Finite Group Databasebeta

← #29833803

Subgroups — #29833803

Order 15616.

1380 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1380 conjugacy classes of subgroups — too many to draw. Only its 195 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
1C23×D488C211yesmaximal, characteristic, complemented
2C2×#16840200C212yesmaximal, complemented
3#21455400C211yesmaximal, characteristic, complemented
4#21447750C214yesmaximal, complemented
5#21447733C214yesmaximal, complemented
6#21447536C212yesmaximal, complemented
7C61×#2112C211yesmaximal, characteristic, complemented
8C23×D244—21no—
9C2×#14519799C2211yescharacteristic, complemented
10C22×D488—28no—
11C22×D488—22no—
12C22×D488—21no—
13C22×D488C2214yes—
14C22×D488C2212yes—
15#16840273—21no—
16#16840200—28no—
17C2×#14519722C2214yescomplemented
18C2×D976C2214yescomplemented
19#16840003C2212yescomplemented
20#16839998C2212yescomplemented
21#16839991—22no—
22#16839800C2211yescharacteristic, complemented
23#16839759—21no—
24#16839564C2218yescomplemented
25C23×C8×C61C2211yescharacteristic
26C2×M32×C61C2212yescomplemented
27C61×C22⋊C16C2214yescomplemented
28C24×D61—41no—
29C2×#11758222—21no—
30C22×D244—28no—
31C22×D244—24no—
32C22×D244—22no—
33C22×D244—44no—
34C22×D244—42no—
35C22×D244—41no—
36#14519799—24no—
37C2×Dic244C2311yescharacteristic, complemented
38C2×#11758190—21no—
39C2×D488—28no—
40C2×D488—216no—
41C2×D488—24no—
42C2×D488—22no—
43C2×D488—24no—
44C2×D488—216no—
45C2×D488D418yescomplemented
46C2×D488C2314yes—
47C2×D488C2311yescharacteristic, complemented
48#14519760—41no—
49#14519756—22no—
50#14519751—22no—
51#14519746—41no—
52C4×C22⋊D61—42no—
53#14519736—21no—
54#14519722—28no—
55D976—28no—
56#14519711C2312yescomplemented
57#14519709—21no—
58#14519702—22no—
59#14519699—22no—
60C23×C4×C61D411yescharacteristic
61C22×C8×C61—24no—
62C22×C8×C61D412yes—
63C22×C8×C61D411yescharacteristic
64C22×C8×C61C2312yes—
65C22×C8×C61C2311yescharacteristic
66M32×C61—24no—
67C2×C16×C61C2314yescomplemented
68C24×C61D811yescharacteristic
69C23×D61—48no—
70C23×D61—44no—
71C23×D61—42no—
72C23×D61—88no—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61D812yes—
76C22×C4×C61D811yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—41no—
80C2×C22⋊D61—42no—
81#11758222—42no—
82#11758222—42no—
83C2×D244—216no—
84C2×D244—28no—
85C2×D244—24no—
86C2×D244—21no—
87C2×D244—48no—
88C2×D244—48no—
89C2×D244—416no—
90C2×D244—44no—
91C2×D244—42no—
92C2×D244—44no—
93C2×C4×D61—41no—
94C2×Dic122—21no—
95C2×C8×C61—22no—
96C2×C8×C61—24no—
97C2×C8×C61—24no—
98C2×C8×C61C2×D412yes—
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61C2×D414yes—
101C2×C8×C61C2411yescharacteristic
102C22⋊Dic61—41no—
103C22⋊Dic61—42no—
104#11758198—42no—
105C4⋊Dic61—22no—
106#11758196—42no—
107C4×Dic61—41no—
108Dic244—22no—
109D488—216no—
110D488—28no—
111D488—24no—
112D488—22no—
113D488C2×D418yes—
114#11758190—42no—
115C16×C61—24no—
116C23×C61—22no—
117C23×C61—24no—
118C23×C61C2×D811yescharacteristic
119C23×C61C2×D812yes—
120C22×D61—416no—
121C22×D61—44no—
122C22×D61—48no—
123C22×D61—41no—
124C22×D61—832no—
125C22×D61—88no—
126C22×D61—88no—
127C22×D61—88no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—24no—
132C2×C4×C61#12012yes—
133C2×C4×C61C2×D812yes—
134C2×C4×C61C2×D811yescharacteristic
135C2×C4×C61AGL(1,ℤ/8)12yes—
136C2×C4×C61C22×D411yescharacteristic
137C22⋊D61—82no—
138C22⋊D61—84no—
139C2×Dic61—42no—
140C2×Dic61—41no—
141D244—28no—
142D244—416no—
143D244—41no—
144D244—44no—
145D244—41no—
146D244—48no—
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#44712yes—
159C22×C61#50914yescomplemented
160C22×C61C22×D811yescharacteristic
161C2×D61—48no—
162C2×D61—816no—
163C2×D61—832no—
164C2×D61—88no—
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×D8—611no—
174C2×#509—612no—
175#3050—611no—
176#2187—614no—
177#2184—614no—
178#2140—612no—
179#2112D6111yescharacteristic, complemented
180C2×C61—22no—
181C2×C61—24no—
182C2×C61C2×#44711yescharacteristic
183C2×C61C2×#50912yes—
184D61—88no—
185D61—161no—
186C23×D4—1221no—
187C2×#135—611no—
188C22×D8—612no—
189C22×D8—614no—
190C22×D8—1228no—
191C22×D8—1221no—
192C22×D8—1222no—
193C23×C8C2×D6111yescharacteristic
194#525—1221no—
195#509—1228no—
196C2×SD32—614no—
197C2×D16—614no—
198C2×M32C2×D6112yescomplemented
199#467—612no—
200#466—612no—
201#465—1222no—
202#427—611no—
203#416—1221no—
204#357—618no—
205C22⋊C16C2×D6114yescomplemented
206C61#3351711yescharacteristic, complemented, Sylow
207C25—2441no—
208C2×#41—1221no—
209C22×D4—1228no—
210C22×D4—1224no—
211C22×D4—1222no—
212C22×D4—2444no—
213C22×D4—2442no—
214C22×D4—2441no—
215C23×C4D24411yescharacteristic
216#135—1224no—
217C2×Q16—611no—
218C2×SD16—1221no—
219C2×D8—618no—
220C2×D8—614no—
221C2×D8—611no—
222C2×D8—12216no—
223C2×D8—1228no—
224C2×D8—12216no—
225C2×D8—1224no—
226C2×D8—1222no—
227C2×D8—1224no—
228C22×C8—24no—
229C22×C8D24411yescharacteristic
230C22×C8D24412yes—
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—24no—
242C2×C16C22×D6114yescomplemented
243#107—612no—
244#106—1221no—
245#103—1222no—
246#102—1222no—
247C24—4888no—
248C24—2448no—
249C24—2442no—
250C24—2444no—
251C24D48811yescharacteristic
252#41—2442no—
253#41—2442no—
254C2×Q8—1221no—
255C2×D4—12216no—
256C2×D4—1224no—
257C2×D4—1221no—
258C2×D4—1228no—
259C2×D4—2448no—
260C2×D4—2448no—
261C2×D4—24416no—
262C2×D4—2442no—
263C2×D4—2441no—
264C2×D4—2444no—
265C2×D4—2442no—
266C2×D4—2444no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—2441no—
270C22×C4D48812yes—
271C22×C4D48811yescharacteristic
272C22×C4C2×D24412yes—
273C22×C4C2×D24411yescharacteristic
274Q16—1222no—
275SD16—2442no—
276D8—618no—
277D8—12216no—
278D8—1224no—
279D8—1222no—
280D8—1228no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8C2×D24412yes—
285C2×C8C2×D24411yescharacteristic
286C2×C8D4×D6114yes—
287C2×C8C23×D6111yescharacteristic
288C4⋊C4—1222no—
289C4⋊C4—2442no—
290C22⋊C4—2441no—
291C22⋊C4—2442no—
292C22⋊C4—2442no—
293C42—2441no—
294C16—24no—
295C23—24no—
296C23—22no—
297C23—48832no—
298C23—4888no—
299C23—4888no—
300C23—4888no—
301C23—24416no—
302C23—2441no—
303C23—2444no—
304C23—2448no—
305C23C2×D48812yes—
306C23C2×D48811yescharacteristic
307Q8—2441no—
308Q8—2441no—
309D4—1228no—
310D4—4884no—
311D4—4882no—
312D4—24416no—
313D4—2441no—
314D4—2441no—
315D4—2444no—
316D4—2448no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—4882no—
322C2×C4—2442no—
323C2×C4—2441no—
324C2×C4#1451974712yes—
325C2×C4C2×D48812yes—
326C2×C4C2×D48811yescharacteristic
327C2×C4#1451980512yes—
328C2×C4C22×D24411yescharacteristic
329C8—22no—
330C8#1451974712yes—
331C8C2×D4×D6112yes—
332C22—28no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—48832no—
337C22—48816no—
338C22—4882no—
339C22—4881no—
340C22—4888no—
341C22—2448no—
342C22#1683989812yes—
343C22#1684020014yescomplemented
344C22C22×D48811yescharacteristic
345C4—22no—
346C4—4882no—
347C4—4881no—
348C4#1683989812yes—
349C4C2×#1451974711yescharacteristic
350C4C2×#1451980511yescharacteristic
351C2—24no—
352C2—22no—
353C2—9761no—
354C2—4888no—
355C2C2×#1683989811yescharacteristic
356C2C2×#1684020012yes—
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2112C2
socleC22×C61C22×D8
hypercenter#2112D61
hypocenterC61#33517
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
1. C23×D488C2semidirect
2. C2×#16840200C2semidirect
3. #21455400C2semidirect
4. #21447750C2semidirect
5. #21447733C2semidirect
6. #21447536C2semidirect
7. C61×#2112C2semidirect
9. C2×#14519799C22semidirect
17. C2×#14519722C22semidirect
18. C2×D976C22semidirect
19. #16840003C22semidirect
20. #16839998C22semidirect
22. #16839800C22semidirect
24. #16839564C22semidirect
26. C2×M32×C61C22semidirect
27. C61×C22⋊C16C22semidirect
37. C2×Dic244C23semidirect
45. C2×D488D4semidirect
47. C2×D488C23semidirect
56. #14519711C23semidirect
67. C2×C16×C61C23semidirect
159. C22×C61#509semidirect
179. #2112D61semidirect
198. C2×M32C2×D61semidirect
205. C22⋊C16C2×D61semidirect
206. C61#33517semidirect
242. C2×C16C22×D61semidirect
343. C22#16840200semidirect

Maximal subgroups

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

subgroupindex
C23×D4882
C2×#168402002
#214554002
#214477502
#214477332
#214475362
C61×#21122
#3351761