Finite Group Databasebeta

← #29835282

Subgroups — #29835282

Order 15616.

536 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456466C211yesmaximal, characteristic, complemented
2#21454717C211yesmaximal, characteristic, complemented
3#21448032C212yesmaximal, complemented
4#21447996C211yesmaximal, characteristic, complemented
5#21447995C211yesmaximal, characteristic, complemented
6#21447892C211yesmaximal, characteristic, complemented
7#21447763C212yesmaximal, complemented
8#21447720C212yesmaximal, complemented
9#21447718C212yesmaximal, complemented
10#21447634C211yesmaximal, characteristic, complemented
11C61×#2213C211yesmaximal, characteristic, complemented
12#16840559C2211yescharacteristic
13#131×D61—21no—
14#16840366—21no—
15#16840198C2212yes—
16#16840164C2211yescharacteristic, complemented
17#16840126—21no—
18#16840012C2212yes—
19#16839995—21no—
20#16839884C2211yescharacteristic
21#16839883C2211yescharacteristic
22#16839881—21no—
23#16839880C2211yescharacteristic
24#16839877—21no—
25#16839874C2212yescomplemented
26D8×Dic61—21no—
27#16839809C2211yescharacteristic
28#16839777—21no—
29#16839716C2211yescharacteristic
30#16839586C2212yescomplemented
31#16839585C2212yescomplemented
32#16839583C2211yescharacteristic, complemented
33#16839575C2212yescomplemented
34#16839573C2212yes—
35#16839572C2212yes—
36#16839543C2212yes—
37#16839539C2212yescomplemented
38C61×#509C2212yes—
39C61×#493C2211yescharacteristic, complemented
40C61×#443C2211yescharacteristic
41C61×#364C2211yescharacteristic, complemented
42C61×#361C2212yescomplemented
43#41×D61—21no—
44#14519809—21no—
45#14519806—21no—
46#14519804—21no—
47#14519801—21no—
48#14519799C2311yescharacteristic
49#14519797—21no—
50#14519785—21no—
51#14519785—41no—
52#14519784—21no—
53#14519783—21no—
54#14519783—41no—
55C2×#11758189C2311yescharacteristic
56M16×D61—21no—
57M16×D61—41no—
58C2×C8×D61—41no—
59C2×C8×D61D411yescharacteristic
60#14519769—21no—
61#14519758—42no—
62#14519749—42no—
63#14519748—22no—
64D4×Dic61—22no—
65#14519734D411yescharacteristic
66#14519721—22no—
67#14519718—22no—
68#14519717C2312yes—
69#14519713C2311yescharacteristic
70#14519712—21no—
71#14519711—21no—
72#14519709C2312yes—
73#14519705—21no—
74#14519704—21no—
75#14519703—21no—
76#14519698—22no—
77#14519698—22no—
78#14519688D411yescharacteristic
79#14519687C2311yescharacteristic
80C8×Dic61D411yescharacteristic
81#135×C61C2311yescharacteristic
82C2×D8×C61—21no—
83C2×D8×C61C2312yes—
84#131×C61—21no—
85#127×C61—21no—
86SD32×C61—22no—
87D16×C61—22no—
88M32×C61C2312yes—
89#108×C61C2311yescharacteristic
90#104×C61—21no—
91C4×C8×C61C2311yescharacteristic
92#41×C61—21no—
93C2×D4×C61—22no—
94C2×D4×C61—42no—
95C2×C22⋊D61—42no—
96C22×Dic61—42no—
97#11758226—21no—
98Q8×D61—21no—
99#11758224—21no—
100#11758224—41no—
101D4×D61—21no—
102D4×D61—41no—
103#11758222—21no—
104#11758222—41no—
105C2×C4×D61—41no—
106C2×C4×D61C2×D411yescharacteristic
107Q16×C61#4111yescharacteristic
108SD16×C61—21no—
109D8×C61—22no—
110D8×C61—22no—
111D8×C61—41no—
112D8×C61—41no—
113D8×C61#4111yescharacteristic
114M16×C61—21no—
115M16×C61—41no—
116C2×C8×C61—41no—
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C22⋊Dic61—42no—
121C22⋊Dic61—42no—
122#11758202—21no—
123#11758201—21no—
124#11758200—21no—
125#11758199—21no—
126#11758198—21no—
127C4⋊Dic61—22no—
128#11758196—21no—
129#11758196—42no—
130C4×Dic61C2×D411yescharacteristic
131#11758194—21no—
132#11758194—41no—
133C2×#92418—41no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244#4111yescharacteristic
137D488#4111yescharacteristic
138#11758190—21no—
139#11758189—21no—
140#11758189—21no—
141#11758189—41no—
142#11758189—41no—
143#11758189—41no—
144C8×D61—21no—
145C8×D61—21no—
146C8×D61—41no—
147C8×D61—41no—
148C8×D61—41no—
149C16×C61—22no—
150#11758185—22no—
151C23×C61—42no—
152C22×D61—41no—
153C22×D61AGL(1,ℤ/8)11yescharacteristic
154Q8×C61—21no—
155D4×C61—21no—
156D4×C61—41no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—162no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61C22×D411yescharacteristic
163C22⋊D61—41no—
164C22⋊D61—41no—
165C22⋊D61—41no—
166C22⋊D61—162no—
167C2×Dic61—21no—
168C2×Dic61—41no—
169C2×Dic61—42no—
170C2×Dic61—162no—
171C2×Dic61AGL(1,ℤ/8)11yescharacteristic
172D244—21no—
173D244—41no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—41no—
177C4×D61—41no—
178C4×D61—41no—
179Dic122—21no—
180Dic122—41no—
181C8×C61—41no—
182C8×C61AGL(1,ℤ/8)11yescharacteristic
183C8×C61AGL(1,ℤ/8)11yescharacteristic
184C8×C61C2×#4111yescharacteristic
185C8×C61C2×#4111yescharacteristic
186#92418—21no—
187#92418—21no—
188#92418—41no—
189#33692—611nomaximal, Sylow
190C22×C61—41no—
191C22×C61—82no—
192C22×C61—82no—
193C22×C61—82no—
194C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
195C2×D61—21no—
196C2×D61—41no—
197C2×D61—41no—
198C2×D61—82no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61#52511yescharacteristic
202C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
203Dic61—21no—
204Dic61—41no—
205Dic61—41no—
206Dic61—82no—
207#3176—611no—
208#2955—611no—
209#2238—612no—
210#2231—611no—
211#2231—611no—
212#2213—611no—
213#2213D6111yescharacteristic, complemented
214#2190—612no—
215AGL(1,ℤ/16)—612no—
216AGL(1,ℤ/16)—612no—
217#2166—611no—
218C2×C61—21no—
219C2×C61—81no—
220C2×C61—162no—
221C2×C61#310911yescharacteristic
222D61—41no—
223D61—81no—
224#575—611no—
225C2×#131—1221no—
226#540—1221no—
227#509—612no—
228#509C2×D6112yes—
229#499—611no—
230#493C2×D6111yescharacteristic, complemented
231#490—1221no—
232#469—612no—
233#466—1221no—
234#444—611no—
235#444—611no—
236#444—1221no—
237#443—611no—
238#443—1221no—
239#443C2×D6111yescharacteristic
240#442—612no—
241C4×D8—1221no—
242#430—611no—
243#421—1221no—
244#405—611no—
245#365—612no—
246#365—612no—
247#364—611no—
248#364C2×D6111yescharacteristic, complemented
249#361—612no—
250#361C2×D6112yescomplemented
251#360—612no—
252#360—612no—
253#349—612no—
254#347—612no—
255C61#3369211yescharacteristic, complemented, Sylow
256C2×#41—1221no—
257#137—1221no—
258#137—1221no—
259AGL(1,ℤ/8)—1221no—
260AGL(1,ℤ/8)—1221no—
261#135—611no—
262#135—1221no—
263#135C22×D6111yescharacteristic
264C2×D8—21no—
265C2×D8C22×D6112yes—
266#131—21no—
267#131—1221no—
268#131—1221no—
269#131—1221no—
270#131—2441no—
271#131—2441no—
272C2×M16—611no—
273C2×M16—1221no—
274C2×M16—2441no—
275C22×C8—611no—
276C22×C8—2441no—
277#127—21no—
278#125—1221no—
279#123—2442no—
280#121—1222no—
281#121—2442no—
282C4×D4—1222no—
283#117—611no—
284SD32—22no—
285SD32—1222no—
286D16—22no—
287D16—1222no—
288M32—612no—
289M32C22×D6112yes—
290#108—611no—
291#108—1221no—
292#108C22×D6111yescharacteristic
293#107—1221no—
294#106—612no—
295#104—21no—
296#104—1221no—
297#104—1221no—
298#104—1221no—
299#102—1222no—
300#102—1222no—
301#97—611no—
302#97—611no—
303C4×C8—611no—
304C4×C8C22×D6111yescharacteristic
305#41—21no—
306#41—1221no—
307#41—1221no—
308#41—1221no—
309#41—2441no—
310#41—2441no—
311C2×Q8—1221no—
312C2×D4—22no—
313C2×D4—42no—
314C2×D4—1221no—
315C2×D4—2441no—
316C2×D4—2442no—
317C22×C4—611no—
318C22×C4—2441no—
319C22×C4—2442no—
320Q16—611no—
321Q16—1221no—
322Q16#1175822411yescharacteristic
323SD16—21no—
324SD16—1221no—
325SD16—1221no—
326SD16—1221no—
327D8—22no—
328D8—22no—
329D8—41no—
330D8—41no—
331D8—611no—
332D8—1221no—
333D8#1175822411yescharacteristic
334M16—21no—
335M16—41no—
336M16—1221no—
337M16—1221no—
338M16—1221no—
339M16—2441no—
340M16—2441no—
341M16—2441no—
342M16—2441no—
343C2×C8—41no—
344C2×C8—611no—
345C2×C8—611no—
346C2×C8—1221no—
347C2×C8—1221no—
348C2×C8—2441no—
349C2×C8—2441no—
350C2×C8—2441no—
351C2×C8—2441no—
352C2×C8D4×D6111yescharacteristic
353C2×C8C23×D6111yescharacteristic
354C4⋊C4—1221no—
355C4⋊C4—1222no—
356C4⋊C4—2442no—
357C22⋊C4—1221no—
358C22⋊C4—2442no—
359C22⋊C4—2442no—
360C42—611no—
361C42D4×D6111yescharacteristic
362C16—22no—
363C16—1222no—
364C23—42no—
365C23—611no—
366C23—2441no—
367Q8—21no—
368Q8—1221no—
369Q8—2441no—
370D4—21no—
371D4—41no—
372D4—42no—
373D4—42no—
374D4—162no—
375D4—1221no—
376D4—9762no—
377D4—2441no—
378D4—2441no—
379D4—2441no—
380D4—2441no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—611no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—9762no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2441no—
393C2×C4C2×D4×D6111yescharacteristic
394C8—41no—
395C8—1221no—
396C8—1221no—
397C8—2441no—
398C8#1451980111yescharacteristic
399C8#1451980111yescharacteristic
400C8#41×D6111yescharacteristic
401C8C2×#1175822411yescharacteristic
402C22—41no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—1221no—
407C22—4882no—
408C22—2441no—
409C22—2441no—
410C22AGL(1,ℤ/8)×D6111yescharacteristic
411C4—41no—
412C4—41no—
413C4—1221no—
414C4—4882no—
415C4—2441no—
416C4—2441no—
417C4#1684027211yescharacteristic
418C4C2×#1451980111yescharacteristic
419C2—21no—
420C2—81no—
421C2—162no—
422C2—4881no—
423C2—2441no—
424C2#2145590911yescharacteristic
subgroupG/H
centerC2#21455909
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2213C2
socleC2×C61#3109
hypercenter#2213D61
hypocenterC61#33692
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #21456466C2semidirect
2. #21454717C2semidirect
3. #21448032C2semidirect
4. #21447996C2semidirect
5. #21447995C2semidirect
6. #21447892C2semidirect
7. #21447763C2semidirect
8. #21447720C2semidirect
9. #21447718C2semidirect
10. #21447634C2semidirect
11. C61×#2213C2semidirect
16. #16840164C22semidirect
25. #16839874C22semidirect
30. #16839586C22semidirect
31. #16839585C22semidirect
32. #16839583C22semidirect
33. #16839575C22semidirect
37. #16839539C22semidirect
39. C61×#493C22semidirect
41. C61×#364C22semidirect
42. C61×#361C22semidirect
213. #2213D61semidirect
230. #493C2×D61semidirect
248. #364C2×D61semidirect
250. #361C2×D61semidirect
255. C61#33692semidirect

Maximal subgroups

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

subgroupindex
#214564662
#214547172
#214480322
#214479962
#214479952
#214478922
#214477632
#214477202
#214477182
#214476342
C61×#22132
#3369261