Finite Group Databasebeta

← #29685038

Subgroups — #29685038

Order 15616.

1256 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2C4×M16×D61C211yesmaximal, characteristic, complemented
3C4×#14519735C211yesmaximal, characteristic, complemented
4#21446763C211yesmaximal, characteristic, complemented
5#21446762C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446118C214yesmaximal, complemented
8#104×Dic61C214yesmaximal, complemented
9C61×#1980C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840319C2211yescharacteristic
12#16840317C411yescharacteristic, complemented
13#16840317C411yescharacteristic, complemented
14#16840310C411yescharacteristic, complemented
15#16840310C411yescharacteristic, complemented
16#16840294—21no—
17#16840294—21no—
18#16840293—21no—
19#16840293—21no—
20#16840291C412yescomplemented
21#16840291C412yescomplemented
22#16840272—21no—
23#16840272—21no—
24#16840269C2212yes—
25#41×Dic61C2212yes—
26C4×#11758224—21no—
27C4×#11758224—21no—
28C2×#14519735—21no—
29C2×C42×D61C2211yescharacteristic
30C2×#14519704C2214yes—
31C4×#11758194C2211yescharacteristic
32#16839714—21no—
33M16×Dic61C2212yes—
34C4×#11758189—21no—
35#97×D61—21no—
36C4×C8×D61—21no—
37C4⋊C4×Dic61—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39C4×#11758198C2212yes—
40#16839609—21no—
41#16839607—21no—
42C42×Dic61C2212yes—
43#16839523C2214yes—
44#16839522C2214yes—
45C61×#532C2211yescharacteristic
46C2×#104×C61C2214yes—
47C4×M16×C61C2211yescharacteristic
48C4×C4⋊C4×C61C2211yescharacteristic
49C2×#11758224—21no—
50C2×#11758224—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758194C2312yes—
55C2×#11758189—22no—
56M16×D61—24no—
57M16×D61—24no—
58C2×C8×D61—22no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519770—22no—
62#14519770—22no—
63#14519769C2×C412yes—
64#14519768—21no—
65#14519768—21no—
66#14519764—21no—
67#14519764—21no—
68#14519759—22no—
69#14519759—22no—
70#14519758—22no—
71#14519757—21no—
72#14519757—21no—
73#14519756—21no—
74#14519756—21no—
75#14519753C2×C412yes—
76#14519753C2×C412yes—
77#14519748—21no—
78#14519748—21no—
79#14519748C2×C412yes—
80#14519748C2×C412yes—
81#14519745—22no—
82Q8×Dic61C2×C412yes—
83Q8×Dic61C2×C412yes—
84C4×Dic122—21no—
85C4×Dic122—21no—
86#14519742—21no—
87#14519742—21no—
88D4×Dic61—22no—
89D4×Dic61—22no—
90D4×Dic61—21no—
91D4×Dic61—21no—
92D4×Dic61C2×C412yes—
93D4×Dic61C2×C412yes—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96#14519737—22no—
97#14519737—22no—
98#14519736—22no—
99#14519736C2312yes—
100#14519735—28no—
101#14519735—21no—
102#14519735—21no—
103#14519733—22no—
104#14519733—22no—
105C2×C4⋊Dic61—21no—
106C2×#11758198—22no—
107C2×C4×Dic61—22no—
108C2×C4×Dic61—22no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61C2312yes—
113C2×C4×Dic61C2311yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61C2×C411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61D411yescharacteristic
122#14519704—28no—
123#14519688—21no—
124#14519687—22no—
125C4×#92418—21no—
126C8×Dic61—22no—
127#14519684—21no—
128#14519684—22no—
129C2×#41×C61C2312yes—
130C2×M16×C61C2312yes—
131C4⋊Q8×C61C2×C411yescharacteristic
132#127×C61C2×C411yescharacteristic
133#124×C61C2×C412yes—
134#121×C61—21no—
135#121×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61—21no—
138C2×C4⋊C4×C61—21no—
139C2×C42×C61C2312yes—
140C2×C42×C61C2311yescharacteristic
141#104×C61—28no—
142#97×C61—21no—
143C4×C8×C61—21no—
144#95×C61—21no—
145C23×D61C2×D411yescharacteristic
146#41×C61—24no—
147#41×C61—24no—
148C2×Q8×C61C22×C412yes—
149C2×D4×C61—22no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61C22×C412yes—
153C22×C4×C61—22no—
154C22×C4×C61—22no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166#11758224—44no—
167#11758224—44no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—24no—
171C2×C4×D61—42no—
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22⋊C411yescharacteristic
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×Dic122—21no—
185C2×Dic122—21no—
186M16×C61—22no—
187M16×C61#4114yescomplemented
188C2×C8×C61—22no—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C42×C61—24no—
195C42×C61C22×C411yescharacteristic
196C42×C61C22×C411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204#11758198—28no—
205#11758198—22no—
206#11758198—22no—
207C4⋊Dic61—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61C22×C412yes—
213C4⋊Dic61C22×C412yes—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61—28no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—24no—
223C4×Dic61—44no—
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22⋊C411yescharacteristic
231C4×Dic61C22⋊C411yescharacteristic
232C4×Dic61C22×C411yescharacteristic
233C4×Dic61C22×C411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236#11758194—22no—
237#11758194#4114yescomplemented
238C2×#92418—22no—
239#11758189—28no—
240C8×D61—28no—
241C23×C61—22no—
242C23×C61C22×D411yescharacteristic
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—22no—
246C22×D61—41no—
247C22×D61C2×C22⋊C411yescharacteristic
248C22×D61C2×C22⋊C411yescharacteristic
249Q8×C61—24no—
250D4×C61—24no—
251D4×C61—42no—
252D4×C61—42no—
253D4×C61—42no—
254D4×C61—42no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—22no—
258C2×C4×C61—41no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—44no—
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C23×C411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4112yes—
269C2×C4×C61C2×#4112yes—
270C22⋊D61—44no—
271C22⋊D61—44no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—24no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—44no—
282C2×Dic61—44no—
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285C2×Dic61C2×C22⋊C411yescharacteristic
286C2×Dic61C2×C22⋊C411yescharacteristic
287C2×Dic61C2×C22⋊C411yescharacteristic
288C2×Dic61C2×C22⋊C411yescharacteristic
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61#10414yescomplemented
301Dic122—42no—
302Dic122—42no—
303C8×C61—24no—
304#92418—24no—
305#21382—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61C22×C22⋊C411yescharacteristic
312C22×C61#53211yescharacteristic
313C22×C61#53211yescharacteristic
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61C2×#10412yes—
322C4×C61—21no—
323C4×C61—21no—
324C4×C61—21no—
325C4×C61—44no—
326C4×C61—42no—
327C4×C61C2×#10412yes—
328C4×C61C4×#4112yes—
329C4×C61#52512yes—
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—21no—
334Dic61—44no—
335Dic61—42no—
336Dic61—42no—
337Dic61C2×#10412yes—
338Dic61#42112yes—
339#3444—611no—
340C2×C4×M16—611no—
341C4×#117—611no—
342#1986—611no—
343#1986—611no—
344#1980—611no—
345#1980D6111yescharacteristic, complemented
346#1871—614no—
347C4×#104—614no—
348C2×C61—21no—
349C2×C61—21no—
350C2×C61—82no—
351C2×C61#229711yescharacteristic
352C2×C61C22×#10411yescharacteristic
353C2×C61C2×#42111yescharacteristic
354D61—22no—
355D61—41no—
356C22×M16—612no—
357#533—611no—
358#533—611no—
359#533—611no—
360#532—611no—
361#532—611no—
362#532C2×D6111yescharacteristic
363#529—612no—
364#529—612no—
365#529—1221no—
366#529—1221no—
367#529—1221no—
368#529—1221no—
369#525—612no—
370#525—1221no—
371#525—1221no—
372C4×#41—612no—
373C4×#41—1221no—
374C4×#41—1221no—
375C2×#117—1221no—
376C22×C42—611no—
377C2×#104—614no—
378C2×#104C2×D6114yes—
379C4×M16—612no—
380C4×M16—611no—
381C4×M16—1221no—
382C4×M16—1221no—
383C4×M16C2×D6111yescharacteristic
384C2×#97—1221no—
385C2×C4×C8—1221no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4—1221no—
388C4×C4⋊C4C2×D6111yescharacteristic
389C4×C22⋊C4—612no—
390C4×C22⋊C4—1221no—
391#376—1221no—
392C43—612no—
393#339—614no—
394#339—614no—
395C61#2138211yescharacteristic, complemented, Sylow
396C2×#41—1221no—
397C2×#41—1221no—
398C2×#41C22×D6112yes—
399C23×C4—611no—
400C23×C4—611no—
401C23×C4—611no—
402C2×M16—612no—
403C2×M16—1222no—
404C2×M16—1224no—
405C2×M16—1224no—
406C2×M16C22×D6112yes—
407C22×C8—1222no—
408C4⋊Q8—611no—
409C4⋊Q8—611no—
410C4⋊Q8C4×D6111yescharacteristic
411#127C4×D6111yescharacteristic
412#126—1222no—
413#126—1222no—
414#125—612no—
415#125—1221no—
416#125—1221no—
417#124—1221no—
418#124—1221no—
419#124C4×D6112yes—
420#123—1222no—
421#123—1222no—
422#123—1222no—
423C22⋊Q8—612no—
424C22⋊Q8—612no—
425C22⋊Q8—1221no—
426C22⋊Q8—1221no—
427C22⋊Q8—1221no—
428C22⋊Q8—1221no—
429#121—21no—
430#121—21no—
431#121—612no—
432#121—612no—
433#121—1221no—
434#121—1221no—
435#120—1222no—
436C4×Q8—612no—
437C4×Q8—612no—
438C4×Q8—1221no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1221no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—612no—
445C4×D4—612no—
446C4×D4—1221no—
447C4×D4—1221no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4—1222no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1222no—
454#117—612no—
455#117—1222no—
456#117—1222no—
457#117—1222no—
458#117—1221no—
459#117—1221no—
460#117—1228no—
461C2×C4⋊C4—21no—
462C2×C4⋊C4—1221no—
463C2×C22⋊C4—1222no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—612no—
476C2×C42—1222no—
477C2×C42—1222no—
478C2×C42—1222no—
479C2×C42C22×D6112yes—
480C2×C42C22×D6111yescharacteristic
481#104—28no—
482#104—1228no—
483#97—21no—
484#97—1222no—
485#97—1221no—
486C4×C8—21no—
487C4×C8—1222no—
488C4×C8—1221no—
489#95—21no—
490#95—1222no—
491#95—1221no—
492C24—611no—
493#41—24no—
494#41—24no—
495#41—2444no—
496#41—2444no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8C2×C4×D6112yes—
500C2×D4—22no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4C2×C4×D6112yes—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—1224no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—2442no—
530C22×C4D4×D6111yescharacteristic
531C22×C4D4×D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533M16—22no—
534M16—614no—
535M16—1228no—
536M16—1222no—
537M16#1175822414yescomplemented
538C2×C8—22no—
539C2×C8—1228no—
540C2×C8—1222no—
541C4⋊C4—24no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—612no—
545C4⋊C4—612no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—1222no—
550C4⋊C4—1224no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1228no—
560C22⋊C4—1222no—
561C22⋊C4—1222no—
562C22⋊C4—1222no—
563C22⋊C4—1222no—
564C22⋊C4—1222no—
565C42—24no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—1222no—
579C42—1222no—
580C42—1228no—
581C42—1222no—
582C42—1224no—
583C42—2444no—
584C42C2×C4×D6111yescharacteristic
585C42C2×C4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42D4×D6111yescharacteristic
588C23—22no—
589C23—611no—
590C23—611no—
591C23—1222no—
592C23—1221no—
593C23—1221no—
594C23—2441no—
595C23C2×D4×D6111yescharacteristic
596Q8—24no—
597Q8—2442no—
598Q8—2442no—
599D4—24no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—2444no—
605D4—2444no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—614no—
616C2×C4—611no—
617C2×C4—611no—
618C2×C4—611no—
619C2×C4—611no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1224no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2444no—
639C2×C4—2444no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4C22⋊C4×D6111yescharacteristic
643C2×C4C22⋊C4×D6111yescharacteristic
644C2×C4C22⋊C4×D6111yescharacteristic
645C2×C4C22⋊C4×D6111yescharacteristic
646C2×C4C22×C4×D6111yescharacteristic
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4#41×D6112yes—
649C2×C4C2×#1175822412yes—
650C8—24no—
651C8—1224no—
652C22—21no—
653C22—21no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—612no—
658C22—1221no—
659C22—1221no—
660C22—1221no—
661C22—1221no—
662C22—1221no—
663C22—2442no—
664C22—2442no—
665C22C2×C22⋊C4×D6111yescharacteristic
666C22#1684031011yescharacteristic
667C22#1684031111yescharacteristic
668C4—21no—
669C4—21no—
670C4—21no—
671C4—44no—
672C4—42no—
673C4—612no—
674C4—612no—
675C4—1221no—
676C4—1221no—
677C4—1221no—
678C4—1221no—
679C4—2444no—
680C4—2442no—
681C4—2442no—
682C4#104×D6112yes—
683C4C4×#1175822412yes—
684C4#1684027212yes—
685C2—21no—
686C2—21no—
687C2—82no—
688C2—1222no—
689C2—2441no—
690C2#2144830611yescharacteristic
691C2C2×#104×D6111yescharacteristic
692C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1980C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1980D61
hypocenterC61#21382
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. C4×M16×D61C2semidirect
3. C4×#14519735C2semidirect
4. #21446763C2semidirect
5. #21446762C2semidirect
6. #21446739C2semidirect
7. #21446118C2semidirect
8. #104×Dic61C2semidirect
9. C61×#1980C2semidirect
12. #16840317C4semidirect
13. #16840317C4semidirect
14. #16840310C4semidirect
15. #16840310C4semidirect
20. #16840291C4semidirect
21. #16840291C4semidirect
187. M16×C61#41semidirect
237. #11758194#41semidirect
300. C4×D61#104semidirect
345. #1980D61semidirect
395. C61#21382semidirect
537. M16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214591142
C4×M16×D612
C4×#145197352
#214467632
#214467622
#214467392
#214461182
#104×Dic612
C61×#19802
#2138261