Finite Group Databasebeta

← C2×#21446187 #29681502

Subgroups — C2×#21446187 #29681502

Order 15616.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic
2C2×#16840243C211yesmaximal, characteristic
3C22×#14519703C211yesmaximal, characteristic
4C2×#16839772C211yesmaximal, characteristic, complemented
5C2×#16839667C211yesmaximal, characteristic, complemented
6#21446187C218yesmaximal, direct product factor
7C2×#16839523C211yesmaximal, characteristic, complemented
8C2×C61×#343C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519810—22no—
12C2×#14519810—24no—
13C2×#14519810C418yescomplemented
14C2×#14519810C2214yes—
15C22×Dic244—21no—
16C22×#11758190—21no—
17#16840243—22no—
18#16840243—22no—
19#16840243—22no—
20#16840243C2214yes—
21C2×#14519742—21no—
22C2×#14519739—21no—
23C2×C4×C22⋊D61—22no—
24C2×#14519736C2211yescharacteristic
25C2×#14519735—21no—
26C2×C4⋊C4×D61—21no—
27C22×C4×Dic61C2211yescharacteristic
28C2×#14519703—24no—
29C2×#14519703C2214yes—
30#16839772—22no—
31#16839772C2214yescomplemented
32C2×#14519702—21no—
33C2×#14519702—21no—
34#16839667—22no—
35#16839667C2214yescomplemented
36C2×#14519684—22no—
37#16839523C2214yescomplemented
38C22×M16×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#343C2214yescomplemented
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222C2×C414yes—
47C2×#11758222D414yes—
48C2×#11758222D412yes—
49C2×#11758222C2312yes—
50C22×Dic122—21no—
51C22×Dic122—21no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—22no—
55C22×Dic122—22no—
56C22×Dic122—21no—
57C22×Dic122—22no—
58C22×Dic122C2×C414yescomplemented
59C22×Dic122D411yescharacteristic
60C22×Dic122C2312yes—
61C22×D244D411yescharacteristic
62C22×C22⋊D61—21no—
63C23×Dic61—21no—
64C23×Dic61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—22no—
67#14519810—216no—
68#14519810—28no—
69#14519810C2×C4116yescomplemented
70C2×Dic244—28no—
71C2×Dic244—24no—
72C2×#11758190—28no—
73C2×#11758190—24no—
74#14519742—24no—
75#14519742—24no—
76#14519739—24no—
77#14519739—24no—
78C4×C22⋊D61—28no—
79C4×C22⋊D61—28no—
80#14519736—21no—
81#14519736—21no—
82#14519736D412yes—
83#14519736C2312yes—
84#14519735—24no—
85#14519735—24no—
86C2×C4⋊Dic61D411yescharacteristic
87C2×C4⋊Dic61D411yescharacteristic
88C2×#11758196—22no—
89C4⋊C4×D61—24no—
90C4⋊C4×D61—24no—
91C2×C22⋊Dic61—21no—
92C2×#11758198—22no—
93C2×C4×Dic61—22no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—22no—
96C2×C4×Dic61—22no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2312yes—
99#14519703—28no—
100#14519702—24no—
101#14519702—24no—
102#14519684—28no—
103C23×C4×C61D412yes—
104C23×C4×C61C2311yescharacteristic
105C2×M16×C61—22no—
106C2×M16×C61—21no—
107C2×M16×C61—21no—
108C2×M16×C61—22no—
109C2×M16×C61—22no—
110C2×M16×C61C2×C414yescomplemented
111C2×M16×C61C2312yes—
112C2×M16×C61C2312yes—
113C22×C8×C61—21no—
114C22×C8×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61D412yes—
120C2×C4⋊C4×C61C2312yes—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—44no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D414yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—22no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—22no—
141C22×Dic61—21no—
142C22×Dic61—24no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—42no—
149C22×Dic61—41no—
150C22×Dic61—41no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153#11758222—24no—
154#11758222—24no—
155#11758222—24no—
156#11758222—24no—
157#11758222—24no—
158#11758222—22no—
159#11758222—22no—
160#11758222C22⋊C418yes—
161#11758222C22×C414yes—
162#11758222C2×D414yes—
163C2×D244—21no—
164C2×D244—22no—
165C2×D244C22⋊C414yes—
166C2×D244C2×D412yes—
167C2×C4×D61—24no—
168C2×C4×D61—22no—
169C2×C4×D61—24no—
170C2×C4×D61—22no—
171C2×C4×D61—28no—
172C2×C4×D61—84no—
173C2×Dic122—21no—
174C2×Dic122—22no—
175C2×Dic122—24no—
176C2×Dic122—24no—
177C2×Dic122—24no—
178C2×Dic122—24no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—24no—
184C2×Dic122—22no—
185C2×Dic122—41no—
186C2×Dic122—41no—
187C2×Dic122—42no—
188C2×Dic122—42no—
189C2×Dic122—42no—
190C2×Dic122—42no—
191C2×Dic122—42no—
192C2×Dic122C22⋊C414yes—
193C2×Dic122C22×C414yescomplemented
194C2×Dic122C2×D412yes—
195M16×C61—24no—
196M16×C61—22no—
197M16×C61—22no—
198M16×C61—22no—
199M16×C61—42no—
200M16×C61C22×C414yescomplemented
201C2×C8×C61—24no—
202C2×C8×C61—22no—
203C2×C8×C61—22no—
204C2×C8×C61—42no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C22⋊Dic61—22no—
210C22⋊Dic61—22no—
211#11758198—28no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61C2×D412yes—
215C4⋊Dic61C2×D412yes—
216#11758196—28no—
217C4×Dic61—28no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—42no—
223Dic244—216no—
224#11758190—216no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61C4×D412yes—
228C23×C61#12011yescharacteristic
229C23×C61#12011yescharacteristic
230C23×C61#12112yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—22no—
233C22×D61—24no—
234C22×D61—42no—
235C22×D61—42no—
236C2×C4×C61—24no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61—82no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12312yes—
251C2×C4×C61#12312yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—24no—
259C22⋊D61—24no—
260C22⋊D61—28no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—48no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—44no—
273C2×Dic61—44no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276D244—22no—
277D244—24no—
278D244C2×C22⋊C414yes—
279C4×D61—24no—
280C4×D61—24no—
281C4×D61—28no—
282C4×D61—88no—
283Dic122—22no—
284Dic122—24no—
285Dic122—24no—
286Dic122—24no—
287Dic122—42no—
288Dic122—42no—
289Dic122—44no—
290Dic122—44no—
291Dic122C2×C22⋊C414yes—
292C8×C61—24no—
293C8×C61—42no—
294C2×#1882—611nomaximal, Sylow
295C22×C61—24no—
296C22×C61—24no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—21no—
301C22×C61#38612yes—
302C22×C61#38612yes—
303C22×C61#45512yes—
304C22×C61#45612yes—
305C22×C61C2×C4×D411yescharacteristic
306C22×C61C2×#12011yescharacteristic
307C22×C61C2×#12111yescharacteristic
308C2×D61—24no—
309C2×D61—48no—
310C2×D61—42no—
311C2×D61—42no—
312C4×C61—44no—
313C4×C61#38612yes—
314C4×C61#38612yes—
315C4×C61C22×C22⋊C411yescharacteristic
316C4×C61C2×C4×D411yescharacteristic
317C4×C61C2×#12111yescharacteristic
318C4×C61C2×#12311yescharacteristic
319Dic61—24no—
320Dic61—44no—
321Dic61—42no—
322Dic61—42no—
323Dic61—44no—
324C22×#137—611no—
325C2×#520—611no—
326C22×#104—611no—
327C2×#419—611no—
328C2×#391—611no—
329#1882—618no—
330C2×#343D6111yescharacteristic, complemented
331C2×#339—611no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61—22no—
335C2×C61#188214yescomplemented
336C2×C61C2×#38611yescharacteristic
337C2×C61C2×#45511yescharacteristic
338C2×C61C2×#45611yescharacteristic
339D61—44no—
340C22×#41—611no—
341C23×Q8—611no—
342C2×#137—618no—
343C2×#137—614no—
344C2×#137—1222no—
345C2×#137—1224no—
346C22×Q16—1221no—
347C22×SD16—1221no—
348C22×M16C2×D6111yescharacteristic
349C22×M16C2×D6111yescharacteristic
350#520—614no—
351#520—1222no—
352#520—1222no—
353#520—1222no—
354C2×C4×Q8—1221no—
355C2×C4×D4—1221no—
356C2×C4×D4—1222no—
357C2×#117—611no—
358C2×#117—1221no—
359C22×C4⋊C4—1221no—
360C22×C4⋊C4C2×D6111yescharacteristic
361C22×C42—611no—
362C2×#104—614no—
363C2×#104—1224no—
364#419—614no—
365#419—1222no—
366C2×#103—1221no—
367C2×#103—1221no—
368#391—614no—
369#391—1222no—
370C2×#95—1222no—
371#343C2×D6114yescomplemented
372#339—614no—
373C61C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—614no—
375C2×#41—612no—
376C2×#41—614no—
377C2×#41—612no—
378C2×#41—1221no—
379C2×#41—1222no—
380C2×#41—1221no—
381C2×#41—1222no—
382C22×Q8—614no—
383C22×Q8—611no—
384C22×Q8—612no—
385C22×Q8—1221no—
386C22×Q8—1222no—
387C22×Q8—1221no—
388C22×Q8—1221no—
389C22×Q8—1222no—
390C22×Q8—1221no—
391C22×Q8—1221no—
392C22×Q8—1222no—
393C22×D4—611no—
394C22×D4—1221no—
395C23×C4—1222no—
396C23×C4—1221no—
397C23×C4—1221no—
398C23×C4—1221no—
399C23×C4D24412yes—
400C23×C4C22×D6111yescharacteristic
401#137—6116no—
402#137—1228no—
403#137—12216no—
404C2×Q16—1224no—
405C2×Q16—1228no—
406C2×SD16—1224no—
407C2×SD16—1228no—
408C2×M16—21no—
409C2×M16—22no—
410C2×M16—21no—
411C2×M16—22no—
412C2×M16—22no—
413C2×M16C4×D6114yescomplemented
414C2×M16C22×D6112yes—
415C2×M16C22×D6112yes—
416C22×C8—21no—
417C22×C8—21no—
418C4×Q8—1224no—
419C4×Q8—1224no—
420C4×D4—1224no—
421C4×D4—1224no—
422C4×D4—1228no—
423C4×D4—1228no—
424#117—612no—
425#117—612no—
426#117—1224no—
427#117—1224no—
428#117—1221no—
429#117—1221no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1224no—
437C2×C4⋊C4—1224no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4D24412yes—
440C2×C4⋊C4C22×D6112yes—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1221no—
443C2×C42—612no—
444C2×C42—1222no—
445C2×C42—1221no—
446C2×C42—1222no—
447C2×C42—1222no—
448C2×C42—1221no—
449#104—1228no—
450#103—1224no—
451#103—1224no—
452#95—1228no—
453C24—1221no—
454C24C2×D24411yescharacteristic
455#41—614no—
456#41—614no—
457#41—618no—
458#41—1222no—
459#41—1222no—
460#41—1224no—
461#41—1224no—
462#41—1224no—
463#41—1224no—
464#41—1224no—
465C2×Q8—614no—
466C2×Q8—612no—
467C2×Q8—614no—
468C2×Q8—1222no—
469C2×Q8—1224no—
470C2×Q8—1221no—
471C2×Q8—1222no—
472C2×Q8—1222no—
473C2×Q8—1222no—
474C2×Q8—1222no—
475C2×Q8—1222no—
476C2×Q8—1224no—
477C2×Q8—1224no—
478C2×Q8—1224no—
479C2×Q8—1224no—
480C2×Q8—2441no—
481C2×Q8—2441no—
482C2×Q8—2442no—
483C2×Q8—2442no—
484C2×Q8—2442no—
485C2×Q8—2442no—
486C2×Q8—2442no—
487C2×D4—612no—
488C2×D4—614no—
489C2×D4—1222no—
490C2×D4—1224no—
491C2×D4—1222no—
492C2×D4—1224no—
493C2×D4—1221no—
494C2×D4—1222no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—44no—
498C22×C4—1228no—
499C22×C4—1221no—
500C22×C4—1222no—
501C22×C4—1224no—
502C22×C4—1222no—
503C22×C4—1224no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1224no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—4884no—
511C22×C4—2442no—
512C22×C4—2441no—
513C22×C4—2441no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4C2×C4×D6112yes—
517C22×C4C2×D24414yes—
518C22×C4C2×D24411yescharacteristic
519C22×C4#1175822212yes—
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4D4×D6112yes—
523C22×C4D4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525C22×C4#1175822612yes—
526C22×C4C23×D6111yescharacteristic
527Q16—12216no—
528SD16—12216no—
529M16—22no—
530M16—24no—
531M16—22no—
532M16—22no—
533M16—42no—
534M16C2×C4×D6114yescomplemented
535C2×C8—22no—
536C2×C8—24no—
537C2×C8—22no—
538C2×C8—42no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—612no—
544C4⋊C4—612no—
545C4⋊C4—1228no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C22⋊C4—1228no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C42—1228no—
552C42—1222no—
553C42—1222no—
554C42—1222no—
555C42—1222no—
556C42—2442no—
557C23—22no—
558C23—22no—
559C23—1222no—
560C23—1224no—
561C23—2442no—
562C23—2442no—
563C23C4×D24412yes—
564C23#1451974711yescharacteristic
565C23#1451974711yescharacteristic
566C23#1451975212yes—
567C23C22×D24411yescharacteristic
568Q8—614no—
569Q8—1222no—
570Q8—1224no—
571Q8—1224no—
572Q8—1224no—
573Q8—2442no—
574Q8—2442no—
575Q8—2444no—
576Q8—2444no—
577D4—614no—
578D4—1224no—
579D4—1224no—
580D4—1228no—
581D4—1222no—
582D4—1224no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—82no—
587C2×C4—1222no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1228no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—4888no—
597C2×C4—2448no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2444no—
605C2×C4—2444no—
606C2×C4C22⋊C4×D6112yes—
607C2×C4C22⋊C4×D6112yes—
608C2×C4#1451973912yes—
609C2×C4#1451973912yes—
610C2×C4C4×D24412yes—
611C2×C4#1451974711yescharacteristic
612C2×C4#1451974711yescharacteristic
613C2×C4#1451975012yes—
614C2×C4#1451975012yes—
615C2×C4#1451975212yes—
616C2×C4#1451976212yes—
617C2×C4#1451976212yes—
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4C2×#1175822211yescharacteristic
623C2×C4C2×#1175822611yescharacteristic
624C8—24no—
625C8—42no—
626C22—24no—
627C22—24no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—1224no—
633C22—2448no—
634C22—2442no—
635C22—2442no—
636C22#1683964012yes—
637C22#1683964012yes—
638C22#1683994212yes—
639C22#1683994712yes—
640C22C2×C4×D24411yescharacteristic
641C22C2×#1451974711yescharacteristic
642C22C2×#1451975211yescharacteristic
643C4—44no—
644C4—1224no—
645C4—2444no—
646C4—2444no—
647C4—2442no—
648C4—2442no—
649C4#1683964012yes—
650C4#1683964012yes—
651C4C2×C22⋊C4×D6111yescharacteristic
652C4C2×#1451973911yescharacteristic
653C4C2×#1451975011yescharacteristic
654C4C2×#1451976211yescharacteristic
655C2—21no—
656C2—21no—
657C2—22no—
658C2—2444no—
659C2#2144618714yesdirect product factor
660C2C2×#1683964011yescharacteristic
661C2C2×#1683994211yescharacteristic
662C2C2×#1683994711yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC23×C61#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#343D61
hypocenterC61C2×#1882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#16839772C2semidirect
5. C2×#16839667C2semidirect
6. #21446187C2direct (N × H)
7. C2×#16839523C2semidirect
8. C2×C61×#343C2semidirect
13. C2×#14519810C4semidirect
31. #16839772C22semidirect
35. #16839667C22semidirect
37. #16839523C22semidirect
41. C61×#343C22semidirect
58. C22×Dic122C2×C4semidirect
69. #14519810C2×C4semidirect
110. C2×M16×C61C2×C4semidirect
193. C2×Dic122C22×C4semidirect
200. M16×C61C22×C4semidirect
330. C2×#343D61semidirect
335. C2×C61#1882semidirect
371. #343C2×D61semidirect
373. C61C2×#1882semidirect
413. C2×M16C4×D61semidirect
534. M16C2×C4×D61semidirect
659. C2#21446187direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168402432
C22×#145197032
C2×#168397722
C2×#168396672
#214461872
C2×#168395232
C2×C61×#3432
C2×#188261