Finite Group Databasebeta

← #29681252

Subgroups — #29681252

Order 15616.

1934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic
3C2×#16839769C212yesmaximal, complemented
4C2×#16839724C211yesmaximal, characteristic
5#21446132C211yesmaximal, characteristic, complemented
6#21446131C211yesmaximal, characteristic, complemented
7#21446129C211yesmaximal, characteristic, complemented
8#21446125C212yesmaximal, complemented
9#21446123C212yesmaximal, complemented
10#21445537C212yesmaximal, complemented
11C61×#1873C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758189C2212yes—
15C2×#14519757C2211yescharacteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519754—21no—
19C2×#14519753C2211yescharacteristic
20C22⋊Q8×D61—24no—
21C22⋊Q8×D61C418yescomplemented
22C2×#14519736C2211yescharacteristic
23C2×#14519735C2212yes—
24C2×#14519734—21no—
25C2×#14519733—21no—
26#117×D61—28no—
27C2×C4⋊C4×D61—21no—
28C2×C4⋊C4×D61C2212yes—
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×C22⋊C4×D61—21no—
31C2×C22⋊C4×D61—21no—
32C2×C42×D61—21no—
33#16839769—28no—
34C2×#14519702C2212yescomplemented
35C2×#14519701C2212yescomplemented
36C2×#14519700C2212yescomplemented
37#16839724—24no—
38C2×C22⋊#92418C2211yescharacteristic
39C2×#14519689C2212yes—
40#16839525C2212yescomplemented
41#16839524C2212yescomplemented
42#16839524C2212yescomplemented
43C2×C22⋊Q8×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C2×#103×C61C2212yescomplemented
46C2×C22⋊C8×C61C2211yescharacteristic
47C61×#340C2212yescomplemented
48C24×D61D411yescharacteristic
49C22×Dic122—21no—
50C22×Dic122C2311yescharacteristic
51C2×Q8×D61—22no—
52C2×Q8×D61—24no—
53C2×Q8×D61—24no—
54C2×Q8×D61C2×C414yescomplemented
55C23×Dic61D411yescharacteristic
56C22×C4×D61—24no—
57C22×C4×D61—24no—
58C22×C4×D61—22no—
59C22×C4×D61—21no—
60C22×C4×D61—21no—
61C22×C4×D61C2×C414yes—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D414yes—
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C2×#11758189—24no—
68C2×#11758189—24no—
69C2×#11758189—24no—
70C22×#92418C2312yes—
71#14519757—22no—
72#14519757C2×C414yes—
73#14519756—22no—
74#14519756C2×C414yescomplemented
75#14519755—28no—
76#14519754—28no—
77#14519753—22no—
78#14519753C2×C414yes—
79#14519736—24no—
80#14519735—24no—
81#14519735D418yes—
82#14519734—28no—
83#14519733—28no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2312yes—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—22no—
90C2×#11758196—21no—
91C4⋊C4×D61—28no—
92C4⋊C4×D61—22no—
93C4⋊C4×D61—24no—
94C4⋊C4×D61—24no—
95C4⋊C4×D61C2×C414yescomplemented
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×#11758198—22no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C22⋊C4×D61—28no—
102C22⋊C4×D61—28no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—22no—
105C42×D61—28no—
106#14519702—24no—
107#14519701—24no—
108#14519700—24no—
109C22⋊#92418—22no—
110#14519689—24no—
111C22×Q8×C61C2311yescharacteristic
112C23×C4×C61C2311yescharacteristic
113C22×C8×C61C2312yes—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61C2×C414yescomplemented
116#117×C61—24no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2312yes—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61—21no—
123#103×C61—24no—
124C22⋊C8×C61—22no—
125C24×C61C2×D411yescharacteristic
126C23×D61—22no—
127C23×D61—22no—
128C23×D61—24no—
129C23×D61—42no—
130C23×D61C22⋊C414yes—
131C23×D61C2×D411yescharacteristic
132C23×D61C2×D411yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61C22×C412yes—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61C22⋊C414yes—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154Q8×D61—216no—
155Q8×D61—48no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—216no—
163C2×C4×D61—216no—
164C2×C4×D61—28no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—44no—
172C2×C4×D61C22⋊C414yes—
173C2×C4×D61C22⋊C414yes—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C22×C412yes—
176C2×C4×D61C2×D412yes—
177C2×C4×D61C2×D414yes—
178C2×C4×D61C2×D412yes—
179C2×C4×D61#4114yes—
180C2×Dic122—22no—
181C2×Dic122—22no—
182C2×Dic122—21no—
183C2×Dic122—24no—
184C2×Dic122—42no—
185C2×Dic122—42no—
186C2×Dic122C22×C412yes—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C22×C412yes—
194C4⋊C4×C61C2×D414yes—
195C22⋊C4×C61—24no—
196C22⋊C4×C61—24no—
197C42×C61—24no—
198C22⋊Dic61—24no—
199C22⋊Dic61—24no—
200#11758198—24no—
201#11758198—24no—
202#11758198—28no—
203C4⋊Dic61—22no—
204C4⋊Dic61—21no—
205C4⋊Dic61—24no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C2×D414yes—
208#11758196—28no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212C4×Dic61—24no—
213C4×Dic61—28no—
214C2×#92418—22no—
215C2×#92418—22no—
216C2×#92418—22no—
217#11758189—48no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61C2×C22⋊C412yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—28no—
230C22×D61—28no—
231C22×D61—44no—
232C22×D61—48no—
233C22×D61—48no—
234C22×D61C2×C22⋊C412yes—
235C22×D61C2×C22⋊C412yes—
236C22×D61#12012yes—
237C22×D61#12312yes—
238C22×D61#13714yes—
239Q8×C61—24no—
240Q8×C61—42no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D412yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4112yes—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—28no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61#12012yes—
272C2×Dic61#12312yes—
273C4×D61—28no—
274C4×D61—24no—
275C4×D61—24no—
276C4×D61—28no—
277C4×D61—48no—
278C4×D61—416no—
279C4×D61—48no—
280C4×D61C4×D418yes—
281C4×D61#12118yes—
282Dic122—24no—
283Dic122—42no—
284Dic122—48no—
285C8×C61—42no—
286#92418—42no—
287#20911—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61C2×#12011yescharacteristic
295C22×C61C2×#12311yescharacteristic
296C22×C61C2×#13712yes—
297C22×C61#57512yes—
298C2×D61—24no—
299C2×D61—24no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—24no—
304C2×D61—24no—
305C2×D61—22no—
306C2×D61—416no—
307C2×D61#38614yes—
308C2×D61#41914yescomplemented
309C4×C61—22no—
310C4×C61—42no—
311C4×C61—44no—
312C4×C61—42no—
313C4×C61C2×C4×D412yes—
314C4×C61C2×#12112yes—
315Dic61—22no—
316Dic61—42no—
317Dic61—44no—
318Dic61—42no—
319Dic61#38614yes—
320C22×C22⋊Q8—611no—
321C22×#117—611no—
322C2×#419—612no—
323C2×#407—611no—
324#1873—611no—
325#1873—611no—
326#1873—611no—
327#1873D6111yescharacteristic, complemented
328#1872—612no—
329#1872—612no—
330#1736—612no—
331C2×C61—42no—
332C2×C61C2×#38611yescharacteristic
333C2×C61C2×#41911yescharacteristic
334C2×C61#289611yescharacteristic
335C2×C61#300712yes—
336C2×C61#309512yes—
337D61—24no—
338D61—24no—
339C23×Q8—611no—
340C24×C4—611no—
341C22×M16—612no—
342C2×C22⋊Q8—611no—
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8—618no—
345C2×C22⋊Q8—611no—
346C2×C22⋊Q8—1221no—
347C2×C22⋊Q8—1221no—
348C2×C22⋊Q8—1224no—
349C2×C22⋊Q8C2×D6111yescharacteristic
350C2×#117—611no—
351C2×#117—612no—
352C2×#117—1221no—
353C2×#117—1221no—
354C2×#117—1228no—
355C2×#117C2×D6111yescharacteristic
356C22×C4⋊C4—611no—
357C22×C4⋊C4—612no—
358C22×C4⋊C4—1221no—
359C22×C22⋊C4—1221no—
360C22×C22⋊C4—1221no—
361C22×C42—1221no—
362#419—1228no—
363C2×#103—612no—
364C2×#103—612no—
365C2×#103—612no—
366C2×#103C2×D6112yescomplemented
367#407—1224no—
368C2×C22⋊C8—611no—
369C2×C22⋊C8—612no—
370C2×C22⋊C8C2×D6111yescharacteristic
371#340—612no—
372#340—612no—
373#340—612no—
374#340C2×D6112yescomplemented
375C61#2091111yescharacteristic, complemented, Sylow
376C25—611no—
377C22×Q8—614no—
378C22×Q8—611no—
379C22×Q8—1221no—
380C22×Q8—1222no—
381C22×Q8—1224no—
382C22×Q8—1224no—
383C22×Q8C22×D6111yescharacteristic
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—614no—
387C23×C4—611no—
388C23×C4—614no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—1224no—
392C23×C4—1224no—
393C23×C4—1221no—
394C23×C4—1222no—
395C23×C4—1221no—
396C23×C4C22×D6111yescharacteristic
397C2×M16—1224no—
398C2×M16—1224no—
399C2×M16—1224no—
400C22×C8—612no—
401C22×C8C22×D6112yes—
402C22⋊Q8—22no—
403C22⋊Q8—614no—
404C22⋊Q8—614no—
405C22⋊Q8—614no—
406C22⋊Q8—1228no—
407C22⋊Q8—1228no—
408C22⋊Q8—1222no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8C4×D6114yescomplemented
412#117—24no—
413#117—618no—
414#117—1228no—
415#117—1228no—
416#117—1224no—
417#117—1224no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—614no—
422C2×C4⋊C4—1224no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1228no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1224no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6112yes—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1228no—
436C2×C22⋊C4—1228no—
437C2×C22⋊C4—1222no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C42—21no—
443C2×C42—1228no—
444C2×C42—1222no—
445C2×C42—1221no—
446#103—24no—
447#103—1224no—
448#103—1224no—
449#103—1224no—
450C22⋊C8—22no—
451C22⋊C8—1224no—
452C22⋊C8—1222no—
453C24—611no—
454C24—611no—
455C24—614no—
456C24—1222no—
457C24—1222no—
458C24—1224no—
459C24—2442no—
460C24D4×D6111yescharacteristic
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—612no—
465C2×Q8—1224no—
466C2×Q8—12216no—
467C2×Q8—1221no—
468C2×Q8—1222no—
469C2×Q8—1222no—
470C2×Q8—2442no—
471C2×Q8—2442no—
472C2×Q8—2448no—
473C2×Q8C2×C4×D6112yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—614no—
483C22×C4—614no—
484C22×C4—611no—
485C22×C4—614no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—614no—
489C22×C4—614no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—12216no—
493C22×C4—12216no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1228no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2444no—
509C22×C4—2444no—
510C22×C4—2444no—
511C22×C4C2×C4×D6112yes—
512C22×C4D4×D6112yes—
513C22×C4D4×D6111yescharacteristic
514C22×C4C23×D6111yescharacteristic
515M16—2448no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—612no—
526C4⋊C4—614no—
527C4⋊C4—1224no—
528C4⋊C4—1224no—
529C4⋊C4—1224no—
530C4⋊C4—1228no—
531C4⋊C4—1222no—
532C4⋊C4—1221no—
533C4⋊C4—1224no—
534C4⋊C4C2×C4×D6112yes—
535C4⋊C4D4×D6114yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—1228no—
539C22⋊C4—1224no—
540C22⋊C4—1224no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C42—24no—
544C42—1228no—
545C42—1224no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—612no—
551C23—612no—
552C23—612no—
553C23—612no—
554C23—614no—
555C23—1228no—
556C23—1228no—
557C23—1221no—
558C23—1221no—
559C23—1222no—
560C23—1222no—
561C23—1222no—
562C23—2444no—
563C23—2448no—
564C23—2448no—
565C23C22⋊C4×D6112yes—
566C23C2×D4×D6111yescharacteristic
567Q8—24no—
568Q8—42no—
569Q8—1224no—
570Q8—2448no—
571Q8—2442no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—618no—
588C2×C4—618no—
589C2×C4—612no—
590C2×C4—1228no—
591C2×C4—1228no—
592C2×C4—1228no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1224no—
596C2×C4—1224no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1224no—
600C2×C4—2448no—
601C2×C4—24416no—
602C2×C4—2448no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4C22⋊C4×D6112yes—
609C2×C4C22×C4×D6111yescharacteristic
610C2×C4C2×D4×D6112yes—
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4#41×D6112yes—
613C8—42no—
614C8—2442no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—42no—
619C22—42no—
620C22—614no—
621C22—614no—
622C22—1222no—
623C22—1224no—
624C22—1224no—
625C22—1224no—
626C22—1224no—
627C22—1222no—
628C22—1222no—
629C22—1222no—
630C22—24416no—
631C22C2×C22⋊C4×D6111yescharacteristic
632C22#120×D6111yescharacteristic
633C22#123×D6111yescharacteristic
634C22#137×D6112yes—
635C22#1684055912yes—
636C4—22no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—614no—
641C4—1222no—
642C4—2442no—
643C4—2444no—
644C4—2442no—
645C4C4×D4×D6112yes—
646C4#121×D6112yes—
647C2—42no—
648C2—1224no—
649C2—1224no—
650C2#386×D6111yescharacteristic
651C2#419×D6111yescharacteristic
652C2#2145433411yescharacteristic
653C2#2145507612yes—
654C2#2145576612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1873C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1873D61
hypocenterC61#20911
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
3. C2×#16839769C2semidirect
5. #21446132C2semidirect
6. #21446131C2semidirect
7. #21446129C2semidirect
8. #21446125C2semidirect
9. #21446123C2semidirect
10. #21445537C2semidirect
11. C61×#1873C2semidirect
21. C22⋊Q8×D61C4semidirect
34. C2×#14519702C22semidirect
35. C2×#14519701C22semidirect
36. C2×#14519700C22semidirect
40. #16839525C22semidirect
41. #16839524C22semidirect
42. #16839524C22semidirect
45. C2×#103×C61C22semidirect
47. C61×#340C22semidirect
54. C2×Q8×D61C2×C4semidirect
74. #14519756C2×C4semidirect
95. C4⋊C4×D61C2×C4semidirect
115. C22⋊Q8×C61C2×C4semidirect
308. C2×D61#419semidirect
327. #1873D61semidirect
366. C2×#103C2×D61semidirect
374. #340C2×D61semidirect
375. C61#20911semidirect
411. C22⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
C2×#117×D612
C2×#168397692
C2×#168397242
#214461322
#214461312
#214461292
#214461252
#214461232
#214455372
C61×#18732
#2091161