Finite Group Databasebeta

← D61×#2101 #29709911

Subgroups — D61×#2101 #29709911

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#389×D61C211yesmaximal, characteristic
2#382×D61C211yesmaximal, characteristic
3#21447392C211yesmaximal, characteristic, complemented
4#21447391C211yesmaximal, characteristic, complemented
5#21447390C211yesmaximal, characteristic, complemented
6#21447389C211yesmaximal, characteristic, complemented
7#21447388C211yesmaximal, characteristic, complemented
8#21447387C211yesmaximal, characteristic, complemented
9#21447386C211yesmaximal, characteristic, complemented
10#340×D61C211yesmaximal, characteristic
11#340×D61C211yesmaximal, characteristic
12#340×D61C211yesmaximal, characteristic
13#340×D61C211yesmaximal, characteristic
14#336×D61C211yesmaximal, characteristic
15C61×#2101C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4⋊C4×D61—21no—
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839661C2211yescharacteristic
25#16839660C2211yescharacteristic
26#16839659—21no—
27#16839658C2211yescharacteristic
28#16839657—21no—
29#16839628C2211yescharacteristic
30#16839627C2211yescharacteristic
31#16839627C2211yescharacteristic
32#16839626—21no—
33#16839626—21no—
34#95×D61—21no—
35#95×D61—21no—
36#95×D61—21no—
37#16839525C2211yescharacteristic
38#16839525C2211yescharacteristic
39#16839525C2211yescharacteristic
40#16839525C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#382C2211yescharacteristic
54C61×#340C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C2×C8×D61—22no—
69C2×C8×D61—22no—
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61—22no—
73C2×C8×D61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—44no—
93C4⋊C4×D61Q814yes—
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97C42×D61—22no—
98C42×D61—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119C23×D61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×C4×D61—24no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61C2×Q812yes—
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C2×C4×D61#4112yes—
156C2×C4×D61#4112yes—
157C2×C4×D61#4112yes—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61C2×Q812yes—
173C42×C61—21no—
174C42×C61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61C2×Q812yes—
184#11758196—24no—
185#11758196—42no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C8×D61—44no—
201C8×D61—44no—
202C23×C61C22×D411yescharacteristic
203C22×D61C22⋊Q812yes—
204C22×D61#12312yes—
205C22×D61#12412yes—
206C22×D61#13512yes—
207C22×D61#13512yes—
208C22×D61AGL(1,ℤ/8)12yes—
209C22×D61#13712yes—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61C22×Q811yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61C22⋊Q812yes—
253C2×Dic61#12312yes—
254C2×Dic61#12412yes—
255C4×D61—48no—
256C4×D61—44no—
257C4×D61—44no—
258C4×D61—84no—
259C4×D61—84no—
260C4×D61—84no—
261C4×D61#12514yes—
262C4×D61#12514yes—
263C4×D61#12614yes—
264C4×D61#12614yes—
265C8×C61—41no—
266C8×C61—41no—
267#92418—41no—
268#92418—41no—
269C2×#2101—611nomaximal, Sylow
270C22×C61C2×C22⋊Q811yescharacteristic
271C22×C61C2×#12311yescharacteristic
272C22×C61C2×#12411yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61C2×#13511yescharacteristic
275C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C2×D61#40014yes—
278C2×D61#47814yes—
279C2×D61#47914yes—
280C2×D61#48214yes—
281C2×D61#48514yes—
282C2×D61#48814yes—
283C2×D61#48914yes—
284C4×C61—42no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×#12511yescharacteristic
291C4×C61C2×#12511yescharacteristic
292C4×C61C2×#12611yescharacteristic
293C4×C61C2×#12611yescharacteristic
294Dic61—42no—
295Dic61—41no—
296Dic61—41no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300Dic61#40014yes—
301C2×#389—611no—
302C2×#382—611no—
303#2101—611no—
304#2101—611no—
305#2101—611no—
306#2101—611no—
307#2101—611no—
308#2101—611no—
309#2101—611no—
310#2101D6111yescharacteristic, direct product factor
311C2×#340—611no—
312C2×#340—611no—
313C2×#340—611no—
314C2×#340—611no—
315C2×#336—611no—
316C2×C61C2×#40011yescharacteristic
317C2×C61C2×#47811yescharacteristic
318C2×C61C2×#47911yescharacteristic
319C2×C61C2×#48211yescharacteristic
320C2×C61C2×#48511yescharacteristic
321C2×C61C2×#48811yescharacteristic
322C2×C61C2×#48911yescharacteristic
323D61#210118yesdirect product factor
324C23×C8—611no—
325C23×C8—611no—
326C22×C4⋊C4—611no—
327C22×C4⋊C4—611no—
328C22×C4⋊C4—611no—
329C22×C4⋊C4—611no—
330C22×C4⋊C4—1221no—
331C22×C42—611no—
332#389—611no—
333#389—611no—
334#389—611no—
335#389—1221no—
336#389—1221no—
337#389C2×D6111yescharacteristic
338#382—611no—
339#382—611no—
340#382—611no—
341#382—1221no—
342#382—1221no—
343#382C2×D6111yescharacteristic
344C2×#95—1221no—
345C2×#95—1221no—
346C2×#95—1221no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340C2×D6111yescharacteristic
360#340C2×D6111yescharacteristic
361#340C2×D6111yescharacteristic
362#340C2×D6111yescharacteristic
363#336—611no—
364#336—611no—
365#336—611no—
366#336C2×D6111yescharacteristic
367C61C2×#210111yescharacteristic, complemented, Sylow
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—1221no—
372C23×C4—1221no—
373C23×C4—1221no—
374C23×C4—1221no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8—1222no—
378C22×C8—1222no—
379C22×C8—1222no—
380C22×C8—1222no—
381C22×C8—1222no—
382C22×C8—1222no—
383C22×C8C22×D6111yescharacteristic
384C22×C8C22×D6111yescharacteristic
385C2×C4⋊C4—21no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—614no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2444no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—1222no—
414C2×C42—1222no—
415C2×C42C22×D6111yescharacteristic
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428C24—611no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1224no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4C23×D6111yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—2444no—
480C2×C8—2444no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—612no—
490C4⋊C4—1224no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4Q8×D6112yes—
505C42—21no—
506C42—21no—
507C42—1222no—
508C42—1222no—
509C42—1221no—
510C42—1221no—
511C23—612no—
512C23—612no—
513C23—612no—
514C23—612no—
515C23—612no—
516C23—612no—
517C23—612no—
518C23C2×D4×D6111yescharacteristic
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—614no—
541C2×C4—614no—
542C2×C4—614no—
543C2×C4—614no—
544C2×C4—1222no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—4884no—
552C2×C4—4884no—
553C2×C4—4884no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2448no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C2×Q8×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C8—41no—
575C8—41no—
576C8—2441no—
577C8—2441no—
578C22—614no—
579C22—614no—
580C22—614no—
581C22—614no—
582C22—614no—
583C22—614no—
584C22—614no—
585C22C22⋊Q8×D6111yescharacteristic
586C22#123×D6111yescharacteristic
587C22#124×D6111yescharacteristic
588C22#135×D6111yescharacteristic
589C22#135×D6111yescharacteristic
590C22AGL(1,ℤ/8)×D6111yescharacteristic
591C22#137×D6111yescharacteristic
592C4—41no—
593C4—41no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—614no—
599C4—4881no—
600C4—4881no—
601C4—4881no—
602C4—2441no—
603C4—2441no—
604C4—2442no—
605C4#125×D6111yescharacteristic
606C4#125×D6111yescharacteristic
607C4#126×D6111yescharacteristic
608C4#126×D6111yescharacteristic
609C2—618no—
610C2#400×D6111yescharacteristic
611C2#478×D6111yescharacteristic
612C2#479×D6111yescharacteristic
613C2#482×D6111yescharacteristic
614C2#485×D6111yescharacteristic
615C2#488×D6111yescharacteristic
616C2#489×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2101C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2101D61
hypocenterC61C2×#2101
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #21447392C2semidirect
4. #21447391C2semidirect
5. #21447390C2semidirect
6. #21447389C2semidirect
7. #21447388C2semidirect
8. #21447387C2semidirect
9. #21447386C2semidirect
15. C61×#2101C2semidirect
310. #2101D61direct (N × H)
323. D61#2101direct (N × H)
367. C61C2×#2101semidirect

Maximal subgroups

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

subgroupindex
#389×D612
#382×D612
#214473922
#214473912
#214473902
#214473892
#214473882
#214473872
#214473862
#340×D612
#336×D612
C61×#21012
C2×#210161