Finite Group Databasebeta

← #29834913

Subgroups — #29834913

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455465C211yesmaximal, characteristic
2#21454429C211yesmaximal, characteristic, complemented
3#21448035C212yesmaximal, complemented
4#21448034C211yesmaximal, characteristic, complemented
5#21448033C211yesmaximal, characteristic, complemented
6#21448032C212yesmaximal, complemented
7#21448031C211yesmaximal, characteristic, complemented
8#21447590C212yesmaximal, complemented
9#361×D61C212yesmaximal, complemented
10#21447438C211yesmaximal, characteristic, complemented
11C61×#2238C211yesmaximal, characteristic, complemented
12C2×D8×D61C2211yescharacteristic
13#16840528—21no—
14C2×#14519785C2211yescharacteristic
15#16840328—21no—
16M32×D61C2212yes—
17#16840016C2211yescharacteristic
18#16840015C2211yescharacteristic
19#16840012C2211yescharacteristic
20#16839997—21no—
21#16839996—21no—
22#16839811C2211yescharacteristic, complemented
23#16839809—21no—
24#16839809C2212yescomplemented
25C2×#14519712—21no—
26#108×D61—21no—
27#108×D61C2212yes—
28#16839797C2211yescharacteristic
29#16839765—21no—
30#16839586C2212yescomplemented
31#16839585C2212yescomplemented
32#16839584C2212yescomplemented
33#16839576C2212yes—
34#16839575C2212yes—
35#16839574C2212yes—
36#16839544C2211yescharacteristic, complemented
37#16839543C2212yescomplemented
38C61×#469C2211yescharacteristic
39C61×#430C2211yescharacteristic, complemented
40C61×#365C2212yescomplemented
41C61×#361C2212yes—
42C61×#349C2211yescharacteristic, complemented
43C2×#11758222D411yescharacteristic
44C2×D4×D61—21no—
45C2×D488C2311yescharacteristic
46C2×#11758199—21no—
47D8×D61—22no—
48D8×D61—22no—
49#14519785—22no—
50#14519785—22no—
51#14519785—42no—
52#14519783—42no—
53C2×#11758194—21no—
54C2×#11758189D411yescharacteristic
55C2×#11758189C2311yescharacteristic
56M16×D61—22no—
57M16×D61—22no—
58C22×#92418D411yescharacteristic
59C2×C8×D61D411yescharacteristic
60C2×C8×D61C2311yescharacteristic
61#14519773—41no—
62#14519763—41no—
63#14519752—21no—
64#14519751—21no—
65#14519748—21no—
66#14519747—41no—
67#14519745—41no—
68#14519735—21no—
69#14519717C2312yes—
70#14519716—22no—
71C16×D61—22no—
72#14519713—21no—
73#14519713C2312yes—
74#14519712—22no—
75#14519712—22no—
76#14519710—21no—
77#14519709C2311yescharacteristic
78#14519699—21no—
79#14519698—21no—
80#14519697—21no—
81C2×D8×C61C2311yescharacteristic
82C2×M16×C61—21no—
83C2×M16×C61C2311yescharacteristic
84#121×C61—21no—
85M32×C61C2312yes—
86#108×C61—21no—
87#108×C61C2312yes—
88#106×C61C2311yescharacteristic
89#102×C61—21no—
90C23×D61—41no—
91C2×D4×C61—21no—
92C2×D4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C2×C22⋊D61SD1611yescharacteristic
98C2×C22⋊D61SD1611yescharacteristic
99D4×D61—42no—
100D4×D61—42no—
101#11758222—22no—
102#11758222—22no—
103C2×D244—21no—
104C2×D244—41no—
105C2×D244SD1611yescharacteristic
106C2×C4×D61C2×D411yescharacteristic
107C2×C4×D61C2×D411yescharacteristic
108C2×Dic122SD1611yescharacteristic
109D8×C61—22no—
110M16×C61—22no—
111M16×C61—22no—
112M16×C61—41no—
113C2×C8×C61—41no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2411yescharacteristic
116C4⋊C4×C61—21no—
117C22⋊C4×C61—41no—
118C22⋊Dic61—41no—
119#11758199—41no—
120#11758199—41no—
121#11758198—41no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4×Dic61—41no—
126#11758194—22no—
127#11758194—41no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—41no—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133D488—22no—
134#11758189—22no—
135#11758189—22no—
136#11758189—42no—
137#11758189—42no—
138C8×D61—21no—
139C8×D61—21no—
140C8×D61—42no—
141C8×D61—42no—
142C8×D61#4112yes—
143C8×D61#4112yes—
144C16×C61—22no—
145#11758185—22no—
146C23×C61—41no—
147C23×C61C2×SD1611yescharacteristic
148C22×D61—41no—
149C22×D61—82no—
150C22×D61—161no—
151C22×D61—161no—
152C22×D61C2×SD1611yescharacteristic
153C22×D61C2×SD1611yescharacteristic
154D4×C61—41no—
155D4×C61—41no—
156D4×C61—161no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—41no—
160C2×C4×C61C2×SD1611yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C22⋊D61—22no—
163C22⋊D61—22no—
164C22⋊D61—82no—
165C22⋊D61—161no—
166C22⋊D61—161no—
167C2×Dic61—41no—
168C2×Dic61C2×SD1611yescharacteristic
169C2×Dic61C2×SD1611yescharacteristic
170D244—22no—
171D244—41no—
172D244—41no—
173D244—161no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61#12312yes—
177C4×D61AGL(1,ℤ/8)12yes—
178Dic122—22no—
179C8×C61—21no—
180C8×C61—42no—
181C8×C61C2×#4112yes—
182#92418—21no—
183#92418—42no—
184#92418#12311yescharacteristic
185#92418#12311yescharacteristic
186#33657—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—41no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61C22×SD1611yescharacteristic
193C2×D61—21no—
194C2×D61—41no—
195C2×D61—41no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—164no—
200C2×D61#48112yes—
201C4×C61—21no—
202C4×C61—81no—
203C4×C61C2×#12311yescharacteristic
204C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
205Dic61—21no—
206Dic61—81no—
207Dic61#48112yes—
208#3058—611no—
209#2916—611no—
210#2238—611no—
211#2238—611no—
212#2238—612no—
213#2238—612no—
214#2238—611no—
215#2238D6111yescharacteristic, complemented
216#2155—612no—
217C2×#361—612no—
218#2115—611no—
219C2×C61—21no—
220C2×C61—41no—
221C2×C61—161no—
222C2×C61C2×#48111yescharacteristic
223D61—22no—
224D61—41no—
225D61—161no—
226C22×D8—611no—
227#568—1221no—
228C2×#131—611no—
229#535—1221no—
230C2×M32—612no—
231#469—611no—
232#469—611no—
233#469—611no—
234#469C2×D6111yescharacteristic
235#466—1221no—
236#466—1221no—
237#430—611no—
238#430—612no—
239#430—1221no—
240#430C2×D6111yescharacteristic, complemented
241C2×#108—612no—
242C2×#108—1221no—
243C2×#108—1221no—
244#427—611no—
245#418—1221no—
246#365—612no—
247#365—612no—
248#365—612no—
249#365C2×D6112yescomplemented
250#361—612no—
251#361—612no—
252#361—612no—
253#361C2×D6112yes—
254#349—611no—
255#349—612no—
256#349C2×D6111yescharacteristic, complemented
257C61#3365711yescharacteristic, complemented, Sylow
258C2×#41—611no—
259C22×D4—1221no—
260C2×D8—611no—
261C2×D8—1222no—
262C2×D8—1222no—
263C2×D8—1221no—
264C2×D8C22×D6111yescharacteristic
265#131—1222no—
266#131—1222no—
267#131—2442no—
268#131—2442no—
269C2×M16—21no—
270C2×M16—611no—
271C2×M16—611no—
272C2×M16—1222no—
273C2×M16—1222no—
274C2×M16—1221no—
275C2×M16C22×D6111yescharacteristic
276C22×C8—611no—
277C22×C8—611no—
278C22×C8—611no—
279#127—2441no—
280#124—2441no—
281#121—21no—
282#121—1221no—
283#121—1221no—
284#121—1221no—
285#120—2441no—
286#120—2441no—
287#117—1221no—
288M32—612no—
289M32—1222no—
290M32C22×D6112yes—
291C2×C16—1222no—
292#108—21no—
293#108—612no—
294#108—1222no—
295#108—1222no—
296#108—1221no—
297#108C22×D6112yes—
298#107—1221no—
299#106—611no—
300#106C22×D6111yescharacteristic
301#102—21no—
302#102—1221no—
303#102—1221no—
304#102—1221no—
305C24—2441no—
306#41—1222no—
307#41—1222no—
308C2×Q8—611no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—611no—
312C2×D4—611no—
313C2×D4—611no—
314C2×D4—1221no—
315C2×D4—2441no—
316C2×D4—2442no—
317C2×D4—2442no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C22×C4—611no—
322C22×C4—611no—
323C22×C4D4×D6111yescharacteristic
324D8—22no—
325D8—1222no—
326D8—2441no—
327D8—2441no—
328M16—22no—
329M16—22no—
330M16—41no—
331M16—1222no—
332M16—1222no—
333M16—1222no—
334M16—2442no—
335M16—2442no—
336M16—2441no—
337C2×C8—41no—
338C2×C8—611no—
339C2×C8—612no—
340C2×C8—612no—
341C2×C8—611no—
342C2×C8—1221no—
343C2×C8—1221no—
344C2×C8—1221no—
345C2×C8—1221no—
346C2×C8—2442no—
347C2×C8—2442no—
348C2×C8—2441no—
349C2×C8D4×D6111yescharacteristic
350C2×C8C23×D6111yescharacteristic
351C4⋊C4—21no—
352C4⋊C4—1221no—
353C22⋊C4—41no—
354C22⋊C4—2441no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C42—2441no—
359C16—22no—
360C16—1222no—
361C23—41no—
362C23—611no—
363C23—611no—
364C23—9761no—
365C23—9761no—
366C23—4882no—
367C23—2441no—
368C23SD16×D6111yescharacteristic
369Q8—1222no—
370D4—41no—
371D4—41no—
372D4—161no—
373D4—1222no—
374D4—1222no—
375D4—1222no—
376D4—9761no—
377D4—9761no—
378D4—9761no—
379D4—4882no—
380D4—2441no—
381D4—2441no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—611no—
386C2×C4—612no—
387C2×C4—612no—
388C2×C4—611no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—2441no—
392C2×C4SD16×D6111yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C8—21no—
395C8—42no—
396C8—611no—
397C8—611no—
398C8—1221no—
399C8—2442no—
400C8#41×D6112yes—
401C22—21no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—612no—
407C22—1221no—
408C22—9764no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22—2441no—
413C22—2441no—
414C22C2×SD16×D6111yescharacteristic
415C4—21no—
416C4—81no—
417C4—612no—
418C4—1221no—
419C4—4881no—
420C4#123×D6111yescharacteristic
421C4AGL(1,ℤ/8)×D6111yescharacteristic
422C2—21no—
423C2—41no—
424C2—161no—
425C2—1222no—
426C2—9761no—
427C2—2441no—
428C2#481×D6111yescharacteristic
subgroupG/H
centerC2#481×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2238C2
socleC2×C61C2×#481
prime radical#21455465C2
hypercenter#2238D61
hypocenterC61#33657
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2238

Complements

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

NH ≅ G/Ntype
2. #21454429C2semidirect
3. #21448035C2semidirect
4. #21448034C2semidirect
5. #21448033C2semidirect
6. #21448032C2semidirect
7. #21448031C2semidirect
8. #21447590C2semidirect
9. #361×D61C2semidirect
10. #21447438C2semidirect
11. C61×#2238C2semidirect
22. #16839811C22semidirect
24. #16839809C22semidirect
30. #16839586C22semidirect
31. #16839585C22semidirect
32. #16839584C22semidirect
36. #16839544C22semidirect
37. #16839543C22semidirect
39. C61×#430C22semidirect
40. C61×#365C22semidirect
42. C61×#349C22semidirect
215. #2238D61semidirect
240. #430C2×D61semidirect
249. #365C2×D61semidirect
256. #349C2×D61semidirect
257. C61#33657semidirect

Maximal subgroups

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

subgroupindex
#214554652
#214544292
#214480352
#214480342
#214480332
#214480322
#214480312
#214475902
#361×D612
#214474382
C61×#22382
#3365761