Finite Group Databasebeta

← #29695677

Subgroups — #29695677

Order 15616.

1582 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840433C211yesmaximal, characteristic
2C2×#16839764C212yesmaximal, complemented
3D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
4#21446745C211yesmaximal, characteristic, complemented
5#21446625C211yesmaximal, characteristic, complemented
6#21446257C212yesmaximal, complemented
7#21446134C212yesmaximal, complemented
8#21445825C211yesmaximal, characteristic, complemented
9#21445760C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C2×C61×#436C211yesmaximal, characteristic, complemented
12C22×#11758222C2212yes—
13#16840433—24no—
14#16840433C418yescomplemented
15C2×#14519774C2211yescharacteristic
16C2×#14519773—21no—
17C2×#14519751C2212yes—
18C2×#14519750—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21C2×#14519739—21no—
22C22×C4⋊Dic61C2212yes—
23#16839764—28no—
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519702C2211yescharacteristic
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31#16839661—21no—
32#16839637—22no—
33#16839632C2211yescharacteristic
34#16839622C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839525C2212yes—
39C22×M16×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#436—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C2×#11758222—24no—
46C2×#11758222—24no—
47C2×#11758222—24no—
48C2×#11758222—42no—
49C2×#11758222—42no—
50C2×#11758222C2×C418yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244C2312yes—
55C22×C22⋊D61—22no—
56C22×C22⋊D61—22no—
57C23×Dic61—22no—
58C22×C4×D61—22no—
59#14519774—22no—
60#14519774C2×C414yescomplemented
61#14519773—28no—
62#14519751—24no—
63#14519751C2×C418yescomplemented
64#14519750—216no—
65C4×Dic122—22no—
66C4×Dic122C2×C414yescomplemented
67D4×Dic61—44no—
68#14519739—28no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—42no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83#14519702—24no—
84#14519702—22no—
85#14519702D414yescomplemented
86#14519699—24no—
87#14519699—24no—
88#14519699—42no—
89#14519699—42no—
90#14519684—21no—
91#14519684—22no—
92C22×D4×C61D411yescharacteristic
93C23×C4×C61C2312yes—
94C2×M16×C61—24no—
95C2×M16×C61—24no—
96C2×M16×C61—42no—
97C22×C8×C61C2312yes—
98C22×C8×C61C2311yescharacteristic
99C22×C8×C61C2311yescharacteristic
100C4×D4×C61C2×C414yes—
101C4×D4×C61D414yes—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106C22⋊C8×C61—24no—
107C4×C8×C61—22no—
108C24×C61C2×D412yes—
109C23×D61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61C22⋊C414yes—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C22×C414yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D414yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—28no—
124C2×C22⋊D61—28no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—82no—
128C2×C22⋊D61—82no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—44no—
135C22×Dic61—44no—
136C22×Dic61—82no—
137#11758222—216no—
138#11758222—48no—
139#11758222—48no—
140#11758222—48no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—24no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—81no—
151C2×D244—81no—
152C2×D244C22×C414yes—
153C2×C4×D61—28no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122C22×C412yes—
160C2×Dic122C2×D412yes—
161C2×Dic122#4112yes—
162M16×C61—48no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—21no—
169C2×C8×C61—22no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D412yes—
172C4⋊C4×C61C22⋊C414yes—
173C22⋊C4×C61C22⋊C418yes—
174C42×C61C22×C412yes—
175C42×C61C2×D412yes—
176C22⋊Dic61—44no—
177#11758198—28no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—22no—
184C4⋊Dic61—44no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187C4⋊Dic61C22×C412yes—
188C4⋊Dic61C2×D412yes—
189#11758196—24no—
190C4×Dic61—24no—
191C4×Dic61—42no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61C2×C22⋊C414yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—24no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207D4×C61—28no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C414yes—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61AGL(1,ℤ/8)12yes—
220C2×C4×C61#13712yes—
221C2×C4×C61C23×C411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C22⋊D61—416no—
228C22⋊D61—88no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—88no—
245D244—24no—
246D244—24no—
247D244—42no—
248D244—42no—
249D244—48no—
250D244—84no—
251C4×D61—416no—
252Dic122—22no—
253Dic122—22no—
254Dic122—22no—
255Dic122C4×D414yescomplemented
256C8×C61—22no—
257C8×C61—42no—
258C8×C61—41no—
259#22432—611nomaximal, Sylow
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61D4211yescharacteristic
269C22×C61#54611yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57611yescharacteristic
273C22×C61#57711yescharacteristic
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—82no—
278C2×D61—82no—
279C2×D61—82no—
280C2×D61—82no—
281C4×C61—24no—
282C4×C61—22no—
283C4×C61#41714yescomplemented
284C4×C61C2×C4×D411yescharacteristic
285C4×C61#51811yescharacteristic
286C4×C61D4211yescharacteristic
287C4×C61#54611yescharacteristic
288Dic61—24no—
289Dic61—42no—
290Dic61—42no—
291Dic61—81no—
292Dic61—82no—
293C2×#550—611no—
294C2×#436D6111yescharacteristic, complemented
295C2×#417—612no—
296D4×C4⋊C4—611no—
297#1982—611no—
298#1957—611no—
299#1895—612no—
300#1874—612no—
301#1803—611no—
302#1788—612no—
303C4×#103—611no—
304C2×C61—44no—
305C2×C61D4×C22⋊C411yescharacteristic
306C2×C61C2×#41711yescharacteristic
307C2×C61#289711yescharacteristic
308C2×C61#327411yescharacteristic
309C2×C61#327811yescharacteristic
310C2×C61#333211yescharacteristic
311C2×C61#333311yescharacteristic
312D61—84no—
313C22×#41—612no—
314C22×M16C2×D6112yes—
315#550—618no—
316#550—1224no—
317C2×#127—611no—
318C2×#127—1221no—
319C2×#121—612no—
320C2×#121—1222no—
321C2×C4×Q8—611no—
322C2×C4×D4—1221no—
323C2×C4×D4—1221no—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4—612no—
326#436—24no—
327C2×C4⋊C8C2×D6111yescharacteristic
328#417—1228no—
329C2×#103—611no—
330C2×#103—612no—
331C2×#103—611no—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8C2×D6112yes—
337C2×C4×C8C2×D6111yescharacteristic
338#389—1221no—
339#385—1222no—
340#384—611no—
341#380—612no—
342C4×C4⋊C4—611no—
343#340—612no—
344#340—611no—
345#340—611no—
346C61#2243211yescharacteristic, complemented, Sylow
347C2×#41—618no—
348C2×#41—1224no—
349C2×#41—1224no—
350C2×#41—1224no—
351C2×#41—2442no—
352C2×#41—2442no—
353C22×Q8—611no—
354C22×D4—612no—
355C22×D4—1222no—
356C22×D4—1222no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4D24411yescharacteristic
360C23×C4—1222no—
361C23×C4—1222no—
362C23×C4C22×D6112yes—
363C2×M16—24no—
364C2×M16—24no—
365C2×M16—42no—
366C22×C8C22×D6112yes—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#127—614no—
370#127—1228no—
371#127—1222no—
372#121—618no—
373#121—1224no—
374#121—12216no—
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—1228no—
378C4×D4—2444no—
379C4×D4C4×D6114yes—
380C4×D4C22⋊D6114yes—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4D24411yescharacteristic
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4D24412yes—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C4⋊C8—22no—
399#103—614no—
400#103—1222no—
401#103—1224no—
402#102—1224no—
403#102—1224no—
404#102—2442no—
405#102—2442no—
406C22⋊C8—24no—
407C4×C8—22no—
408#95—1221no—
409#95—1222no—
410C24—1222no—
411C24C2×D24412yes—
412#41—12216no—
413#41—2448no—
414#41—2448no—
415#41—2448no—
416C2×Q8—612no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—614no—
425C2×D4—1228no—
426C2×D4—1228no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1224no—
433C2×D4—1224no—
434C2×D4—4881no—
435C2×D4—4881no—
436C2×D4—4882no—
437C2×D4—4882no—
438C2×D4—2444no—
439C2×D4—2444no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4#1175819814yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1228no—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—4882no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4C2×C4×D6114yes—
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24412yes—
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×C22⋊D6114yes—
463C22×C4C23×D6111yescharacteristic
464M16—48no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—21no—
473C2×C8D4×D6112yes—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1224no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1222no—
483C4⋊C4—4881no—
484C4⋊C4—2444no—
485C4⋊C4—2442no—
486C4⋊C4#1175819814yes—
487C22⋊C4—1228no—
488C22⋊C4—2444no—
489C22⋊C4#1175819818yes—
490C42—1224no—
491C42—2442no—
492C42C2×C4×D6112yes—
493C42C2×C22⋊D6112yes—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—1224no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23C2×#1175819814yes—
508C23C22×D24411yescharacteristic
509Q8—614no—
510Q8—1222no—
511Q8—1222no—
512Q8—1222no—
513D4—28no—
514D4—1224no—
515D4—1224no—
516D4—4884no—
517D4—4888no—
518D4—24416no—
519D4—2448no—
520D4—2442no—
521D4—2442no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—4888no—
536C2×C4—24416no—
537C2×C4—2441no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2441no—
541C2×C4—2442no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4C2×#1175819812yes—
548C2×C4C2×#1175819814yes—
549C2×C4C2×#1175819812yes—
550C2×C4#1451980512yes—
551C2×C4#1451981012yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982311yescharacteristic
558C8—22no—
559C8—42no—
560C8—41no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—4882no—
569C22—4882no—
570C22—4882no—
571C22—4882no—
572C22—2444no—
573C22—2444no—
574C22—2444no—
575C22C22×#1175819811yescharacteristic
576C22D4×D24411yescharacteristic
577C22#1684040711yescharacteristic
578C22C2×#1451980511yescharacteristic
579C22C2×#1451981011yescharacteristic
580C22#1684056711yescharacteristic
581C22#1684057411yescharacteristic
582C4—22no—
583C4—24no—
584C4—1224no—
585C4—4882no—
586C4—4881no—
587C4—2442no—
588C4—2442no—
589C4#1683976414yescomplemented
590C4C4×D4×D6111yescharacteristic
591C4#1684023411yescharacteristic
592C4D4×C22⋊D6111yescharacteristic
593C4#1684040011yescharacteristic
594C2—44no—
595C2—4884no—
596C2D4×#1175819811yescharacteristic
597C2C2×#1683976411yescharacteristic
598C2#2145434611yescharacteristic
599C2#2145744111yescharacteristic
600C2#2145748711yescharacteristic
601C2#2145814711yescharacteristic
602C2#2145815811yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#436C2
socleC23×C61C22×D4
prime radicalC2×#16840433C2
cosocleC22×C4×C61C24
hypercenterC2×#436D61
hypocenterC61#22432
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. D4×C4⋊Dic61C2semidirect
4. #21446745C2semidirect
5. #21446625C2semidirect
6. #21446257C2semidirect
7. #21446134C2semidirect
8. #21445825C2semidirect
9. #21445760C2semidirect
10. C4×#14519702C2semidirect
11. C2×C61×#436C2semidirect
14. #16840433C4semidirect
50. C2×#11758222C2×C4semidirect
60. #14519774C2×C4semidirect
63. #14519751C2×C4semidirect
66. C4×Dic122C2×C4semidirect
85. #14519702D4semidirect
255. Dic122C4×D4semidirect
283. C4×C61#417semidirect
294. C2×#436D61semidirect
346. C61#22432semidirect
589. C4#16839764semidirect

Maximal subgroups

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

subgroupindex
C2×#168404332
C2×#168397642
D4×C4⋊Dic612
#214467452
#214466252
#214462572
#214461342
#214458252
#214457602
C4×#145197022
C2×C61×#4362
#2243261