Finite Group Databasebeta

← #29695673

Subgroups — #29695673

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×#16840434C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic, complemented
3C2×#16839762C212yesmaximal, complemented
4#21448719C211yesmaximal, characteristic, complemented
5#21446745C211yesmaximal, characteristic, complemented
6#21446624C211yesmaximal, characteristic, complemented
7#21446256C212yesmaximal, complemented
8#21446135C212yesmaximal, complemented
9#21445758C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758226C2212yes—
13C22×#11758189C2212yes—
14#16840434—24no—
15#16840434C418yescomplemented
16C2×#14519774—21no—
17C2×#14519774C2211yescharacteristic
18C2×#14519752—22no—
19C2×#14519752C2212yes—
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519739—21no—
23C2×C4⋊C4×D61C2212yes—
24#16839839—24no—
25#16839762—28no—
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839658—21no—
35#16839633—22no—
36#16839619C2212yes—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C2×#11758226—24no—
46C2×#11758226—24no—
47C2×#11758226—24no—
48C2×#11758226—42no—
49C2×#11758226—42no—
50C2×#11758226C2×C418yescomplemented
51C22×D244—22no—
52C22×D244—21no—
53C22×D244—22no—
54C22×D244—21no—
55C22×D244D411yescharacteristic
56C22×D244C2312yes—
57C22×C4×D61—22no—
58C22×C4×D61—22no—
59C22×C4×D61C2312yes—
60C2×#11758189—24no—
61C2×#11758189—24no—
62C2×#11758189—42no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519774—28no—
66#14519774—22no—
67#14519774C2×C414yescomplemented
68#14519752—24no—
69#14519752—216no—
70#14519752C2×C418yescomplemented
71C4×D244—28no—
72C4×D244C2×C414yes—
73C4×D244D414yes—
74#14519739—44no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—22no—
79C4⋊C4×D61—24no—
80C4⋊C4×D61—42no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84C2×C4×Dic61—21no—
85#14519701—22no—
86#14519701D414yescomplemented
87#14519699—24no—
88#14519699—42no—
89#14519697—24no—
90#14519697—42no—
91#14519689—24no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C22×C8×C61C2312yes—
97C4×Q8×C61—22no—
98C4×Q8×C61C2×C414yescomplemented
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107#103×C61—24no—
108C23×D61—22no—
109C23×D61C2×D412yes—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61C22×C412yes—
114C2×Q8×C61C2×D412yes—
115C2×Q8×C61#4112yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61C2×D412yes—
125#11758226—216no—
126#11758226—48no—
127#11758226—48no—
128#11758226—48no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—28no—
136C2×D244—24no—
137C2×D244—22no—
138C2×D244—28no—
139C2×D244—24no—
140C2×D244—44no—
141C2×D244—44no—
142C2×D244—42no—
143C2×D244—42no—
144C2×D244—81no—
145C2×D244—81no—
146C2×D244—82no—
147C2×D244—82no—
148C2×D244C22⋊C414yes—
149C2×D244C22×C414yes—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—28no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61—44no—
158C2×C4×D61—44no—
159C2×C4×D61—82no—
160C2×C4×D61C22×C414yes—
161C2×C4×D61C2×D414yes—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—81no—
171C4⋊C4×C61C22×C412yes—
172C42×C61—24no—
173C42×C61C22×C412yes—
174C42×C61C2×D412yes—
175#11758198—28no—
176#11758198—44no—
177#11758198C22⋊C418yes—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22⋊C414yes—
182C4⋊Dic61C2×D412yes—
183#11758196—44no—
184C4×Dic61—42no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418C2×D412yes—
191#11758189—48no—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61C2×C22⋊C414yes—
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61C4×D414yescomplemented
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61AGL(1,ℤ/8)12yes—
225C2×C4×C61#13712yes—
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C2×C22⋊C414yes—
241D244—24no—
242D244—24no—
243D244—28no—
244D244—416no—
245D244—42no—
246D244—42no—
247D244—48no—
248D244—84no—
249D244—88no—
250C4×D61—24no—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—24no—
254C4×D61—416no—
255C4×D61—88no—
256C8×C61—42no—
257#92418—22no—
258#92418—41no—
259#22432—611nomaximal, Sylow
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61D4211yescharacteristic
262C22×C61#54611yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57711yescharacteristic
267C2×D61—24no—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
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—42no—
284C4×C61—82no—
285C4×C61#41714yescomplemented
286C4×C61C2×C4×D411yescharacteristic
287C4×C61#51811yescharacteristic
288C4×C61D4211yescharacteristic
289C4×C61#54611yescharacteristic
290Dic61—24no—
291Dic61—42no—
292Dic61—81no—
293C2×#550—611no—
294C2×#436—611no—
295C2×#417—612no—
296D4×C4⋊C4—611no—
297#1982—611no—
298#1957—611no—
299#1895—612no—
300#1874—612no—
301#1803D6111yescharacteristic, complemented
302#1788—612no—
303C4×#103—611no—
304C2×C61D4×C22⋊C411yescharacteristic
305C2×C61C2×#41711yescharacteristic
306C2×C61#289711yescharacteristic
307C2×C61#327411yescharacteristic
308C2×C61#327811yescharacteristic
309C2×C61#333211yescharacteristic
310C2×C61#333311yescharacteristic
311D61—44no—
312D61—84no—
313C22×#41—612no—
314C22×M16—612no—
315#550—618no—
316#550—1224no—
317C2×#127—611no—
318C2×#127—1221no—
319C2×#121—612no—
320C2×#121—1222no—
321C2×C4×Q8C2×D6111yescharacteristic
322C2×C4×D4—611no—
323C2×C4×D4—1221no—
324C2×C4×D4—1221no—
325C22×C4⋊C4—612no—
326#436—1224no—
327C2×C4⋊C8C2×D6111yescharacteristic
328#417—1228no—
329C2×#103—611no—
330C2×#103—611no—
331C2×#103C2×D6112yes—
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8—612no—
337C2×C4×C8—611no—
338#389—1221no—
339#385—1222no—
340#384C2×D6111yescharacteristic
341#380—612no—
342C4×C4⋊C4—611no—
343#340—611no—
344#340—611no—
345#340C2×D6112yes—
346C61#2243211yescharacteristic, complemented, Sylow
347C2×#41—618no—
348C2×#41—1224no—
349C2×#41—1224no—
350C2×#41—1224no—
351C2×#41—2442no—
352C2×#41—2442no—
353C22×Q8C22×D6111yescharacteristic
354C22×D4—611no—
355C22×D4—612no—
356C22×D4—1222no—
357C22×D4—1222no—
358C22×D4—1221no—
359C22×D4—1221no—
360C23×C4—612no—
361C23×C4—1222no—
362C23×C4—1222no—
363C2×M16—1224no—
364C2×M16—1224no—
365C2×M16—2442no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8C22×D6112yes—
369#127—614no—
370#127—1228no—
371#127—1222no—
372#121—618no—
373#121—12216no—
374#121—1224no—
375C4×Q8—22no—
376C4×Q8C4×D6114yescomplemented
377C4×D4—614no—
378C4×D4—614no—
379C4×D4—1228no—
380C4×D4—2444no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—2442no—
390C2×C4⋊C4C22×D6112yes—
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—612no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C42—21no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398C4⋊C8—22no—
399#103—24no—
400#103—614no—
401#103—1222no—
402#102—1224no—
403#102—1224no—
404#102—2442no—
405#102—2442no—
406C22⋊C8—1224no—
407C4×C8—1222no—
408#95—1222no—
409#95—1221no—
410C24—612no—
411C24—1222no—
412#41—12216no—
413#41—2448no—
414#41—2448no—
415#41—2448no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8C2×C4×D6112yes—
420C2×Q8D4×D6112yes—
421C2×Q8#1175822612yes—
422C2×D4—614no—
423C2×D4—614no—
424C2×D4—1228no—
425C2×D4—1228no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1222no—
432C2×D4—1224no—
433C2×D4—1224no—
434C2×D4—1222no—
435C2×D4—4881no—
436C2×D4—4881no—
437C2×D4—4882no—
438C2×D4—4882no—
439C2×D4—2444no—
440C2×D4—2444no—
441C2×D4—2442no—
442C2×D4—2442no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—614no—
449C22×C4—1228no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
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—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—614no—
481C4⋊C4—612no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—2444no—
486C4⋊C4C2×C4×D6112yes—
487C22⋊C4—618no—
488C22⋊C4—1228no—
489C22⋊C4—2444no—
490C42—24no—
491C42—2442no—
492C42C2×C4×D6112yes—
493C42D4×D6112yes—
494C23—614no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1224no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23C2×D4×D6111yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—22no—
512Q8#1451973914yescomplemented
513D4—1224no—
514D4—1224no—
515D4—1228no—
516D4—4884no—
517D4—4888no—
518D4—24416no—
519D4—2448no—
520D4—2442no—
521D4—2442no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—614no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—4888no—
542C2×C4—24416no—
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#1451980412yes—
551C2×C4#1451980912yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C2×#1175822611yescharacteristic
556C2×C4#1451982311yescharacteristic
557C2×C4#1451982311yescharacteristic
558C8—42no—
559C8—1222no—
560C8—2441no—
561C22—1224no—
562C22—1224no—
563C22—1224no—
564C22—4882no—
565C22—4882no—
566C22—4882no—
567C22—4882no—
568C22—2444no—
569C22—2444no—
570C22—2444no—
571C22—2442no—
572C22—2442no—
573C22—2442no—
574C22—2442no—
575C22C2×C22⋊C4×D6111yescharacteristic
576C22#1684039611yescharacteristic
577C22#1684040311yescharacteristic
578C22C2×#1451980411yescharacteristic
579C22C2×#1451980911yescharacteristic
580C22#1684056611yescharacteristic
581C22#1684057111yescharacteristic
582C4—22no—
583C4—24no—
584C4—42no—
585C4—82no—
586C4—1224no—
587C4—4881no—
588C4—2442no—
589C4#1683976214yescomplemented
590C4C2×#1451973911yescharacteristic
591C4#1684023611yescharacteristic
592C4#1684039611yescharacteristic
593C4#1684040311yescharacteristic
594C2—4884no—
595C2—2444no—
596C2#2144863911yescharacteristic
597C2C2×#1683976211yescharacteristic
598C2#2145434311yescharacteristic
599C2#2145743411yescharacteristic
600C2#2145748011yescharacteristic
601C2#2145814311yescharacteristic
602C2#2145815411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC2×#16840434C2
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22432
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16839839C2semidirect
3. C2×#16839762C2semidirect
4. #21448719C2semidirect
5. #21446745C2semidirect
6. #21446624C2semidirect
7. #21446256C2semidirect
8. #21446135C2semidirect
9. #21445758C2semidirect
10. C4×#14519701C2semidirect
11. C61×#1803C2semidirect
15. #16840434C4semidirect
50. C2×#11758226C2×C4semidirect
67. #14519774C2×C4semidirect
70. #14519752C2×C4semidirect
86. #14519701D4semidirect
98. C4×Q8×C61C2×C4semidirect
210. Q8×C61C4×D4semidirect
285. C4×C61#417semidirect
301. #1803D61semidirect
346. C61#22432semidirect
376. C4×Q8C4×D61semidirect
512. Q8#14519739semidirect
589. C4#16839762semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
C2×#168398392
C2×#168397622
#214487192
#214467452
#214466242
#214462562
#214461352
#214457582
C4×#145197012
C61×#18032
#2243261