Finite Group Databasebeta

← #29709933

Subgroups — #29709933

Order 15616.

902 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448300C211yesmaximal, characteristic
2#384×D61C211yesmaximal, characteristic
3#21447385C212yesmaximal, complemented
4#21447384C212yesmaximal, complemented
5#21447378C211yesmaximal, characteristic, complemented
6#21447377C211yesmaximal, characteristic, complemented
7#21447376C211yesmaximal, characteristic, complemented
8#21445537C212yesmaximal
9#21445537C212yesmaximal
10#21445511C211yesmaximal, characteristic
11C61×#2099C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13C2×#14519735C2211yescharacteristic
14C2×#14519735C2211yescharacteristic
15C2×C4⋊C4×D61—21no—
16C2×C4⋊C4×D61—21no—
17C2×C4⋊C4×D61C2212yes—
18C2×C42×D61C2211yescharacteristic
19#16839661C2211yescharacteristic
20#16839660C2211yescharacteristic
21#16839658C2211yescharacteristic
22#16839636—21no—
23#16839633—21no—
24#16839632C2211yescharacteristic
25#16839631—21no—
26#16839631—21no—
27#16839631C2212yes—
28C4⋊C4×Dic61—21no—
29#16839613—21no—
30#16839525C2212yes—
31#16839525C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839517C2211yescharacteristic
37#16839516C2212yes—
38C61×#389C2211yescharacteristic
39C61×#384C2211yescharacteristic
40C61×#340C2212yes—
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C2×#11758189—24no—
48C2×#11758189—24no—
49C2×#11758189—24no—
50C22×#92418C2312yes—
51#14519735—22no—
52#14519735—24no—
53#14519735Q814yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758196—22no—
64C2×#11758196—21no—
65C4⋊C4×D61—28no—
66C4⋊C4×D61—44no—
67C4⋊C4×D61—44no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61—21no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75C42×D61—24no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79C22×C8×C61C2312yes—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C42×C61C2311yescharacteristic
87#95×C61—21no—
88C23×D61C2×D411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2411yescharacteristic
95C22×Dic61—21no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61C2×D411yescharacteristic
99C22×Dic61C2×D411yescharacteristic
100C22×Dic61C2×D411yescharacteristic
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—44no—
108C2×C4×D61—44no—
109C2×C4×D61—44no—
110C2×C4×D61C2×Q812yes—
111C2×C4×D61#4114yes—
112C2×C4×D61#4114yes—
113C2×C4×D61#4112yes—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—42no—
121C4⋊C4×C61—42no—
122C4⋊C4×C61—42no—
123C4⋊C4×C61C2×Q812yes—
124C42×C61—22no—
125#11758198—24no—
126#11758198—42no—
127C4⋊Dic61—21no—
128C4⋊Dic61—24no—
129C4⋊Dic61—22no—
130C4⋊Dic61—42no—
131C4⋊Dic61—42no—
132C4⋊Dic61—42no—
133C4⋊Dic61C2×Q812yes—
134#11758196—44no—
135#11758196—44no—
136#11758196—42no—
137#11758196—44no—
138#11758196—42no—
139C4×Dic61—22no—
140C4×Dic61—24no—
141C4×Dic61—22no—
142C4×Dic61—22no—
143C4×Dic61—42no—
144C2×#92418—22no—
145C2×#92418—22no—
146C2×#92418—22no—
147#11758189—48no—
148C23×C61C22×D411yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61C22⋊Q812yes—
154C22×D61#12312yes—
155C22×D61#12412yes—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—22no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61AGL(1,ℤ/8)12yes—
170C2×C4×C61AGL(1,ℤ/8)12yes—
171C2×C4×C61C22×Q811yescharacteristic
172C2×C4×C61C2×#4112yes—
173C2×C4×C61C2×#4112yes—
174C2×C4×C61C2×#4111yescharacteristic
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—22no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61C22⋊Q812yes—
189C2×Dic61#12312yes—
190C2×Dic61#12412yes—
191C2×Dic61#13712yes—
192C2×Dic61#13712yes—
193C4×D61—28no—
194C4×D61—28no—
195C4×D61—88no—
196C4×D61#12518yes—
197C4×D61#12618yes—
198C8×C61—42no—
199#92418—42no—
200#23745—611nomaximal, Sylow
201C22×C61C2×C22⋊Q811yescharacteristic
202C22×C61C2×#12311yescharacteristic
203C22×C61C2×#12411yescharacteristic
204C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C61C2×#13711yescharacteristic
207C22×C61C2×#13711yescharacteristic
208C2×D61—22no—
209C2×D61—22no—
210C2×D61—22no—
211C2×D61—22no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61#40014yes—
215C4×C61—22no—
216C4×C61—22no—
217C4×C61—42no—
218C4×C61—81no—
219C4×C61—82no—
220C4×C61C2×#12512yes—
221C4×C61C2×#12612yes—
222Dic61—22no—
223Dic61—22no—
224Dic61—42no—
225Dic61—81no—
226Dic61—82no—
227Dic61#40014yes—
228#2296—611no—
229C2×#384—611no—
230#2100—612no—
231#2100—612no—
232#2099—611no—
233#2099—611no—
234#2099—611no—
235#2099D6111yescharacteristic, complemented
236#1736—612no—
237#1736—612no—
238#1728—611no—
239C2×C61C2×#40011yescharacteristic
240C2×C61#307911yescharacteristic
241C2×C61#307911yescharacteristic
242C2×C61#309211yescharacteristic
243C2×C61#309211yescharacteristic
244C2×C61#313811yescharacteristic
245C2×C61#313911yescharacteristic
246D61—24no—
247C22×M16—612no—
248C2×#117—611no—
249C2×#117—611no—
250C22×C4⋊C4—612no—
251C22×C4⋊C4—1221no—
252C22×C4⋊C4—1221no—
253C22×C42—611no—
254#389—611no—
255#389—611no—
256#389—611no—
257#389C2×D6111yescharacteristic
258#385—1221no—
259#385—1221no—
260#384—612no—
261#384—611no—
262#384—1221no—
263#384—1221no—
264#384C2×D6111yescharacteristic
265C4×C4⋊C4—1221no—
266C4×C4⋊C4—1221no—
267#340—612no—
268#340—612no—
269#340—612no—
270#340—612no—
271#340—612no—
272#340—612no—
273#340C2×D6112yes—
274#340C2×D6112yes—
275#336—612no—
276#336—611no—
277#336C2×D6111yescharacteristic
278C61#2374511yescharacteristic, complemented, Sylow
279C23×C4—611no—
280C23×C4—611no—
281C23×C4—611no—
282C23×C4—1222no—
283C2×M16—1224no—
284C2×M16—1224no—
285C2×M16—1224no—
286C22×C8—612no—
287C22×C8C22×D6112yes—
288#117—614no—
289#117—1222no—
290#117—1224no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—611no—
295C2×C4⋊C4—612no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—1228no—
298C2×C4⋊C4—1222no—
299C2×C4⋊C4—1222no—
300C2×C4⋊C4—1222no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—2444no—
307C2×C4⋊C4—2444no—
308C2×C4⋊C4C22×D6112yes—
309C2×C4⋊C4C22×D6111yescharacteristic
310C2×C4⋊C4C22×D6111yescharacteristic
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C42—611no—
314C2×C42—611no—
315C2×C42—611no—
316C2×C42—1224no—
317C2×C42—1221no—
318C2×C42—1221no—
319C2×C42C22×D6111yescharacteristic
320#95—21no—
321#95—1221no—
322#95—1221no—
323#95—1221no—
324C24—611no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—611no—
329C22×C4—612no—
330C22×C4—614no—
331C22×C4—614no—
332C22×C4—612no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—1222no—
336C22×C4—1222no—
337C22×C4—1222no—
338C22×C4—1222no—
339C22×C4—1222no—
340C22×C4—1222no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1222no—
344C22×C4—2444no—
345C22×C4—2444no—
346C22×C4—2444no—
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349C22×C4C23×D6111yescharacteristic
350M16—2448no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—1222no—
355C2×C8—1222no—
356C2×C8—1222no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—612no—
364C4⋊C4—1221no—
365C4⋊C4—1224no—
366C4⋊C4—1222no—
367C4⋊C4—2444no—
368C4⋊C4—2442no—
369C4⋊C4—2442no—
370C4⋊C4—2444no—
371C4⋊C4—2444no—
372C4⋊C4—2442no—
373C4⋊C4—2442no—
374C4⋊C4—2442no—
375C4⋊C4Q8×D6112yes—
376C22⋊C4—1224no—
377C22⋊C4—2442no—
378C42—22no—
379C42—1222no—
380C42—1222no—
381C42—1224no—
382C42—1222no—
383C42—2442no—
384C23—612no—
385C23—612no—
386C23—612no—
387C23—1221no—
388C23—1221no—
389C23—1221no—
390C23—1221no—
391C23C2×D4×D6111yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—612no—
406C2×C4—612no—
407C2×C4—612no—
408C2×C4—618no—
409C2×C4—618no—
410C2×C4—612no—
411C2×C4—612no—
412C2×C4—1228no—
413C2×C4—1228no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1222no—
419C2×C4—4888no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4#1451980112yes—
429C2×C4#1451980412yes—
430C2×C4C2×Q8×D6111yescharacteristic
431C2×C4#41×D6112yes—
432C2×C4#41×D6112yes—
433C2×C4#41×D6111yescharacteristic
434C8—42no—
435C8—2442no—
436C22—614no—
437C22—1222no—
438C22—1222no—
439C22—1222no—
440C22—1222no—
441C22—1222no—
442C22—1222no—
443C22C22⋊Q8×D6111yescharacteristic
444C22#123×D6111yescharacteristic
445C22#124×D6111yescharacteristic
446C22C2×#1451980111yescharacteristic
447C22C2×#1451980411yescharacteristic
448C22#137×D6111yescharacteristic
449C22#137×D6111yescharacteristic
450C4—22no—
451C4—22no—
452C4—42no—
453C4—82no—
454C4—81no—
455C4—614no—
456C4—1222no—
457C4—1222no—
458C4—4882no—
459C4—4881no—
460C4—2442no—
461C4#125×D6112yes—
462C4#126×D6112yes—
463C2—1224no—
464C2#400×D6111yescharacteristic
465C2#2145562711yescharacteristic
466C2#2145562911yescharacteristic
467C2#2145573811yescharacteristic
468C2#2145574311yescharacteristic
469C2#2145619011yescharacteristic
470C2#2145619811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2099C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2099D61
hypocenterC61#23745
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
3. #21447385C2semidirect
4. #21447384C2semidirect
5. #21447378C2semidirect
6. #21447377C2semidirect
7. #21447376C2semidirect
11. C61×#2099C2semidirect
235. #2099D61semidirect
278. C61#23745semidirect

Maximal subgroups

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

subgroupindex
#214483002
#384×D612
#214473852
#214473842
#214473782
#214473772
#214473762
#214455372
#214455112
C61×#20992
#2374561