Finite Group Databasebeta

← D61×#1804 #29679462

Subgroups — D61×#1804 #29679462

Order 15616.

1942 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D61C211yesmaximal, characteristic
2C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
3C2×#102×D61C212yesmaximal
4#382×D61C211yesmaximal, characteristic, complemented
5#21445830C211yesmaximal, characteristic, complemented
6#21445829C211yesmaximal, characteristic, complemented
7#21445828C212yesmaximal, complemented
8#21445827C212yesmaximal, complemented
9#21445826C211yesmaximal, characteristic, complemented
10#340×D61C212yesmaximal, complemented
11C61×#1804C211yesmaximal, characteristic, complemented
12C22×D4×D61C2211yescharacteristic
13C23×C4×D61—21no—
14C22×C8×D61C2212yes—
15C2×C4×D244C2211yescharacteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×C4×C22⋊D61—21no—
19C2×#14519737—21no—
20C4×D4×D61—24no—
21C4×D4×D61C418yescomplemented
22C2×C4⋊C4×D61C2212yes—
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C22⋊C4×D61—21no—
25C2×C42×D61C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic, complemented
27C2×#14519706C2212yescomplemented
28C4⋊C8×D61—24no—
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32#102×D61—28no—
33#16839628C2211yescharacteristic, complemented
34#16839627C2212yescomplemented
35#16839626—21no—
36#16839626—21no—
37#95×D61—21no—
38#95×D61—21no—
39#16839525C2212yescomplemented
40#16839524C2212yescomplemented
41#16839524C2212yescomplemented
42C2×C4×D4×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic, complemented
44C2×#102×C61C2212yes—
45C61×#382C2211yescharacteristic, complemented
46C61×#340C2212yescomplemented
47C24×D61—21no—
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C2×D4×D61—22no—
51C2×D4×D61—22no—
52C2×D4×D61—22no—
53C2×D4×D61C2×C414yes—
54C2×D4×D61D414yes—
55C2×D4×D61Q814yes—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58C22×C4×D61—21no—
59C22×C4×D61—21no—
60C22×C4×D61—28no—
61C22×C4×D61—22no—
62C22×C4×D61—48no—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2312yes—
67C2×C8×D61—24no—
68C2×C8×D61—24no—
69C2×C8×D61—24no—
70C4×D244—22no—
71C4×D244C2×C414yescomplemented
72D4×Dic61—22no—
73D4×Dic61C2×C414yescomplemented
74#14519739—22no—
75#14519739C2×C414yescomplemented
76C4×C22⋊D61—28no—
77#14519737—28no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—22no—
82C4⋊C4×D61—22no—
83C4⋊C4×D61—24no—
84C4⋊C4×D61—24no—
85C4⋊C4×D61C2×C414yes—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C22⋊C4×D61—28no—
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C42×D61C2×C414yes—
94C42×D61D414yes—
95C4⋊#92418—22no—
96#14519706—24no—
97#14519699—24no—
98#14519698—24no—
99#14519697—24no—
100#14519684—21no—
101#14519684—21no—
102#14519684—22no—
103#14519684—22no—
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C22×C8×C61C2312yes—
107C4×D4×C61—22no—
108C4×D4×C61C2×C414yescomplemented
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61—22no—
114#102×C61—24no—
115#95×C61—21no—
116#95×C61—21no—
117C24×C61—21no—
118C23×D61—21no—
119C23×D61—24no—
120C23×D61—24no—
121C23×D61—24no—
122C23×D61—44no—
123C23×D61—44no—
124C23×D61C2×D411yescharacteristic
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61C22×C412yes—
129C2×D4×C61C2×D412yes—
130C2×D4×C61C2×Q812yes—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—44no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—24no—
141C22×Dic61—21no—
142C22×Dic61—24no—
143C22×Dic61—22no—
144C22×Dic61—44no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148D4×D61—28no—
149D4×D61—28no—
150D4×D61—28no—
151D4×D61C4⋊C4116yes—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C22×C412yes—
156C2×D244C2×D412yes—
157C2×D244C2×Q812yes—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—432no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×C4×D61C22⋊C414yes—
171C2×C4×D61C22⋊C414yes—
172C2×C4×D61C22×C412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×Q812yes—
176C2×C4×D61#4112yes—
177C2×C4×D61#4112yes—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—24no—
186C42×C61C22×C412yes—
187C42×C61C2×D412yes—
188C22⋊Dic61—24no—
189#11758198—24no—
190#11758198—24no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61C22×C412yes—
195#11758196—24no—
196#11758196—44no—
197C4×Dic61C22⋊C414yes—
198C4×Dic61C22⋊C414yes—
199C4×Dic61C22×C412yes—
200C4×Dic61C2×D412yes—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C8×D61—48no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—216no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—48no—
218C22×D61—48no—
219C22×D61—416no—
220C22×D61—416no—
221C22×D61C2×C22⋊C412yes—
222C22×D61#12112yes—
223C22×D61C22⋊Q812yes—
224C22×D61#13514yes—
225C22×D61AGL(1,ℤ/8)14yes—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61C2×C4⋊C414yes—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—216no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—48no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61#12112yes—
259C2×Dic61C22⋊Q812yes—
260D244—22no—
261D244—22no—
262D244—22no—
263D244C2×C4⋊C414yes—
264C4×D61—28no—
265C4×D61—28no—
266C4×D61—48no—
267C4×D61—48no—
268C4×D61—48no—
269C4×D61—88no—
270C4×D61C2×C4⋊C414yes—
271C4×D61#11714yes—
272C4×D61#12114yes—
273C4×D61C22⋊Q814yes—
274C8×C61—42no—
275#92418—42no—
276C2×#1804—611nomaximal, Sylow
277C22×C61—24no—
278C22×C61—44no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—44no—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61C2×#12111yescharacteristic
284C22×C61C2×C22⋊Q811yescharacteristic
285C22×C61C2×#13512yes—
286C22×C61C2×AGL(1,ℤ/8)12yes—
287C2×D61—24no—
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—44no—
292C2×D61—432no—
293C2×D61#38014yes—
294C2×D61#41614yes—
295C2×D61#41814yes—
296C2×D61#46318yes—
297C2×D61#47818yes—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—82no—
302C4×C61C22×C4⋊C411yescharacteristic
303C4×C61C2×#11711yescharacteristic
304C4×C61C2×#12111yescharacteristic
305C4×C61C2×C22⋊Q811yescharacteristic
306Dic61—22no—
307Dic61—22no—
308Dic61—42no—
309Dic61—82no—
310Dic61#38014yes—
311C22×C4×D4—611no—
312C22×C4⋊C8—611no—
313C22×#102—612no—
314C2×#382—611no—
315#1804—611no—
316#1804—612no—
317#1804—611no—
318#1804—612no—
319#1804—611no—
320#1804D6111yescharacteristic, direct product factor
321C2×#340—612no—
322C2×C61—44no—
323C2×C61C2×#38011yescharacteristic
324C2×C61C2×#41611yescharacteristic
325C2×C61C2×#41811yescharacteristic
326C2×C61C2×#46312yes—
327C2×C61C2×#47812yes—
328D61—44no—
329D61#180418yesdirect product factor
330C23×D4—611no—
331C24×C4—1221no—
332C23×C8—612no—
333C2×C4×D4—611no—
334C2×C4×D4—618no—
335C2×C4×D4—611no—
336C2×C4×D4—611no—
337C2×C4×D4—1221no—
338C2×C4×D4—1224no—
339C2×C4×D4—1221no—
340C2×C4×D4C2×D6111yescharacteristic
341C22×C4⋊C4—611no—
342C22×C4⋊C4—612no—
343C22×C22⋊C4—1221no—
344C22×C42—611no—
345C2×C4⋊C8—612no—
346C2×C4⋊C8—611no—
347C2×C4⋊C8—1224no—
348C2×C4⋊C8C2×D6111yescharacteristic, complemented
349C2×#102—612no—
350C2×#102—612no—
351C2×#102—612no—
352C2×#102—1228no—
353C2×#102C2×D6112yes—
354#382—612no—
355#382—611no—
356#382—1221no—
357#382—1221no—
358#382C2×D6111yescharacteristic, complemented
359C2×#95—1221no—
360C2×#95—1221no—
361#340—612no—
362#340—612no—
363#340—612no—
364#340C2×D6112yescomplemented
365C61C2×#180411yescharacteristic, complemented, Sylow
366C25—1221no—
367C22×D4—614no—
368C22×D4—614no—
369C22×D4—614no—
370C22×D4—611no—
371C22×D4—1222no—
372C22×D4—1222no—
373C22×D4—1222no—
374C22×D4—1221no—
375C22×D4C22×D6111yescharacteristic
376C23×C4—21no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—1221no—
381C23×C4—1228no—
382C23×C4—1221no—
383C23×C4—1222no—
384C23×C4—1221no—
385C23×C4—1221no—
386C23×C4—2448no—
387C22×C8—612no—
388C22×C8—1224no—
389C22×C8—1224no—
390C22×C8—1224no—
391C22×C8C22×D6112yes—
392C4×D4—22no—
393C4×D4—614no—
394C4×D4—614no—
395C4×D4—614no—
396C4×D4—1228no—
397C4×D4—1222no—
398C4×D4—1228no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4C4×D6114yescomplemented
402C2×C4⋊C4—611no—
403C2×C4⋊C4—614no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1224no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4C22×D6112yes—
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1228no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—614no—
421C2×C42—614no—
422C2×C42C22×D6111yescharacteristic
423C4⋊C8—22no—
424C4⋊C8—1224no—
425C4⋊C8—1222no—
426#102—24no—
427#102—1224no—
428#102—1224no—
429#102—1224no—
430#95—21no—
431#95—21no—
432#95—1222no—
433#95—1222no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C24—611no—
438C24—1224no—
439C24—1224no—
440C24—1224no—
441C24—1221no—
442C24—2444no—
443C24—2444no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—6116no—
448C2×D4—612no—
449C2×D4—612no—
450C2×D4—612no—
451C2×D4—1228no—
452C2×D4—1228no—
453C2×D4—1228no—
454C2×D4—1224no—
455C2×D4—1224no—
456C2×D4—1224no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4C2×C4×D6112yes—
461C2×D4D4×D6112yes—
462C2×D4Q8×D6112yes—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—611no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—614no—
477C22×C4—614no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1224no—
482C22×C4—1224no—
483C22×C4—1221no—
484C22×C4—1224no—
485C22×C4—1222no—
486C22×C4—2444no—
487C22×C4—24432no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4—2444no—
492C22×C4—2444no—
493C22×C4—2444no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—2448no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—612no—
508C4⋊C4—1224no—
509C4⋊C4—1221no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—2444no—
513C4⋊C4C2×C4×D6112yes—
514C22⋊C4—24no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C42—612no—
519C42—612no—
520C42—614no—
521C42—614no—
522C42C2×C4×D6112yes—
523C42D4×D6112yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—612no—
530C23—612no—
531C23—612no—
532C23—614no—
533C23—614no—
534C23—12216no—
535C23—1222no—
536C23—1222no—
537C23—1222no—
538C23—2448no—
539C23—24416no—
540C23—24416no—
541C23—2448no—
542C23—2442no—
543C23—2442no—
544C23C2×D4×D6111yescharacteristic
545D4—22no—
546D4—22no—
547D4—22no—
548D4—614no—
549D4—12216no—
550D4—1222no—
551D4—1222no—
552D4—1222no—
553D4C4⋊C4×D6114yes—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—614no—
566C2×C4—614no—
567C2×C4—614no—
568C2×C4—614no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1228no—
572C2×C4—1228no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—4888no—
577C2×C4—2448no—
578C2×C4—2448no—
579C2×C4—2448no—
580C2×C4—2448no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4C22⋊C4×D6112yes—
586C2×C4C22⋊C4×D6112yes—
587C2×C4C22×C4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4C2×Q8×D6111yescharacteristic
591C2×C4#41×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C8—42no—
594C8—2442no—
595C22—24no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—618no—
601C22—618no—
602C22—614no—
603C22—614no—
604C22—614no—
605C22—1224no—
606C22—24432no—
607C22—2444no—
608C22—2444no—
609C22—2442no—
610C22—2442no—
611C22C2×C22⋊C4×D6111yescharacteristic
612C22#121×D6111yescharacteristic
613C22C22⋊Q8×D6111yescharacteristic
614C22#135×D6112yes—
615C22AGL(1,ℤ/8)×D6112yes—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—82no—
620C4—614no—
621C4—1222no—
622C4—1222no—
623C4—4882no—
624C4—2442no—
625C4C2×C4⋊C4×D6111yescharacteristic
626C4#117×D6111yescharacteristic
627C4#121×D6111yescharacteristic
628C4C22⋊Q8×D6111yescharacteristic
629C2—44no—
630C2—618no—
631C2—2444no—
632C2#380×D6111yescharacteristic
633C2#416×D6111yescharacteristic
634C2#418×D6111yescharacteristic
635C2#463×D6112yes—
636C2#478×D6112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1804D61
hypocenterC61C2×#1804
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C8×D61C2semidirect
4. #382×D61C2semidirect
5. #21445830C2semidirect
6. #21445829C2semidirect
7. #21445828C2semidirect
8. #21445827C2semidirect
9. #21445826C2semidirect
10. #340×D61C2semidirect
11. C61×#1804C2semidirect
21. C4×D4×D61C4semidirect
26. C2×C4⋊#92418C22semidirect
27. C2×#14519706C22semidirect
33. #16839628C22semidirect
34. #16839627C22semidirect
39. #16839525C22semidirect
40. #16839524C22semidirect
41. #16839524C22semidirect
43. C2×C4⋊C8×C61C22semidirect
45. C61×#382C22semidirect
46. C61×#340C22semidirect
71. C4×D244C2×C4semidirect
73. D4×Dic61C2×C4semidirect
75. #14519739C2×C4semidirect
108. C4×D4×C61C2×C4semidirect
320. #1804D61direct (N × H)
329. D61#1804direct (N × H)
348. C2×C4⋊C8C2×D61semidirect
358. #382C2×D61semidirect
364. #340C2×D61semidirect
365. C61C2×#1804semidirect
401. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×D612
C2×C4⋊C8×D612
C2×#102×D612
#382×D612
#214458302
#214458292
#214458282
#214458272
#214458262
#340×D612
C61×#18042
C2×#180461