Finite Group Databasebeta

← #29580558

Subgroups — #29580558

Order 15616.

944 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16839903C212yesmaximal, complemented
2#21451921C211yesmaximal, characteristic
3C2×#16839497C212yesmaximal, complemented
4#21444089C211yesmaximal, characteristic, complemented
5C61×#1280C211yesmaximal, characteristic, complemented
6C23×Dic122C2211yescharacteristic
7C22×#11758190—22no—
8C2×#14519751—22no—
9C2×#14519751C2212yes—
10#16839903—24no—
11#16839903—24no—
12C2×#14519702—22no—
13#16839699—22no—
14#16839672—22no—
15C2×#14519684—22no—
16C2×#14519684C2211yescharacteristic
17#16839497—24no—
18C2×C22⋊C8×C61C2212yes—
19C2×#95×C61C2211yescharacteristic
20C22×Dic122—22no—
21C22×Dic122—22no—
22C22×Dic122—21no—
23C22×Dic122—44no—
24C22×Dic122D412yescomplemented
25C22×D244—22no—
26C22×D244D412yes—
27C22×C22⋊D61—22no—
28C23×Dic61—21no—
29C23×Dic61—21no—
30C2×#11758190—28no—
31C2×#11758190—48no—
32#14519751—22no—
33#14519751—22no—
34#14519751—22no—
35#14519751—22no—
36#14519751—44no—
37#14519751—44no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61D412yes—
40C2×#11758198—24no—
41C2×#11758198—22no—
42#14519702—28no—
43#14519684—22no—
44#14519684—44no—
45C23×C4×C61D412yes—
46C23×C4×C61C2311yescharacteristic
47C22×C8×C61—22no—
48C22⋊C8×C61—22no—
49C22⋊C8×C61D414yescomplemented
50#95×C61—22no—
51C24×C61C2×D411yescharacteristic
52C23×D61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—42no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61SD1614yescomplemented
59C22×C4×C61C2×D412yes—
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×C4×C61#4111yescharacteristic
63C2×C22⋊D61—42no—
64C2×C22⋊D61—42no—
65C2×C22⋊D61—44no—
66C2×C22⋊D61—44no—
67C22×Dic61—24no—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—42no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—42no—
76C22×Dic61—41no—
77C22×Dic61—42no—
78C2×D244—22no—
79C2×D244—22no—
80C2×D244—44no—
81C2×D244—44no—
82C2×D244—82no—
83C2×D244—82no—
84C2×Dic122—28no—
85C2×Dic122—24no—
86C2×Dic122—44no—
87C2×Dic122—48no—
88C2×Dic122—44no—
89C2×Dic122—44no—
90C2×Dic122—42no—
91C2×C8×C61—24no—
92C2×C8×C61—44no—
93#11758198—48no—
94#11758198—42no—
95#11758198—42no—
96C4⋊Dic61—22no—
97C4⋊Dic61—22no—
98C4⋊Dic61—44no—
99#11758190—416no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—22no—
103C23×C61#12011yescharacteristic
104C23×C61#12111yescharacteristic
105C23×C61#12711yescharacteristic
106C23×C61C2×SD1612yes—
107C23×C61AGL(1,ℤ/8)12yes—
108C22×D61—42no—
109C22×D61—44no—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61—44no—
113C2×C4×C61—24no—
114C2×C4×C61—24no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—44no—
121C2×C4×C61—82no—
122C2×C4×C61#12012yes—
123C2×C4×C61#12112yes—
124C22⋊D61—84no—
125C22⋊D61—82no—
126C22⋊D61—82no—
127C2×Dic61—24no—
128C2×Dic61—42no—
129C2×Dic61—44no—
130C2×Dic61—42no—
131C2×Dic61—44no—
132C2×Dic61—44no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—44no—
136C2×Dic61—82no—
137C2×Dic61—81no—
138C2×Dic61—81no—
139D244—44no—
140D244—82no—
141D244—88no—
142Dic122—28no—
143Dic122—44no—
144Dic122—48no—
145C8×C61—44no—
146#9964—611nomaximal, Sylow
147C22×C61—22no—
148C22×C61—22no—
149C22×C61—22no—
150C22×C61—22no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61#39211yescharacteristic
156C22×C61#45012yes—
157C22×C61#45511yescharacteristic
158C22×C61#45611yescharacteristic
159C22×C61B(4,𝔽2)12yes—
160C22×C61#46012yes—
161C22×C61#49211yescharacteristic
162C22×C61#49611yescharacteristic
163C2×D61—84no—
164C2×D61—84no—
165C2×D61—82no—
166C2×D61—82no—
167C2×D61—82no—
168C4×C61—22no—
169C4×C61—22no—
170C4×C61—44no—
171Dic61—44no—
172Dic61—82no—
173Dic61—82no—
174C2×#448—612no—
175#2656—611no—
176C2×#327—612no—
177#1446—611no—
178#1280D6111yescharacteristic, complemented
179C2×C61—21no—
180C2×C61—22no—
181C2×C61—21no—
182C2×C61#160112yes—
183C2×C61#160412yes—
184C2×C61#200711yescharacteristic
185C2×C61#201411yescharacteristic
186C2×C61#202111yescharacteristic
187D61—162no—
188C23×Q8—611no—
189C22×SD16—1222no—
190C2×#121—612no—
191C2×#121—1222no—
192#448—1224no—
193#448—1224no—
194C2×#103—1222no—
195C2×C22⋊C8C2×D6112yes—
196#399—1222no—
197#392—1222no—
198C2×#95—611no—
199C2×#95—1222no—
200C2×#95C2×D6111yescharacteristic
201#327—1224no—
202C61#996411yescharacteristic, complemented, Sylow
203C22×Q8—612no—
204C22×Q8—1221no—
205C22×Q8—1222no—
206C22×Q8—1222no—
207C22×Q8—2444no—
208C22×D4—612no—
209C22×D4—1222no—
210C22×D4—1222no—
211C23×C4—1221no—
212C23×C4—1221no—
213C23×C4D24412yes—
214C23×C4C22×D6111yescharacteristic
215C2×SD16—1228no—
216C2×SD16—2448no—
217C22×C8—22no—
218#121—1222no—
219#121—1222no—
220#121—1222no—
221#121—1222no—
222#121—2444no—
223#121—2444no—
224C2×C4⋊C4—612no—
225C2×C4⋊C4—1222no—
226C2×C22⋊C4—1224no—
227C2×C22⋊C4—1222no—
228#103—1228no—
229C22⋊C8—22no—
230C22⋊C8D24414yescomplemented
231#95—22no—
232#95—1222no—
233#95—2444no—
234C24—1222no—
235C24C2×D24411yescharacteristic
236C2×Q8—1228no—
237C2×Q8—1224no—
238C2×Q8—2444no—
239C2×Q8—2444no—
240C2×Q8—2444no—
241C2×Q8—2448no—
242C2×Q8—2442no—
243C2×D4—1222no—
244C2×D4—1222no—
245C2×D4—4882no—
246C2×D4—4882no—
247C2×D4—2444no—
248C2×D4—2444no—
249C2×D4—2442no—
250C2×D4—2442no—
251C2×D4—2444no—
252C2×D4—2444no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—1224no—
259C22×C4—1222no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4—2442no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4#1175819014yescomplemented
270C22×C4C2×D24412yes—
271C22×C4D4×D6112yes—
272C22×C4D4×D6111yescharacteristic
273C22×C4#1175822611yescharacteristic
274SD16—24416no—
275C2×C8—24no—
276C2×C8—44no—
277C4⋊C4—1222no—
278C4⋊C4—1222no—
279C4⋊C4—2444no—
280C22⋊C4—2448no—
281C22⋊C4—2442no—
282C22⋊C4—2442no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—2442no—
287C23—2442no—
288C23—2444no—
289C23—2444no—
290C23—2442no—
291C23#1451974711yescharacteristic
292C23#1451975211yescharacteristic
293C23#1451977411yescharacteristic
294C23C2×#1175819012yes—
295C23#1451980512yes—
296Q8—1228no—
297Q8—2444no—
298Q8—2448no—
299D4—4882no—
300D4—4888no—
301D4—4884no—
302D4—4882no—
303D4—4882no—
304D4—2444no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—1224no—
315C2×C4—4881no—
316C2×C4—4882no—
317C2×C4—4881no—
318C2×C4—2444no—
319C2×C4—2442no—
320C2×C4—2444no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2444no—
324C2×C4—2444no—
325C2×C4—2442no—
326C2×C4#1451974712yes—
327C2×C4#1451975212yes—
328C8—44no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—21no—
336C22—21no—
337C22—4882no—
338C22—4884no—
339C22—4884no—
340C22—4882no—
341C22—4882no—
342C22#1683967211yescharacteristic
343C22#1683991512yes—
344C22#1683994211yescharacteristic
345C22#1683994711yescharacteristic
346C22#1683995012yes—
347C22#1683996312yes—
348C22#1684013511yescharacteristic
349C22#1684015111yescharacteristic
350C4—22no—
351C4—22no—
352C4—44no—
353C4—4882no—
354C4—4882no—
355C4—2444no—
356C2—21no—
357C2—22no—
358C2—21no—
359C2—9762no—
360C2#2144477312yes—
361C2#2144479012yes—
362C2#2144686811yescharacteristic
363C2#2144690911yescharacteristic
364C2#2144694611yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1280C2
socleC23×C61#120
prime radical#21451921C2
hypercenter#1280D61
hypocenterC61#9964
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
1. C2×#16839903C2semidirect
3. C2×#16839497C2semidirect
4. #21444089C2semidirect
5. C61×#1280C2semidirect
24. C22×Dic122D4semidirect
49. C22⋊C8×C61D4semidirect
58. C22×C4×C61SD16semidirect
178. #1280D61semidirect
202. C61#9964semidirect
230. C22⋊C8D244semidirect
269. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168399032
#214519212
C2×#168394972
#214440892
C61×#12802
#996461