Finite Group Databasebeta

← C2×#21461825 #30615667

Subgroups — C2×#21461825 #30615667

Order 15616.

1750 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840830C211yesmaximal, characteristic
2C2×#16840755C212yesmaximal
3C2×D61⋊#101C211yesmaximal, characteristic
4#21461825C218yesmaximal, direct product factor
5C2×#16840633C212yesmaximal
6C2×C4⋊Q8×D61C211yesmaximal, characteristic
7C2×D61⋊M16C2212yes—
8#16840830—24no—
9#16840830—22no—
10#16840830C2214yes—
11C2×Q8×#6923—24no—
12C2×#14519856—22no—
13C2×#14519856C2212yes—
14#16840755—24no—
15#16840755C2218yes—
16C2×#14519840—24no—
17D61⋊#101—22no—
18D61⋊#101C2214yes—
19C2×#14519838—22no—
20#16840633C2218yes—
21C22×Q8×D61C2211yescharacteristic
22C22×Q8×D61C2211yescharacteristic
23C2×#14519777C411yescharacteristic, complemented
24C2×#14519776—22no—
25C2×#14519775C411yescharacteristic, complemented
26C2×#14519775C411yescharacteristic, complemented
27C4⋊Q8×D61—22no—
28C4⋊Q8×D61—22no—
29C4⋊Q8×D61C414yescomplemented
30C4⋊Q8×D61C2214yes—
31C2×C4⋊C4×D61—22no—
32C2×C42×D61C2211yescharacteristic
33C2×C4⋊Q8×C61C411yescharacteristic, complemented
34C2×#11758213—22no—
35D61⋊M16—22no—
36D61⋊M16—22no—
37D61⋊M16—22no—
38D61⋊M16C2314yes—
39C2×#11758217—22no—
40C2×D61⋊C8—22no—
41Q8×#6923—216no—
42Q8×#6923—216no—
43#14519856—28no—
44#14519856—28no—
45#14519856—22no—
46#14519856—22no—
47#14519856D414yes—
48#14519856C2314yes—
49C2×C4⋊#6923—24no—
50C2×C4⋊#6923D414yes—
51C2×C22⋊#6923—22no—
52C2×C4×#6923—24no—
53C2×C4×#6923—22no—
54#14519840—216no—
55#14519838—28no—
56C22×Dic122—21no—
57C22×Dic122C2×C411yescharacteristic
58C22×Dic122C2×C411yescharacteristic
59C22×Dic122D412yes—
60C2×Q8×D61—22no—
61C2×Q8×D61—22no—
62C2×Q8×D61—21no—
63C2×Q8×D61—21no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—22no—
68C2×Q8×D61—22no—
69C2×Q8×D61—22no—
70C2×Q8×D61C2×C412yes—
71C2×Q8×D61C2×C412yes—
72C2×Q8×D61D412yes—
73C2×Q8×D61D412yes—
74C2×Q8×D61C2312yes—
75C2×Q8×D61C2312yes—
76C22×C4×D61—21no—
77C22×C4×D61—21no—
78C22×C4×D61D412yes—
79C22×C4×D61C2311yescharacteristic
80#14519777—22no—
81#14519777C2×C414yescomplemented
82#14519776—28no—
83#14519776—28no—
84#14519775—22no—
85#14519775—22no—
86#14519775C2×C414yescomplemented
87#14519775C2×C414yescomplemented
88C2×C4⋊Dic61—22no—
89C2×#11758196—22no—
90C2×#11758196—22no—
91C4⋊C4×D61—28no—
92C4⋊C4×D61—28no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2×C411yescharacteristic
95C42×D61—21no—
96C42×D61—21no—
97C42×D61C2×C412yes—
98C42×D61C2312yes—
99C22×Q8×C61C2×C411yescharacteristic
100C22×Q8×C61C2×C411yescharacteristic
101C4⋊Q8×C61—22no—
102C4⋊Q8×C61C2×C414yescomplemented
103C2×C4⋊C4×C61—22no—
104C2×C42×C61C2×C411yescharacteristic
105C23×D61C2×D411yescharacteristic
106C22×#6923—24no—
107C2×Q8×C61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61C22⋊C412yes—
113C2×Q8×C61C22×C412yes—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61C22⋊C412yes—
118C22×C4×C61C22×C411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61C22⋊C412yes—
122C22×Dic61C2×D411yescharacteristic
123Q8×D61—24no—
124Q8×D61—24no—
125Q8×D61—24no—
126Q8×D61—24no—
127Q8×D61—24no—
128Q8×D61—24no—
129Q8×D61—24no—
130Q8×D61—24no—
131Q8×D61—24no—
132Q8×D61—24no—
133Q8×D61—24no—
134Q8×D61—24no—
135Q8×D61—42no—
136Q8×D61—42no—
137Q8×D61—42no—
138Q8×D61—42no—
139C2×C4×D61—28no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61C22⋊C414yes—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D414yes—
155C2×C4×D61C2×D412yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61#4112yes—
160C2×C4×D61#4112yes—
161C2×C4×D61C2411yescharacteristic
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—24no—
165C2×Dic122—24no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122C22⋊C414yes—
174C2×Dic122C22⋊C412yes—
175C2×Dic122C22×C412yes—
176C2×Dic122C22×C412yes—
177C2×Dic122C2×D414yes—
178C22⋊#6923—24no—
179C22⋊#6923—24no—
180#11758217—24no—
181#11758217—42no—
182C2×#92420—42no—
183C4⋊#6923—216no—
184C4⋊#6923—24no—
185C4⋊#6923C2×D418yes—
186C4×#6923—216no—
187C4×#6923—24no—
188C4×#6923—24no—
189#11758213—24no—
190#11758213—42no—
191D61⋊C8—24no—
192D61⋊C8—42no—
193C4⋊C4×C61—28no—
194C42×C61—21no—
195C42×C61C22×C412yes—
196C4⋊Dic61—28no—
197#11758196—28no—
198#11758196—28no—
199C4×Dic61—21no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61C22×C412yes—
203C23×C61C2×C22⋊C411yescharacteristic
204C22×D61—24no—
205C22×D61C2×C22⋊C411yescharacteristic
206C22×D61#12011yescharacteristic
207C22×D61#12011yescharacteristic
208C22×D61#12311yescharacteristic
209C22×D61#12311yescharacteristic
210C22×D61C22×D411yescharacteristic
211C2×#6923—28no—
212C2×#6923—48no—
213Q8×C61—24no—
214Q8×C61—24no—
215Q8×C61—24no—
216Q8×C61—42no—
217C2×C4×C61—24no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C2×C22⋊C414yes—
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C4×D412yes—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×Dic61—24no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61C2×C22⋊C414yes—
237C2×Dic61C2×C22⋊C411yescharacteristic
238C2×Dic61#12112yes—
239C2×Dic61#12112yes—
240C2×Dic61C22×D411yescharacteristic
241C4×D61—28no—
242C4×D61—28no—
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61C4×D412yes—
252C4×D61C4×D412yes—
253C4×D61#12012yes—
254C4×D61#12112yes—
255C4×D61#12112yes—
256C4×D61#12312yes—
257C4×D61C22×D412yes—
258C4×D61C2×#4112yes—
259Dic122—24no—
260Dic122—24no—
261Dic122—28no—
262Dic122—28no—
263Dic122—24no—
264Dic122—42no—
265Dic122—44no—
266Dic122—44no—
267#92420—44no—
268C2×#1886—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61#38611yescharacteristic
271C22×C61#38611yescharacteristic
272C22×C61C22×C22⋊C411yescharacteristic
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61#38611yescharacteristic
279C2×D61#38611yescharacteristic
280C2×D61#45614yes—
281C2×D61C2×#12011yescharacteristic
282C2×D61C2×#12311yescharacteristic
283#6923—48no—
284C4×C61—24no—
285C4×C61—24no—
286C4×C61—42no—
287C4×C61#38612yes—
288C4×C61C2×C4×D412yes—
289Dic61—24no—
290Dic61—24no—
291Dic61—42no—
292Dic61#38612yes—
293Dic61C2×#12112yes—
294C22×C4⋊Q8—611no—
295C2×#519—611no—
296C2×#419—612no—
297C22×#101—611no—
298#1886—618no—
299C2×#339—612no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61#188612yescomplemented
303C2×C61C2×#38611yescharacteristic
304D61—21no—
305D61—21no—
306D61#188612yescomplemented
307D61C2×#45612yes—
308C23×Q8—611no—
309C23×Q8—611no—
310C22×M16—612no—
311C2×C4⋊Q8—611no—
312C2×C4⋊Q8—611no—
313C2×C4⋊Q8—614no—
314C2×C4⋊Q8—614no—
315C2×C4⋊Q8—611no—
316C2×C4⋊Q8—1222no—
317C2×C4⋊Q8—1222no—
318C2×C4⋊Q8—1222no—
319C2×C4⋊Q8#692311yescharacteristic, complemented
320#519—614no—
321#519—1224no—
322#519—1222no—
323C2×C4×Q8—1224no—
324C2×#117—612no—
325C2×#117—1222no—
326C22×C4⋊C4—1222no—
327C22×C42—611no—
328#419—618no—
329#419—1224no—
330C2×#103—1224no—
331C2×#101—614no—
332C2×#101—1222no—
333C2×#101—1222no—
334#339—618no—
335C61C2×#188611yescharacteristic, complemented, Sylow
336C22×Q8—612no—
337C22×Q8—612no—
338C22×Q8—612no—
339C22×Q8—612no—
340C22×Q8—612no—
341C22×Q8—611no—
342C22×Q8—612no—
343C22×Q8—612no—
344C22×Q8—611no—
345C22×Q8—1221no—
346C22×Q8—1222no—
347C22×Q8—1222no—
348C22×Q8—1222no—
349C22×Q8—1222no—
350C22×Q8—1221no—
351C22×Q8—1221no—
352C22×Q8—1222no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8C2×#692311yescharacteristic
357C22×Q8C2×#692311yescharacteristic
358C23×C4—611no—
359C23×C4—612no—
360C23×C4—1221no—
361C23×C4—1221no—
362C2×M16—614no—
363C2×M16—1222no—
364C2×M16—1222no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1222no—
368C22×C8—1222no—
369C4⋊Q8—22no—
370C4⋊Q8—614no—
371C4⋊Q8—614no—
372C4⋊Q8—614no—
373C4⋊Q8—1222no—
374C4⋊Q8—1228no—
375C4⋊Q8—1228no—
376C4⋊Q8—1222no—
377C4⋊Q8—1222no—
378C4⋊Q8C2×#692314yescomplemented
379C4×Q8—12216no—
380C4×Q8—12216no—
381#117—614no—
382#117—614no—
383#117—1222no—
384#117—1222no—
385#117—1228no—
386#117—1228no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—614no—
389C2×C4⋊C4—1228no—
390C2×C4⋊C4—1228no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C42—611no—
397C2×C42—612no—
398C2×C42—612no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1222no—
403C2×C42—1224no—
404C2×C42C2×#692311yescharacteristic
405#103—12216no—
406#101—1228no—
407C24—611no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—614no—
414C2×Q8—614no—
415C2×Q8—612no—
416C2×Q8—612no—
417C2×Q8—612no—
418C2×Q8—1224no—
419C2×Q8—1224no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1224no—
423C2×Q8—1224no—
424C2×Q8—1224no—
425C2×Q8—1224no—
426C2×Q8—1224no—
427C2×Q8—1224no—
428C2×Q8—1224no—
429C2×Q8—1224no—
430C2×Q8—1224no—
431C2×Q8—1224no—
432C2×Q8—1224no—
433C2×Q8—1224no—
434C2×Q8—1221no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8—1222no—
439C2×Q8—2442no—
440C2×Q8—2442no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C2×Q8—2442no—
444C2×Q8—2442no—
445C2×Q8C22⋊#692312yes—
446C2×Q8C22×#692312yes—
447C2×Q8C22×#692312yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—612no—
460C22×C4—614no—
461C22×C4—614no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1228no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—1221no—
474C22×C4—2441no—
475C22×C4—2441no—
476C22×C4—2441no—
477C22×C4—2441no—
478C22×C4C22⋊#692312yes—
479C22×C4C22×#692311yescharacteristic
480M16—1224no—
481M16—1224no—
482M16—2442no—
483M16—2442no—
484C2×C8—1224no—
485C2×C8—2442no—
486C2×C8—2442no—
487C4⋊C4—28no—
488C4⋊C4—618no—
489C4⋊C4—1228no—
490C4⋊C4—1228no—
491C4⋊C4—1224no—
492C4⋊C4—12216no—
493C4⋊C4—1228no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C42—21no—
497C42—612no—
498C42—1222no—
499C42—1222no—
500C42—1224no—
501C42—1224no—
502C42—12216no—
503C42—1221no—
504C42C22×#692312yes—
505C23—611no—
506C23—611no—
507C23—611no—
508C23—611no—
509C23—611no—
510C23—611no—
511C23—1224no—
512C23C2×C22⋊#692311yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—24no—
516Q8—42no—
517Q8—1228no—
518Q8—1228no—
519Q8—1224no—
520Q8—1224no—
521Q8—1224no—
522Q8—2444no—
523Q8—2444no—
524Q8—2442no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—614no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1228no—
550C2×C4—1228no—
551C2×C4—1224no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1228no—
556C2×C4—1222no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2448no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4C2×C22⋊#692314yes—
565C2×C4C2×C22⋊#692311yescharacteristic
566C2×C4D4×#692312yes—
567C2×C4D4×#692312yes—
568C2×C4C23×#692311yescharacteristic
569C8—2444no—
570C22—22no—
571C22—614no—
572C22—611no—
573C22—611no—
574C22—611no—
575C22—611no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22#1684071011yescharacteristic
582C22#1684071011yescharacteristic
583C22C22×C22⋊#692311yescharacteristic
584C4—24no—
585C4—24no—
586C4—42no—
587C4—612no—
588C4—612no—
589C4—1224no—
590C4—1224no—
591C4—2448no—
592C4—2442no—
593C4#1684071012yes—
594C4C2×D4×#692312yes—
595C2—21no—
596C2—21no—
597C2—612no—
598C2—612no—
599C2—1221no—
600C2—1221no—
601C2#2146182512yesdirect product factor
602C2C2×#1684071011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61C2×#1886
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #21461825C2direct (N × H)
23. C2×#14519777C4semidirect
25. C2×#14519775C4semidirect
26. C2×#14519775C4semidirect
29. C4⋊Q8×D61C4semidirect
33. C2×C4⋊Q8×C61C4semidirect
81. #14519777C2×C4semidirect
86. #14519775C2×C4semidirect
87. #14519775C2×C4semidirect
102. C4⋊Q8×C61C2×C4semidirect
302. C2×C61#1886semidirect
306. D61#1886semidirect
319. C2×C4⋊Q8#6923semidirect
335. C61C2×#1886semidirect
378. C4⋊Q8C2×#6923semidirect
601. C2#21461825direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168408302
C2×#168407552
C2×D61⋊#1012
#214618252
C2×#168406332
C2×C4⋊Q8×D612
C2×#188661