Finite Group Databasebeta

← #29577982

Subgroups — #29577982

Order 15616.

524 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 107 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#21450136C211yesmaximal, characteristic
2#21445897C212yesmaximal, complemented
3#21444157C211yesmaximal, characteristic
4#21443724C212yesmaximal, complemented
5C61×#2973C211yesmaximal, characteristic
6C22×C4⋊Dic61C2211yescharacteristic
7#16839726C2212yes—
8#16839701—21no—
9#16839698—21no—
10#16839660—21no—
11#16839642C412yes—
12#16839634C412yes—
13#16839622—21no—
14#16839622C2211yescharacteristic
15#16839530—24no—
16#16839488—21no—
17#16839486—21no—
18C2×C4×D4×C61C2211yescharacteristic
19C61×#435—24no—
20C61×#432—21no—
21C61×#407C2212yes—
22C2×C22⋊C8×C61—21no—
23C23×Dic61C2×C412yes—
24C2×#11758194—22no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—22no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—22no—
30C2×#11758196—22no—
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61C2×C412yes—
33C22⋊#92418—22no—
34C22⋊#92418—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—21no—
38C22×D4×C61C2×C411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C2×M16×C61D414yes—
42C22×C8×C61—22no—
43C4×D4×C61—22no—
44C4×D4×C61—22no—
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C22⋊C4×C61C2×C411yescharacteristic
47C2×C22⋊C4×C61D411yescharacteristic
48C2×C42×C61D411yescharacteristic
49C4⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53#97×C61—22no—
54C24×C61C22⋊C411yescharacteristic
55C24×C61C22×C411yescharacteristic
56C2×D4×C61—22no—
57C2×D4×C61—22no—
58C2×D4×C61—41no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C2×D412yes—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61#4112yes—
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—24no—
73C22×Dic61—82no—
74M16×C61—44no—
75C2×C8×C61—24no—
76C2×C8×C61—82no—
77C4⋊C4×C61—22no—
78C22⋊C4×C61—22no—
79C22⋊C4×C61—22no—
80C42×C61—21no—
81C42×C61—21no—
82C22⋊Dic61—42no—
83C22⋊Dic61—82no—
84C4⋊Dic61—42no—
85C4⋊Dic61—41no—
86C4⋊Dic61—42no—
87C4⋊Dic61—41no—
88#11758196—82no—
89#11758194—44no—
90C2×#92418—42no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—41no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61#10112yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61C4×D412yes—
102D4×C61—44no—
103C2×C4×C61—24no—
104C2×C4×C61—22no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—41no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—81no—
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61#12112yes—
118C2×C4×C61#12311yescharacteristic
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—84no—
126C8×C61—44no—
127#92418—82no—
128#9509—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—42no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—81no—
141C22×C61#38611yescharacteristic
142C22×C61C2×#10111yescharacteristic
143C22×C61#41311yescharacteristic
144C4×C61—22no—
145C4×C61—21no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61#45812yes—
150Dic61—82no—
151Dic61—82no—
152#2973D6111yescharacteristic
153#2487—611no—
154#1821—612no—
155#1465—611no—
156#1305—612no—
157C2×C61—21no—
158C2×C61—21no—
159C2×C61—42no—
160C2×C61—42no—
161C2×C61#189011yescharacteristic
162C2×C61#190111yescharacteristic
163C2×C61#191411yescharacteristic
164C2×C4×D4C2×D6111yescharacteristic
165C22×C4⋊C4—611no—
166#435—24no—
167#432—21no—
168#407—612no—
169#407C2×D6112yes—
170C2×C22⋊C8—21no—
171#400—1221no—
172#399—1221no—
173#389—1221no—
174#386—612no—
175#385—612no—
176#380—611no—
177#380—1221no—
178#343—1224no—
179#324—1221no—
180#323—1221no—
181C61#950911yescharacteristic, complemented, Sylow
182C22×D4C4×D6111yescharacteristic
183C23×C4—612no—
184C23×C4C22⋊D6111yescharacteristic
185C23×C4C22×D6111yescharacteristic
186C2×M16—1222no—
187C2×M16C22⋊D6114yes—
188C22×C8—22no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—1222no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—1222no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4D24411yescharacteristic
198C2×C22⋊C4—612no—
199C2×C22⋊C4—1222no—
200C2×C22⋊C4C4×D6111yescharacteristic
201C2×C22⋊C4D24411yescharacteristic
202C2×C42C22⋊D6111yescharacteristic
203C4⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—1222no—
208C22⋊C8—1222no—
209#97—22no—
210#95—1222no—
211#95—1222no—
212#95—1221no—
213C24#1175819811yescharacteristic
214C24C2×C4×D6111yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—1224no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—4882no—
226C22×C4#1175819811yescharacteristic
227C22×C4#1175819811yescharacteristic
228C22×C4#1175819811yescharacteristic
229C22×C4C2×D24411yescharacteristic
230C22×C4#1175822212yes—
231C22×C4C2×C22⋊D6112yes—
232C22×C4C2×C22⋊D6111yescharacteristic
233M16—44no—
234M16—2444no—
235C2×C8—24no—
236C2×C8—82no—
237C2×C8—2442no—
238C4⋊C4—22no—
239C4⋊C4—4882no—
240C4⋊C4—2442no—
241C4⋊C4—2441no—
242C4⋊C4—2442no—
243C4⋊C4—2441no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—4882no—
247C22⋊C4—2442no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1451969512yes—
259C23C2×#1175819811yescharacteristic
260C23C4×C22⋊D6112yes—
261D4—44no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—4884no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4#1451974611yescharacteristic
283C2×C4#1451975112yes—
284C2×C4#1451976011yescharacteristic
285C8—44no—
286C8—4882no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1683964111yescharacteristic
300C22C2×#1451969511yescharacteristic
301C22#1683974411yescharacteristic
302C4—22no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—4882no—
308C4—4882no—
309C4#1683995212yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#2144623211yescharacteristic
315C2#2144628711yescharacteristic
316C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2973C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#2973D61
hypocenterC61#9509
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21445897C2semidirect
4. #21443724C2semidirect
181. C61#9509semidirect

Maximal subgroups

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

subgroupindex
#214501362
#214458972
#214441572
#214437242
C61×#29732
#950961