Finite Group Databasebeta

← #29685665

Subgroups — #29685665

Order 15616.

1868 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
1C22×C4×D244C211yesmaximal, characteristic
2C2×#16839912C211yesmaximal, characteristic
3C2×#16839860C212yesmaximal, complemented
4#21446156C212yesmaximal, complemented
5#21446127C211yesmaximal, characteristic, complemented
6#21446065C212yesmaximal, complemented
7#21446006C211yesmaximal, characteristic, complemented
8#21445749C211yesmaximal, characteristic
9#21445647C212yesmaximal, complemented
10#21445581C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61—21no—
14C22×#11758201C2212yes—
15C2×C4×D244—28no—
16C2×C4×D244—21no—
17C2×C4×D244—21no—
18C2×C4×D244C2212yes—
19C2×C4×D244C2212yes—
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758198—21no—
22#16839912C418yescomplemented
23#16839860—28no—
24C2×#14519710C2212yescomplemented
25C2×#14519701C2212yescomplemented
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519688C2212yescomplemented
32#16839524C2212yescomplemented
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C2×C22⋊Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic, complemented
37C22×C42×C61C2211yescharacteristic
38C61×#391C2211yescharacteristic, complemented
39C61×#389C2211yescharacteristic, complemented
40C61×#387—21no—
41C61×#385—21no—
42C61×#380C2211yescharacteristic
43C4×C4⋊C4×C61—21no—
44C4×C4⋊C4×C61C2211yescharacteristic, complemented
45C4×C22⋊C4×C61—21no—
46C24×D61—21no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—24no—
51C22×D244C2×C414yes—
52C22×D244C2312yes—
53C23×Dic61—21no—
54C22×C4×D61—28no—
55C22×C4×D61—22no—
56C22×C4×D61—22no—
57C22×C4×D61—44no—
58C22×C4×D61—44no—
59C2×#11758201—24no—
60C2×#11758201—24no—
61C2×#11758201C2×C418yescomplemented
62C22×#92418C2312yes—
63C22×#92418C2312yes—
64C4×D244—28no—
65C4×D244—28no—
66C4×D244—24no—
67C4×D244—24no—
68C4×D244—24no—
69C4×D244D418yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—24no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—28no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76#14519710—24no—
77#14519701—24no—
78#14519697—24no—
79#14519697—24no—
80#14519697C2×C418yescomplemented
81C22⋊#92418—22no—
82C22⋊#92418C2×C414yescomplemented
83#14519688—24no—
84C22×Q8×C61C2311yescharacteristic
85C23×C4×C61D411yescharacteristic
86C23×C4×C61D411yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61C2×C414yescomplemented
90C4×Q8×C61—24no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—24no—
100C2×C42×C61—21no—
101C2×C42×C61—21no—
102C2×C42×C61D411yescharacteristic
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106#95×C61—21no—
107#95×C61—21no—
108#95×C61—21no—
109#95×C61—21no—
110#95×C61—21no—
111C24×C61C2×D411yescharacteristic
112C23×D61—24no—
113C23×D61—24no—
114C23×D61—24no—
115C23×D61—22no—
116C23×D61—42no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—42no—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61C22×C412yescomplemented
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—41no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C2×D244—24no—
149C2×D244—216no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—216no—
156C2×D244C22×C414yes—
157C2×D244C2×D414yes—
158C2×D244#4114yes—
159C2×C4×D61—28no—
160C2×C4×D61—28no—
161C2×C4×D61—432no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61C22×C412yescomplemented
173C22⋊C4×C61—24no—
174C22⋊C4×C61—42no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61—42no—
179C42×C61—42no—
180C42×C61C2×D414yes—
181#11758201—216no—
182#11758198—28no—
183#11758198—28no—
184C4⋊Dic61—24no—
185C4⋊Dic61—22no—
186C4⋊Dic61C2×D414yes—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418C22×C414yescomplemented
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61C4×D412yes—
198C23×C61AGL(1,ℤ/8)12yes—
199C23×C61#13712yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—216no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—48no—
206C22×D61—48no—
207C22×D61—48no—
208C22×D61—48no—
209C22×D61—48no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—48no—
215Q8×C61—24no—
216Q8×C61—42no—
217C2×C4×C61—28no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—44no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12012yes—
247C2×C4×C61AGL(1,ℤ/8)12yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D412yes—
250C2×C4×C61C2×#4112yes—
251C2×C4×C61C2×#4111yescharacteristic
252C2×Dic61—24no—
253C2×Dic61—48no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256D244—216no—
257D244—24no—
258D244—24no—
259D244—24no—
260D244C4×D418yes—
261C4×D61—416no—
262C4×D61—416no—
263#92418—24no—
264#92418—42no—
265#21453—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61#44014yescomplemented
273C22×C61C2×C4×D411yescharacteristic
274C22×C61C2×#12011yescharacteristic
275C22×C61#52511yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—28no—
281C2×D61—432no—
282C2×D61—44no—
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—44no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61C2×C4×D412yes—
298C4×C61#52512yes—
299Dic61—44no—
300C22×C4×D4—611no—
301C2×#450—611no—
302C2×#440—612no—
303C4×C22⋊Q8D6111yescharacteristic, complemented
304#1877—612no—
305#1872—611no—
306#1862—612no—
307#1852—611no—
308#1786—611no—
309#1762—612no—
310#1749—611no—
311C2×C61—24no—
312C2×C61C4×#12011yescharacteristic
313C2×C61C2×#44011yescharacteristic
314C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C61#300311yescharacteristic
316C2×C61#300711yescharacteristic
317C2×C61#311011yescharacteristic
318C2×C61#311411yescharacteristic
319D61—48no—
320C23×D4—611no—
321C24×C4—1221no—
322C22×SD16—612no—
323C2×C22⋊Q8C2×D6111yescharacteristic
324C2×C4×Q8C2×D6111yescharacteristic, complemented
325C2×C4×D4—612no—
326C2×C4×D4—612no—
327C2×C4×D4—1228no—
328C2×C4×D4—1221no—
329C2×C4×D4—1221no—
330C22×C4⋊C4—611no—
331C22×C22⋊C4—1221no—
332C22×C42C2×D6111yescharacteristic
333#450—618no—
334#440—1228no—
335C2×#107—612no—
336C2×#103—612no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102—612no—
340C2×C22⋊C8—611no—
341C2×C22⋊C8—611no—
342C2×#97—612no—
343#391C2×D6111yescharacteristic, complemented
344#389C2×D6111yescharacteristic, complemented
345#387—21no—
346#385—21no—
347#380C2×D6111yescharacteristic
348C4×C4⋊C4—21no—
349C4×C4⋊C4C2×D6111yescharacteristic, complemented
350C4×C22⋊C4—21no—
351#340—612no—
352#340—612no—
353#336—612no—
354C61#2145311yescharacteristic, complemented, Sylow
355C25—1221no—
356C22×Q8C22×D6111yescharacteristic
357C22×D4—614no—
358C22×D4—612no—
359C22×D4—1224no—
360C22×D4—1224no—
361C22×D4—1222no—
362C22×D4—1221no—
363C23×C4—1228no—
364C23×C4—1221no—
365C23×C4—1222no—
366C23×C4—1222no—
367C23×C4—2444no—
368C23×C4—2444no—
369C23×C4C22⋊D6111yescharacteristic
370C23×C4C22⋊D6111yescharacteristic
371C23×C4C22×D6111yescharacteristic
372C2×SD16—618no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C22×C8—612no—
376C22×C8—612no—
377C22⋊Q8—22no—
378C22⋊Q8C4×D6114yescomplemented
379C4×Q8—24no—
380C4×D4—618no—
381C4×D4—1224no—
382C4×D4—1224no—
383C4×D4—1228no—
384C4×D4—1228no—
385C4×D4—1224no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1228no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C42—24no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22⋊D6111yescharacteristic
404C2×C42C22⋊D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406C2×C42C22×D6111yescharacteristic
407#107—1224no—
408#103—1224no—
409#102—618no—
410#102—1224no—
411#102—1224no—
412C22⋊C8—614no—
413C22⋊C8—1222no—
414#97—1224no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—1224no—
421C24—1224no—
422C24—1222no—
423C24—1224no—
424C24—2442no—
425C24—2442no—
426C24—2442no—
427C24—2442no—
428C24C2×C22⋊D6111yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C4×D6112yescomplemented
433C2×D4—614no—
434C2×D4—614no—
435C2×D4—614no—
436C2×D4—12216no—
437C2×D4—1224no—
438C2×D4—1224no—
439C2×D4—12216no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1224no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—1228no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—1228no—
460C22×C4—24432no—
461C22×C4—2444no—
462C22×C4—2444no—
463C22×C4—2444no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4—2444no—
467C22×C4C2×C4×D6112yes—
468C22×C4#1175822212yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C23×D6111yescharacteristic
475SD16—12216no—
476C2×C8—614no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—614no—
489C4⋊C4—1224no—
490C4⋊C4—1222no—
491C4⋊C4C2×C4×D6112yescomplemented
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—1228no—
495C22⋊C4—1228no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42D4×D6114yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—12216no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—2448no—
511C23—2448no—
512C23—2448no—
513C23—2448no—
514C23—2448no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2448no—
520C23C4×C22⋊D6112yes—
521C23#1451980212yes—
522C23#1451980812yes—
523C23C22×C22⋊D6111yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—618no—
527D4—12216no—
528D4—1224no—
529D4—1224no—
530D4—1224no—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—1224no—
559C2×C4—24416no—
560C2×C4—24416no—
561C2×C4—2448no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4C4×C22⋊D6112yes—
565C2×C4#1451974512yes—
566C2×C4#1451974512yes—
567C2×C4#1451980212yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C2×D4×D6112yes—
570C2×C4#41×D6112yes—
571C2×C4C2×#1175822211yescharacteristic
572C8—1224no—
573C8—2442no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—1228no—
581C22—24432no—
582C22—2444no—
583C22—2444no—
584C22—2444no—
585C22—2444no—
586C22—2444no—
587C22—2444no—
588C22#1683986014yescomplemented
589C22C2×C4×C22⋊D6111yescharacteristic
590C22C2×#1451974511yescharacteristic
591C22#1684026911yescharacteristic
592C22C2×#1451980211yescharacteristic
593C22C2×#1451980211yescharacteristic
594C22C2×#1451980811yescharacteristic
595C22#1684056011yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—2444no—
606C4C4×D4×D6112yes—
607C4#1684027112yes—
608C2—24no—
609C2—2448no—
610C2C4×#1451974511yescharacteristic
611C2C2×#1683986011yescharacteristic
612C2C4×#1451980211yescharacteristic
613C2#2145504511yescharacteristic
614C2#2145507711yescharacteristic
615C2#2145591411yescharacteristic
616C2#2145595311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC23×D244C22
cosocleC22×C4×C61C24
hypercenterC4×C22⋊Q8D61
hypocenterC61#21453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16839860C2semidirect
4. #21446156C2semidirect
5. #21446127C2semidirect
6. #21446065C2semidirect
7. #21446006C2semidirect
9. #21445647C2semidirect
10. #21445581C2semidirect
11. C4×C22⋊Q8×C61C2semidirect
22. #16839912C4semidirect
24. C2×#14519710C22semidirect
25. C2×#14519701C22semidirect
31. C2×#14519688C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
36. C2×C4×Q8×C61C22semidirect
38. C61×#391C22semidirect
39. C61×#389C22semidirect
44. C4×C4⋊C4×C61C22semidirect
61. C2×#11758201C2×C4semidirect
80. #14519697C2×C4semidirect
82. C22⋊#92418C2×C4semidirect
89. C22⋊Q8×C61C2×C4semidirect
123. C2×Q8×C61C22×C4semidirect
172. C4⋊C4×C61C22×C4semidirect
192. C2×#92418C22×C4semidirect
272. C22×C61#440semidirect
303. C4×C22⋊Q8D61semidirect
324. C2×C4×Q8C2×D61semidirect
343. #391C2×D61semidirect
344. #389C2×D61semidirect
349. C4×C4⋊C4C2×D61semidirect
354. C61#21453semidirect
378. C22⋊Q8C4×D61semidirect
432. C2×Q8C2×C4×D61semidirect
491. C4⋊C4C2×C4×D61semidirect
588. C22#16839860semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2442
C2×#168399122
C2×#168398602
#214461562
#214461272
#214460652
#214460062
#214457492
#214456472
#214455812
C4×C22⋊Q8×C612
#2145361