Finite Group Databasebeta

← #29658151

Subgroups — #29658151

Order 15616.

1132 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 163 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#21460224C211yesmaximal, characteristic, complemented
2#21456737C212yesmaximal, complemented
3#21444900C214yesmaximal, complemented
4#21444871C214yesmaximal, complemented
5#21444481C211yesmaximal, characteristic, complemented
6#21444237C212yesmaximal, complemented
7C61×#3151C211yesmaximal, characteristic, complemented
8#16840504—22no—
9#16840478C2212yescomplemented
10#16840477—22no—
11D4×D244—22no—
12#16840397C2212yescomplemented
13#16840328—24no—
14#16840326—22no—
15#16840312C2212yes—
16#16840138—22no—
17#16840108—22no—
18#16839954C2218yescomplemented
19#16839896—24no—
20#16839815C2212yescomplemented
21#16839511C2212yescomplemented
22#16839506C2212yescomplemented
23#16839491C2218yescomplemented
24C2×AGL(1,ℤ/8)×C61—22no—
25C61×#532C2211yescharacteristic
26C2×#127×C61C2211yescharacteristic, complemented
27C61×#496—22no—
28C61×#493C2212yes—
29C61×#492C2212yescomplemented
30C4×M16×C61C2211yescharacteristic, complemented
31C2×#11758222—22no—
32C22×D244—22no—
33C2×D4×D61—24no—
34C2×#11758199—28no—
35#14519777D412yes—
36#14519774D412yes—
37#14519773—24no—
38#14519766—24no—
39#14519765—22no—
40#14519763—28no—
41#14519752—24no—
42#14519751—24no—
43#14519751—22no—
44#14519751D414yes—
45#14519748—24no—
46#14519747—28no—
47#14519745—24no—
48#14519745—24no—
49C4×D244—22no—
50C4×D244C2314yes—
51#14519736—22no—
52#14519735—24no—
53C4⋊#92418C2314yes—
54#14519698—28no—
55C22⋊#92418—22no—
56C4×#92418—22no—
57C2×#41×C61D412yes—
58C22×D4×C61—21no—
59C22×D4×C61—22no—
60AGL(1,ℤ/8)×C61—48no—
61C2×SD16×C61—24no—
62C2×D8×C61—24no—
63C2×M16×C61—21no—
64C4⋊Q8×C61C2311yescharacteristic
65#127×C61—22no—
66#127×C61—21no—
67#127×C61C2312yes—
68#127×C61C2311yescharacteristic
69#124×C61D412yes—
70#121×C61—22no—
71C4×D4×C61—22no—
72C2×C42×C61C2311yescharacteristic
73#97×C61—21no—
74C4×C8×C61C2312yes—
75C24×C61—41no—
76C23×D61—24no—
77#41×C61—44no—
78C2×Q8×C61C2×D412yes—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61—24no—
83C2×D4×C61—44no—
84C2×D4×C61—42no—
85C2×D4×C61—41no—
86C2×D4×C61—84no—
87C2×D4×C61C2×D412yes—
88C22×C4×C61—22no—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D411yescharacteristic
91C2×C22⋊D61—28no—
92C2×C22⋊D61—24no—
93D4×D61—88no—
94#11758222—84no—
95C2×D244—22no—
96C2×D244—24no—
97C2×D244—22no—
98C2×D244—84no—
99C2×D244C2×D414yes—
100C2×C4×D61—24no—
101C2×Dic122—22no—
102SD16×C61—48no—
103D8×C61—48no—
104M16×C61—24no—
105C2×C8×C61—22no—
106C4⋊C4×C61—22no—
107C22⋊C4×C61—24no—
108C42×C61C2×D412yes—
109C42×C61C2×D411yescharacteristic
110C42×C61C2411yescharacteristic
111C22⋊Dic61—24no—
112#11758199—88no—
113#11758198—28no—
114#11758198—24no—
115#11758198—24no—
116C4⋊Dic61—22no—
117C4⋊Dic61C2×D414yes—
118C4×Dic61—24no—
119C2×#92418—24no—
120C23×C61—22no—
121C23×C61—42no—
122C23×C61—82no—
123C23×C61—84no—
124C23×C61#12011yescharacteristic
125C22×D61—24no—
126C22×D61—84no—
127C22×D61—88no—
128Q8×C61—42no—
129D4×C61—42no—
130D4×C61—44no—
131D4×C61—44no—
132D4×C61—48no—
133D4×C61—84no—
134D4×C61—82no—
135D4×C61—82no—
136C2×C4×C61—24no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—44no—
140C2×C4×C61#12012yes—
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61C22×D411yescharacteristic
143C2×C4×C61C22×D412yes—
144C22⋊D61—84no—
145C22⋊D61—88no—
146C2×Dic61—24no—
147D244—28no—
148D244—44no—
149D244—82no—
150D244—84no—
151C4×D61—48no—
152Dic122—82no—
153C8×C61—24no—
154#92418—44no—
155#18714—611nomaximal, Sylow
156C22×C61—22no—
157C22×C61—21no—
158C22×C61—44no—
159C22×C61—42no—
160C22×C61—84no—
161C22×C61—82no—
162C22×C61—164no—
163C22×C61C2×#12011yescharacteristic
164C2×D61—44no—
165C2×D61—44no—
166C2×D61—44no—
167C2×D61—168no—
168C4×C61—22no—
169C4×C61—21no—
170C4×C61—42no—
171C4×C61#45312yes—
172C4×C61#57614yes—
173Dic61—44no—
174#3549—611no—
175#3205—612no—
176#3151D6111yescharacteristic, complemented
177#1624—614no—
178#1619—614no—
179#1543—611no—
180#1484—612no—
181C2×C61—41no—
182C2×C61—82no—
183C2×C61—162no—
184C2×C61#300812yes—
185C2×C61C2×#45311yescharacteristic
186D61—84no—
187C2×AGL(1,ℤ/8)—22no—
188H(3,𝔽4)—1222no—
189#555—612no—
190#555—1222no—
191D42—612no—
192D42—1222no—
193#535—1224no—
194#535—1222no—
195#532—612no—
196#532C2×D6111yescharacteristic
197C2×#127C2×D6111yescharacteristic, complemented
198#496—22no—
199#493—1222no—
200#493C2×D6112yes—
201#492C2×D6112yescomplemented
202#486—1222no—
203#459—618no—
204#447—1224no—
205#431—612no—
206C4×M16C2×D6111yescharacteristic, complemented
207#333—612no—
208#331—612no—
209D4⋊C8—618no—
210C61#1871411yescharacteristic, complemented, Sylow
211C2×#41—1222no—
212C2×#41C22⋊D6112yes—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—1222no—
216C22×D4—1224no—
217AGL(1,ℤ/8)—48no—
218C2×SD16—24no—
219C2×D8—24no—
220C2×D8—1228no—
221C2×M16—21no—
222C4⋊Q8—612no—
223C4⋊Q8C22×D6111yescharacteristic
224#127—22no—
225#127—21no—
226#127—612no—
227#127—1224no—
228#127C22×D6112yes—
229#127C22×D6111yescharacteristic
230#124—1228no—
231#124—1224no—
232#124—1222no—
233#124C22⋊D6112yes—
234#121—22no—
235#121—614no—
236#121—1224no—
237#121—1224no—
238#121—1224no—
239#121—1222no—
240#120—1228no—
241#120—1224no—
242#120—1224no—
243C4×D4—22no—
244C4×D4—614no—
245C4×D4—1222no—
246#117—1224no—
247#117—1222no—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—614no—
250#102—1228no—
251C22⋊C8—1222no—
252#97—21no—
253C4×C8—1222no—
254C4×C8C22×D6112yes—
255C24—41no—
256C24—1224no—
257#41—44no—
258#41—4884no—
259C2×Q8—1222no—
260C2×Q8C2×C22⋊D6112yes—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—84no—
269C2×D4—614no—
270C2×D4—1224no—
271C2×D4—1228no—
272C2×D4—1224no—
273C2×D4—1222no—
274C2×D4—1222no—
275C2×D4—4884no—
276C2×D4—4888no—
277C2×D4C2×C22⋊D6112yes—
278C22×C4—22no—
279C22×C4—1224no—
280C22×C4D4×D6112yes—
281C22×C4C2×C22⋊D6111yescharacteristic
282SD16—48no—
283D8—48no—
284D8—4888no—
285M16—24no—
286C2×C8—22no—
287C2×C8—1224no—
288C4⋊C4—22no—
289C4⋊C4—614no—
290C4⋊C4—1222no—
291C22⋊C4—24no—
292C22⋊C4—1228no—
293C22⋊C4—1224no—
294C22⋊C4—1224no—
295C22⋊C4—1224no—
296C42—1224no—
297C42D4×D6112yes—
298C42C2×C22⋊D6111yescharacteristic
299C42C23×D6111yescharacteristic
300C23—22no—
301C23—42no—
302C23—82no—
303C23—84no—
304C23—1224no—
305C23—4884no—
306C23—4888no—
307C23#1451974511yescharacteristic
308Q8—42no—
309Q8—4882no—
310D4—42no—
311D4—44no—
312D4—44no—
313D4—48no—
314D4—84no—
315D4—82no—
316D4—82no—
317D4—1228no—
318D4—4882no—
319D4—4884no—
320D4—4884no—
321D4—4888no—
322D4—2444no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—44no—
327C2×C4—1224no—
328C2×C4—2448no—
329C2×C4#1451974512yes—
330C2×C4#1451974511yescharacteristic
331C2×C4C2×D4×D6112yes—
332C2×C4C22×C22⋊D6111yescharacteristic
333C8—24no—
334C8—2444no—
335C22—21no—
336C22—22no—
337C22—42no—
338C22—44no—
339C22—82no—
340C22—84no—
341C22—164no—
342C22—9768no—
343C22—2444no—
344C22—2444no—
345C22—2444no—
346C22C2×#1451974511yescharacteristic
347C4—21no—
348C4—22no—
349C4—42no—
350C4—2444no—
351C4#1683992912yes—
352C4#1684056614yes—
353C2—41no—
354C2—82no—
355C2—162no—
356C2—4884no—
357C2#2145508612yes—
358C2C2×#1683992911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3151C2
socleC22×C61C2×#120
hypercenter#3151D61
hypocenterC61#18714
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #21460224C2semidirect
2. #21456737C2semidirect
3. #21444900C2semidirect
4. #21444871C2semidirect
5. #21444481C2semidirect
6. #21444237C2semidirect
7. C61×#3151C2semidirect
9. #16840478C22semidirect
12. #16840397C22semidirect
18. #16839954C22semidirect
20. #16839815C22semidirect
21. #16839511C22semidirect
22. #16839506C22semidirect
23. #16839491C22semidirect
26. C2×#127×C61C22semidirect
29. C61×#492C22semidirect
30. C4×M16×C61C22semidirect
176. #3151D61semidirect
197. C2×#127C2×D61semidirect
201. #492C2×D61semidirect
206. C4×M16C2×D61semidirect
210. C61#18714semidirect

Maximal subgroups

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

subgroupindex
#214602242
#214567372
#214449002
#214448712
#214444812
#214442372
C61×#31512
#1871461