Finite Group Databasebeta

← #29681713

Subgroups — #29681713

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
1C23×#11758194C211yesmaximal, characteristic, complemented
2C2×#16840273C211yesmaximal, characteristic
3C2×#16839761C212yesmaximal, complemented
4#21446255C212yesmaximal, complemented
5#21446250C212yesmaximal, complemented
6#21446245C212yesmaximal, complemented
7#21446240C212yesmaximal, complemented
8#21445538C212yesmaximal, complemented
9C2×C61×#380C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194—28no—
12C22×#11758194—24no—
13C22×#11758194C2212yes—
14C22×#11758194C2212yes—
15C23×#92418C2212yes—
16#16840273—24no—
17#16840273C418yescomplemented
18C2×#14519760—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519751C2211yescharacteristic
22C2×#14519751C2211yescharacteristic
23C2×#14519746—21no—
24C2×C4×C22⋊D61—21no—
25C2×C4×C22⋊D61—21no—
26C2×#14519736C2211yescharacteristic
27#16839761—28no—
28C2×#14519700C2214yescomplemented
29C2×#14519697C2214yescomplemented
30#16839524C2214yescomplemented
31#16839524C2214yescomplemented
32C24×C4×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2212yes—
34C22×C22⋊C4×C61—21no—
35C61×#380—21no—
36C61×#380—21no—
37C61×#380C2212yes—
38C61×#380C2212yes—
39C2×#95×C61—21no—
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—
49C22×C4×D61—21no—
50C2×#11758194—232no—
51C2×#11758194—24no—
52C2×#11758194—24no—
53C2×#11758194—24no—
54C2×#11758194—28no—
55C2×#11758194D418yescomplemented
56C22×#92418—24no—
57C22×#92418—28no—
58C22×#92418C2314yes—
59#14519760—28no—
60#14519756—22no—
61#14519756—22no—
62#14519756C2×C414yescomplemented
63#14519756C2×C414yescomplemented
64#14519751—22no—
65#14519751—22no—
66#14519751C2×C414yescomplemented
67#14519751C2×C414yescomplemented
68#14519746—28no—
69C4×C22⋊D61—28no—
70C4×C22⋊D61—28no—
71#14519736—22no—
72#14519736C2×C414yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83#14519700—28no—
84#14519697—28no—
85C25×C61D411yescharacteristic
86C23×C4×C61—24no—
87C23×C4×C61—24no—
88C23×C4×C61—22no—
89C23×C4×C61C2×C414yes—
90C23×C4×C61D414yes—
91C23×C4×C61D411yescharacteristic
92C23×C4×C61D411yescharacteristic
93C23×C4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61C2314yes—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—44no—
104#95×C61—22no—
105#95×C61—22no—
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×C61—24no—
126C22×C4×C61—44no—
127C22×C4×C61—44no—
128C22×C4×C61—44no—
129C22×C4×C61C22⋊C414yes—
130C22×C4×C61C22⋊C414yes—
131C22×C4×C61C22⋊C414yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D414yes—
142C22×C4×C61#4114yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—24no—
146C2×C22⋊D61—24no—
147C2×C22⋊D61—44no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—42no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
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—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—44no—
170C22⋊C4×C61—44no—
171C22⋊C4×C61—44no—
172C22⋊Dic61—24no—
173C22⋊Dic61—24no—
174C22⋊Dic61—24no—
175#11758198—24no—
176#11758198—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61C22×C412yes—
180C4⋊Dic61C22×C412yes—
181#11758196—24no—
182#11758196—24no—
183C4×Dic61—24no—
184#11758194—216no—
185#11758194—216no—
186C2×#92418—216no—
187C2×#92418—24no—
188C2×#92418—24no—
189C2×#92418C2×D418yes—
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×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—88no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C4×D418yes—
235C2×C4×C61#12014yes—
236C2×C4×C61#12114yes—
237C2×C4×C61#12314yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D412yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4112yes—
242C22⋊D61—48no—
243C22⋊D61—48no—
244C22⋊D61—48no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—42no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252D244—24no—
253D244—41no—
254D244—41no—
255C4×D61—48no—
256Dic122—24no—
257Dic122—41no—
258Dic122—41no—
259#92418—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—84no—
290C4×C61#38614yes—
291C4×C61C2×C4×D412yes—
292C4×C61C2×#12011yescharacteristic
293C4×C61C2×#12311yescharacteristic
294Dic61—42no—
295Dic61—42no—
296Dic61—42no—
297C23×M16—611no—
298C2×#525—611no—
299C2×#417—612no—
300C2×#380D6111yescharacteristic, complemented
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—612no—
318C22×M16—612no—
319C22×M16—1228no—
320C22×M16—1224no—
321C23×C8—612no—
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⋊C4C2×D6112yes—
334C22×C22⋊C4—21no—
335#417—1228no—
336C2×#103—614no—
337C2×#102—614no—
338#380—21no—
339#380—21no—
340#380C2×D6112yes—
341#380C2×D6112yes—
342C2×#95—21no—
343#340—614no—
344#340—614no—
345C61#2097111yescharacteristic, complemented, Sylow
346C25C22⋊D6111yescharacteristic
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—22no—
359C23×C4—1221no—
360C23×C4C4×D6114yes—
361C23×C4D24414yes—
362C23×C4C22⋊D6111yescharacteristic
363C23×C4C22⋊D6111yescharacteristic
364C23×C4C22⋊D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C23×C4C22×D6111yescharacteristic
368C2×M16—618no—
369C2×M16—1228no—
370C2×M16—12232no—
371C2×M16—1224no—
372C2×M16—1224no—
373C2×M16—1224no—
374C22×C8—614no—
375C22×C8—1228no—
376C22×C8—1224no—
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—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22×D6114yes—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—44no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C42—1221no—
408#103—1228no—
409#102—1228no—
410#95—22no—
411#95—22no—
412C24—22no—
413C24—24no—
414C24—22no—
415C24—42no—
416C24—1221no—
417C24#1175819814yes—
418C24C2×C22⋊D6111yescharacteristic
419C24C2×C22⋊D6111yescharacteristic
420C24C2×C22⋊D6111yescharacteristic
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—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1224no—
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×D24412yes—
465C22×C4C2×D24412yes—
466C22×C4C2×D24412yes—
467C22×C4D4×D6114yes—
468C22×C4#1175822414yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C23×D6111yescharacteristic
473M16—12216no—
474M16—12216no—
475C2×C8—618no—
476C2×C8—12216no—
477C2×C8—1224no—
478C2×C8—1224no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—44no—
482C4⋊C4—612no—
483C4⋊C4—612no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
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#1451974914yes—
519C23#1451980312yes—
520C23#1451980812yes—
521C23C22×C22⋊D6111yescharacteristic
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—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
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×C4#1451973718yes—
558C2×C4#1451974714yes—
559C2×C4#1451974914yes—
560C2×C4#1451976114yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6112yes—
563C2×C4C22×D24411yescharacteristic
564C2×C4C2×#1175822412yes—
565C8—1228no—
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#1683963914yes—
586C22#1683976114yescomplemented
587C22C22×#1175819811yescharacteristic
588C22C2×#1451974912yes—
589C22C2×#1451980311yescharacteristic
590C22C2×#1451980811yescharacteristic
591C22#1684056012yes—
592C4—22no—
593C4—22no—
594C4—84no—
595C4—2442no—
596C4—2442no—
597C4—2442no—
598C4#1683963914yes—
599C4C2×#1451973712yes—
600C4C2×#1451974711yescharacteristic
601C4C2×#1451976111yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—4882no—
606C2C2×#1683963911yescharacteristic
607C2C2×#1683976111yescharacteristic
608C2#2145433611yescharacteristic
609C2#2145543412yes—
610C2#2145544512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#380C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC2×#380D61
hypocenterC61#20971
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C23×#11758194C2semidirect
3. C2×#16839761C2semidirect
4. #21446255C2semidirect
5. #21446250C2semidirect
6. #21446245C2semidirect
7. #21446240C2semidirect
8. #21445538C2semidirect
9. C2×C61×#380C2semidirect
17. #16840273C4semidirect
28. C2×#14519700C22semidirect
29. C2×#14519697C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
55. C2×#11758194D4semidirect
62. #14519756C2×C4semidirect
63. #14519756C2×C4semidirect
66. #14519751C2×C4semidirect
67. #14519751C2×C4semidirect
276. C22×C61#417semidirect
300. C2×#380D61semidirect
345. C61#20971semidirect
586. C22#16839761semidirect

Maximal subgroups

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

subgroupindex
C23×#117581942
C2×#168402732
C2×#168397612
#214462552
#214462502
#214462452
#214462402
#214455382
C2×C61×#3802
#2097161