Finite Group Databasebeta

← #29835293

Subgroups — #29835293

Order 15616.

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

normal subgroups only

This group has 512 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#21456479C211yesmaximal, characteristic, complemented
2#21454717C211yesmaximal, characteristic
3#21448040C212yesmaximal, complemented
4#21447992C211yesmaximal, characteristic, complemented
5#21447901C211yesmaximal, characteristic, complemented
6#21447900C211yesmaximal, characteristic, complemented
7#21447771C212yesmaximal
8#21447709C212yesmaximal, complemented
9#21447707C212yesmaximal, complemented
10#21447634C211yesmaximal, characteristic, complemented
11C61×#2231C211yesmaximal, characteristic, complemented
12#16840559C2211yescharacteristic
13#131×D61—21no—
14#16840375—21no—
15#16840205C2212yes—
16#16840202C2212yes—
17#16840142C2211yescharacteristic, complemented
18#16840132—21no—
19#16840021C2212yes—
20#16840004—21no—
21#16839884C2211yescharacteristic
22#16839883C2211yescharacteristic
23#16839881—21no—
24#16839880C2211yescharacteristic
25#16839877—21no—
26#16839868C2212yescomplemented
27#16839856—21no—
28#16839809C2211yescharacteristic
29#16839777—21no—
30#16839716C2211yescharacteristic
31#16839584C2212yescomplemented
32#16839583C2211yescharacteristic, complemented
33#16839579C2212yescomplemented
34#16839577C2212yescomplemented
35#16839571C2212yes—
36#16839543C2212yes—
37#16839539C2212yescomplemented
38C61×#499C2211yescharacteristic, complemented
39C61×#443C2211yescharacteristic
40C61×#365C2212yescomplemented
41C61×#364C2211yescharacteristic, complemented
42C61×#360C2212yes—
43#41×D61—21no—
44#14519809—21no—
45#14519806—21no—
46#14519804—21no—
47#14519801—21no—
48#14519799C2311yescharacteristic
49#14519797—21no—
50C2×Dic244—21no—
51C2×Dic244C2312yes—
52#14519785—21no—
53#14519785—41no—
54#14519784—21no—
55#14519783—21no—
56#14519783—41no—
57C2×#11758189C2311yescharacteristic
58M16×D61—21no—
59M16×D61—41no—
60C2×C8×D61—41no—
61C2×C8×D61D411yescharacteristic
62#14519777—21no—
63#14519772—42no—
64#14519768—42no—
65#14519757—22no—
66#14519742—22no—
67#14519734D411yescharacteristic
68Dic488—22no—
69#14519723—22no—
70#14519722—22no—
71#14519721—22no—
72#14519717C2312yes—
73#14519713C2311yescharacteristic
74#14519712—21no—
75#14519705—21no—
76#14519704—21no—
77#14519703—21no—
78#14519700—22no—
79#14519700—22no—
80#14519688D411yescharacteristic
81#14519687C2311yescharacteristic
82C8×Dic61D411yescharacteristic
83#135×C61C2311yescharacteristic
84#131×C61—21no—
85#125×C61—21no—
86M32×C61C2312yes—
87#108×C61C2311yescharacteristic
88#107×C61—21no—
89#106×C61C2312yes—
90#104×C61—21no—
91C4×C8×C61C2311yescharacteristic
92#41×C61—21no—
93#11758226—21no—
94Q8×D61—21no—
95#11758224—21no—
96#11758224—41no—
97D4×D61—21no—
98D4×D61—41no—
99#11758222—21no—
100#11758222—41no—
101C2×C4×D61—41no—
102C2×C4×D61C2×D411yescharacteristic
103C2×Dic122—22no—
104Q16×C61#4111yescharacteristic
105SD16×C61—21no—
106D8×C61#4111yescharacteristic
107M16×C61—21no—
108M16×C61—41no—
109C2×C8×C61—41no—
110C2×C8×C61C2×D411yescharacteristic
111C2×C8×C61C2411yescharacteristic
112C4⋊C4×C61—22no—
113C4⋊C4×C61—42no—
114C42×C61C2×D411yescharacteristic
115#11758202—21no—
116#11758201—21no—
117#11758200—21no—
118#11758199—21no—
119#11758198—21no—
120#11758198—42no—
121C4⋊Dic61—42no—
122C4⋊Dic61—42no—
123#11758196—21no—
124#11758196—42no—
125C4×Dic61—42no—
126C4×Dic61C2×D411yescharacteristic
127#11758194—21no—
128#11758194—41no—
129C2×#92418—41no—
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132Dic244—22no—
133Dic244—22no—
134Dic244—41no—
135Dic244—41no—
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—
151C22×D61—41no—
152C22×D61AGL(1,ℤ/8)11yescharacteristic
153Q8×C61—21no—
154D4×C61—21no—
155D4×C61—41no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61C22×D411yescharacteristic
160C22⋊D61—41no—
161C22⋊D61—41no—
162C22⋊D61—41no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—42no—
166C2×Dic61AGL(1,ℤ/8)11yescharacteristic
167D244—21no—
168D244—41no—
169C4×D61—21no—
170C4×D61—21no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61—41no—
174Dic122—21no—
175Dic122—41no—
176Dic122—42no—
177Dic122—42no—
178C8×C61—41no—
179C8×C61AGL(1,ℤ/8)11yescharacteristic
180C8×C61AGL(1,ℤ/8)11yescharacteristic
181C8×C61C2×#4111yescharacteristic
182C8×C61C2×#4111yescharacteristic
183#92418—21no—
184#92418—21no—
185#92418—41no—
186#33693—611nomaximal, Sylow
187C22×C61—41no—
188C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
189C2×D61—21no—
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—82no—
193C4×C61—41no—
194C4×C61—41no—
195C4×C61—82no—
196C4×C61#52511yescharacteristic
197C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
198Dic61—21no—
199Dic61—41no—
200Dic61—41no—
201Dic61—82no—
202#3177—611no—
203#2955—611no—
204#2239—612no—
205#2231—611no—
206#2231D6111yescharacteristic, complemented
207#2214—611no—
208#2214—611no—
209#2191—612no—
210#2180—612no—
211#2180—612no—
212#2166—611no—
213C2×C61—21no—
214C2×C61—81no—
215C2×C61#310911yescharacteristic
216D61—41no—
217D61—81no—
218#575—611no—
219C2×#131—1221no—
220#541—1221no—
221#510—612no—
222#510—612no—
223#499C2×D6111yescharacteristic, complemented
224#494—611no—
225#491—1221no—
226#470—612no—
227#467—1221no—
228#444—611no—
229#444—611no—
230#444—1221no—
231#443—611no—
232#443—1221no—
233#443C2×D6111yescharacteristic
234#441—612no—
235C4×Q16—1221no—
236#430—611no—
237#421—1221no—
238#405—611no—
239#365—612no—
240#365C2×D6112yescomplemented
241#364—611no—
242#364C2×D6111yescharacteristic, complemented
243#362—612no—
244#362—612no—
245#360—612no—
246#360C2×D6112yes—
247#349—612no—
248#347—612no—
249C61#3369311yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251#137—1221no—
252#137—1221no—
253AGL(1,ℤ/8)—1221no—
254AGL(1,ℤ/8)—1221no—
255#135—611no—
256#135—1221no—
257#135C22×D6111yescharacteristic
258C2×Q16—612no—
259C2×Q16—1221no—
260#131—21no—
261#131—1221no—
262#131—1221no—
263#131—1221no—
264#131—2441no—
265#131—2441no—
266C2×M16—611no—
267C2×M16—1221no—
268C2×M16—2441no—
269C22×C8—611no—
270C22×C8—2441no—
271C4⋊Q8—1221no—
272#126—2442no—
273#125—21no—
274#125—2442no—
275C22⋊Q8—1222no—
276C4×Q8—1222no—
277#117—611no—
278Q32—1222no—
279Q32—1222no—
280SD32—1222no—
281SD32—1222no—
282M32—612no—
283M32C22×D6112yes—
284#108—611no—
285#108—1221no—
286#108C22×D6111yescharacteristic
287#107—21no—
288#106C22×D6112yes—
289#104—21no—
290#104—1221no—
291#104—1221no—
292#104—1221no—
293#103—1222no—
294#103—1222no—
295#97—611no—
296#97—611no—
297C4×C8—611no—
298C4×C8C22×D6111yescharacteristic
299#41—21no—
300#41—1221no—
301#41—1221no—
302#41—1221no—
303#41—2441no—
304#41—2441no—
305C2×Q8—1221no—
306C2×Q8—1222no—
307C2×D4—1221no—
308C2×D4—2441no—
309C22×C4—611no—
310C22×C4—2441no—
311Q16—611no—
312Q16—1221no—
313Q16—1222no—
314Q16—1222no—
315Q16—2441no—
316Q16—2441no—
317Q16#1175822611yescharacteristic
318SD16—21no—
319SD16—1221no—
320SD16—1221no—
321SD16—1221no—
322D8—611no—
323D8—1221no—
324D8#1175822611yescharacteristic
325M16—21no—
326M16—41no—
327M16—1221no—
328M16—1221no—
329M16—1221no—
330M16—2441no—
331M16—2441no—
332M16—2441no—
333M16—2441no—
334C2×C8—41no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—1221no—
338C2×C8—1221no—
339C2×C8—2441no—
340C2×C8—2441no—
341C2×C8—2441no—
342C2×C8—2441no—
343C2×C8D4×D6111yescharacteristic
344C2×C8C23×D6111yescharacteristic
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—1221no—
348C4⋊C4—2442no—
349C4⋊C4—2442no—
350C4⋊C4—2442no—
351C22⋊C4—1221no—
352C22⋊C4—2442no—
353C42—611no—
354C42—2442no—
355C42D4×D6111yescharacteristic
356C16—22no—
357C16—1222no—
358C23—611no—
359C23—2441no—
360Q8—21no—
361Q8—1221no—
362Q8—2441no—
363Q8—2442no—
364Q8—2442no—
365D4—21no—
366D4—41no—
367D4—1221no—
368D4—2441no—
369D4—2441no—
370D4—2441no—
371D4—2441no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—611no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2442no—
383C2×C4—2441no—
384C2×C4C2×D4×D6111yescharacteristic
385C8—41no—
386C8—1221no—
387C8—1221no—
388C8—2441no—
389C8#1451980411yescharacteristic
390C8#1451980411yescharacteristic
391C8#41×D6111yescharacteristic
392C8C2×#1175822611yescharacteristic
393C22—41no—
394C22—1221no—
395C22—4882no—
396C22—2441no—
397C22—2441no—
398C22AGL(1,ℤ/8)×D6111yescharacteristic
399C4—41no—
400C4—41no—
401C4—82no—
402C4—1221no—
403C4—4882no—
404C4—2441no—
405C4—2441no—
406C4#1684027511yescharacteristic
407C4C2×#1451980411yescharacteristic
408C2—21no—
409C2—81no—
410C2—4881no—
411C2—2441no—
412C2#2145591211yescharacteristic
subgroupG/H
centerC2#21455912
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2231C2
socleC2×C61#3109
prime radical#16840559C22
hypercenter#2231D61
hypocenterC61#33693
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2231

Complements

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

NH ≅ G/Ntype
1. #21456479C2semidirect
3. #21448040C2semidirect
4. #21447992C2semidirect
5. #21447901C2semidirect
6. #21447900C2semidirect
8. #21447709C2semidirect
9. #21447707C2semidirect
10. #21447634C2semidirect
11. C61×#2231C2semidirect
17. #16840142C22semidirect
26. #16839868C22semidirect
31. #16839584C22semidirect
32. #16839583C22semidirect
33. #16839579C22semidirect
34. #16839577C22semidirect
37. #16839539C22semidirect
38. C61×#499C22semidirect
40. C61×#365C22semidirect
41. C61×#364C22semidirect
206. #2231D61semidirect
223. #499C2×D61semidirect
240. #365C2×D61semidirect
242. #364C2×D61semidirect
249. C61#33693semidirect

Maximal subgroups

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

subgroupindex
#214564792
#214547172
#214480402
#214479922
#214479012
#214479002
#214477712
#214477092
#214477072
#214476342
C61×#22312
#3369361