Finite Group Databasebeta

← #29709172

Subgroups — #29709172

Order 15616.

1016 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839847C212yesmaximal, complemented
2#21448381C211yesmaximal, characteristic
3#21448294C211yesmaximal, characteristic, complemented
4#21447296C211yesmaximal, characteristic, complemented
5#21447194C212yesmaximal, complemented
6#21447133C212yesmaximal, complemented
7#21447054C211yesmaximal, characteristic
8#21446906C211yesmaximal, characteristic
9#21445667C211yesmaximal, characteristic, complemented
10#21445633C212yesmaximal, complemented
11C61×#2072C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×Q8×Dic61C2211yescharacteristic
15C2×#14519739—21no—
16C2×#14519739—21no—
17C2×#14519739C2212yes—
18#16839847—28no—
19C2×#14519708C2212yes—
20C2×#14519706C2212yes—
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23C2×#14519697C2212yes—
24C2×C8×Dic61C2212yes—
25#16839669C2211yescharacteristic
26#16839663—21no—
27#16839663—21no—
28#16839663C2212yes—
29#16839660C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839647—21no—
32#16839645—21no—
33C4⋊C4×Dic61C2211yescharacteristic
34#16839613—21no—
35#16839607—21no—
36C42×Dic61C2211yescharacteristic
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39C2×#103×C61C2212yes—
40C61×#391C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#336C2211yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C4×D61—22no—
48C2×#11758201—24no—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C22×#92418C2312yes—
52#14519774—24no—
53Q8×Dic61—22no—
54Q8×Dic61D414yescomplemented
55#14519739—24no—
56#14519739—24no—
57#14519739—44no—
58#14519739—44no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75#14519708—24no—
76#14519706—24no—
77#14519699—24no—
78#14519699—24no—
79#14519697—24no—
80C8×Dic61—24no—
81#14519684—21no—
82#14519684—21no—
83#14519684—21no—
84#14519684—21no—
85#14519684—21no—
86C22×Q8×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2312yes—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C42×C61C2311yescharacteristic
93#103×C61—24no—
94#95×C61—21no—
95#95×C61—21no—
96C23×D61—22no—
97C2×Q8×C61—21no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C2×Q8×C61C2×D412yes—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112C22×Dic61C2×D411yescharacteristic
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—24no—
118C2×D244—24no—
119C2×D244—22no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—42no—
123C2×D244—42no—
124C2×D244#4114yes—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61—44no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—42no—
136C42×C61—22no—
137#11758201—216no—
138#11758198—44no—
139#11758198—44no—
140#11758198—44no—
141C4⋊Dic61—21no—
142C4⋊Dic61—24no—
143C4⋊Dic61—22no—
144C4⋊Dic61C2×D412yes—
145#11758196—42no—
146#11758196—42no—
147C4×Dic61—22no—
148C4×Dic61—22no—
149C4×Dic61—24no—
150C4×Dic61—22no—
151C4×Dic61—41no—
152C4×Dic61—42no—
153C4×Dic61SD1614yescomplemented
154C4×Dic61C2×D412yes—
155C4×Dic61#4112yes—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418#4114yes—
159C23×C61C22×D411yescharacteristic
160C22×D61—24no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167Q8×C61—24no—
168Q8×C61—42no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—22no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61C2×SD1612yes—
183C2×C4×C61#13512yes—
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C2×#4112yes—
186C2×C4×C61C2×#4112yes—
187C2×C4×C61C2×#4111yescharacteristic
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—22no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61#12112yes—
202C2×Dic61#12112yes—
203C2×Dic61C2×SD1612yes—
204C2×Dic61C2×SD1612yes—
205C2×Dic61AGL(1,ℤ/8)12yes—
206D244—24no—
207D244—24no—
208D244—48no—
209D244—48no—
210D244—42no—
211D244—42no—
212C4×D61—88no—
213C8×C61—42no—
214#92418—24no—
215#23678—611nomaximal, Sylow
216C22×C61C2×#12111yescharacteristic
217C22×C61#52511yescharacteristic
218C22×C61#53211yescharacteristic
219C22×C61C22×SD1611yescharacteristic
220C22×C61C2×#13511yescharacteristic
221C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C61#57511yescharacteristic
223C2×D61—44no—
224C2×D61—44no—
225C2×D61—44no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—82no—
230C4×C61—22no—
231C4×C61—22no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61—82no—
235C4×C61#52512yes—
236C4×C61#52912yes—
237Dic61—22no—
238Dic61—22no—
239Dic61—41no—
240Dic61—42no—
241Dic61—81no—
242Dic61#46014yes—
243C2×C4×SD16—612no—
244C4×#127—611no—
245#2295—611no—
246#2084—611no—
247#2072D6111yescharacteristic, complemented
248#2065—612no—
249#2055—612no—
250#2041—611no—
251#2014—611no—
252C4×C4⋊C8—611no—
253C4×#102—612no—
254C2×C61#239211yescharacteristic
255C2×C61C2×#46011yescharacteristic
256C2×C61#303511yescharacteristic
257C2×C61#310111yescharacteristic
258C2×C61#310511yescharacteristic
259C2×C61#316711yescharacteristic
260C2×C61#317111yescharacteristic
261D61—84no—
262C22×SD16—612no—
263C2×#127—611no—
264C2×C4×Q8—611no—
265C2×C4×D4—612no—
266C2×C4×D4—1221no—
267C2×C4×D4—1221no—
268C4×SD16—1228no—
269C2×#106—612no—
270C2×C4⋊C8—612no—
271C2×#103C2×D6112yes—
272C2×#102—612no—
273C2×#102—612no—
274C2×#102—612no—
275C2×C4×C8—612no—
276#391—611no—
277#391C2×D6111yescharacteristic
278#390—612no—
279#390—1221no—
280#390—1221no—
281#389—611no—
282#389—611no—
283#387—1221no—
284#387—1221no—
285#384C2×D6111yescharacteristic
286C4×C4⋊C4—611no—
287C4×C4⋊C4—1221no—
288#376—1221no—
289C43—611no—
290#340—612no—
291#340C2×D6112yes—
292#336—611no—
293#336C2×D6111yescharacteristic
294C61#2367811yescharacteristic, complemented, Sylow
295C22×Q8C22×D6111yescharacteristic
296C22×D4—612no—
297C22×D4—1221no—
298C22×D4—1221no—
299C23×C4—1222no—
300C2×SD16—1224no—
301C2×SD16—1224no—
302C2×SD16—1224no—
303C22×C8—612no—
304C22×C8C22×D6112yes—
305#127—1224no—
306C4×Q8—614no—
307C4×Q8—1222no—
308C4×D4—1224no—
309C4×D4—1224no—
310C4×D4—2444no—
311C4×D4—2444no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22×D6112yes—
320C2×C4⋊C4C22×D6111yescharacteristic
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C42—611no—
325C2×C42—611no—
326C2×C42—611no—
327C2×C42—611no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—1221no—
331C2×C42—1221no—
332C2×C42C22×D6111yescharacteristic
333#106—1224no—
334C4⋊C8—1224no—
335#103—24no—
336#102—1224no—
337#102—1224no—
338#102—1224no—
339C4×C8—1224no—
340#95—21no—
341#95—21no—
342#95—1221no—
343#95—1221no—
344#95—1221no—
345#95—1221no—
346#95—1221no—
347C24—1222no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8D4×D6112yes—
352C2×D4—614no—
353C2×D4—1224no—
354C2×D4—1224no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—1222no—
359C2×D4—1222no—
360C2×D4—2442no—
361C2×D4—2442no—
362C2×D4—2442no—
363C2×D4—2442no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—611no—
368C22×C4—611no—
369C22×C4—611no—
370C22×C4—611no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—2444no—
374C22×C4—2444no—
375C22×C4—2444no—
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4C23×D6111yescharacteristic
379SD16—12216no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—614no—
384C2×C8—1222no—
385C2×C8—1222no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—612no—
392C4⋊C4—1224no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C22⋊C4—2444no—
398C22⋊C4—2444no—
399C22⋊C4—2444no—
400C42—22no—
401C42—612no—
402C42—612no—
403C42—614no—
404C42—1222no—
405C42—1224no—
406C42—1222no—
407C42—1222no—
408C42—2441no—
409C42—2442no—
410C23—1224no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23—2442no—
417C23C2×D4×D6111yescharacteristic
418Q8—24no—
419Q8—42no—
420D4—1224no—
421D4—1224no—
422D4—2448no—
423D4—2448no—
424D4—2442no—
425D4—2442no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—612no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1222no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—4888no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4SD16×D6112yes—
459C2×C4#1451980012yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4#41×D6112yes—
462C2×C4#41×D6112yes—
463C2×C4#41×D6111yescharacteristic
464C8—42no—
465C8—1224no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—2444no—
471C22—2444no—
472C22—2444no—
473C22#121×D6111yescharacteristic
474C22#1684027011yescharacteristic
475C22#1684031111yescharacteristic
476C22C2×SD16×D6111yescharacteristic
477C22C2×#1451980011yescharacteristic
478C22AGL(1,ℤ/8)×D6111yescharacteristic
479C22#1684056111yescharacteristic
480C4—22no—
481C4—22no—
482C4—42no—
483C4—81no—
484C4—82no—
485C4—614no—
486C4—1222no—
487C4—1222no—
488C4—4881no—
489C4—2441no—
490C4—2442no—
491C4#1684027012yes—
492C4#1684029512yes—
493C2—4884no—
494C2#2144905611yescharacteristic
495C2#460×D6111yescharacteristic
496C2#2145527711yescharacteristic
497C2#2145582411yescharacteristic
498C2#2145586911yescharacteristic
499C2#2145638111yescharacteristic
500C2#2145641711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2072C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2072D61
hypocenterC61#23678
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
1. C2×#16839847C2semidirect
3. #21448294C2semidirect
4. #21447296C2semidirect
5. #21447194C2semidirect
6. #21447133C2semidirect
9. #21445667C2semidirect
10. #21445633C2semidirect
11. C61×#2072C2semidirect
54. Q8×Dic61D4semidirect
153. C4×Dic61SD16semidirect
247. #2072D61semidirect
294. C61#23678semidirect

Maximal subgroups

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

subgroupindex
C2×#168398472
#214483812
#214482942
#214472962
#214471942
#214471332
#214470542
#214469062
#214456672
#214456332
C61×#20722
#2367861