Finite Group Databasebeta

← C2×#21461860 #30615776

Subgroups — C2×#21461860 #30615776

Order 15616.

1686 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840840C211yesmaximal, characteristic
2C2×#16840755C212yesmaximal, complemented
3C2×#16840742C211yesmaximal, characteristic
4#21461860C218yesmaximal, direct product factor
5C2×#16840629C212yesmaximal, complemented
6C2×#16840332C211yesmaximal, characteristic
7#16840840—24no—
8#16840840—22no—
9#16840840C2214yes—
10C2×#14519864—24no—
11C2×D61⋊M16C2212yes—
12C22×#11758217—22no—
13C2×#14519856C2212yes—
14#16840755—24no—
15#16840755C2218yescomplemented
16C2×#14519840—24no—
17#16840742—22no—
18#16840742C2214yes—
19C2×#14519835—22no—
20#16840629C2218yescomplemented
21C22×#11758224C2211yescharacteristic
22C22×Q8×D61C2211yescharacteristic
23#16840332—22no—
24#16840332—22no—
25#16840332C414yescomplemented
26#16840332C2214yes—
27C2×#14519777C411yescharacteristic, complemented
28C2×#14519775C411yescharacteristic, complemented
29C2×#14519764C411yescharacteristic, complemented
30C2×#14519759—22no—
31C2×#14519754—22no—
32C2×#14519734C2211yescharacteristic
33C2×#124×C61C411yescharacteristic, complemented
34#14519864—216no—
35#14519864—216no—
36C2×#11758213—22no—
37D61⋊M16—22no—
38D61⋊M16—22no—
39D61⋊M16D414yes—
40D61⋊M16C2314yes—
41C2×#11758217—24no—
42C2×#11758217—28no—
43C2×#11758217—22no—
44C2×#11758217—28no—
45C2×D61⋊C8D414yes—
46C22×#92420—24no—
47#14519856—22no—
48#14519856—22no—
49#14519856—22no—
50#14519856C2314yes—
51C2×C4⋊#6923—22no—
52C2×C22⋊#6923—22no—
53C2×C4×#6923—22no—
54#14519840—216no—
55#14519835—28no—
56C2×#11758224—22no—
57C2×#11758224—22no—
58C2×#11758224—22no—
59C2×#11758224—22no—
60C2×#11758224C2×C412yes—
61C2×#11758224D412yes—
62C2×#11758224D412yes—
63C2×#11758224C2312yes—
64C22×Dic122—21no—
65C22×Dic122C2×C411yescharacteristic
66C22×Dic122C2×C411yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61—21no—
69C2×Q8×D61—22no—
70C2×Q8×D61—22no—
71C2×Q8×D61—22no—
72C2×Q8×D61—22no—
73C2×Q8×D61C2×C412yes—
74C2×Q8×D61C2312yes—
75C22×C22⋊D61D412yes—
76C23×Dic61D412yes—
77C22×C4×D61—21no—
78C22×C4×D61C2311yescharacteristic
79#14519777—22no—
80#14519777C2×C414yescomplemented
81#14519775—22no—
82#14519775C2×C414yescomplemented
83#14519764—22no—
84#14519764C2×C414yescomplemented
85#14519759—28no—
86#14519759—28no—
87#14519754—28no—
88#14519754—28no—
89#14519734—21no—
90#14519734—21no—
91#14519734C2×C412yes—
92#14519734C2312yes—
93C2×C4⋊Dic61—22no—
94C2×#11758196—22no—
95C2×#11758196—21no—
96C2×C22⋊Dic61—22no—
97C2×#11758198—21no—
98C2×C4×Dic61C2×C411yescharacteristic
99C22×Q8×C61C2×C411yescharacteristic
100C22×D4×C61C2×C411yescharacteristic
101#124×C61—22no—
102#124×C61C2×C414yescomplemented
103C2×C22⋊C4×C61—22no—
104C2×C42×C61C2×C411yescharacteristic
105C24×C61C22⋊C412yes—
106C23×D61C2×D411yescharacteristic
107C22×#6923—42no—
108C2×Q8×C61—21no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61C22×C412yes—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61C22⋊C412yes—
115C2×D4×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C22×C411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61C22⋊C414yes—
122C2×C22⋊D61C2×D414yes—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61C22⋊C412yes—
130C22×Dic61C22⋊C414yes—
131C22×Dic61C2×D414yes—
132C22×Dic61C2×D411yescharacteristic
133Q8×D61—24no—
134Q8×D61—24no—
135Q8×D61—24no—
136Q8×D61—24no—
137Q8×D61—42no—
138Q8×D61—42no—
139Q8×D61—42no—
140Q8×D61—42no—
141#11758224—24no—
142#11758224—24no—
143#11758224—24no—
144#11758224—24no—
145#11758224—24no—
146#11758224—24no—
147#11758224—24no—
148#11758224—24no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×D412yes—
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61#4112yes—
159C2×C4×D61#4112yes—
160C2×C4×D61C2411yescharacteristic
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—21no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—41no—
169C2×Dic122—41no—
170C2×Dic122—41no—
171C2×Dic122—41no—
172C2×Dic122C22⋊C412yes—
173C2×Dic122C22×C412yes—
174C2×Dic122C22×C412yes—
175C22⋊#6923—24no—
176C22⋊#6923—42no—
177#11758217—216no—
178#11758217—24no—
179#11758217—24no—
180C2×#92420—216no—
181C2×#92420—24no—
182C4⋊#6923—24no—
183C4⋊#6923—42no—
184C4×#6923—24no—
185C4×#6923—42no—
186#11758213—24no—
187#11758213—24no—
188D61⋊C8—24no—
189D61⋊C8C2×D418yes—
190C22⋊C4×C61—28no—
191C42×C61—21no—
192C42×C61C22×C412yes—
193C22⋊Dic61—28no—
194#11758198—22no—
195#11758198—22no—
196C4⋊Dic61—28no—
197#11758196—22no—
198#11758196—28no—
199#11758196—22no—
200C4×Dic61—21no—
201C4×Dic61C22×C412yes—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—44no—
205C23×C61C2×C22⋊C411yescharacteristic
206C23×C61C2×C22⋊C414yes—
207C22×D61—42no—
208C22×D61C2×C22⋊C411yescharacteristic
209C22×D61#12112yes—
210C22×D61#12112yes—
211C22×D61C22×D411yescharacteristic
212C2×#6923—44no—
213C2×#6923—84no—
214Q8×C61—24no—
215Q8×C61—42no—
216D4×C61—24no—
217D4×C61—24no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C4×D412yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C23×C411yescharacteristic
227C22⋊D61—28no—
228C22⋊D61—28no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C2×C22⋊C414yes—
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61#12011yescharacteristic
243C2×Dic61#12011yescharacteristic
244C2×Dic61#12311yescharacteristic
245C2×Dic61#12311yescharacteristic
246C2×Dic61C22×D411yescharacteristic
247C4×D61—44no—
248C4×D61—44no—
249C4×D61C4×D412yes—
250C4×D61C4×D412yes—
251C4×D61#12012yes—
252C4×D61#12112yes—
253C4×D61#12112yes—
254C4×D61#12312yes—
255C4×D61C22×D412yes—
256C4×D61C2×#4112yes—
257Dic122—24no—
258Dic122—24no—
259Dic122—24no—
260Dic122—42no—
261Dic122—42no—
262Dic122—42no—
263Dic122—42no—
264Dic122—42no—
265#92420—28no—
266C2×#1909—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—24no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—82no—
272C22×C61#38611yescharacteristic
273C22×C61#38611yescharacteristic
274C22×C61C22×C22⋊C411yescharacteristic
275C2×D61—24no—
276C2×D61—41no—
277C2×D61—42no—
278C2×D61—41no—
279C2×D61#38612yes—
280C2×D61C2×#12112yes—
281#6923—84no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61#38612yes—
285C4×C61C2×C4×D412yes—
286Dic61—24no—
287Dic61—42no—
288Dic61—42no—
289Dic61#38611yescharacteristic
290Dic61#38611yescharacteristic
291Dic61C2×#12011yescharacteristic
292Dic61C2×#12311yescharacteristic
293C2×#568—611no—
294C2×#536—611no—
295C2×#419—612no—
296C2×#410—611no—
297#1909—618no—
298C2×H(3,ℤ/4)—612no—
299C2×C61—21no—
300C2×C61—21no—
301C2×C61—44no—
302C2×C61#190912yescomplemented
303C2×C61C2×#38611yescharacteristic
304D61—22no—
305D61—22no—
306C22×#41—611no—
307C23×Q8—611no—
308#568—614no—
309#568—1224no—
310#568—1222no—
311C2×#131—1224no—
312C22×M16—612no—
313C22×M16—1222no—
314#536—614no—
315#536—614no—
316#536—1222no—
317#536—1222no—
318C2×C4⋊Q8—611no—
319C2×C4⋊Q8—611no—
320C2×#124—611no—
321C2×#124#692311yescharacteristic, complemented
322C2×#123—1222no—
323C2×C22⋊Q8—1222no—
324C2×#117—611no—
325C2×#117—612no—
326#419—618no—
327#419—1224no—
328C2×#103—1224no—
329#410—614no—
330#410—1222no—
331C2×#99—1222no—
332H(3,ℤ/4)—618no—
333C61C2×#190911yescharacteristic, complemented, Sylow
334C2×#41—612no—
335C2×#41—612no—
336C2×#41—612no—
337C2×#41—612no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C2×#41—1222no—
342C22×Q8—611no—
343C22×Q8—612no—
344C22×Q8—612no—
345C22×Q8—611no—
346C22×Q8—1221no—
347C22×Q8—1221no—
348C22×Q8—1221no—
349C22×Q8—1222no—
350C22×Q8—1222no—
351C22×Q8—1222no—
352C22×Q8—1222no—
353C22×Q8C2×#692311yescharacteristic
354C22×D4—612no—
355C22×D4C2×#692311yescharacteristic
356C23×C4—611no—
357C23×C4—612no—
358C23×C4—1221no—
359#131—12216no—
360#131—12216no—
361C2×M16—614no—
362C2×M16—614no—
363C2×M16—1222no—
364C2×M16—1228no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1224no—
369C2×M16—1228no—
370C22×C8—614no—
371C22×C8—1224no—
372C4⋊Q8—614no—
373C4⋊Q8—614no—
374C4⋊Q8—1222no—
375C4⋊Q8—1222no—
376#124—22no—
377#124—614no—
378#124—1222no—
379#124C2×#692314yescomplemented
380#123—1228no—
381#123—1228no—
382C22⋊Q8—1228no—
383C22⋊Q8—1228no—
384#117—612no—
385#117—612no—
386#117—614no—
387#117—1221no—
388#117—1221no—
389#117—1222no—
390#117—1222no—
391#117—1222no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1222no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C42—611no—
401C2×C42—1222no—
402C2×C42C2×#692311yescharacteristic
403#103—12216no—
404#99—1228no—
405C24—611no—
406C24C22⋊#692312yes—
407#41—1224no—
408#41—1224no—
409#41—1224no—
410#41—1224no—
411#41—1224no—
412#41—1224no—
413#41—1224no—
414#41—1224no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—612no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1224no—
424C2×Q8—1224no—
425C2×Q8—1224no—
426C2×Q8—1224no—
427C2×Q8—1221no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—2441no—
433C2×Q8—2441no—
434C2×Q8—2441no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×Q8—2442no—
438C2×Q8—2442no—
439C2×Q8—2442no—
440C2×Q8C22×#692312yes—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—614no—
444C2×D4—614no—
445C2×D4—1224no—
446C2×D4—1224no—
447C2×D4C22⋊#692312yes—
448C2×D4C22×#692312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—612no—
461C22×C4—614no—
462C22×C4—614no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4C22×#692311yescharacteristic
475M16—1224no—
476M16—1224no—
477M16—1224no—
478M16—1224no—
479M16—12216no—
480C2×C8—618no—
481C2×C8—1224no—
482C2×C8—1224no—
483C2×C8—12216no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1228no—
487C4⋊C4—1228no—
488C4⋊C4—1224no—
489C4⋊C4—2442no—
490C22⋊C4—28no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1228no—
494C22⋊C4—1224no—
495C22⋊C4—2442no—
496C42—21no—
497C42—612no—
498C42—1224no—
499C42—1221no—
500C42—2442no—
501C42C22×#692312yes—
502C23—22no—
503C23—22no—
504C23—44no—
505C23—612no—
506C23—612no—
507C23—611no—
508C23—611no—
509C23—2442no—
510C23C2×C22⋊#692311yescharacteristic
511C23C2×C22⋊#692314yes—
512Q8—24no—
513Q8—42no—
514Q8—1224no—
515Q8—1224no—
516Q8—1224no—
517Q8—2442no—
518Q8—2442no—
519Q8—2442no—
520Q8—2442no—
521Q8—2442no—
522D4—24no—
523D4—24no—
524D4—1228no—
525D4—1228no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—614no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—4884no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4D4×#692312yes—
563C2×C4D4×#692312yes—
564C2×C4C23×#692311yescharacteristic
565C8—1228no—
566C22—22no—
567C22—24no—
568C22—44no—
569C22—44no—
570C22—82no—
571C22—612no—
572C22—612no—
573C22—1224no—
574C22—2441no—
575C22—2441no—
576C22—2442no—
577C22#1684071011yescharacteristic
578C22#1684071011yescharacteristic
579C22C22×C22⋊#692311yescharacteristic
580C4—42no—
581C4—42no—
582C4—611no—
583C4—611no—
584C4—611no—
585C4—611no—
586C4—1224no—
587C4—4884no—
588C4—2442no—
589C4—2442no—
590C4#1684071012yes—
591C4C2×D4×#692312yes—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—1222no—
596C2—1222no—
597C2#2146186012yesdirect product factor
598C2C2×#1684071011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#124×C61C4
socleC22×C61#386
prime radicalC22×#11758224C22
cosocleC2×C4×D61C24
hypercenterC2×#124#6923
hypocenterC61C2×#1909
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#16840755C2semidirect
4. #21461860C2direct (N × H)
5. C2×#16840629C2semidirect
15. #16840755C22semidirect
20. #16840629C22semidirect
25. #16840332C4semidirect
27. C2×#14519777C4semidirect
28. C2×#14519775C4semidirect
29. C2×#14519764C4semidirect
33. C2×#124×C61C4semidirect
80. #14519777C2×C4semidirect
82. #14519775C2×C4semidirect
84. #14519764C2×C4semidirect
102. #124×C61C2×C4semidirect
302. C2×C61#1909semidirect
321. C2×#124#6923semidirect
333. C61C2×#1909semidirect
379. #124C2×#6923semidirect
597. C2#21461860direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168408402
C2×#168407552
C2×#168407422
#214618602
C2×#168406292
C2×#168403322
C2×#190961