Finite Group Databasebeta

← #29833800

Subgroups — #29833800

Order 15616.

1380 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1380 conjugacy classes of subgroups — too many to draw. Only its 211 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×#16840198C212yesmaximal, complemented
2#21455400C211yesmaximal, characteristic, complemented
3#21447748C214yesmaximal, complemented
4#21447731C214yesmaximal, complemented
5#21447535C212yesmaximal, complemented
6#21447430C211yesmaximal, characteristic, complemented
7C23×D8×C61C211yesmaximal, characteristic, complemented
8C2×#14519799C2211yescharacteristic, complemented
9#16840273—21no—
10#16840198—28no—
11C2×#14519721C2214yescomplemented
12C2×#14519718C2214yescomplemented
13C2×#14519717C2212yescomplemented
14#16840003C2212yescomplemented
15#16839998C2212yescomplemented
16#16839991—22no—
17#16839800C2211yescharacteristic, complemented
18#16839759—21no—
19#16839563C2218yescomplemented
20C22⋊#11758185C2214yescomplemented
21C23×D4×C61—21no—
22C22×D8×C61—28no—
23C22×D8×C61—22no—
24C22×D8×C61—21no—
25C22×D8×C61C2214yes—
26C22×D8×C61C2212yes—
27C23×C8×C61C2211yescharacteristic
28C2×#11758222—21no—
29#14519799—24no—
30C2×Dic244C2311yescharacteristic, complemented
31C2×#11758190—21no—
32C2×D488C2311yescharacteristic, complemented
33#14519760—41no—
34#14519756—22no—
35#14519751—22no—
36#14519746—41no—
37C4×C22⋊D61—42no—
38#14519736—21no—
39#14519721—28no—
40#14519718—28no—
41#14519717—24no—
42C2×#11758185C2314yescomplemented
43#14519711C2312yescomplemented
44#14519709—21no—
45#14519702—22no—
46#14519699—22no—
47C25×C61—41no—
48C22×D4×C61—28no—
49C22×D4×C61—24no—
50C22×D4×C61—22no—
51C22×D4×C61—44no—
52C22×D4×C61—42no—
53C22×D4×C61—41no—
54C23×C4×C61D411yescharacteristic
55C2×D8×C61—28no—
56C2×D8×C61—216no—
57C2×D8×C61—216no—
58C2×D8×C61—24no—
59C2×D8×C61—22no—
60C2×D8×C61—24no—
61C2×D8×C61D418yescomplemented
62C2×D8×C61C2314yes—
63C22×C8×C61—24no—
64C22×C8×C61D412yes—
65C22×C8×C61D411yescharacteristic
66C22×C8×C61C2312yes—
67C22×C8×C61C2311yescharacteristic
68C24×C61—48no—
69C24×C61—44no—
70C24×C61—42no—
71C24×C61—88no—
72C24×C61D811yescharacteristic
73C2×D4×C61—216no—
74C2×D4×C61—28no—
75C2×D4×C61—24no—
76C2×D4×C61—48no—
77C2×D4×C61—48no—
78C2×D4×C61—416no—
79C2×D4×C61—44no—
80C2×D4×C61—42no—
81C2×D4×C61—44no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61D812yes—
85C22×C4×C61D811yescharacteristic
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D411yescharacteristic
88C2×C22⋊D61—41no—
89C2×C22⋊D61—42no—
90#11758222—42no—
91#11758222—42no—
92C2×D244—21no—
93C2×C4×D61—41no—
94C2×Dic122—21no—
95D8×C61—216no—
96D8×C61—28no—
97D8×C61—24no—
98D8×C61C2×D418yes—
99C2×C8×C61—22no—
100C2×C8×C61—24no—
101C2×C8×C61—24no—
102C2×C8×C61C2×D412yes—
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61C2×D414yes—
105C2×C8×C61C2411yescharacteristic
106C22⋊Dic61—41no—
107C22⋊Dic61—42no—
108#11758198—42no—
109C4⋊Dic61—22no—
110#11758196—42no—
111C4×Dic61—41no—
112Dic244—22no—
113D488—22no—
114#11758190—42no—
115#11758185—24no—
116C23×C61—22no—
117C23×C61—24no—
118C23×C61—416no—
119C23×C61—44no—
120C23×C61—48no—
121C23×C61—832no—
122C23×C61—88no—
123C23×C61—88no—
124C23×C61—88no—
125C23×C61C2×D811yescharacteristic
126C23×C61C2×D812yes—
127C22×D61—41no—
128D4×C61—28no—
129D4×C61—416no—
130D4×C61—44no—
131D4×C61—48no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—24no—
136C2×C4×C61#12012yes—
137C2×C4×C61C2×D812yes—
138C2×C4×C61C2×D811yescharacteristic
139C2×C4×C61AGL(1,ℤ/8)12yes—
140C2×C4×C61C22×D411yescharacteristic
141C22⋊D61—82no—
142C22⋊D61—84no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145D244—41no—
146D244—41no—
147C4×D61—82no—
148Dic122—41no—
149Dic122—41no—
150C8×C61—22no—
151C8×C61#12012yes—
152C8×C61C22×D412yes—
153#33517—611nomaximal, Sylow
154C22×C61—28no—
155C22×C61—22no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—48no—
159C22×C61—816no—
160C22×C61—832no—
161C22×C61—88no—
162C22×C61#44712yes—
163C22×C61#50914yescomplemented
164C22×C61C22×D811yescharacteristic
165C2×D61—82no—
166C2×D61—81no—
167C4×C61—22no—
168C4×C61#44712yes—
169C4×C61C2×#12011yescharacteristic
170C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
171Dic61—82no—
172Dic61—81no—
173C23×D8D6111yescharacteristic, complemented
174C2×#509—612no—
175#3050—611no—
176#2187—614no—
177#2184—614no—
178#2140—612no—
179#2112—611no—
180C2×C61—22no—
181C2×C61—24no—
182C2×C61—88no—
183C2×C61C2×#44711yescharacteristic
184C2×C61C2×#50912yes—
185D61—161no—
186C23×D4—21no—
187C2×#135—611no—
188C22×D8—28no—
189C22×D8—21no—
190C22×D8—22no—
191C22×D8C2×D6114yes—
192C22×D8C2×D6112yes—
193C23×C8C2×D6111yescharacteristic
194#525—1221no—
195#509—1228no—
196C2×SD32—614no—
197C2×D16—614no—
198C2×M32—612no—
199#467—612no—
200#466—612no—
201#465—1222no—
202#427—611no—
203#416—1221no—
204#357—618no—
205C22⋊C16—614no—
206C61#3351711yescharacteristic, complemented, Sylow
207C25—41no—
208C2×#41—1221no—
209C22×D4—28no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—44no—
213C22×D4—42no—
214C22×D4—41no—
215C23×C4C22⋊D6111yescharacteristic
216#135—1224no—
217C2×Q16—611no—
218C2×SD16—1221no—
219C2×D8—28no—
220C2×D8—216no—
221C2×D8—216no—
222C2×D8—24no—
223C2×D8—24no—
224C2×D8—22no—
225C2×D8—611no—
226C2×D8C22⋊D6118yescomplemented
227C2×D8C22×D6114yes—
228C22×C8—24no—
229C22×C8C22⋊D6111yescharacteristic
230C22×C8C22⋊D6112yes—
231C22×C8C22×D6112yes—
232C22×C8C22×D6111yescharacteristic
233#123—2441no—
234C22⋊Q8—1222no—
235#121—1222no—
236#120—2441no—
237C4×D4—2442no—
238#117—1221no—
239SD32—1228no—
240D16—1228no—
241M32—1224no—
242C2×C16—614no—
243#107—612no—
244#106—1221no—
245#103—1222no—
246#102—1222no—
247C24—48no—
248C24—44no—
249C24—42no—
250C24—88no—
251C24#1175819911yescharacteristic
252#41—2442no—
253#41—2442no—
254C2×Q8—1221no—
255C2×D4—216no—
256C2×D4—24no—
257C2×D4—28no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—416no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—1221no—
265C2×D4—2442no—
266C2×D4—2441no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—2441no—
270C22×C4#1175819912yes—
271C22×C4#1175819911yescharacteristic
272C22×C4C2×C22⋊D6112yes—
273C22×C4C2×C22⋊D6111yescharacteristic
274Q16—1222no—
275SD16—2442no—
276D8—216no—
277D8—24no—
278D8—28no—
279D8—1222no—
280D8C2×C22⋊D6118yes—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8C2×C22⋊D6112yes—
285C2×C8C2×C22⋊D6114yes—
286C2×C8C2×C22⋊D6111yescharacteristic
287C2×C8C23×D6111yescharacteristic
288C4⋊C4—1222no—
289C4⋊C4—2442no—
290C22⋊C4—2441no—
291C22⋊C4—2442no—
292C22⋊C4—2442no—
293C42—2441no—
294C16—1224no—
295C23—24no—
296C23—22no—
297C23—416no—
298C23—44no—
299C23—48no—
300C23—832no—
301C23—88no—
302C23—88no—
303C23—88no—
304C23—2441no—
305C23C2×#1175819912yes—
306C23C2×#1175819911yescharacteristic
307Q8—2441no—
308Q8—2441no—
309D4—28no—
310D4—416no—
311D4—44no—
312D4—48no—
313D4—4884no—
314D4—4882no—
315D4—2441no—
316D4—2441no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—4882no—
322C2×C4—2442no—
323C2×C4—2441no—
324C2×C4#1451974612yes—
325C2×C4C2×#1175819912yes—
326C2×C4C2×#1175819911yescharacteristic
327C2×C4#1451980312yes—
328C2×C4C22×C22⋊D6111yescharacteristic
329C8—22no—
330C8#1451974612yes—
331C8C22×C22⋊D6112yes—
332C22—28no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—48no—
337C22—832no—
338C22—816no—
339C22—88no—
340C22—4881no—
341C22—4882no—
342C22#1683989612yes—
343C22#1684019814yescomplemented
344C22C22×#1175819911yescharacteristic
345C4—22no—
346C4—4881no—
347C4—4882no—
348C4#1683989612yes—
349C4C2×#1451974611yescharacteristic
350C4C2×#1451980311yescharacteristic
351C2—24no—
352C2—22no—
353C2—88no—
354C2—9761no—
355C2C2×#1683989611yescharacteristic
356C2C2×#1684019812yes—
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC23×D8×C61C2
socleC22×C61C22×D8
hypercenterC23×D8D61
hypocenterC61#33517
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. #21455400C2semidirect
3. #21447748C2semidirect
4. #21447731C2semidirect
5. #21447535C2semidirect
6. #21447430C2semidirect
7. C23×D8×C61C2semidirect
8. C2×#14519799C22semidirect
11. C2×#14519721C22semidirect
12. C2×#14519718C22semidirect
13. C2×#14519717C22semidirect
14. #16840003C22semidirect
15. #16839998C22semidirect
17. #16839800C22semidirect
19. #16839563C22semidirect
20. C22⋊#11758185C22semidirect
30. C2×Dic244C23semidirect
32. C2×D488C23semidirect
42. C2×#11758185C23semidirect
43. #14519711C23semidirect
61. C2×D8×C61D4semidirect
163. C22×C61#509semidirect
173. C23×D8D61semidirect
206. C61#33517semidirect
226. C2×D8C22⋊D61semidirect
343. C22#16840198semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
#214554002
#214477482
#214477312
#214475352
#214474302
C23×D8×C612
#3351761