Finite Group Databasebeta

← #29834504

Subgroups — #29834504

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×#16840202C211yesmaximal, characteristic, complemented
2C2×M32×D61C211yesmaximal, characteristic, complemented
3#21455396C211yesmaximal, characteristic
4C2×#16840004C211yesmaximal, characteristic, complemented
5#21447931C212yesmaximal, complemented
6#21447926C212yesmaximal, complemented
7#21447914C212yesmaximal, complemented
8#21447909C212yesmaximal, complemented
9#21447605C211yesmaximal, characteristic, complemented
10#21447540C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840269—21no—
15C2×#14519757—21no—
16#16840202—24no—
17C2×#14519723C2212yescomplemented
18C2×#14519721C2212yescomplemented
19C2×#14519717C2211yescharacteristic
20C2×#14519716C2212yes—
21M32×D61—24no—
22C2×C16×D61C2212yes—
23#16840004—21no—
24#16840004—21no—
25#16840004C2212yes—
26#16840004C2212yes—
27#16840000C2211yescharacteristic
28#16840000C2211yescharacteristic
29#16839995C2211yescharacteristic
30#16839995C2211yescharacteristic
31#16839988—21no—
32#16839988—21no—
33#16839800C2211yescharacteristic
34#16839758—21no—
35C2×#14519700—21no—
36#16839589C2212yescomplemented
37#16839587C2212yescomplemented
38#16839567C2212yescomplemented
39#16839567C2212yescomplemented
40C2×#135×C61C2211yescharacteristic
41C2×M32×C61C2211yescharacteristic
42C2×#107×C61C2211yescharacteristic
43C61×#358C2212yescomplemented
44C61×#357C2212yescomplemented
45C22×Dic122—21no—
46C22×C4×D61D411yescharacteristic
47C2×Dic244—22no—
48C2×Dic244—22no—
49C2×Dic244—21no—
50C2×Dic244C2312yes—
51C22×#92418D411yescharacteristic
52C2×C8×D61—22no—
53C2×C8×D61—22no—
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56C2×C8×D61C2311yescharacteristic
57C2×C8×D61C2311yescharacteristic
58#14519758—41no—
59#14519757—22no—
60#14519757—22no—
61#14519757—41no—
62#14519757—41no—
63#14519753—21no—
64#14519753—21no—
65#14519748—21no—
66#14519748—21no—
67#14519745—41no—
68C4×C22⋊D61—41no—
69C4×C22⋊D61—41no—
70#14519736—21no—
71C2×C4⋊Dic61—41no—
72C2×#11758198—41no—
73#14519723—24no—
74#14519721—24no—
75#14519717D414yescomplemented
76#14519716—24no—
77C2×#11758185C2312yes—
78C16×D61—24no—
79#14519711C2311yescharacteristic
80#14519711C2311yescharacteristic
81#14519709—21no—
82#14519701—21no—
83#14519701—21no—
84#14519700—22no—
85#14519700—22no—
86#14519698—21no—
87#14519698—21no—
88C2×#41×C61—21no—
89#135×C61—22no—
90#135×C61—22no—
91C2×Q16×C61C2311yescharacteristic
92C2×SD16×C61—21no—
93C2×D8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95C2×C4⋊C4×C61—21no—
96M32×C61—22no—
97C2×C16×C61C2312yes—
98#107×C61—21no—
99#107×C61C2312yes—
100C23×D61D811yescharacteristic
101#41×C61—42no—
102#41×C61—42no—
103C2×Q8×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C2×C22⋊D61—41no—
110C2×C22⋊D61—41no—
111C22×Dic61—41no—
112C22×Dic61D811yescharacteristic
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61D811yescharacteristic
116C2×C4×D61D811yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×Dic122—22no—
120C2×Dic122—42no—
121C2×Dic122—42no—
122C2×Dic122—41no—
123Q16×C61—21no—
124Q16×C61—21no—
125SD16×C61—42no—
126D8×C61—21no—
127D8×C61—21no—
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D412yes—
130C2×C8×C61#4112yes—
131C2×C8×C61C2411yescharacteristic
132C4⋊C4×C61—22no—
133C4⋊C4×C61—41no—
134C22⋊Dic61—41no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137#11758198—42no—
138#11758198—42no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—42no—
144C4⋊Dic61—42no—
145#11758196—41no—
146#11758196—41no—
147C4×Dic61—41no—
148C2×#92418—22no—
149C2×#92418C2×D411yescharacteristic
150C2×#92418C2×D411yescharacteristic
151Dic244—22no—
152Dic244—22no—
153Dic244—42no—
154C8×D61—22no—
155C8×D61—22no—
156C8×D61—21no—
157C8×D61—21no—
158C8×D61—21no—
159C8×D61—21no—
160C16×C61—22no—
161#11758185C2×D414yes—
162C23×C61—41no—
163C23×C61C2×D811yescharacteristic
164C22×D61—22no—
165C22×D61—22no—
166C22×D61—41no—
167C22×D61C2×D811yescharacteristic
168C22×D61C2×D811yescharacteristic
169Q8×C61—41no—
170Q8×C61—41no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—82no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—82no—
177C2×C4×C61—82no—
178C2×C4×C61#12112yes—
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61#13512yes—
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C22⋊D61—82no—
184C2×Dic61—22no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—82no—
189C2×Dic61C2×D811yescharacteristic
190C2×Dic61C2×D811yescharacteristic
191C4×D61—21no—
192C4×D61—21no—
193C4×D61—21no—
194C4×D61—21no—
195C4×D61—22no—
196C4×D61—22no—
197Dic122—42no—
198Dic122—42no—
199Dic122—82no—
200C8×C61#12112yes—
201C8×C61C22×D411yescharacteristic
202C8×C61C2×#4111yescharacteristic
203#92418—21no—
204#92418—21no—
205#33606—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61#46612yes—
212C22×C61C22×D811yescharacteristic
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—21no—
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—44no—
219C2×D61#51012yescomplemented
220C4×C61—82no—
221C4×C61—81no—
222C4×C61#46612yes—
223C4×C61C2×#12111yescharacteristic
224C4×C61C2×#13511yescharacteristic
225Dic61—21no—
226Dic61—21no—
227Dic61—81no—
228Dic61—81no—
229Dic61—82no—
230C2×#510—611no—
231C22×M32—611no—
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#2139D6111yescharacteristic, complemented
241C2×C61—22no—
242C2×C61—161no—
243C2×C61C2×#46611yescharacteristic
244C2×C61C2×#51011yescharacteristic
245C2×C61#341411yescharacteristic
246D61—22no—
247D61—41no—
248C2×#135C2×D6111yescharacteristic
249C22×Q16—611no—
250C23×C8—611no—
251#525—1221no—
252C2×C22⋊Q8—1221no—
253#510—1224no—
254C2×Q32—612no—
255C2×SD32—612no—
256C2×M32—611no—
257C2×M32—612no—
258C2×M32—1224no—
259C2×M32C2×D6111yescharacteristic
260C22×C16—612no—
261#467—611no—
262#467—611no—
263#467—612no—
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×#107C2×D6111yescharacteristic
273#416—1221no—
274C2×#103—1221no—
275#367—612no—
276#366—612no—
277#358—612no—
278#358—612no—
279#358C2×D6112yescomplemented
280#357C2×D6112yescomplemented
281C61#3360611yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×Q8—1221no—
284C23×C4—611no—
285#135—22no—
286#135—22no—
287C2×Q16—612no—
288C2×Q16—1222no—
289C2×Q16—1222no—
290C2×Q16—1221no—
291C2×Q16C22×D6111yescharacteristic
292C2×SD16—21no—
293C2×D8C22×D6111yescharacteristic
294C22×C8—611no—
295C22×C8—611no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—611no—
299C22×C8—1222no—
300C22×C8—1222no—
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—21no—
316C2×C4⋊C4—2441no—
317C2×C22⋊C4—2441no—
318Q32—1224no—
319SD32—1224no—
320M32—22no—
321M32—614no—
322M32—1224no—
323C2×C16—612no—
324C2×C16—1224no—
325C2×C16C22×D6112yes—
326#107—21no—
327#107—611no—
328#107—611no—
329#107C22×D6112yes—
330#106—1221no—
331#103—1222no—
332#103—1222no—
333#103—1221no—
334#103—1221no—
335#102—1221no—
336#102—1221no—
337C24—611no—
338#41—42no—
339#41—42no—
340C2×Q8—21no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×Q8—2441no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—2441no—
359C22×C4D4×D6111yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—1222no—
363Q16—1222no—
364Q16—2442no—
365SD16—42no—
366D8—21no—
367D8—21no—
368C2×C8—611no—
369C2×C8—611no—
370C2×C8—1222no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8#1175822212yes—
378C2×C8D4×D6112yes—
379C2×C8D4×D6111yescharacteristic
380C2×C8C23×D6111yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—41no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
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—
398C16—614no—
399C23—41no—
400C23—611no—
401C23—611no—
402C23—1222no—
403C23—1222no—
404C23—2441no—
405C23D8×D6111yescharacteristic
406Q8—41no—
407Q8—41no—
408Q8—4882no—
409Q8—2442no—
410Q8—2442no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—4882no—
415D4—4882no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—611no—
421C2×C4—611no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1221no—
426C2×C4—1221no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—4882no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4#1451975012yes—
434C2×C4D8×D6111yescharacteristic
435C2×C4#1451979812yes—
436C2×C4C2×D4×D6111yescharacteristic
437C8—1221no—
438C8—1221no—
439C8#1451975012yes—
440C8C2×D4×D6111yescharacteristic
441C8C2×#1175822211yescharacteristic
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—612no—
448C22—1221no—
449C22—1221no—
450C22—1221no—
451C22—1221no—
452C22—1221no—
453C22—2444no—
454C22#1683999712yes—
455C22C2×D8×D6111yescharacteristic
456C4—82no—
457C4—81no—
458C4—1221no—
459C4—1221no—
460C4—4881no—
461C4—4881no—
462C4—4882no—
463C4#1683999712yes—
464C4C2×#1451975011yescharacteristic
465C4C2×#1451979811yescharacteristic
466C2—22no—
467C2—161no—
468C2—1222no—
469C2—2441no—
470C2C2×#1683999711yescharacteristic
471C2#510×D6111yescharacteristic
472C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radical#21455396C2
hypercenter#2139D61
hypocenterC61#33606
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. C2×M32×D61C2semidirect
4. C2×#16840004C2semidirect
5. #21447931C2semidirect
6. #21447926C2semidirect
7. #21447914C2semidirect
8. #21447909C2semidirect
9. #21447605C2semidirect
10. #21447540C2semidirect
11. C61×#2139C2semidirect
17. C2×#14519723C22semidirect
18. C2×#14519721C22semidirect
36. #16839589C22semidirect
37. #16839587C22semidirect
38. #16839567C22semidirect
39. #16839567C22semidirect
43. C61×#358C22semidirect
44. C61×#357C22semidirect
75. #14519717D4semidirect
219. C2×D61#510semidirect
240. #2139D61semidirect
279. #358C2×D61semidirect
280. #357C2×D61semidirect
281. C61#33606semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C2×M32×D612
#214553962
C2×#168400042
#214479312
#214479262
#214479142
#214479092
#214476052
#214475402
C61×#21392
#3360661