Finite Group Databasebeta

← #29681717

Subgroups — #29681717

Order 15616.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2C2×#16839764C212yesmaximal, complemented
3C2×#16839622C211yesmaximal, characteristic, complemented
4#21446257C212yesmaximal, complemented
5#21446252C212yesmaximal, complemented
6#21446247C212yesmaximal, complemented
7#21446242C212yesmaximal, complemented
8#21445539C212yesmaximal, complemented
9C23×M16×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840273—24no—
12#16840273C418yescomplemented
13C2×#14519760—21no—
14C2×#14519756C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C2×#14519751C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519746—21no—
19C2×C4×C22⋊D61—21no—
20C2×C4×C22⋊D61—21no—
21C2×#14519736C2211yescharacteristic
22C22×C4⋊Dic61C2212yes—
23C22×C22⋊Dic61—21no—
24#16839764—28no—
25C2×#14519702C2214yescomplemented
26C2×#14519699C2214yescomplemented
27#16839622—21no—
28#16839622—21no—
29#16839622C2212yes—
30#16839622C2212yes—
31C2×#14519684—21no—
32#16839525C2214yescomplemented
33#16839525C2214yescomplemented
34C24×C4×C61C2211yescharacteristic
35C22×M16×C61—24no—
36C22×M16×C61—28no—
37C22×M16×C61C2212yes—
38C22×M16×C61C2212yes—
39C23×C8×C61C2212yes—
40C2×#11758222—22no—
41C2×#11758222—24no—
42C2×#11758222—24no—
43C2×#11758222C2×C414yes—
44C22×Dic122C2311yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C22×C22⋊D61—21no—
48C22×C22⋊D61—21no—
49C23×Dic61—22no—
50C22×C4×D61—21no—
51#14519760—28no—
52#14519756—22no—
53#14519756—22no—
54#14519756C2×C414yescomplemented
55#14519756C2×C414yescomplemented
56#14519751—22no—
57#14519751—22no—
58#14519751C2×C414yescomplemented
59#14519751C2×C414yescomplemented
60#14519746—28no—
61C4×C22⋊D61—28no—
62C4×C22⋊D61—28no—
63#14519736—22no—
64#14519736C2×C414yes—
65C2×C4⋊Dic61—24no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2314yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—44no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82#14519702—28no—
83#14519699—28no—
84#14519684—22no—
85#14519684—22no—
86C25×C61D411yescharacteristic
87C23×C4×C61—24no—
88C23×C4×C61—24no—
89C23×C4×C61C2×C414yes—
90C23×C4×C61D411yescharacteristic
91C23×C4×C61D411yescharacteristic
92C23×C4×C61D411yescharacteristic
93C23×C4×C61D414yes—
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—232no—
98C2×M16×C61—24no—
99C2×M16×C61—24no—
100C2×M16×C61—24no—
101C2×M16×C61—28no—
102C2×M16×C61D418yescomplemented
103C22×C8×C61—24no—
104C22×C8×C61—28no—
105C22×C8×C61C2314yes—
106C24×C61—24no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—42no—
110C24×C61C22⋊C414yes—
111C24×C61C2×D411yescharacteristic
112C24×C61C2×D411yescharacteristic
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C22×C4×C61—24no—
116C22×C4×C61—216no—
117C22×C4×C61—28no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—28no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61C22⋊C414yes—
126C22×C4×C61C22⋊C414yes—
127C22×C4×C61C22⋊C414yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D414yes—
138C22×C4×C61#4114yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—44no—
144C2×C22⋊D61—44no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—44no—
152C22×Dic61—44no—
153C22×Dic61—44no—
154#11758222—216no—
155#11758222—44no—
156#11758222—44no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244C22×C412yes—
161C2×C4×D61—24no—
162C2×C4×D61—44no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—21no—
166C2×Dic122C22×C412yes—
167M16×C61—216no—
168M16×C61—216no—
169C2×C8×C61—216no—
170C2×C8×C61—24no—
171C2×C8×C61—24no—
172C2×C8×C61C2×D418yes—
173C22⋊Dic61—24no—
174C22⋊Dic61—24no—
175C22⋊Dic61—24no—
176C22⋊Dic61—44no—
177C22⋊Dic61—44no—
178#11758198—24no—
179#11758198—24no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—24no—
183C4⋊Dic61—24no—
184C4⋊Dic61—44no—
185C4⋊Dic61C22×C412yes—
186C4⋊Dic61C22×C412yes—
187#11758196—24no—
188#11758196—24no—
189C4×Dic61—24no—
190C23×C61—22no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—28no—
200C23×C61—28no—
201C23×C61—44no—
202C23×C61—48no—
203C23×C61—48no—
204C23×C61C2×C22⋊C412yes—
205C23×C61C2×C22⋊C412yes—
206C23×C61C2×C22⋊C412yes—
207C23×C61#12114yes—
208C23×C61AGL(1,ℤ/8)12yes—
209C23×C61#13712yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—42no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—42no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—28no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—28no—
226C2×C4×C61—28no—
227C2×C4×C61C2×C22⋊C412yes—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61C2×C22⋊C412yes—
230C2×C4×C61C4×D418yes—
231C2×C4×C61#12014yes—
232C2×C4×C61#12114yes—
233C2×C4×C61#12314yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D412yes—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4112yes—
238C22⋊D61—48no—
239C22⋊D61—48no—
240C22⋊D61—48no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—44no—
249C2×Dic61—44no—
250C2×Dic61—44no—
251C2×Dic61—88no—
252D244—24no—
253D244—41no—
254D244—41no—
255C4×D61—48no—
256Dic122—24no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—28no—
260#20971—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—416no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61#38614yes—
276C22×C61#41714yescomplemented
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61C2×#12112yes—
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61C2×#13711yescharacteristic
281C22×C61#57512yes—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61#38614yes—
290C4×C61C2×C4×D412yes—
291C4×C61C2×#12011yescharacteristic
292C4×C61C2×#12311yescharacteristic
293Dic61—42no—
294Dic61—42no—
295Dic61—42no—
296Dic61—84no—
297C23×M16D6111yescharacteristic, complemented
298C2×#525—611no—
299C2×#417—612no—
300C2×#380—611no—
301#1895—612no—
302#1894—612no—
303#1893—612no—
304#1892—612no—
305#1736—612no—
306C2×C61—24no—
307C2×C61—24no—
308C2×C61—42no—
309C2×C61C2×#38611yescharacteristic
310C2×C61C2×#41711yescharacteristic
311C2×C61#289611yescharacteristic
312C2×C61#305512yes—
313C2×C61#305612yes—
314D61—82no—
315C22×#41—611no—
316C24×C4C2×D6111yescharacteristic
317C22×M16—24no—
318C22×M16—28no—
319C22×M16C2×D6112yes—
320C22×M16C2×D6112yes—
321C23×C8C2×D6112yes—
322#525—618no—
323#525—1224no—
324C2×#123—1221no—
325C2×C22⋊Q8—611no—
326C2×C22⋊Q8—611no—
327C2×#121—611no—
328C2×#121—611no—
329C2×#120—1221no—
330C2×C4×D4—1221no—
331C2×C4×D4—1221no—
332C2×#117—611no—
333C22×C4⋊C4—612no—
334C22×C22⋊C4—1221no—
335#417—1228no—
336C2×#103—614no—
337C2×#102—614no—
338#380—612no—
339#380—612no—
340#380—1221no—
341#380—1221no—
342C2×#95—1221no—
343#340—614no—
344#340—614no—
345C61#2097111yescharacteristic, complemented, Sylow
346C25D24411yescharacteristic
347C2×#41—614no—
348C2×#41—1222no—
349C2×#41—1224no—
350C2×#41—1224no—
351C22×Q8—611no—
352C22×D4—611no—
353C22×D4—1221no—
354C22×D4—1221no—
355C22×D4—1221no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—1221no—
359C23×C4—1222no—
360C23×C4C4×D6114yes—
361C23×C4D24411yescharacteristic
362C23×C4D24411yescharacteristic
363C23×C4D24411yescharacteristic
364C23×C4C22⋊D6114yes—
365C23×C4C22×D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C23×C4C22×D6111yescharacteristic
368C2×M16—232no—
369C2×M16—24no—
370C2×M16—24no—
371C2×M16—24no—
372C2×M16—28no—
373C2×M16C22⋊D6118yescomplemented
374C22×C8—24no—
375C22×C8—28no—
376C22×C8C22×D6114yes—
377#123—1228no—
378C22⋊Q8—614no—
379C22⋊Q8—614no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382#121—614no—
383#121—614no—
384#121—1222no—
385#121—1222no—
386#120—1228no—
387C4×D4—1228no—
388C4×D4—1228no—
389#117—614no—
390#117—1222no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—614no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1222no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—2444no—
407C2×C42—1221no—
408#103—1228no—
409#102—1228no—
410#95—1222no—
411#95—1222no—
412C24—22no—
413C24—24no—
414C24—22no—
415C24—42no—
416C24—1221no—
417C24#1175819814yes—
418C24C2×D24411yescharacteristic
419C24C2×D24411yescharacteristic
420C24C2×D24411yescharacteristic
421#41—12216no—
422#41—2444no—
423#41—2444no—
424C2×Q8—612no—
425C2×Q8—1221no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×D4—612no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1224no—
432C2×D4—1222no—
433C2×D4—1221no—
434C2×D4—1222no—
435C2×D4—2444no—
436C2×D4—2442no—
437C2×D4—2444no—
438C2×D4—2442no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—216no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1224no—
453C22×C4—1224no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4#1175819814yes—
459C22×C4#1175819814yes—
460C22×C4#1175819814yes—
461C22×C4C2×C4×D6112yes—
462C22×C4C2×C4×D6112yes—
463C22×C4C2×C4×D6112yes—
464C22×C4C2×D24411yescharacteristic
465C22×C4C2×D24411yescharacteristic
466C22×C4C2×D24411yescharacteristic
467C22×C4#1175822214yes—
468C22×C4C2×C22⋊D6112yes—
469C22×C4C2×C22⋊D6112yes—
470C22×C4C2×C22⋊D6112yes—
471C22×C4C2×C22⋊D6114yes—
472C22×C4C23×D6111yescharacteristic
473M16—216no—
474M16—216no—
475C2×C8—216no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8C2×C22⋊D6118yes—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1224no—
484C4⋊C4—1224no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—2444no—
488C22⋊C4—1224no—
489C22⋊C4—1224no—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—2444no—
494C22⋊C4—2444no—
495C42—1224no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—28no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—1222no—
511C23—2442no—
512C23—2441no—
513C23—2441no—
514C23—2442no—
515C23C2×#1175819812yes—
516C23C2×#1175819812yes—
517C23C2×#1175819812yes—
518C23#1451975114yes—
519C23#1451980512yes—
520C23#1451981012yes—
521C23C22×D24411yescharacteristic
522Q8—1224no—
523Q8—2441no—
524Q8—2441no—
525D4—1224no—
526D4—2448no—
527D4—2448no—
528D4—2448no—
529D4—2441no—
530D4—2441no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—4888no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2444no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4—2448no—
554C2×C4C2×#1175819812yes—
555C2×C4C2×#1175819812yes—
556C2×C4C2×#1175819812yes—
557C2×C4C4×C22⋊D6118yes—
558C2×C4#1451974614yes—
559C2×C4#1451975114yes—
560C2×C4#1451976014yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C22×C22⋊D6112yes—
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C2×#1175822212yes—
565C8—28no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—4882no—
581C22—4882no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22#1683964114yes—
586C22#1683976414yescomplemented
587C22C22×#1175819811yescharacteristic
588C22C2×#1451975112yes—
589C22C2×#1451980511yescharacteristic
590C22C2×#1451981011yescharacteristic
591C22#1684056212yes—
592C4—22no—
593C4—22no—
594C4—4884no—
595C4—2442no—
596C4—2442no—
597C4—2442no—
598C4#1683964114yes—
599C4C2×C4×C22⋊D6112yes—
600C4C2×#1451974611yescharacteristic
601C4C2×#1451976011yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—4882no—
606C2C2×#1683964111yescharacteristic
607C2C2×#1683976411yescharacteristic
608C2#2145433911yescharacteristic
609C2#2145543812yes—
610C2#2145544912yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC23×M16×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC23×M16D61
hypocenterC61#20971
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C2×#16839622C2semidirect
4. #21446257C2semidirect
5. #21446252C2semidirect
6. #21446247C2semidirect
7. #21446242C2semidirect
8. #21445539C2semidirect
9. C23×M16×C61C2semidirect
12. #16840273C4semidirect
25. C2×#14519702C22semidirect
26. C2×#14519699C22semidirect
32. #16839525C22semidirect
33. #16839525C22semidirect
54. #14519756C2×C4semidirect
55. #14519756C2×C4semidirect
58. #14519751C2×C4semidirect
59. #14519751C2×C4semidirect
102. C2×M16×C61D4semidirect
276. C22×C61#417semidirect
297. C23×M16D61semidirect
345. C61#20971semidirect
373. C2×M16C22⋊D61semidirect
586. C22#16839764semidirect

Maximal subgroups

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

subgroupindex
C2×#168402732
C2×#168397642
C2×#168396222
#214462572
#214462522
#214462472
#214462422
#214455392
C23×M16×C612
#2097161