Finite Group Databasebeta

← #29709001

Subgroups — #29709001

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839801C212yesmaximal
2#389×D61C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4#21447306C212yesmaximal, complemented
5#21447304C212yesmaximal, complemented
6#21447296C211yesmaximal, characteristic, complemented
7#21447292C211yesmaximal, characteristic, complemented
8#21447290C211yesmaximal, characteristic, complemented
9#21445537C212yesmaximal
10#21445511C211yesmaximal, characteristic
11C61×#2084C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13C2×#14519735C2212yes—
14C2×C4⋊C4×D61—21no—
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C4⋊C4×D61C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839801—28no—
19C2×#14519711C2212yes—
20C2×#14519710C2212yes—
21C2×#14519708C2212yes—
22#16839661C2211yescharacteristic
23#16839660C2211yescharacteristic
24#16839659—21no—
25#16839658C2211yescharacteristic
26#16839657—21no—
27#16839632C2211yescharacteristic
28#16839627C2212yes—
29#16839619—21no—
30#16839619—21no—
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61—21no—
33#95×D61—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38#16839516C2212yes—
39C2×#106×C61C2212yes—
40C61×#389C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#336C2211yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C2×#11758189—24no—
50C2×#11758189—24no—
51C2×#11758189Q818yes—
52C22×#92418C2312yes—
53#14519735—24no—
54#14519735—24no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196—21no—
65C4⋊C4×D61—22no—
66C4⋊C4×D61—24no—
67C4⋊C4×D61—44no—
68C4⋊C4×D61D414yes—
69C2×#11758198—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74C42×D61—24no—
75#14519711—24no—
76#14519710—24no—
77#14519708—24no—
78#14519684—21no—
79#14519684—21no—
80#14519684—21no—
81#14519684—22no—
82#14519684—22no—
83C22×C8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C42×C61C2311yescharacteristic
91#106×C61—24no—
92#95×C61—21no—
93C23×D61C2×D411yescharacteristic
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—22no—
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C2×C4×D61—24no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×C4×D61C2×D412yes—
119C2×C4×D61C2×Q814yes—
120C2×C4×D61#4114yes—
121C2×C4×D61#4112yes—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61C2×Q814yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—42no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61C2×D412yes—
133C42×C61—22no—
134#11758198—44no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61—22no—
139C4⋊Dic61—42no—
140C4⋊Dic61—42no—
141C4⋊Dic61—42no—
142C4⋊Dic61C2×D412yes—
143#11758196—24no—
144#11758196—42no—
145#11758196—42no—
146#11758196—42no—
147C4×Dic61—22no—
148C4×Dic61—22no—
149C4×Dic61—22no—
150C4×Dic61—44no—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418C2×Q814yes—
154#11758189—216no—
155C23×C61C22×D411yescharacteristic
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61#12112yes—
161C22×D61#12312yes—
162C22×D61C4⋊Q812yes—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61AGL(1,ℤ/8)12yes—
177C2×C4×C61#13712yes—
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×Q812yes—
180C2×C4×C61C2×#4112yes—
181C2×C4×C61C2×#4111yescharacteristic
182C2×Dic61—22no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61#12112yes—
198C2×Dic61#12312yes—
199C2×Dic61C4⋊Q812yes—
200C4×D61—44no—
201C4×D61—44no—
202C4×D61—48no—
203C4×D61—84no—
204C4×D61C22⋊Q818yes—
205C4×D61#12518yes—
206C8×C61—24no—
207#92418—24no—
208#23657—611nomaximal, Sylow
209C22×C61C2×#12111yescharacteristic
210C22×C61C2×#12311yescharacteristic
211C22×C61C2×C4⋊Q811yescharacteristic
212C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
213C22×C61C2×#13711yescharacteristic
214C22×C61#57511yescharacteristic
215C22×C61#57511yescharacteristic
216C2×D61—22no—
217C2×D61—22no—
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—22no—
221C2×D61—22no—
222C2×D61#39814yes—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—42no—
226C4×C61—81no—
227C4×C61—82no—
228C4×C61C2×C22⋊Q812yes—
229C4×C61C2×#12512yes—
230Dic61—41no—
231Dic61—41no—
232Dic61—42no—
233Dic61—81no—
234Dic61—82no—
235Dic61#39814yes—
236C2×#428—612no—
237C2×#389—611no—
238#2285—611no—
239#2085—612no—
240#2085—612no—
241#2084—611no—
242#2084D6111yescharacteristic, complemented
243#2083—611no—
244#2083—611no—
245#1736—612no—
246#1728—611no—
247C2×C61C2×#39811yescharacteristic
248C2×C61#303611yescharacteristic
249C2×C61#303711yescharacteristic
250C2×C61#309411yescharacteristic
251C2×C61#309511yescharacteristic
252C2×C61#316511yescharacteristic
253C2×C61#316611yescharacteristic
254D61—24no—
255C22×M16—612no—
256C2×#117—612no—
257C22×C4⋊C4—611no—
258C22×C4⋊C4—611no—
259C22×C4⋊C4—1221no—
260C22×C42—611no—
261#428—1228no—
262C2×#107—612no—
263C2×#107—612no—
264C2×#106—612no—
265C2×#106C2×D6112yes—
266#389—611no—
267#389—611no—
268#389—611no—
269#389—1221no—
270#389—1221no—
271#389C2×D6111yescharacteristic
272#384—611no—
273#384C2×D6111yescharacteristic
274#382—612no—
275#380—1221no—
276#380—1221no—
277C4×C4⋊C4—1221no—
278C4×C4⋊C4—1221no—
279C2×#95—1221no—
280#340—612no—
281#340—612no—
282#340—612no—
283#340C2×D6112yes—
284#336—612no—
285#336—611no—
286#336C2×D6111yescharacteristic
287C61#2365711yescharacteristic, complemented, Sylow
288C23×C4—611no—
289C23×C4—611no—
290C23×C4—611no—
291C23×C4—1221no—
292C23×C4—1221no—
293C2×M16—618no—
294C2×M16—1224no—
295C2×M16—1224no—
296C22×C8—612no—
297C22×C8C22×D6112yes—
298#117—1224no—
299#117—1224no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—614no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1224no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—2444no—
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C4⋊C4C22×D6112yes—
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C22⋊C4—1222no—
321C2×C42—611no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—1224no—
325C2×C42—1222no—
326C2×C42C22×D6111yescharacteristic
327#107—1224no—
328#107—1224no—
329#106—24no—
330#106—1224no—
331#95—21no—
332#95—1221no—
333#95—1221no—
334#95—1222no—
335#95—1222no—
336#95—1221no—
337C24—611no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—611no—
342C22×C4—612no—
343C22×C4—614no—
344C22×C4—614no—
345C22×C4—612no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1224no—
351C22×C4—1221no—
352C22×C4—1222no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1221no—
356C22×C4—2441no—
357C22×C4—2441no—
358C22×C4—2442no—
359C22×C4—2442no—
360C22×C4—2442no—
361C22×C4—2442no—
362C22×C4—2442no—
363C22×C4D4×D6111yescharacteristic
364C22×C4D4×D6111yescharacteristic
365C22×C4C23×D6111yescharacteristic
366M16—12216no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—614no—
370C2×C8—1222no—
371C2×C8—1222no—
372C2×C8Q8×D6114yes—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—612no—
381C4⋊C4—1224no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—1222no—
385C4⋊C4—1222no—
386C4⋊C4—2442no—
387C4⋊C4—2442no—
388C4⋊C4—2442no—
389C4⋊C4—2442no—
390C4⋊C4—2442no—
391C4⋊C4—2442no—
392C4⋊C4D4×D6112yes—
393C22⋊C4—2444no—
394C42—22no—
395C42—1222no—
396C42—1222no—
397C42—1222no—
398C42—2444no—
399C23—612no—
400C23—612no—
401C23—612no—
402C23—1221no—
403C23—1221no—
404C23—1221no—
405C23—1221no—
406C23C2×D4×D6111yescharacteristic
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—612no—
421C2×C4—612no—
422C2×C4—612no—
423C2×C4—618no—
424C2×C4—618no—
425C2×C4—1222no—
426C2×C4—1221no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—4884no—
433C2×C4—2444no—
434C2×C4—2444no—
435C2×C4—2448no—
436C2×C4—2442no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4#1451980412yes—
445C2×C4#1451980612yes—
446C2×C4C2×D4×D6111yescharacteristic
447C2×C4C2×Q8×D6112yes—
448C2×C4#41×D6112yes—
449C2×C4#41×D6111yescharacteristic
450C8—24no—
451C8—1224no—
452C22—614no—
453C22—1222no—
454C22—1222no—
455C22—1222no—
456C22—1222no—
457C22—1222no—
458C22—1222no—
459C22#121×D6111yescharacteristic
460C22#123×D6111yescharacteristic
461C22C4⋊Q8×D6111yescharacteristic
462C22C2×#1451980411yescharacteristic
463C22C2×#1451980611yescharacteristic
464C22#1684055811yescharacteristic
465C22#1684056111yescharacteristic
466C4—22no—
467C4—22no—
468C4—42no—
469C4—81no—
470C4—82no—
471C4—614no—
472C4—4881no—
473C4—4882no—
474C4—2441no—
475C4—2441no—
476C4—2442no—
477C4C22⋊Q8×D6112yes—
478C4#125×D6112yes—
479C2—1224no—
480C2#398×D6111yescharacteristic
481C2#2145528411yescharacteristic
482C2#2145528611yescharacteristic
483C2#2145575611yescharacteristic
484C2#2145576211yescharacteristic
485C2#2145636011yescharacteristic
486C2#2145637011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2084C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2084D61
hypocenterC61#23657
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
4. #21447306C2semidirect
5. #21447304C2semidirect
6. #21447296C2semidirect
7. #21447292C2semidirect
8. #21447290C2semidirect
11. C61×#2084C2semidirect
242. #2084D61semidirect
287. C61#23657semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
#389×D612
#214482242
#214473062
#214473042
#214472962
#214472922
#214472902
#214455372
#214455112
C61×#20842
#2365761