Finite Group Databasebeta

← C2×#21461770 #30615467

Subgroups — C2×#21461770 #30615467

Order 15616.

1662 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519856C211yesmaximal, characteristic
2C22×#14519846C211yesmaximal, characteristic, complemented
3C22×#14519845C211yesmaximal, characteristic, complemented
4C2×#16840704C211yesmaximal, characteristic
5#21461770C218yesmaximal, direct product factor
6C2×#16840636C211yesmaximal, characteristic, complemented
7C2×#16840636C211yesmaximal, characteristic, complemented
8C2×C22⋊C8×D61C211yesmaximal, characteristic
9C2×#14519856—24no—
10C2×#14519856—24no—
11C2×#14519856—22no—
12C2×#14519856C2214yes—
13C22×C4⋊#6923C2211yescharacteristic
14C22×C4⋊#6923C2211yescharacteristic
15C22×C4⋊#6923C2211yescharacteristic
16C22×C4⋊#6923C2211yescharacteristic
17C22×C22⋊#6923—21no—
18C22×C22⋊#6923—21no—
19C22×C4×#6923—21no—
20C2×#14519846—24no—
21C2×#14519846C2214yescomplemented
22C2×#14519845—24no—
23C2×#14519845C2214yescomplemented
24#16840704—22no—
25#16840704C2214yes—
26C2×(C2×#6923)⋊C4—21no—
27#16840636C2214yescomplemented
28#16840636C2214yescomplemented
29C23×C4×D61C2211yescharacteristic
30C22×C8×D61C2211yescharacteristic
31C22×C8×D61C2211yescharacteristic
32C2×C22⋊#92418C411yescharacteristic, complemented
33C2×#14519689C411yescharacteristic, complemented
34C2×#14519689C411yescharacteristic, complemented
35C22⋊C8×D61C414yescomplemented
36C22⋊C8×D61C2214yes—
37C2×C22⋊C8×C61C411yescharacteristic, complemented
38C23×#6923—21no—
39C23×#6923—21no—
40C23×#6923—21no—
41C23×#6923—21no—
42#14519856—216no—
43#14519856—28no—
44#14519856—28no—
45C2×C4⋊#6923—21no—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923—22no—
48C2×C4⋊#6923—22no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923D414yes—
56C2×C4⋊#6923D414yes—
57C2×C4⋊#6923C2312yes—
58C2×C4⋊#6923C2312yes—
59C2×C4⋊#6923C2312yes—
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—28no—
62C2×C22⋊#6923—24no—
63C2×C22⋊#6923—24no—
64C2×C22⋊#6923—44no—
65C2×C4×#6923—28no—
66C2×C4×#6923—24no—
67#14519846—28no—
68#14519845—28no—
69(C2×#6923)⋊C4—24no—
70C24×D61D411yescharacteristic
71C23×Dic61Q811yescharacteristic
72C22×C4×D61—22no—
73C22×C4×D61—22no—
74C22×C4×D61—22no—
75C22×C4×D61—22no—
76C22×C4×D61C2×C412yes—
77C22×C4×D61C2×C411yescharacteristic
78C22×C4×D61D412yes—
79C22×C4×D61D411yescharacteristic
80C22×C4×D61Q811yescharacteristic
81C22×C4×D61Q812yes—
82C22×C4×D61C2312yes—
83C22×C4×D61C2311yescharacteristic
84C22×#92418C2×C411yescharacteristic
85C22×#92418C2×C411yescharacteristic
86C2×C8×D61—22no—
87C2×C8×D61—22no—
88C2×C8×D61—22no—
89C2×C8×D61—22no—
90C2×C8×D61C2×C412yes—
91C2×C8×D61C2×C412yes—
92C2×C8×D61C2312yes—
93C2×C8×D61C2312yes—
94C22⋊#92418C2×C414yescomplemented
95#14519689C2×C414yescomplemented
96#14519689C2×C414yescomplemented
97C23×C4×C61C2×C411yescharacteristic
98C22×C8×C61C2×C411yescharacteristic
99C22×C8×C61C2×C411yescharacteristic
100C22⋊C8×C61C2×C414yescomplemented
101C24×C61C4⋊C411yescharacteristic
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61C4⋊C411yescharacteristic
109C23×D61C4⋊C412yes—
110C23×D61C2×D412yes—
111C23×D61C2×D411yescharacteristic
112C22×#6923—24no—
113C22×#6923—24no—
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923—22no—
118C22×#6923—42no—
119C22×#6923—42no—
120C22×#6923—42no—
121C22×#6923—42no—
122C22×#6923—42no—
123C22×#6923—42no—
124C22×#6923—42no—
125C22×#6923—42no—
126C22×#6923—42no—
127C22×#6923—42no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61C4⋊C411yescharacteristic
131C22×C4×C61C4⋊C412yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C22×C411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61C4⋊C411yescharacteristic
137C22×Dic61C4⋊C412yes—
138C22×Dic61C2×Q812yes—
139C22×Dic61C2×Q811yescharacteristic
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—24no—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61C4⋊C412yes—
157C2×C4×D61C4⋊C412yes—
158C2×C4×D61C4⋊C412yes—
159C2×C4×D61C22×C412yes—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×Q812yes—
166C2×C4×D61C2×Q811yescharacteristic
167C2×C4×D61#4112yes—
168C2×C4×D61#4112yes—
169C2×C4×D61C2411yescharacteristic
170C22⋊#6923—216no—
171C22⋊#6923—48no—
172C4⋊#6923—24no—
173C4⋊#6923—24no—
174C4⋊#6923—22no—
175C4⋊#6923—22no—
176C4⋊#6923—22no—
177C4⋊#6923—22no—
178C4⋊#6923—22no—
179C4⋊#6923—22no—
180C4⋊#6923—42no—
181C4⋊#6923—42no—
182C4⋊#6923C2×D414yes—
183C4⋊#6923C2×D414yes—
184C4×#6923—216no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C2×C8×C61C22×C412yes—
188C2×C8×C61C22×C412yes—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418C22×C412yes—
192C2×#92418C22×C412yes—
193C8×D61—24no—
194C8×D61—24no—
195C8×D61—22no—
196C8×D61—22no—
197C8×D61—22no—
198C8×D61—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61C2×C4⋊C412yes—
203C23×C61C2×C4⋊C411yescharacteristic
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61C2×C4⋊C412yes—
221C22×D61C2×C4⋊C411yescharacteristic
222C22×D61#12012yes—
223C22×D61#12312yes—
224C22×D61#13512yes—
225C22×D61#13512yes—
226C22×D61AGL(1,ℤ/8)12yes—
227C22×D61#13712yes—
228C22×D61C22×D411yescharacteristic
229C2×#6923—24no—
230C2×#6923—24no—
231C2×#6923—42no—
232C2×#6923—42no—
233C2×#6923—42no—
234C2×#6923—42no—
235C2×#6923—48no—
236C2×#6923—48no—
237C2×#6923—42no—
238C2×#6923—42no—
239C2×#6923—42no—
240C2×#6923—42no—
241C2×#6923—42no—
242C2×#6923—42no—
243C2×#6923—84no—
244C2×#6923—84no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—24no—
249C2×C4×C61C2×C4⋊C412yes—
250C2×C4×C61C2×C4⋊C411yescharacteristic
251C2×C4×C61C4×D412yes—
252C2×C4×C61C4×D412yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61C2×C4⋊C412yes—
259C2×Dic61C2×C4⋊C411yescharacteristic
260C2×Dic61C22⋊Q812yes—
261C2×Dic61C22⋊Q812yes—
262C2×Dic61C22×Q811yescharacteristic
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—24no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—24no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—24no—
272C4×D61C4×D412yes—
273C4×D61C4×D412yes—
274C4×D61#12012yes—
275C4×D61C22⋊Q812yes—
276C4×D61C22⋊Q812yes—
277C4×D61#12312yes—
278C4×D61C22×D411yescharacteristic
279C4×D61C22×D411yescharacteristic
280C4×D61C2×#4111yescharacteristic
281C4×D61C2×#4111yescharacteristic
282C8×C61—22no—
283C8×C61—22no—
284#92418—22no—
285#92418—22no—
286C2×#1854—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—44no—
291C22×C61#38512yes—
292C22×C61#42712yes—
293C22×C61#42812yes—
294C22×C61C22×C4⋊C411yescharacteristic
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—44no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—48no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61#38512yes—
305C2×D61#42712yes—
306C2×D61#42812yes—
307C2×D61#44914yes—
308C2×D61#45214yes—
309C2×D61#48214yes—
310C2×D61#48514yes—
311C2×D61C2×#12011yescharacteristic
312C2×D61C2×#12311yescharacteristic
313C2×D61C2×#13511yescharacteristic
314C2×D61C2×#13511yescharacteristic
315C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
316C2×D61C2×#13711yescharacteristic
317#6923—44no—
318#6923—44no—
319#6923—82no—
320#6923—82no—
321C4×C61—22no—
322C4×C61#38512yes—
323C4×C61C2×C4×D411yescharacteristic
324C4×C61C2×C4×D411yescharacteristic
325Dic61—22no—
326Dic61#38512yes—
327Dic61C2×C22⋊Q811yescharacteristic
328Dic61C2×C22⋊Q811yescharacteristic
329C22×#117—611no—
330C22×#107—611no—
331C22×#106—611no—
332C22×C22⋊C8—611no—
333C2×#380—611no—
334#1854—618no—
335C2×#340—611no—
336C2×#340—611no—
337C2×C61—42no—
338C2×C61#185414yescomplemented
339C2×C61C2×#38511yescharacteristic
340C2×C61C2×#42711yescharacteristic
341C2×C61C2×#42811yescharacteristic
342D61—42no—
343D61#185414yescomplemented
344D61C2×#44911yescharacteristic
345D61C2×#45211yescharacteristic
346D61C2×#48211yescharacteristic
347D61C2×#48511yescharacteristic
348C24×C4—611no—
349C23×C8—611no—
350C23×C8—611no—
351C2×#117—614no—
352C2×#117—1224no—
353C2×#117—1224no—
354C2×#117—1222no—
355C22×C4⋊C4—611no—
356C22×C4⋊C4—611no—
357C22×C4⋊C4—611no—
358C22×C4⋊C4—611no—
359C22×C22⋊C4—1221no—
360C22×C22⋊C4—1221no—
361C22×C42—1221no—
362C2×#107—614no—
363C2×#107—1224no—
364C2×#106—614no—
365C2×#106—1224no—
366C2×C22⋊C8—611no—
367C2×C22⋊C8—611no—
368C2×C22⋊C8—611no—
369C2×C22⋊C8—614no—
370C2×C22⋊C8—614no—
371C2×C22⋊C8#692311yescharacteristic, complemented
372#380—614no—
373#380—1222no—
374C2×#95—1221no—
375#340—614no—
376#340—614no—
377C61C2×#185411yescharacteristic, complemented, Sylow
378C25—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—612no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—612no—
385C23×C4—612no—
386C23×C4—612no—
387C23×C4—611no—
388C23×C4—1222no—
389C23×C4—1222no—
390C23×C4—1222no—
391C23×C4—1222no—
392C23×C4—1221no—
393C23×C4—1221no—
394C23×C4—1221no—
395C23×C4—1221no—
396C23×C4C2×#692311yescharacteristic
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—612no—
400C22×C8—612no—
401C22×C8—612no—
402C22×C8—612no—
403C22×C8—1222no—
404C22×C8—1222no—
405C22×C8—1222no—
406C22×C8—1222no—
407C22×C8C2×#692311yescharacteristic
408C22×C8C2×#692311yescharacteristic
409#117—1228no—
410#117—1228no—
411#117—12216no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1222no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1222no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C22⋊C4—1224no—
429C2×C22⋊C4—1228no—
430C2×C22⋊C4—1224no—
431C2×C22⋊C4—2444no—
432C2×C42—1224no—
433C2×C42—1228no—
434#107—1228no—
435#106—1228no—
436C22⋊C8—614no—
437C22⋊C8—614no—
438C22⋊C8—614no—
439C22⋊C8C2×#692314yescomplemented
440#95—1224no—
441C24—611no—
442C24—612no—
443C24—612no—
444C24—611no—
445C24—1222no—
446C24—1222no—
447C24—1222no—
448C24—1222no—
449C24—1222no—
450C24—1222no—
451C24C4⋊#692311yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—611no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1224no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1224no—
491C22×C4—1224no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4C4⋊#692312yes—
507C22×C4C4⋊#692311yescharacteristic
508C22×C4C22×#692312yes—
509C22×C4C22×#692311yescharacteristic
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—612no—
513C2×C8—612no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—1224no—
517C2×C8—1224no—
518C2×C8—1222no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8C22×#692312yes—
523C2×C8C22×#692312yes—
524C4⋊C4—614no—
525C4⋊C4—614no—
526C4⋊C4—1224no—
527C4⋊C4—1224no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C22⋊C4—12216no—
537C22⋊C4—2448no—
538C42—12216no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—612no—
543C23—612no—
544C23—612no—
545C23—612no—
546C23—612no—
547C23—612no—
548C23—611no—
549C23—611no—
550C23—612no—
551C23—1222no—
552C23—1222no—
553C23—1222no—
554C23—1224no—
555C23—1222no—
556C23—1222no—
557C23—1224no—
558C23—1222no—
559C23—1224no—
560C23—1222no—
561C23—1222no—
562C23—1222no—
563C23—2444no—
564C23—2444no—
565C23—2444no—
566C23—2444no—
567C23C2×C4⋊#692312yes—
568C23C2×C4⋊#692311yescharacteristic
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—612no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1224no—
597C2×C4—1222no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1224no—
602C2×C4—1224no—
603C2×C4—4884no—
604C2×C4—4884no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2448no—
612C2×C4—2448no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4C2×C4⋊#692312yes—
618C2×C4C2×C4⋊#692311yescharacteristic
619C2×C4D4×#692312yes—
620C2×C4D4×#692312yes—
621C2×C4C23×#692311yescharacteristic
622C8—22no—
623C8—22no—
624C8—1222no—
625C8—1222no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—44no—
630C22—614no—
631C22—614no—
632C22—614no—
633C22—614no—
634C22—611no—
635C22—611no—
636C22—612no—
637C22—611no—
638C22—611no—
639C22—612no—
640C22—611no—
641C22—611no—
642C22—612no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—2442no—
647C22—2442no—
648C22—2442no—
649C22—2442no—
650C22—2448no—
651C22—2444no—
652C22#1684070912yes—
653C22#1684077212yes—
654C22#1684077312yes—
655C22C22×C4⋊#692311yescharacteristic
656C4—22no—
657C4—611no—
658C4—611no—
659C4—612no—
660C4—1222no—
661C4—4882no—
662C4—4882no—
663C4—2444no—
664C4—2444no—
665C4#1684070912yes—
666C4C2×D4×#692311yescharacteristic
667C4C2×D4×#692311yescharacteristic
668C2—42no—
669C2—611no—
670C2—611no—
671C2—611no—
672C2—611no—
673C2—614no—
674C2—2442no—
675C2#2146177014yesdirect product factor
676C2C2×#1684070911yescharacteristic
677C2C2×#1684077211yescharacteristic
678C2C2×#1684077311yescharacteristic
subgroupG/H
centerC23C2×C4⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊C8×C61C4
socleC23×C61C2×C4⋊C4
prime radicalC24×D61D4
cosocleC2×C4×D61C24
hypercenterC2×C22⋊C8#6923
hypocenterC61C2×#1854
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#14519846C2semidirect
3. C22×#14519845C2semidirect
5. #21461770C2direct (N × H)
6. C2×#16840636C2semidirect
7. C2×#16840636C2semidirect
21. C2×#14519846C22semidirect
23. C2×#14519845C22semidirect
27. #16840636C22semidirect
28. #16840636C22semidirect
32. C2×C22⋊#92418C4semidirect
33. C2×#14519689C4semidirect
34. C2×#14519689C4semidirect
35. C22⋊C8×D61C4semidirect
37. C2×C22⋊C8×C61C4semidirect
94. C22⋊#92418C2×C4semidirect
95. #14519689C2×C4semidirect
96. #14519689C2×C4semidirect
100. C22⋊C8×C61C2×C4semidirect
338. C2×C61#1854semidirect
343. D61#1854semidirect
371. C2×C22⋊C8#6923semidirect
377. C61C2×#1854semidirect
439. C22⋊C8C2×#6923semidirect
675. C2#21461770direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198562
C22×#145198462
C22×#145198452
C2×#168407042
#214617702
C2×#168406362
C2×C22⋊C8×D612
C2×#185461