Finite Group Databasebeta

← #29656077

Subgroups — #29656077

Order 15616.

1716 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#16840397C211yesmaximal, characteristic, complemented
2#21455906C212yesmaximal, complemented
3#21444933C214yesmaximal, complemented
4#21444871C214yesmaximal, complemented
5#21444533C211yesmaximal, characteristic, complemented
6#21444263C212yesmaximal, complemented
7C61×#3151C211yesmaximal, characteristic, complemented
8C22×D4×D61—22no—
9#16840397—24no—
10#16840397—22no—
11#16840397C2214yes—
12C2×#14519748—22no—
13C2×#14519745—22no—
14C2×C4×D244C2212yes—
15#16840078—22no—
16#16840049C2214yescomplemented
17#16839954C2214yescomplemented
18#16839895—22no—
19#16839872—24no—
20#16839819C2212yescomplemented
21#16839506C2214yescomplemented
22#16839491C2218yescomplemented
23C2×AGL(1,ℤ/8)×C61—22no—
24C61×#532C2211yescharacteristic, complemented
25C2×#127×C61C2211yescharacteristic
26C61×#496—22no—
27C61×#493C2212yescomplemented
28C61×#492C2212yescomplemented
29C4×M16×C61C2211yescharacteristic, complemented
30C24×D61—22no—
31C22×D244D412yes—
32C22×C22⋊D61—24no—
33C2×D4×D61—28no—
34C2×D4×D61—28no—
35C2×D4×D61—416no—
36C22×C4×D61—22no—
37C2×#11758199—24no—
38#14519748—28no—
39#14519748—28no—
40#14519745—28no—
41#14519745—28no—
42C4×D244—22no—
43C4×D244—22no—
44C4×D244D414yes—
45C4×D244C2314yes—
46C2×C4⋊Dic61D412yes—
47C2×C22⋊Dic61—22no—
48C2×#11758198—22no—
49#14519708—24no—
50C4⋊#92418C2314yescomplemented
51#14519698—24no—
52#14519697—24no—
53C22⋊#92418—22no—
54#14519688—22no—
55C2×#41×C61—21no—
56C22×D4×C61—22no—
57C22×D4×C61D412yes—
58AGL(1,ℤ/8)×C61—48no—
59C2×SD16×C61—24no—
60C2×D8×C61—24no—
61C2×M16×C61—21no—
62C4⋊Q8×C61C2311yescharacteristic, complemented
63#127×C61—22no—
64#127×C61D412yes—
65#127×C61C2312yes—
66#127×C61C2311yescharacteristic, complemented
67#124×C61—21no—
68#121×C61—22no—
69C4×D4×C61—22no—
70C2×C42×C61C2311yescharacteristic
71#97×C61—21no—
72C4×C8×C61C2312yescomplemented
73C24×C61—22no—
74C23×D61—22no—
75C23×D61—28no—
76C23×D61—416no—
77C23×D61—84no—
78#41×C61—44no—
79C2×Q8×C61—21no—
80C2×D4×C61—24no—
81C2×D4×C61—22no—
82C2×D4×C61—24no—
83C2×D4×C61—22no—
84C2×D4×C61—21no—
85C2×D4×C61—48no—
86C2×D4×C61—44no—
87C2×D4×C61—41no—
88C2×D4×C61C2×D414yes—
89C22×C4×C61—41no—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—216no—
93C2×C22⋊D61—416no—
94C22×Dic61—22no—
95D4×D61—432no—
96D4×D61—416no—
97D4×D61—416no—
98C2×D244—22no—
99C2×D244—28no—
100C2×D244C2×D414yes—
101C2×C4×D61—24no—
102C2×C4×D61—24no—
103C2×C4×D61—48no—
104SD16×C61—48no—
105D8×C61—48no—
106M16×C61—24no—
107C2×C8×C61—22no—
108C4⋊C4×C61—22no—
109C22⋊C4×C61—42no—
110C42×C61C2×D412yes—
111C42×C61C2×D411yescharacteristic
112C42×C61C2411yescharacteristic
113C22⋊Dic61—28no—
114#11758199—48no—
115#11758198—24no—
116#11758198—24no—
117C4⋊Dic61—22no—
118C4⋊Dic61C2×D414yes—
119C2×#92418—24no—
120C23×C61—24no—
121C23×C61—48no—
122C23×C61—44no—
123C23×C61—41no—
124C23×C61#12011yescharacteristic
125C22×D61—28no—
126C22×D61—24no—
127C22×D61—432no—
128C22×D61—44no—
129C22×D61—44no—
130C22×D61—88no—
131C22×D61—832no—
132Q8×C61—42no—
133D4×C61—44no—
134D4×C61—48no—
135D4×C61—42no—
136D4×C61—48no—
137D4×C61—44no—
138D4×C61—82no—
139D4×C61—82no—
140C2×C4×C61—22no—
141C2×C4×C61—24no—
142C2×C4×C61—41no—
143C2×C4×C61—82no—
144C2×C4×C61#12012yes—
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61C22×D411yescharacteristic
147C2×C4×C61C22×D412yes—
148C22⋊D61—432no—
149C2×Dic61—24no—
150C2×Dic61—44no—
151D244—28no—
152D244—24no—
153D244AGL(1,ℤ/8)18yescomplemented
154C4×D61—48no—
155C4×D61—48no—
156C8×C61—24no—
157#92418—44no—
158#18524—611nomaximal, Sylow
159C22×C61—22no—
160C22×C61—48no—
161C22×C61—44no—
162C22×C61—88no—
163C22×C61—82no—
164C22×C61—81no—
165C22×C61#45312yes—
166C22×C61C2×#12011yescharacteristic
167C2×D61—28no—
168C2×D61—416no—
169C2×D61—44no—
170C2×D61—44no—
171C2×D61—832no—
172C4×C61—22no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—81no—
176C4×C61C2×AGL(1,ℤ/8)14yes—
177Dic61—44no—
178C2×D42—611no—
179#3151D6111yescharacteristic, complemented
180#3109—612no—
181#1629—614no—
182#1619—614no—
183#1555—611no—
184#1490—612no—
185C2×C61—22no—
186C2×C61—84no—
187C2×C61—161no—
188C2×C61#300312yes—
189C2×C61C2×#45311yescharacteristic
190D61—48no—
191C23×D4—1222no—
192C2×AGL(1,ℤ/8)—22no—
193D42—614no—
194D42—1224no—
195D42—1222no—
196#532C2×D6111yescharacteristic, complemented
197C2×#127C2×D6111yescharacteristic
198C2×#121—1222no—
199C2×#120—1222no—
200C2×C4×D4—612no—
201#496—22no—
202#493C2×D6112yescomplemented
203#492C2×D6112yescomplemented
204#481—1222no—
205#476—614no—
206#459—614no—
207#447—1222no—
208#442—1224no—
209#432—612no—
210C4×M16C2×D6111yescharacteristic, complemented
211#331—614no—
212D4⋊C8—618no—
213C61#1852411yescharacteristic, complemented, Sylow
214C25—1222no—
215C2×#41—21no—
216C22×D4—22no—
217C22×D4—612no—
218C22×D4—1228no—
219C22×D4—1228no—
220C22×D4—1224no—
221C22×D4—24416no—
222C22×D4C22⋊D6112yes—
223C23×C4—1222no—
224AGL(1,ℤ/8)—48no—
225C2×SD16—24no—
226C2×D8—24no—
227C2×D8—1224no—
228C2×M16—21no—
229C4⋊Q8C22×D6111yescharacteristic, complemented
230#127—22no—
231#127C22⋊D6112yes—
232#127C22×D6112yes—
233#127C22×D6111yescharacteristic, complemented
234#124—21no—
235#121—22no—
236#121—1228no—
237#121—1228no—
238#120—1228no—
239#120—1228no—
240C4×D4—22no—
241C4×D4—614no—
242C4×D4—614no—
243C4×D4—1222no—
244C4×D4—1222no—
245C2×C4⋊C4—612no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C42C22×D6111yescharacteristic
249#106—1224no—
250C4⋊C8—614no—
251#102—1224no—
252#102—1224no—
253C22⋊C8—1222no—
254#97—21no—
255#97—1222no—
256C4×C8C22×D6112yescomplemented
257C24—22no—
258C24—1228no—
259C24—1222no—
260C24—4884no—
261C24—24416no—
262#41—44no—
263C2×Q8—21no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—614no—
273C2×D4—1228no—
274C2×D4—12216no—
275C2×D4—1222no—
276C2×D4—24416no—
277C2×D4—24432no—
278C2×D4—24416no—
279C2×D4—24416no—
280C2×D4C2×C22⋊D6114yes—
281C22×C4—41no—
282C22×C4—1222no—
283C22×C4—1224no—
284C22×C4—1224no—
285C22×C4—2448no—
286C22×C4D4×D6112yes—
287C22×C4C2×C22⋊D6111yescharacteristic
288SD16—48no—
289D8—48no—
290D8—2448no—
291M16—24no—
292C2×C8—22no—
293C2×C8—1224no—
294C4⋊C4—22no—
295C4⋊C4—614no—
296C4⋊C4—1222no—
297C22⋊C4—42no—
298C22⋊C4—1228no—
299C22⋊C4—1224no—
300C22⋊C4—1224no—
301C42D4×D6112yes—
302C42C2×C22⋊D6111yescharacteristic
303C42C23×D6111yescharacteristic
304C23—24no—
305C23—48no—
306C23—44no—
307C23—41no—
308C23—1228no—
309C23—1224no—
310C23—4888no—
311C23—48832no—
312C23—2444no—
313C23—24432no—
314C23—2444no—
315C23#1451974511yescharacteristic
316Q8—42no—
317D4—44no—
318D4—48no—
319D4—42no—
320D4—48no—
321D4—44no—
322D4—82no—
323D4—82no—
324D4—618no—
325D4—1228no—
326D4—1224no—
327D4—24432no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—1224no—
333C2×C4—2448no—
334C2×C4—2444no—
335C2×C4—2448no—
336C2×C4#1451974511yescharacteristic
337C2×C4#1451974512yes—
338C2×C4C2×D4×D6112yes—
339C2×C4C22×C22⋊D6111yescharacteristic
340C8—24no—
341C8—2444no—
342C22—22no—
343C22—44no—
344C22—48no—
345C22—88no—
346C22—81no—
347C22—82no—
348C22—1228no—
349C22—48832no—
350C22—24416no—
351C22—2444no—
352C22—2444no—
353C22#1683992912yes—
354C22C2×#1451974511yescharacteristic
355C4—21no—
356C4—21no—
357C4—22no—
358C4—81no—
359C4—2444no—
360C4AGL(1,ℤ/8)×D6114yes—
361C2—22no—
362C2—84no—
363C2—161no—
364C2—2448no—
365C2#2145504412yes—
366C2C2×#1683992911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3151C2
socleC22×C61C2×#120
hypercenter#3151D61
hypocenterC61#18524
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. C2×#16840397C2semidirect
2. #21455906C2semidirect
3. #21444933C2semidirect
4. #21444871C2semidirect
5. #21444533C2semidirect
6. #21444263C2semidirect
7. C61×#3151C2semidirect
16. #16840049C22semidirect
17. #16839954C22semidirect
20. #16839819C22semidirect
21. #16839506C22semidirect
22. #16839491C22semidirect
24. C61×#532C22semidirect
27. C61×#493C22semidirect
28. C61×#492C22semidirect
29. C4×M16×C61C22semidirect
50. C4⋊#92418C23semidirect
62. C4⋊Q8×C61C23semidirect
66. #127×C61C23semidirect
72. C4×C8×C61C23semidirect
153. D244AGL(1,ℤ/8)semidirect
179. #3151D61semidirect
196. #532C2×D61semidirect
202. #493C2×D61semidirect
203. #492C2×D61semidirect
210. C4×M16C2×D61semidirect
213. C61#18524semidirect
229. C4⋊Q8C22×D61semidirect
233. #127C22×D61semidirect
256. C4×C8C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403972
#214559062
#214449332
#214448712
#214445332
#214442632
C61×#31512
#1852461