Finite Group Databasebeta

← #29834501

Subgroups — #29834501

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840205C211yesmaximal, characteristic, complemented
2#21455400C211yesmaximal, characteristic
3C2×#16840003C211yesmaximal, characteristic, complemented
4#21447932C212yesmaximal, complemented
5#21447927C212yesmaximal, complemented
6#21447915C212yesmaximal, complemented
7#21447910C212yesmaximal, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447540C211yesmaximal, characteristic, complemented
10#21447532C211yesmaximal, characteristic, complemented
11C22×M32×C61C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×Dic244C2211yescharacteristic
14#16840273—21no—
15C2×#14519756—21no—
16#16840205—24no—
17C2×Dic488C2212yescomplemented
18C2×#14519722C2212yescomplemented
19#16840003—21no—
20#16840003—21no—
21#16840003C2212yes—
22#16840003C2212yes—
23#16840003C2211yescharacteristic
24#16840003C2211yescharacteristic
25#16839998C2211yescharacteristic
26#16839998C2211yescharacteristic
27#16839991—21no—
28#16839991—21no—
29#16839800C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31#16839759—21no—
32C2×#14519702—21no—
33#16839590C2212yescomplemented
34#16839588C2212yescomplemented
35#16839567C2212yescomplemented
36#16839567C2212yescomplemented
37#16839567C2212yescomplemented
38#16839564C2212yescomplemented
39C23×C8×C61C2211yescharacteristic
40C2×M32×C61—24no—
41C2×M32×C61C2211yescharacteristic
42C2×M32×C61C2212yes—
43C2×M32×C61C2211yescharacteristic
44C22×C16×C61C2212yes—
45C2×#11758222—21no—
46C22×Dic122—21no—
47#14519799—22no—
48#14519799—22no—
49C2×Dic244—22no—
50C2×Dic244—22no—
51C2×Dic244—21no—
52C2×Dic244C2312yes—
53C2×Dic244C2311yescharacteristic
54C2×#11758190—21no—
55C2×D488C2311yescharacteristic
56#14519760—41no—
57#14519756—21no—
58#14519756—21no—
59#14519756—22no—
60#14519756—22no—
61#14519756—41no—
62#14519756—41no—
63#14519751—21no—
64#14519751—21no—
65#14519746—41no—
66C4×C22⋊D61—41no—
67C4×C22⋊D61—41no—
68#14519736—21no—
69C2×C4⋊Dic61—21no—
70C2×#11758196—41no—
71C2×C22⋊Dic61—41no—
72Dic488—24no—
73#14519722—24no—
74#14519711—21no—
75#14519711C2311yescharacteristic
76#14519711C2311yescharacteristic
77#14519711C2312yes—
78#14519709—21no—
79#14519702—21no—
80#14519702—21no—
81#14519702—22no—
82#14519702—22no—
83#14519699—21no—
84#14519699—21no—
85C23×C4×C61D411yescharacteristic
86C22×C8×C61—22no—
87C22×C8×C61—22no—
88C22×C8×C61D411yescharacteristic
89C22×C8×C61D411yescharacteristic
90C22×C8×C61D411yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94M32×C61—22no—
95M32×C61—24no—
96M32×C61D414yescomplemented
97C2×C16×C61—24no—
98C2×C16×C61C2312yes—
99C2×C16×C61C2312yes—
100C24×C61D811yescharacteristic
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61D811yescharacteristic
104C22×C4×C61D811yescharacteristic
105C22×C4×C61D811yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C2×C22⋊D61—41no—
110C2×C22⋊D61—41no—
111C2×C22⋊D61—41no—
112C22×Dic61—41no—
113C22×Dic61—41no—
114#11758222—42no—
115#11758222—42no—
116C2×D244—21no—
117C2×C4×D61—41no—
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—42no—
121C2×Dic122—42no—
122C2×Dic122—41no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61—21no—
127C2×C8×C61—21no—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D412yes—
134C2×C8×C61#4112yes—
135C2×C8×C61C2411yescharacteristic
136C22⋊Dic61—42no—
137C22⋊Dic61—42no—
138C22⋊Dic61—41no—
139C22⋊Dic61—41no—
140C22⋊Dic61—41no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—41no—
147#11758196—42no—
148#11758196—42no—
149#11758196—41no—
150#11758196—41no—
151C4×Dic61—41no—
152Dic244—22no—
153Dic244—21no—
154Dic244—22no—
155Dic244—21no—
156Dic244—42no—
157D488—21no—
158D488—21no—
159#11758190—42no—
160C16×C61—22no—
161C16×C61C2×D414yes—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—41no—
165C23×C61C2×D811yescharacteristic
166C23×C61C2×D811yescharacteristic
167C23×C61C2×D811yescharacteristic
168C22×D61—41no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61#12112yes—
177C2×C4×C61C2×D811yescharacteristic
178C2×C4×C61C2×D811yescharacteristic
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61#13512yes—
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C22⋊D61—82no—
184C22⋊D61—82no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—82no—
191C2×Dic61—82no—
192D244—41no—
193D244—41no—
194C4×D61—82no—
195Dic122—41no—
196Dic122—42no—
197Dic122—42no—
198Dic122—41no—
199Dic122—82no—
200C8×C61—21no—
201C8×C61—21no—
202C8×C61#12112yes—
203C8×C61C22×D411yescharacteristic
204C8×C61C2×#4111yescharacteristic
205#33606—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—44no—
214C22×C61#46612yes—
215C22×C61#51012yescomplemented
216C22×C61C22×D811yescharacteristic
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61#46612yes—
223C4×C61C2×#12111yescharacteristic
224C4×C61C2×#13511yescharacteristic
225Dic61—81no—
226Dic61—82no—
227Dic61—81no—
228Dic61—82no—
229Dic61—81no—
230C2×#510—611no—
231C22×M32D6111yescharacteristic, complemented
232#3050—611no—
233C2×#467—611no—
234#2220—612no—
235#2219—612no—
236#2217—612no—
237#2216—612no—
238#2159—611no—
239#2141—611no—
240#2139—611no—
241C2×C61—22no—
242C2×C61—22no—
243C2×C61—41no—
244C2×C61C2×#46611yescharacteristic
245C2×C61C2×#51011yescharacteristic
246C2×C61#341411yescharacteristic
247D61—161no—
248C2×#135—611no—
249C22×Q16—611no—
250C23×C8C2×D6111yescharacteristic
251#525—1221no—
252C2×C22⋊Q8—1221no—
253#510—1224no—
254C2×Q32—612no—
255C2×SD32—612no—
256C2×M32—24no—
257C2×M32C2×D6111yescharacteristic
258C2×M32C2×D6112yes—
259C2×M32C2×D6111yescharacteristic
260C22×C16C2×D6112yes—
261#467—611no—
262#467—612no—
263#467—611no—
264#467—612no—
265#467—1221no—
266#467—1221no—
267#466—611no—
268#466—611no—
269#465—1221no—
270#465—1221no—
271#427—611no—
272C2×#107—611no—
273#416—1221no—
274C2×#103—1221no—
275#367—612no—
276#366—612no—
277#358—612no—
278#358—612no—
279#358—612no—
280#357—612no—
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—1221no—
283C22×Q8—1221no—
284C23×C4D24411yescharacteristic
285#135—1222no—
286#135—1222no—
287C2×Q16—612no—
288C2×Q16—611no—
289C2×Q16—1221no—
290C2×Q16—1222no—
291C2×Q16—1222no—
292C2×SD16—1221no—
293C2×D8—611no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8D24411yescharacteristic
297C22×C8D24411yescharacteristic
298C22×C8D24411yescharacteristic
299C22×C8C22×D6111yescharacteristic
300C22×C8C22×D6111yescharacteristic
301C22×C8C22×D6111yescharacteristic
302#123—2441no—
303C22⋊Q8—1222no—
304C22⋊Q8—1222no—
305C22⋊Q8—1221no—
306C22⋊Q8—1221no—
307C22⋊Q8—2441no—
308C22⋊Q8—2441no—
309#121—1221no—
310#121—1221no—
311#120—2441no—
312C4×D4—2441no—
313C4×D4—2441no—
314#117—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—2441no—
317C2×C22⋊C4—2441no—
318Q32—1224no—
319SD32—1224no—
320M32—22no—
321M32—24no—
322M32C22⋊D6114yescomplemented
323C2×C16—24no—
324C2×C16C22×D6112yes—
325C2×C16C22×D6112yes—
326#107—612no—
327#107—611no—
328#107—611no—
329#107—1221no—
330#106—1221no—
331#103—1222no—
332#103—1222no—
333#103—1221no—
334#103—1221no—
335#102—1221no—
336#102—1221no—
337C24D48811yescharacteristic
338#41—2442no—
339#41—2442no—
340C2×Q8—1221no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×Q8—2441no—
345C2×D4—1221no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—2441no—
352C22×C4—2441no—
353C22×C4—2441no—
354C22×C4D48811yescharacteristic
355C22×C4D48811yescharacteristic
356C22×C4D48811yescharacteristic
357C22×C4C2×D24411yescharacteristic
358C22×C4C2×D24411yescharacteristic
359C22×C4C2×D24411yescharacteristic
360Q16—1222no—
361Q16—1221no—
362Q16—1222no—
363Q16—1221no—
364Q16—2442no—
365SD16—2442no—
366D8—1221no—
367D8—1221no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8C2×D24411yescharacteristic
376C2×C8C2×D24411yescharacteristic
377C2×C8C2×D24411yescharacteristic
378C2×C8#1175822212yes—
379C2×C8C2×C22⋊D6112yes—
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—2441no—
385C4⋊C4—2442no—
386C4⋊C4—2442no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C22⋊C4—2441no—
396C42—2441no—
397C16—22no—
398C16C2×C22⋊D6114yes—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—2441no—
403C23C2×D48811yescharacteristic
404C23C2×D48811yescharacteristic
405C23C2×D48811yescharacteristic
406Q8—4882no—
407Q8—2441no—
408Q8—2442no—
409Q8—2442no—
410Q8—2441no—
411D4—4882no—
412D4—4882no—
413D4—4882no—
414D4—2441no—
415D4—2441no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—4882no—
424C2×C4—4882no—
425C2×C4—4882no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4—2441no—
430C2×C4—2442no—
431C2×C4#1451975112yes—
432C2×C4C2×D48811yescharacteristic
433C2×C4C2×D48811yescharacteristic
434C2×C4C2×D48811yescharacteristic
435C2×C4#1451979912yes—
436C2×C4C22×D24411yescharacteristic
437C8—21no—
438C8—21no—
439C8#1451975112yes—
440C8C22×C22⋊D6111yescharacteristic
441C8C2×#1175822211yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22#1683999812yes—
454C22#1684020512yescomplemented
455C22C22×D48811yescharacteristic
456C4—21no—
457C4—21no—
458C4—4881no—
459C4—4882no—
460C4—4881no—
461C4—4881no—
462C4—4882no—
463C4#1683999812yes—
464C4C2×#1451975111yescharacteristic
465C4C2×#1451979911yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—9761no—
470C2C2×#1683999811yescharacteristic
471C2C2×#1684020511yescharacteristic
472C2#2145895511yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC22×M32×C61C2
socleC22×C61C22×D8
prime radical#21455400C2
hypercenterC22×M32D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#16840205C2semidirect
3. C2×#16840003C2semidirect
4. #21447932C2semidirect
5. #21447927C2semidirect
6. #21447915C2semidirect
7. #21447910C2semidirect
8. #21447607C2semidirect
9. #21447540C2semidirect
10. #21447532C2semidirect
11. C22×M32×C61C2semidirect
17. C2×Dic488C22semidirect
18. C2×#14519722C22semidirect
33. #16839590C22semidirect
34. #16839588C22semidirect
35. #16839567C22semidirect
36. #16839567C22semidirect
37. #16839567C22semidirect
38. #16839564C22semidirect
96. M32×C61D4semidirect
215. C22×C61#510semidirect
231. C22×M32D61semidirect
281. C61#33606semidirect
322. M32C22⋊D61semidirect
454. C22#16840205semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
#214554002
C2×#168400032
#214479322
#214479272
#214479152
#214479102
#214476072
#214475402
#214475322
C22×M32×C612
#3360661