Finite Group Databasebeta

← #29695675

Subgroups — #29695675

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×#16840432C211yesmaximal, characteristic
2C2×#16839841C211yesmaximal, characteristic, complemented
3C2×#16839760C212yesmaximal, complemented
4#21448719C211yesmaximal, characteristic, complemented
5#21446747C211yesmaximal, characteristic, complemented
6#21446254C212yesmaximal, complemented
7#21446133C212yesmaximal, complemented
8#21445821C211yesmaximal, characteristic, complemented
9#21445756C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C61×#1957C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yes—
13C22×#11758189C2212yes—
14#16840432—24no—
15#16840432C418yescomplemented
16C2×#14519773—21no—
17C2×#14519749—22no—
18C2×#14519748C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×D4×Dic61—21no—
22C2×#14519739—21no—
23C2×C4⋊C4×D61C2212yes—
24#16839841—24no—
25C2×C4⋊#92418C2211yescharacteristic
26#16839760—28no—
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2212yes—
30C2×#14519698C2212yes—
31C2×#14519698C2212yes—
32C2×#14519689C2212yes—
33#16839658—21no—
34#16839633—22no—
35#16839619C2212yes—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839525C2211yescharacteristic
39#16839524C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C2×#102×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#384C2211yescharacteristic
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—24no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C418yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×D244D411yescharacteristic
53C22×C22⋊D61—22no—
54C22×C22⋊D61—22no—
55C23×Dic61—22no—
56C22×C4×D61—22no—
57C22×C4×D61C2312yes—
58C2×#11758189—24no—
59C2×#11758189—24no—
60C2×#11758189—42no—
61C22×#92418C2312yes—
62#14519773—28no—
63#14519749—216no—
64#14519748—24no—
65#14519748C2×C418yescomplemented
66C4×Dic122—22no—
67C4×Dic122C2×C414yescomplemented
68C4×D244C2×C414yes—
69C4×D244D414yes—
70D4×Dic61—28no—
71#14519739—44no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—22no—
78C4⋊C4×D61—24no—
79C4⋊C4×D61—42no—
80C2×C22⋊Dic61—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C4⋊#92418—22no—
86#14519702—22no—
87#14519702D414yescomplemented
88#14519700—24no—
89#14519698—24no—
90#14519698—42no—
91#14519689—24no—
92#14519684—21no—
93#14519684—22no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61C2312yes—
97C22×C8×C61C2311yescharacteristic
98C22×C8×C61C2311yescharacteristic
99#127×C61—22no—
100#127×C61C2×C414yescomplemented
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C42×C61C2311yescharacteristic
105#102×C61—24no—
106#102×C61—42no—
107C4×C8×C61—22no—
108C24×C61—22no—
109C23×D61C2×D412yes—
110C2×D4×C61—24no—
111C2×D4×C61—24no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—81no—
120C2×D4×C61—81no—
121C2×D4×C61C22×C414yes—
122C22×C4×C61—22no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—28no—
127C2×C22⋊D61—28no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—82no—
131C2×C22⋊D61—82no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—28no—
136C22×Dic61—21no—
137C22×Dic61—44no—
138C22×Dic61—44no—
139C22×Dic61C2×D412yes—
140#11758224—216no—
141#11758224—48no—
142#11758224—48no—
143#11758224—48no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244C22⋊C414yes—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C2×C4×D61—82no—
154C2×C4×D61C22×C414yes—
155C2×C4×D61C2×D414yes—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122C22×C412yes—
160C2×Dic122C2×D412yes—
161C2×Dic122#4112yes—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—81no—
172C42×C61C22×C412yes—
173C42×C61C2×D412yes—
174C22⋊Dic61—28no—
175#11758198—44no—
176#11758198C22⋊C418yes—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22⋊C414yes—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61C2×D412yes—
184#11758196—24no—
185#11758196—44no—
186C4×Dic61—24no—
187C4×Dic61—42no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191#11758189—48no—
192C23×C61—24no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61C2×C22⋊C414yes—
207D4×C61—24no—
208D4×C61—24no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—48no—
212D4×C61—84no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61AGL(1,ℤ/8)12yes—
221C2×C4×C61#13712yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C22⋊D61—416no—
229C22⋊D61—88no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—416no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61C2×C22⋊C414yes—
246D244—28no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—24no—
251C4×D61—88no—
252Dic122—22no—
253Dic122—22no—
254Dic122—22no—
255Dic122C4×D414yescomplemented
256C8×C61—22no—
257C8×C61—41no—
258#92418—42no—
259#22432—611nomaximal, Sylow
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61—82no—
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—24no—
275C2×D61—24no—
276C2×D61—24no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C4×C61—22no—
282C4×C61—82no—
283C4×C61#41714yescomplemented
284C4×C61C2×C4×D411yescharacteristic
285C4×C61#51811yescharacteristic
286C4×C61D4211yescharacteristic
287C4×C61#54611yescharacteristic
288Dic61—24no—
289Dic61—24no—
290Dic61—42no—
291Dic61—42no—
292Dic61—81no—
293C2×#550—611no—
294C2×#436—611no—
295C2×#417—612no—
296D4×C4⋊C4—611no—
297#1982—611no—
298#1957D6111yescharacteristic, complemented
299#1895—612no—
300#1874—612no—
301#1803—611no—
302#1788—612no—
303C4×#103—611no—
304C2×C61—84no—
305C2×C61D4×C22⋊C411yescharacteristic
306C2×C61C2×#41711yescharacteristic
307C2×C61#289711yescharacteristic
308C2×C61#327411yescharacteristic
309C2×C61#327811yescharacteristic
310C2×C61#333211yescharacteristic
311C2×C61#333311yescharacteristic
312D61—44no—
313C22×#41—612no—
314C22×M16—612no—
315#550—618no—
316#550—1224no—
317C2×#127—1221no—
318C2×#127C2×D6111yescharacteristic
319C2×#121—612no—
320C2×#121—1222no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—1221no—
324C2×C4×D4—1221no—
325C22×C4⋊C4—612no—
326#436—1224no—
327C2×C4⋊C8—611no—
328#417—1228no—
329C2×#103—611no—
330C2×#103—611no—
331C2×#103—612no—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102C2×D6112yes—
335C2×#102C2×D6112yes—
336C2×C22⋊C8—612no—
337C2×C4×C8C2×D6111yescharacteristic
338#389—1221no—
339#385—1222no—
340#384C2×D6111yescharacteristic
341#380—612no—
342C4×C4⋊C4—611no—
343#340—611no—
344#340—611no—
345#340—612no—
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—21no—
355C22×D4—21no—
356C22×D4—611no—
357C22×D4—1222no—
358C22×D4—1222no—
359C22×D4C22×D6112yes—
360C23×C4—612no—
361C23×C4—1222no—
362C23×C4—1222no—
363C2×M16—1224no—
364C2×M16—1224no—
365C2×M16—2442no—
366C22×C8—612no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#127—22no—
370#127—1228no—
371#127C4×D6114yescomplemented
372#121—618no—
373#121—12216no—
374#121—1224no—
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—614no—
378C4×D4—614no—
379C4×D4—1228no—
380C4×D4—2444no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4C22×D6112yes—
392C2×C22⋊C4—612no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C4⋊C8—1222no—
399#103—614no—
400#103—1224no—
401#103—1222no—
402#102—24no—
403#102—42no—
404#102—1224no—
405#102—2442no—
406C22⋊C8—1224no—
407C4×C8—22no—
408#95—1222no—
409#95—1221no—
410C24—22no—
411C24—612no—
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—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—81no—
432C2×D4—81no—
433C2×D4—614no—
434C2×D4—1228no—
435C2×D4—1228no—
436C2×D4—1222no—
437C2×D4—1222no—
438C2×D4—4882no—
439C2×D4—4882no—
440C2×D4—2444no—
441C2×D4—2444no—
442C2×D4C2×C4×D6114yes—
443C22×C4—22no—
444C22×C4—612no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—1228no—
448C22×C4—1221no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—4882no—
457C22×C4—2444no—
458C22×C4—2444no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464M16—2448no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8D4×D6112yes—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—614no—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2444no—
487C22⋊C4—618no—
488C22⋊C4—1228no—
489C22⋊C4—2444no—
490C42—1224no—
491C42—2442no—
492C42C2×C4×D6112yes—
493C42D4×D6112yes—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—614no—
502C23—1222no—
503C23—1222no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23C2×D4×D6111yescharacteristic
509Q8—614no—
510Q8—1222no—
511Q8—1222no—
512Q8—1222no—
513D4—24no—
514D4—24no—
515D4—48no—
516D4—42no—
517D4—42no—
518D4—84no—
519D4—1228no—
520D4—4888no—
521D4—24416no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—614no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—4888no—
540C2×C4—24416no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4C22⋊C4×D6112yes—
549C2×C4C22⋊C4×D6112yes—
550C2×C4#1451980112yes—
551C2×C4#1451980612yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982311yescharacteristic
558C8—22no—
559C8—41no—
560C8—2442no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—1224no—
569C22—1224no—
570C22—1224no—
571C22—2442no—
572C22—2442no—
573C22—2442no—
574C22—2442no—
575C22C2×C22⋊C4×D6111yescharacteristic
576C22#1684039711yescharacteristic
577C22#1684040511yescharacteristic
578C22C2×#1451980111yescharacteristic
579C22C2×#1451980611yescharacteristic
580C22#1684056311yescharacteristic
581C22#1684056811yescharacteristic
582C4—22no—
583C4—82no—
584C4—1224no—
585C4—1224no—
586C4—4881no—
587C4—2442no—
588C4—2442no—
589C4#1683976014yescomplemented
590C4C4×D4×D6111yescharacteristic
591C4#1684023411yescharacteristic
592C4#1684039611yescharacteristic
593C4#1684040911yescharacteristic
594C2—84no—
595C2—2444no—
596C2#2144864011yescharacteristic
597C2C2×#1683976011yescharacteristic
598C2#2145434011yescharacteristic
599C2#2145743811yescharacteristic
600C2#2145748411yescharacteristic
601C2#2145814511yescharacteristic
602C2#2145815611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1957C2
socleC23×C61C22×D4
prime radicalC2×#16840432C2
cosocleC22×C4×C61C24
hypercenter#1957D61
hypocenterC61#22432
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×#16839841C2semidirect
3. C2×#16839760C2semidirect
4. #21448719C2semidirect
5. #21446747C2semidirect
6. #21446254C2semidirect
7. #21446133C2semidirect
8. #21445821C2semidirect
9. #21445756C2semidirect
10. C4×#14519702C2semidirect
11. C61×#1957C2semidirect
15. #16840432C4semidirect
50. C2×#11758224C2×C4semidirect
65. #14519748C2×C4semidirect
67. C4×Dic122C2×C4semidirect
87. #14519702D4semidirect
100. #127×C61C2×C4semidirect
255. Dic122C4×D4semidirect
283. C4×C61#417semidirect
298. #1957D61semidirect
346. C61#22432semidirect
371. #127C4×D61semidirect
589. C4#16839760semidirect

Maximal subgroups

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

subgroupindex
C2×#168404322
C2×#168398412
C2×#168397602
#214487192
#214467472
#214462542
#214461332
#214458212
#214457562
C4×#145197022
C61×#19572
#2243261