—24no—
261Dic122—44no—
262C8×C61—24no—
263#92418—24no—
264C2×#3248—611nomaximal, Sylow
265C22×C61#53712yes—
266C22×C61C2×#13512yes—
267C22×C61C2×#13712yes—
268C22×C61C23×D411yescharacteristic
269C2×D61—22no—
270C2×D61—28no—
271C2×D61—22no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—84no—
276C2×D61C2×#13714yes—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—44no—
280C4×C61—42no—
281C4×C61#53712yes—
282C4×C61C2×#13514yes—
283C4×C61C2×H(5,𝔽2)11yescharacteristic
284C4×C61C2×2−1+411yescharacteristic
285Dic61—22no—
286Dic61—22no—
287Dic61—22no—
288Dic61—44no—
289Dic61—42no—
290C2×#533—611no—
291C2×#529—611no—
292#3248—6116no—
293C2×#479—612no—
294C2×#475D6111yescharacteristic, complemented
295C2×#474—611no—
296C2×#467—612no—
297C2×#465—612no—
298C2×#464—612no—
299C2×#462—611no—
300C2×#461—611no—
301C22×C4⋊C8—611no—
302C2×C61#324814yescomplemented
303C2×C61C2×#53711yescharacteristic
304C2×C61C22×#13511yescharacteristic
305C2×C61C22×#13711yescharacteristic
306D61—24no—
307D61—82no—
308C22×Q16—612no—
309C22×SD16—612no—
310C23×C8—612no—
311#533—618no—
312#533—1224no—
313C2×C4⋊Q8—611no—
314C2×C4⋊Q8C2×D6111yescharacteristic
315#529—618no—
316#529—1224no—
317C2×#126—1221no—
318C2×#125—611no—
319C2×#125—1221no—
320C2×#124—611no—
321C2×#123—1221no—
322C2×C22⋊Q8—611no—
323C2×C22⋊Q8—612no—
324C2×C22⋊Q8—1221no—
325C2×#121—611no—
326C2×C4×Q8—612no—
327C2×C4×Q8—1221no—
328C2×C4×Q8C2×D6111yescharacteristic
329C2×C4×D4—611no—
330C2×C4×D4—1221no—
331C2×#117—1221no—
332C2×#117—1221no—
333C22×C42—611no—
334#479—6116no—
335#475C2×D6118yescomplemented
336#474—618no—
337#467—6116no—
338#465—6116no—
339#464—6116no—
340#462—618no—
341#461—618no—
342C2×#107—612no—
343C2×#106C2×D6112yes—
344C2×C4⋊C8—612no—
345C2×C4⋊C8—611no—
346C2×C4⋊C8—618no—
347C2×C4⋊C8C2×D6111yescharacteristic
348C2×#103—612no—
349C2×#103—612no—
350C2×#103C2×D6112yes—
351C2×#102—612no—
352C61C2×#324811yescharacteristic, complemented, Sylow
353C22×Q8—21no—
354C22×Q8—612no—
355C22×Q8C22×D6111yescharacteristic
356C22×D4—611no—
357C22×D4—1221no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—1221no—
362C2×Q16—618no—
363C2×Q16—2444no—
364C2×SD16—618no—
365C2×SD16—2444no—
366C22×C8—612no—
367C22×C8—618no—
368C22×C8—2444no—
369C22×C8C22×D6112yes—
370C4⋊Q8—22no—
371C4⋊Q8—614no—
372C4⋊Q8—1222no—
373C4⋊Q8C22×D6114yes—
374#126—1228no—
375#125—614no—
376#125—1222no—
377#125—1228no—
378#124—614no—
379#124—1222no—
380#123—1228no—
381C22⋊Q8—614no—
382C22⋊Q8—618no—
383C22⋊Q8—1224no—
384C22⋊Q8—1222no—
385C22⋊Q8—1228no—
386#121—614no—
387#121—1222no—
388C4×Q8—22no—
389C4×Q8—618no—
390C4×Q8—1224no—
391C4×Q8—1228no—
392C4×Q8C22×D6114yes—
393C4×D4—614no—
394C4×D4—1228no—
395C4×D4—1222no—
396#117—1228no—
397#117—1228no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4C22×D6112yes—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—21no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—614no—
421C2×C42—614no—
422C2×C42—1222no—
423C2×C42C22×D6111yescharacteristic
424#107—618no—
425#106C22×D6118yes—
426C4⋊C8—618no—
427C4⋊C8—614no—
428C4⋊C8C22×D6114yes—
429#103—618no—
430#103—618no—
431#103C22×D6118yes—
432#102—618no—
433C24—611no—
434C24—1221no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—614no—
441C2×Q8—1222no—
442C2×Q8—1224no—
443C2×Q8—1224no—
444C2×Q8C23×D6112yes—
445C2×D4—612no—
446C2×D4—1224no—
447C2×D4—1221no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—2444no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—614no—
458C22×C4—614no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—1224no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1224no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1224no—
468C22×C4—2444no—
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472Q16—2448no—
473SD16—2448no—
474C2×C8—24no—
475C2×C8—614no—
476C2×C8—1224no—
477C2×C8—2448no—
478C2×C8C23×D6114yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—612no—
484C4⋊C4—1224no—
485C4⋊C4—1224no—
486C4⋊C4—1228no—
487C4⋊C4—1224no—
488C4⋊C4—1228no—
489C4⋊C4—1221no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4C23×D6114yes—
493C4⋊C4C23×D6112yes—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4—1228no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C42—24no—
501C42—614no—
502C42—612no—
503C42—614no—
504C42—612no—
505C42—1228no—
506C42D4×D6112yes—
507C42C23×D6112yes—
508C23—614no—
509C23—612no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23C2×D4×D6111yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—48no—
520Q8—42no—
521Q8—1224no—
522Q8—1224no—
523Q8—2444no—
524D4—1222no—
525D4—1222no—
526D4—2448no—
527D4—2442no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1228no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1228no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—2442no—
549C2×C4—2444no—
550C2×C4—2448no—
551C2×C4#1451979814yes—
552C2×C4C2×D4×D6112yes—
553C2×C4C2×D4×D6112yes—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4#1451982312yes—
556C2×C4#1451982512yes—
557C2×C4C24×D6111yescharacteristic
558C8—24no—
559C8—1224no—
560C22—614no—
561C22—1228no—
562C22—1222no—
563C22—1222no—
564C22—4884no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22#1684034212yes—
569C22C2×#1451979812yes—
570C22#137×D6112yes—
571C22C22×D4×D6111yescharacteristic
572C4—22no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—1222no—
577C4—1222no—
578C4—1222no—
579C4—2442no—
580C4—2444no—
581C4#1684034212yes—
582C4C2×#1451979814yes—
583C4C2×#1451982311yescharacteristic
584C4C2×#1451982511yescharacteristic
585C2—1224no—
586C2—4882no—
587C2#2145715914yesdirect product factor
588C2C2×#1684034211yescharacteristic
589C2C22×#1451979811yescharacteristic
590C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#475C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenterC2×#475D61
hypocenterC61C2×#3248
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
3. #21457159C2direct (N × H)
4. C2×#16840072C2semidirect
5. C2×#16840039C2semidirect
6. C2×#16840002C2semidirect
8. C2×#16839987C2semidirect
9. C2×#16839972C2semidirect
10. C2×#16839968C2semidirect
11. C2×C4⋊C8×D61C2semidirect
12. C2×C61×#475C2semidirect
17. #16840317C22semidirect
37. #16840072C22semidirect
38. #16840039C22semidirect
39. #16840002C22semidirect
41. #16839987C22semidirect
42. #16839972C22semidirect
43. #16839968C22semidirect
47. C4⋊C8×D61C22semidirect
53. C61×#475C22semidirect
294. C2×#475D61semidirect
302. C2×C61#3248semidirect
335. #475C2×D61semidirect
352. C61C2×#3248semidirect
587. C2#21457159direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168402992
#214571592
C2×#168400722
C2×#168400392
C2×#168400022
C2×#168399922
C2×#168399872
C2×#168399722
C2×#168399682
C2×C4⋊C8×D612
C2×C61×#4752
C2×#324861