Finite Group Databasebeta

← #29719004

Subgroups — #29719004

Order 15616.

1182 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519698C211yesmaximal, characteristic
2#21450305C211yesmaximal, characteristic, complemented
3D4×C22⋊Dic61C211yesmaximal, characteristic
4#21447235C212yesmaximal, complemented
5#21447132C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic, complemented
7#21446155C212yesmaximal, complemented
8#21445996C211yesmaximal, characteristic, complemented
9#21445906C211yesmaximal, characteristic, complemented
10#21445827C212yesmaximal, complemented
11C2×C61×#450C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×D4×Dic61C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61—21no—
16C2×#14519706C2212yescomplemented
17C2×#14519701C2212yescomplemented
18C2×#14519698—24no—
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21#16839701—22no—
22#16839669C2212yescomplemented
23#16839665C2212yes—
24#16839642—22no—
25#16839627C2212yescomplemented
26#16839623—21no—
27#16839623—21no—
28#16839622C2211yescharacteristic
29#16839618—21no—
30C4×C22⋊Dic61C2211yescharacteristic
31#16839525C2212yescomplemented
32#16839525C2212yescomplemented
33#16839524C2212yescomplemented
34#16839516C2212yescomplemented
35C23×D4×C61C2211yescharacteristic
36C22×SD16×C61C2211yescharacteristic
37C22×SD16×C61C2211yescharacteristic
38C2×C22⋊Q8×C61C2211yescharacteristic
39C61×#450—24no—
40C2×#102×C61C2211yescharacteristic
41C2×#102×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C23×Dic61—21no—
44C23×Dic61—22no—
45C22×#92418—22no—
46C22×#92418—42no—
47C22×#92418C2312yes—
48D4×Dic61—24no—
49D4×Dic61—42no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—44no—
60C2×C22⋊Dic61D411yescharacteristic
61C2×C22⋊Dic61D411yescharacteristic
62C2×C4×Dic61C2312yes—
63#14519706—42no—
64#14519701—42no—
65#14519698—24no—
66#14519698—24no—
67#14519684—21no—
68#14519684—22no—
69#14519684—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684D412yes—
73C25×C61—21no—
74C22×Q8×C61C2311yescharacteristic
75C22×D4×C61—24no—
76C22×D4×C61—21no—
77C22×D4×C61—22no—
78C22×D4×C61—42no—
79C22×D4×C61D414yes—
80C22×D4×C61C2311yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C2×SD16×C61—22no—
84C2×SD16×C61—22no—
85C2×SD16×C61—24no—
86C2×SD16×C61—42no—
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94#102×C61—22no—
95#102×C61—22no—
96C22⋊C8×C61—22no—
97C24×C61—21no—
98C24×C61—21no—
99C24×C61—24no—
100C24×C61—44no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61—82no—
104C24×C61C2×D411yescharacteristic
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×D4×C61—28no—
110C2×D4×C61—24no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—44no—
117C2×D4×C61—48no—
118C2×D4×C61—44no—
119C2×D4×C61C2×D412yes—
120C2×D4×C61C2×D412yes—
121C2×D4×C61#4114yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61#4112yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—44no—
133C22×Dic61—44no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61—42no—
137C22×Dic61—82no—
138C22×Dic61C2×D412yes—
139C22×Dic61C2×D412yes—
140SD16×C61—44no—
141SD16×C61—44no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—42no—
149C22⋊C4×C61—42no—
150C22⋊Dic61—22no—
151C22⋊Dic61—22no—
152C22⋊Dic61—44no—
153C22⋊Dic61—44no—
154C22⋊Dic61—42no—
155C22⋊Dic61—82no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—42no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—44no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—24no—
171C23×C61—22no—
172C23×C61—48no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—48no—
181C23×C61—84no—
182C23×C61—88no—
183C23×C61—88no—
184C23×C61#12112yes—
185C23×C61#13512yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61C22×D411yescharacteristic
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—42no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—24no—
195D4×C61—48no—
196D4×C61—42no—
197D4×C61#12314yes—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61#12112yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)12yes—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—88no—
224C2×Dic61—84no—
225C8×C61—41no—
226C8×C61—41no—
227#92418—42no—
228#24527—611nomaximal, Sylow
229C22×C61—22no—
230C22×C61—24no—
231C22×C61—22no—
232C22×C61—24no—
233C22×C61—24no—
234C22×C61—42no—
235C22×C61—44no—
236C22×C61—42no—
237C22×C61—44no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—816no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61#46314yes—
245C22×C61C2×#12111yescharacteristic
246C22×C61#54612yes—
247C22×C61C2×#13511yescharacteristic
248C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C61#57712yes—
250C4×C61—22no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61C2×#12311yescharacteristic
254C4×C61#53411yescharacteristic
255C4×C61#55112yes—
256Dic61—42no—
257Dic61—42no—
258Dic61—82no—
259Dic61—82no—
260C2×#450D6111yescharacteristic
261C22×#102—611no—
262#2503—611no—
263D4×C22⋊C4—611no—
264#2073—612no—
265#2055—612no—
266#1892—611no—
267#1877—612no—
268#1850—611no—
269#1823—611no—
270#1804—612no—
271C2×C61—24no—
272C2×C61—44no—
273C2×C61—82no—
274C2×C61#241211yescharacteristic
275C2×C61C2×#46311yescharacteristic
276C2×C61#304211yescharacteristic
277C2×C61#329612yes—
278C2×C61#331012yes—
279C23×D4C2×D6111yescharacteristic
280C22×SD16C2×D6111yescharacteristic
281C22×SD16C2×D6111yescharacteristic
282C23×C8—611no—
283C2×C22⋊Q8C2×D6111yescharacteristic
284C2×C4×D4—612no—
285C22×C4⋊C4—611no—
286C22×C22⋊C4—1221no—
287#450—24no—
288C2×C4⋊C8—612no—
289C2×#103—612no—
290C2×#102—612no—
291C2×#102—612no—
292C2×#102—1224no—
293C2×#102C2×D6111yescharacteristic
294C2×#102C2×D6111yescharacteristic
295C2×C22⋊C8C2×D6111yescharacteristic
296#400—1222no—
297#391—612no—
298#390—612no—
299#386—1222no—
300#382—612no—
301#381—1221no—
302#381—1221no—
303#380—611no—
304#379—1221no—
305C4×C22⋊C4—611no—
306#340—612no—
307#340—612no—
308#340—612no—
309#336—612no—
310C61#2452711yescharacteristic, complemented, Sylow
311C25—21no—
312C22×Q8C22×D6111yescharacteristic
313C22×D4—24no—
314C22×D4—21no—
315C22×D4—22no—
316C22×D4—42no—
317C22×D4C22⋊D6114yes—
318C22×D4C22×D6111yescharacteristic
319C22×D4C22×D6111yescharacteristic
320C23×C4—1222no—
321C23×C4—1221no—
322C23×C4C22×D6111yescharacteristic
323C2×SD16—24no—
324C2×SD16—22no—
325C2×SD16—22no—
326C2×SD16—42no—
327C22×C8—612no—
328C22×C8—1222no—
329C22×C8—2442no—
330C22×C8C22×D6111yescharacteristic
331C22×C8C22×D6111yescharacteristic
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C4×D4—1224no—
335C4×D4—2442no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—611no—
345C2×C22⋊C4—611no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—2444no—
351C2×C42—612no—
352C4⋊C8—2442no—
353#103—2442no—
354#102—22no—
355#102—22no—
356#102—1224no—
357#102—1224no—
358C22⋊C8—22no—
359#95—612no—
360#95—1222no—
361#95—1222no—
362#95—1222no—
363#95—1222no—
364#95—1221no—
365C24—21no—
366C24—21no—
367C24—24no—
368C24—44no—
369C24—42no—
370C24—42no—
371C24—82no—
372C24D4×D6111yescharacteristic
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×D4—28no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4#1175822414yes—
388C2×D4C2×C22⋊D6112yes—
389C2×D4C2×C22⋊D6112yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—4882no—
399C22×C4—2444no—
400C22×C4—2444no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4D4×D6111yescharacteristic
405C22×C4#1175822412yes—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C23×D6111yescharacteristic
408SD16—44no—
409SD16—44no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—2444no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—42no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—2442no—
426C22⋊C4—42no—
427C22⋊C4—1222no—
428C22⋊C4—1222no—
429C22⋊C4—4882no—
430C22⋊C4—2444no—
431C22⋊C4—2444no—
432C22⋊C4—2442no—
433C42—1222no—
434C42—1222no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—48no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—48no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—84no—
450C23—88no—
451C23—88no—
452C23#1451974912yes—
453C23#1451979812yes—
454C23#1451980412yes—
455C23C2×D4×D6111yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—42no—
459D4—22no—
460D4—22no—
461D4—22no—
462D4—24no—
463D4—48no—
464D4—42no—
465D4#1451975814yes—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—4884no—
481C2×C4—4888no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4#1451974912yes—
489C2×C4C22×C22⋊D6111yescharacteristic
490C2×C4C2×#1175822412yes—
491C2×C4C2×#1175822411yescharacteristic
492C2×C4#1451982212yes—
493C8—41no—
494C8—41no—
495C8—2442no—
496C22—22no—
497C22—24no—
498C22—22no—
499C22—24no—
500C22—24no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—816no—
509C22—82no—
510C22—82no—
511C22#1683997914yes—
512C22C2×#1451974911yescharacteristic
513C22#1684039912yes—
514C22C2×#1451979811yescharacteristic
515C22C2×#1451980411yescharacteristic
516C22#1684057212yes—
517C4—22no—
518C4—81no—
519C4—81no—
520C4—4882no—
521C4—4882no—
522C4—2442no—
523C4—2442no—
524C4C2×#1451975811yescharacteristic
525C4#1684032111yescharacteristic
526C4#1684044312yes—
527C2—24no—
528C2—44no—
529C2—82no—
530C2#2144921611yescharacteristic
531C2C2×#1683997911yescharacteristic
532C2#2145533311yescharacteristic
533C2#2145765812yes—
534C2#2145784412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#450C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenterC2×#450D61
hypocenterC61#24527
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
2. #21450305C2semidirect
4. #21447235C2semidirect
5. #21447132C2semidirect
6. #21446243C2semidirect
7. #21446155C2semidirect
8. #21445996C2semidirect
9. #21445906C2semidirect
10. #21445827C2semidirect
16. C2×#14519706C22semidirect
17. C2×#14519701C22semidirect
22. #16839669C22semidirect
25. #16839627C22semidirect
31. #16839525C22semidirect
32. #16839525C22semidirect
33. #16839524C22semidirect
34. #16839516C22semidirect
310. C61#24527semidirect

Maximal subgroups

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

subgroupindex
C22×#145196982
#214503052
D4×C22⋊Dic612
#214472352
#214471322
#214462432
#214461552
#214459962
#214459062
#214458272
C2×C61×#4502
#2452761