Finite Group Databasebeta

← #29834484

Subgroups — #29834484

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D61C211yesmaximal, characteristic, complemented
2#21455396C211yesmaximal, characteristic, complemented
3C2×#16839995C211yesmaximal, characteristic, complemented
4#21447929C212yesmaximal, complemented
5#21447919C212yesmaximal, complemented
6#21447912C212yesmaximal, complemented
7#21447902C212yesmaximal, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447535C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic, complemented
11C2×C61×#509C211yesmaximal, characteristic, complemented
12C22×C8×D61C2211yescharacteristic
13#16840269—21no—
14C2×#14519748—21no—
15C2×#14519717C2211yescharacteristic
16C2×#14519716C2212yescomplemented
17M32×D61—24no—
18C2×C16×D61C2212yescomplemented
19#16840000C2211yescharacteristic, complemented
20#16840000C2211yescharacteristic, complemented
21#16839995—21no—
22#16839995—21no—
23#16839995C2212yes—
24#16839995C2212yes—
25#16839995C2211yescharacteristic, complemented
26#16839995C2211yescharacteristic, complemented
27#16839988—21no—
28#16839988—21no—
29#16839800C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31#16839758—21no—
32C2×#14519698—21no—
33#16839590C2212yescomplemented
34#16839588C2212yescomplemented
35#16839566C2212yescomplemented
36#16839563C2212yescomplemented
37#16839563C2212yescomplemented
38#16839563C2212yescomplemented
39C2×#135×C61C2211yescharacteristic
40C22×D8×C61C2211yescharacteristic
41C61×#509—24no—
42C2×SD32×C61C2212yescomplemented
43C2×D16×C61C2212yescomplemented
44C2×M32×C61C2211yescharacteristic
45C22×C22⋊D61—41no—
46C22×C4×D61D411yescharacteristic
47C22×#92418D411yescharacteristic
48C2×C8×D61—22no—
49C2×C8×D61—22no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61D411yescharacteristic
52C2×C8×D61C2311yescharacteristic
53C2×C8×D61C2311yescharacteristic
54#14519758—41no—
55#14519753—21no—
56#14519753—21no—
57#14519748—21no—
58#14519748—21no—
59#14519748—22no—
60#14519748—22no—
61#14519748—41no—
62#14519748—41no—
63#14519745—41no—
64C4×C22⋊D61—41no—
65C4×C22⋊D61—41no—
66#14519736—21no—
67C2×C4⋊Dic61—21no—
68C2×C22⋊Dic61—41no—
69#14519717—22no—
70#14519716—24no—
71C2×#11758185C2312yes—
72C16×D61—24no—
73#14519711—21no—
74#14519711C2312yes—
75#14519711C2311yescharacteristic
76#14519711C2311yescharacteristic
77#14519709—21no—
78#14519701—21no—
79#14519701—21no—
80#14519698—21no—
81#14519698—21no—
82#14519698—22no—
83#14519698—22no—
84C2×#41×C61—21no—
85C22×D4×C61—21no—
86#135×C61—22no—
87#135×C61—22no—
88C2×Q16×C61C2311yescharacteristic
89C2×SD16×C61—21no—
90C2×D8×C61—22no—
91C2×D8×C61—21no—
92C2×D8×C61—22no—
93C2×D8×C61C2312yes—
94C2×D8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96SD32×C61—24no—
97D16×C61—24no—
98M32×C61D414yescomplemented
99C2×C16×C61C2312yes—
100C24×C61—41no—
101C23×D61D811yescharacteristic
102#41×C61—42no—
103#41×C61—42no—
104C2×Q8×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—41no—
108C2×D4×C61—42no—
109C2×D4×C61—41no—
110C2×D4×C61—42no—
111C22×C4×C61—41no—
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—42no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—41no—
116C2×C22⋊D61—41no—
117C2×C22⋊D61—84no—
118C22×Dic61—41no—
119C22×Dic61D811yescharacteristic
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61D811yescharacteristic
123C2×C4×D61D811yescharacteristic
124C2×C4×D61C2×D411yescharacteristic
125C2×C4×D61C2×D411yescharacteristic
126Q16×C61—21no—
127Q16×C61—21no—
128SD16×C61—42no—
129D8×C61—21no—
130D8×C61—22no—
131D8×C61—22no—
132D8×C61—21no—
133D8×C61—42no—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D412yes—
136C2×C8×C61#4112yes—
137C2×C8×C61C2411yescharacteristic
138C22⋊Dic61—41no—
139C22⋊Dic61—42no—
140C22⋊Dic61—42no—
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143#11758198—41no—
144#11758198—41no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—41no—
149#11758196—41no—
150#11758196—41no—
151C4×Dic61—41no—
152C2×#92418—22no—
153C2×#92418C2×D411yescharacteristic
154C2×#92418C2×D411yescharacteristic
155C8×D61—22no—
156C8×D61—22no—
157C8×D61—21no—
158C8×D61—21no—
159C8×D61—21no—
160C8×D61—21no—
161C16×C61C2×D414yescomplemented
162#11758185—22no—
163C23×C61—41no—
164C23×C61—42no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61—82no—
168C23×C61C2×D811yescharacteristic
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—41no—
172C22×D61C2×D811yescharacteristic
173C22×D61C2×D811yescharacteristic
174Q8×C61—41no—
175Q8×C61—41no—
176D4×C61—42no—
177D4×C61—41no—
178D4×C61—42no—
179D4×C61—41no—
180D4×C61—82no—
181D4×C61—82no—
182C2×C4×C61—41no—
183C2×C4×C61—82no—
184C2×C4×C61#12112yes—
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61#13512yes—
187C2×C4×C61C22×D411yescharacteristic
188C22⋊D61—82no—
189C22⋊D61—84no—
190C22⋊D61—82no—
191C22⋊D61—84no—
192C2×Dic61—22no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—82no—
197C2×Dic61C2×D811yescharacteristic
198C2×Dic61C2×D811yescharacteristic
199C4×D61—21no—
200C4×D61—21no—
201C4×D61—21no—
202C4×D61—21no—
203C4×D61—22no—
204C4×D61—22no—
205C8×C61#12112yes—
206C8×C61C22×D411yescharacteristic
207C8×C61C2×#4111yescharacteristic
208#92418—21no—
209#92418—21no—
210#33605—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—82no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61—82no—
217C22×C61—81no—
218C22×C61—82no—
219C22×C61—164no—
220C22×C61#46612yes—
221C22×C61C22×D811yescharacteristic
222C2×D61—21no—
223C2×D61—21no—
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—21no—
227C2×D61—44no—
228C2×D61#50912yescomplemented
229C4×C61—81no—
230C4×C61#46612yes—
231C4×C61C2×#12111yescharacteristic
232C4×C61C2×#13511yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—82no—
236Dic61—81no—
237Dic61—81no—
238C2×#509D6111yescharacteristic, complemented
239C22×M32—611no—
240#3050—611no—
241C2×#466—611no—
242#2220—612no—
243#2218—612no—
244#2217—612no—
245#2215—612no—
246#2159—611no—
247#2140—611no—
248#2139—611no—
249C2×C61—22no—
250C2×C61—161no—
251C2×C61—162no—
252C2×C61C2×#46611yescharacteristic
253C2×C61C2×#50911yescharacteristic
254C2×C61#341411yescharacteristic
255D61—22no—
256D61—41no—
257C2×#135C2×D6111yescharacteristic
258C22×D8C2×D6111yescharacteristic
259C23×C8—611no—
260#525—1221no—
261C2×#121—1221no—
262#509—24no—
263C2×SD32C2×D6112yescomplemented
264C2×D16C2×D6112yescomplemented
265C2×M32—611no—
266C2×M32—612no—
267C2×M32—1224no—
268C2×M32C2×D6111yescharacteristic
269C22×C16—612no—
270#467—611no—
271#467—611no—
272#466—612no—
273#466—612no—
274#466—611no—
275#466—611no—
276#466—1221no—
277#466—1221no—
278#465—1221no—
279#465—1221no—
280#427—611no—
281C2×#107—611no—
282#416—1221no—
283C2×#102—1221no—
284#367—612no—
285#366—612no—
286#358—612no—
287#357—612no—
288#357—612no—
289#357—612no—
290C61#3360511yescharacteristic, complemented, Sylow
291C2×#41—21no—
292C22×D4—21no—
293C22×D4—2441no—
294C23×C4—611no—
295#135—22no—
296#135—22no—
297C2×Q16C22×D6111yescharacteristic
298C2×SD16—21no—
299C2×D8—21no—
300C2×D8—22no—
301C2×D8—22no—
302C2×D8C22×D6112yes—
303C2×D8C22×D6111yescharacteristic
304C22×C8—611no—
305C22×C8—611no—
306C22×C8—611no—
307C22×C8—611no—
308C22×C8—611no—
309C22×C8—1222no—
310C22×C8—1222no—
311C22×C8C22×D6111yescharacteristic
312#123—2441no—
313C22⋊Q8—1221no—
314C22⋊Q8—1221no—
315#121—1222no—
316#121—1222no—
317#121—1221no—
318#121—1221no—
319#121—2441no—
320#121—2441no—
321#120—2441no—
322C4×D4—2441no—
323C4×D4—2441no—
324#117—1221no—
325C2×C4⋊C4—1221no—
326C2×C22⋊C4—2441no—
327SD32—24no—
328D16—24no—
329M32—1222no—
330M32—1224no—
331M32C22⋊D6114yescomplemented
332C2×C16—612no—
333C2×C16—1224no—
334C2×C16C22×D6112yes—
335#107—611no—
336#107—611no—
337#107—612no—
338#107—1221no—
339#106—1221no—
340#103—1221no—
341#103—1221no—
342#102—1222no—
343#102—1222no—
344#102—1221no—
345#102—1221no—
346C24—41no—
347C24—611no—
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—4884no—
358C2×D4—2441no—
359C2×D4—2441no—
360C2×D4—2442no—
361C2×D4—2442no—
362C22×C4—41no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—611no—
368C22×C4—1222no—
369C22×C4—1222no—
370C22×C4—2441no—
371C22×C4D4×D6111yescharacteristic
372Q16—21no—
373Q16—21no—
374SD16—42no—
375D8—21no—
376D8—22no—
377D8—22no—
378D8—21no—
379D8—42no—
380C2×C8—611no—
381C2×C8—611no—
382C2×C8—1222no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1221no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8D4×D6111yescharacteristic
390C2×C8#1175822412yes—
391C2×C8C2×C22⋊D6112yes—
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—1222no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2442no—
402C22⋊C4—2442no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42—2441no—
407C16—1222no—
408C16C2×C22⋊D6114yescomplemented
409C23—41no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—611no—
415C23—611no—
416C23—1222no—
417C23—1222no—
418C23—2441no—
419C23D8×D6111yescharacteristic
420Q8—41no—
421Q8—41no—
422D4—42no—
423D4—41no—
424D4—42no—
425D4—41no—
426D4—82no—
427D4—82no—
428D4—4884no—
429D4—4882no—
430D4—4884no—
431D4—4882no—
432C2×C4—41no—
433C2×C4—82no—
434C2×C4—611no—
435C2×C4—611no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—4882no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2442no—
447C2×C4#1451974812yes—
448C2×C4D8×D6111yescharacteristic
449C2×C4#1451979612yes—
450C2×C4C2×D4×D6111yescharacteristic
451C8—1221no—
452C8—1221no—
453C8#1451974812yes—
454C8C22×C22⋊D6111yescharacteristic
455C8C2×#1175822411yescharacteristic
456C22—21no—
457C22—21no—
458C22—81no—
459C22—82no—
460C22—81no—
461C22—82no—
462C22—82no—
463C22—81no—
464C22—164no—
465C22—612no—
466C22—1221no—
467C22—1221no—
468C22—1221no—
469C22—1221no—
470C22—1221no—
471C22—2444no—
472C22#1683999512yes—
473C22C2×D8×D6111yescharacteristic
474C4—81no—
475C4—1221no—
476C4—1221no—
477C4—4881no—
478C4—4882no—
479C4—4881no—
480C4#1683999512yes—
481C4C2×#1451974811yescharacteristic
482C4C2×#1451979611yescharacteristic
483C2—22no—
484C2—162no—
485C2—161no—
486C2—1222no—
487C2—2441no—
488C2C2×#1683999511yescharacteristic
489C2#509×D6111yescharacteristic
490C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33605
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×M32×D61C2semidirect
2. #21455396C2semidirect
3. C2×#16839995C2semidirect
4. #21447929C2semidirect
5. #21447919C2semidirect
6. #21447912C2semidirect
7. #21447902C2semidirect
8. #21447607C2semidirect
9. #21447535C2semidirect
10. #21447531C2semidirect
11. C2×C61×#509C2semidirect
16. C2×#14519716C22semidirect
18. C2×C16×D61C22semidirect
19. #16840000C22semidirect
20. #16840000C22semidirect
25. #16839995C22semidirect
26. #16839995C22semidirect
33. #16839590C22semidirect
34. #16839588C22semidirect
35. #16839566C22semidirect
36. #16839563C22semidirect
37. #16839563C22semidirect
38. #16839563C22semidirect
42. C2×SD32×C61C22semidirect
43. C2×D16×C61C22semidirect
98. M32×C61D4semidirect
161. C16×C61C2×D4semidirect
228. C2×D61#509semidirect
238. C2×#509D61semidirect
263. C2×SD32C2×D61semidirect
264. C2×D16C2×D61semidirect
290. C61#33605semidirect
331. M32C22⋊D61semidirect
408. C16C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D612
#214553962
C2×#168399952
#214479292
#214479192
#214479122
#214479022
#214476072
#214475352
#214475312
C2×C61×#5092
#3360561