Finite Group Databasebeta

← C2×#21461648 #30614848

Subgroups — C2×#21461648 #30614848

Order 15616.

4062 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519836C212yesmaximal, complemented
2C22×#14519835C212yesmaximal, complemented
3C2×#16840705C212yesmaximal, complemented
4#21461648C218yesmaximal, direct product factor
5C23×#11758222C211yesmaximal, characteristic
6C22×C22⋊#6923C2214yes—
7C2×#14519836—28no—
8C2×#14519836C2218yescomplemented
9C2×#14519835—28no—
10C2×#14519835C2218yescomplemented
11#16840705—24no—
12#16840705C2218yescomplemented
13C2×(C2×#6923)⋊C4—24no—
14C22×#11758222—28no—
15C22×#11758222—28no—
16C22×#11758222—24no—
17C22×#11758222—24no—
18C22×#11758222C414yescomplemented
19C22×#11758222C2214yes—
20C23×Dic122C411yescharacteristic, complemented
21C23×D244C2211yescharacteristic
22C23×C22⋊D61C411yescharacteristic, complemented
23C23×C22⋊D61C2211yescharacteristic
24C23×C4×D61C411yescharacteristic, complemented
25C23×C4×D61C2211yescharacteristic
26C24×C4×C61C411yescharacteristic, complemented
27C23×#6923—24no—
28C2×C22⋊#6923—28no—
29C2×C22⋊#6923—28no—
30C2×C22⋊#6923—24no—
31C2×C22⋊#6923C2318yes—
32#14519836—216no—
33#14519835—216no—
34(C2×#6923)⋊C4—216no—
35C24×D61C2×C411yescharacteristic
36C24×D61C2311yescharacteristic
37C2×#11758222—264no—
38C2×#11758222—264no—
39C2×#11758222—216no—
40C2×#11758222—216no—
41C2×#11758222—216no—
42C2×#11758222—216no—
43C2×#11758222—216no—
44C2×#11758222—216no—
45C22×Dic122—24no—
46C22×Dic122—24no—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122C2×C414yescomplemented
50C22×D244—22no—
51C22×D244—24no—
52C22×D244—22no—
53C22×D244C2×C412yes—
54C22×D244D418yes—
55C22×D244C2312yes—
56C22×C22⋊D61—24no—
57C22×C22⋊D61—24no—
58C22×C22⋊D61—22no—
59C22×C22⋊D61—24no—
60C22×C22⋊D61—22no—
61C22×C22⋊D61—22no—
62C22×C22⋊D61—22no—
63C22×C22⋊D61C2×C412yes—
64C22×C22⋊D61C2×C414yescomplemented
65C22×C22⋊D61D418yes—
66C22×C22⋊D61C2312yes—
67C23×Dic61C2×C411yescharacteristic
68C23×Dic61C2×C411yescharacteristic
69C22×C4×D61—24no—
70C22×C4×D61—24no—
71C22×C4×D61—24no—
72C22×C4×D61—22no—
73C22×C4×D61—22no—
74C22×C4×D61—24no—
75C22×C4×D61C2×C412yes—
76C22×C4×D61C2×C414yescomplemented
77C22×C4×D61D414yes—
78C22×C4×D61D414yes—
79C22×C4×D61C2312yes—
80C25×C61C2×C411yescharacteristic
81C23×C4×C61—24no—
82C23×C4×C61—24no—
83C23×C4×C61—22no—
84C23×C4×C61—22no—
85C23×C4×C61C2×C411yescharacteristic
86C23×C4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C414yescomplemented
88C24×C61—22no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—44no—
92C24×C61C22⋊C414yes—
93C24×C61C22×C411yescharacteristic
94C24×C61C22×C412yes—
95C23×D61—22no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—44no—
99C23×D61—44no—
100C23×D61C22⋊C414yes—
101C23×D61C22×C412yes—
102C23×D61C22×C411yescharacteristic
103C23×D61C2×D412yes—
104C23×D61C2×D414yes—
105C23×D61C2×D414yes—
106C23×D61C2×D412yes—
107C23×D61C2411yescharacteristic
108C22×#6923—28no—
109C22×#6923—416no—
110C22×#6923—48no—
111C22×C4×C61—216no—
112C22×C4×C61—216no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—28no—
119C22×C4×C61—28no—
120C22×C4×C61—28no—
121C22×C4×C61C22⋊C412yes—
122C22×C4×C61C22⋊C412yes—
123C22×C4×C61C22⋊C414yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C2×C22⋊D61—216no—
127C2×C22⋊D61—216no—
128C2×C22⋊D61—28no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—28no—
132C2×C22⋊D61—216no—
133C2×C22⋊D61—216no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—24no—
136C2×C22⋊D61—28no—
137C2×C22⋊D61C22⋊C418yes—
138C2×C22⋊D61C2×D418yes—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C22⋊C414yes—
145C22×Dic61C22⋊C412yes—
146C22×Dic61C22⋊C412yes—
147C22×Dic61C22×C412yes—
148C22×Dic61C22×C412yes—
149#11758222—2128no—
150#11758222—2128no—
151C2×D244—24no—
152C2×D244—24no—
153C2×D244—216no—
154C2×D244—216no—
155C2×D244—24no—
156C2×D244—24no—
157C2×D244C22⋊C418yes—
158C2×D244C2×D418yes—
159C2×C4×D61—216no—
160C2×C4×D61—216no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—28no—
164C2×C4×D61—216no—
165C2×C4×D61—216no—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×C4×D61—28no—
169C2×C4×D61—28no—
170C2×C4×D61C22⋊C418yes—
171C2×C4×D61C2×D418yes—
172C2×Dic122—216no—
173C2×Dic122—216no—
174C2×Dic122—28no—
175C2×Dic122—28no—
176C2×Dic122—28no—
177C22⋊#6923—28no—
178C22⋊#6923—28no—
179C22⋊#6923—48no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—24no—
184C23×C61—24no—
185C23×C61—44no—
186C23×C61—48no—
187C23×C61—48no—
188C23×C61—44no—
189C23×C61—44no—
190C23×C61C2×C22⋊C412yes—
191C23×C61C2×C22⋊C412yes—
192C23×C61C2×C22⋊C411yescharacteristic
193C23×C61C2×C22⋊C414yes—
194C23×C61C2×C22⋊C411yescharacteristic
195C23×C61C23×C411yescharacteristic
196C22×D61—28no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—42no—
202C22×D61—44no—
203C22×D61—48no—
204C22×D61—48no—
205C22×D61—42no—
206C22×D61—44no—
207C22×D61—48no—
208C22×D61—44no—
209C22×D61—48no—
210C22×D61—44no—
211C22×D61C2×C22⋊C412yes—
212C22×D61C2×C22⋊C414yes—
213C22×D61C2×C22⋊C412yes—
214C22×D61C2×C22⋊C412yes—
215C22×D61#12018yes—
216C22×D61#12018yes—
217C22×D61C22×D412yes—
218C22×D61C22×D412yes—
219C22×D61C22×D412yes—
220C2×#6923—416no—
221C2×#6923—48no—
222C2×#6923—816no—
223C2×C4×C61—232no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61C2×C22⋊C414yes—
233C2×C4×C61C2×C22⋊C414yes—
234C22⋊D61—232no—
235C22⋊D61—216no—
236C22⋊D61—216no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61C2×C22⋊C414yes—
246C2×Dic61C2×C22⋊C414yes—
247D244—216no—
248D244—216no—
249C4×D61—232no—
250C4×D61—216no—
251C4×D61—216no—
252Dic122—232no—
253C2×#1782—611nomaximal, Sylow
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—24no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—48no—
262C22×C61—44no—
263C22×C61—48no—
264C22×C61#37912yes—
265C22×C61#37912yes—
266C22×C61#41014yes—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61C22×C22⋊C411yescharacteristic
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—24no—
271C2×D61—24no—
272C2×D61—216no—
273C2×D61—24no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—48no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—44no—
280C2×D61—48no—
281C2×D61—44no—
282C2×D61—48no—
283C2×D61—44no—
284C2×D61#37914yes—
285C2×D61C2×#12014yes—
286#6923—88no—
287C4×C61—24no—
288C4×C61—24no—
289Dic61—24no—
290Dic61—24no—
291C23×#41—611no—
292C22×#99—612no—
293C22×#99—612no—
294C2×#381—612no—
295#1782—618no—
296C2×C61—21no—
297C2×C61—21no—
298C2×C61—22no—
299C2×C61—44no—
300C2×C61#178214yescomplemented
301C2×C61C2×#37911yescharacteristic
302C2×C61C2×#41012yes—
303D61—24no—
304D61—24no—
305D61—44no—
306C22×#41—614no—
307C22×#41—614no—
308C22×#41—1224no—
309C22×#41—1224no—
310C22×#41—1228no—
311C22×#41—1228no—
312C23×Q8—611no—
313C23×D4—611no—
314C23×D4—611no—
315C23×D4—611no—
316C24×C4—611no—
317C24×C4—611no—
318C24×C4#692311yescharacteristic, complemented
319C22×C22⋊C4—614no—
320C2×#99—618no—
321C2×#99—618no—
322C2×#99—1228no—
323C2×#99—1228no—
324#381—618no—
325#381—1224no—
326C2×#95—1224no—
327C61C2×#178211yescharacteristic, complemented, Sylow
328C25—611no—
329C25—611no—
330C25C2×#692311yescharacteristic
331C2×#41—12216no—
332C2×#41—12216no—
333C2×#41—12216no—
334C2×#41—12216no—
335C2×#41—12216no—
336C2×#41—12216no—
337C2×#41—12264no—
338C2×#41—12264no—
339C22×Q8—614no—
340C22×Q8—1222no—
341C22×Q8—1224no—
342C22×Q8—1222no—
343C22×Q8—1224no—
344C22×D4—618no—
345C22×D4—612no—
346C22×D4—612no—
347C22×D4—614no—
348C22×D4—618no—
349C22×D4—612no—
350C22×D4—612no—
351C22×D4—1222no—
352C22×D4—1224no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4—1222no—
356C22×D4—1222no—
357C22×D4—1224no—
358C22×D4—1224no—
359C22×D4—1222no—
360C22×D4—1224no—
361C23×C4—24no—
362C23×C4—24no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—614no—
368C23×C4—614no—
369C23×C4—612no—
370C23×C4—612no—
371C23×C4—614no—
372C23×C4—1224no—
373C23×C4—1224no—
374C23×C4—1224no—
375C23×C4—1224no—
376C23×C4—1222no—
377C23×C4—1222no—
378C23×C4—1224no—
379C23×C4C2×#692311yescharacteristic
380C23×C4C2×#692311yescharacteristic
381C23×C4C2×#692314yescomplemented
382C2×C22⋊C4—618no—
383C2×C22⋊C4—1228no—
384C2×C22⋊C4—1228no—
385C2×C22⋊C4—1224no—
386#99—12216no—
387#99—12216no—
388#95—12216no—
389C24—21no—
390C24—22no—
391C24—21no—
392C24—44no—
393C24—612no—
394C24—612no—
395C24—614no—
396C24—614no—
397C24—611no—
398C24—611no—
399C24—614no—
400C24—612no—
401C24—1222no—
402C24—1221no—
403C24—1221no—
404C24—2444no—
405C24—2444no—
406C24C22⋊#692314yes—
407C24C22×#692311yescharacteristic
408C24C22×#692312yes—
409#41—122128no—
410#41—122128no—
411C2×Q8—1228no—
412C2×Q8—1228no—
413C2×Q8—12216no—
414C2×Q8—12216no—
415C2×Q8—1228no—
416C2×D4—618no—
417C2×D4—618no—
418C2×D4—618no—
419C2×D4—618no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—12216no—
425C2×D4—12216no—
426C2×D4—1228no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1228no—
432C2×D4—12216no—
433C2×D4—12216no—
434C2×D4—12216no—
435C2×D4—12216no—
436C2×D4—1228no—
437C22×C4—216no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—28no—
444C22×C4—21no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—612no—
448C22×C4—614no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—618no—
453C22×C4—618no—
454C22×C4—1228no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1228no—
460C22×C4—12216no—
461C22×C4—12216no—
462C22×C4—12216no—
463C22×C4—12216no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1228no—
470C22×C4—1228no—
471C22×C4—2448no—
472C22×C4—24416no—
473C22×C4C22⋊#692312yes—
474C22×C4C22⋊#692314yes—
475C22×C4C22⋊#692312yes—
476C22×C4C22×#692312yes—
477C22×C4C22×#692312yes—
478C22⋊C4—1228no—
479C22⋊C4—1228no—
480C22⋊C4—2448no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—48no—
487C23—44no—
488C23—44no—
489C23—48no—
490C23—44no—
491C23—612no—
492C23—612no—
493C23—618no—
494C23—612no—
495C23—612no—
496C23—612no—
497C23—612no—
498C23—618no—
499C23—614no—
500C23—1228no—
501C23—1222no—
502C23—1222no—
503C23—1224no—
504C23—1224no—
505C23—2444no—
506C23—2442no—
507C23—2442no—
508C23—2448no—
509C23—2448no—
510C23—2444no—
511C23—2448no—
512C23—2444no—
513C23—2448no—
514C23—2444no—
515C23C2×C22⋊#692312yes—
516C23C2×C22⋊#692311yescharacteristic
517C23C2×C22⋊#692312yes—
518C23C2×C22⋊#692314yes—
519C23C2×C22⋊#692311yescharacteristic
520C23C23×#692311yescharacteristic
521Q8—12232no—
522D4—12216no—
523D4—12216no—
524D4—12216no—
525D4—12216no—
526D4—12232no—
527C2×C4—232no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—614no—
537C2×C4—614no—
538C2×C4—12216no—
539C2×C4—12216no—
540C2×C4—12232no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—48816no—
550C2×C4—2448no—
551C2×C4—24416no—
552C2×C4C2×C22⋊#692314yes—
553C2×C4C2×C22⋊#692314yes—
554C22—22no—
555C22—22no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—44no—
563C22—48no—
564C22—614no—
565C22—614no—
566C22—1224no—
567C22—1224no—
568C22—12216no—
569C22—1224no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22—2442no—
574C22—2448no—
575C22—2444no—
576C22—2444no—
577C22—2448no—
578C22—2444no—
579C22—2448no—
580C22#1684070312yes—
581C22#1684070312yes—
582C22#1684074114yes—
583C22C22×C22⋊#692311yescharacteristic
584C22C22×C22⋊#692311yescharacteristic
585C22C22×C22⋊#692311yescharacteristic
586C4—24no—
587C4—24no—
588C4—1224no—
589C4—1224no—
590C4—4888no—
591C2—21no—
592C2—21no—
593C2—22no—
594C2—44no—
595C2—1224no—
596C2—1224no—
597C2—2444no—
598C2#2146164814yesdirect product factor
599C2C2×#1684070311yescharacteristic
600C2C2×#1684074112yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC24×C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×#11758222C2
cosocleC23×D61C24
hypercenterC24×C4#6923
hypocenterC61C2×#1782
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C22×#14519836C2semidirect
2. C22×#14519835C2semidirect
3. C2×#16840705C2semidirect
4. #21461648C2direct (N × H)
8. C2×#14519836C22semidirect
10. C2×#14519835C22semidirect
12. #16840705C22semidirect
18. C22×#11758222C4semidirect
20. C23×Dic122C4semidirect
22. C23×C22⋊D61C4semidirect
24. C23×C4×D61C4semidirect
26. C24×C4×C61C4semidirect
49. C22×Dic122C2×C4semidirect
64. C22×C22⋊D61C2×C4semidirect
76. C22×C4×D61C2×C4semidirect
87. C23×C4×C61C2×C4semidirect
300. C2×C61#1782semidirect
318. C24×C4#6923semidirect
327. C61C2×#1782semidirect
381. C23×C4C2×#6923semidirect
598. C2#21461648direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198362
C22×#145198352
C2×#168407052
#214616482
C23×#117582222
C2×#178261