Finite Group Databasebeta

← #29577740

Subgroups — #29577740

Order 15616.

528 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450293C211yesmaximal, characteristic
2#21448422C211yesmaximal, characteristic
3#21444353C212yesmaximal
4#21443673C212yesmaximal
5C61×#1821C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7#16839693—22no—
8#16839667C412yescomplemented
9#16839648—24no—
10#16839635—21no—
11#16839635—21no—
12#16839631C412yescomplemented
13#16839625C2212yes—
14#16839622—21no—
15C4×C22⋊Dic61—21no—
16C4×C22⋊Dic61C2211yescharacteristic
17#16839500—24no—
18C22×C4⋊C4×C61C2211yescharacteristic
19C61×#407C2212yes—
20C61×#343—22no—
21C22×Dic122C2×C411yescharacteristic
22#14519756—22no—
23#14519756—22no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×#11758196—21no—
27C2×#11758196—22no—
28C2×#11758196—22no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C2×C22⋊Dic61—22no—
32C2×C4×Dic61—22no—
33C2×C4×Dic61C2×C412yes—
34#14519684—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684—21no—
39#14519684—22no—
40#14519684D414yes—
41C23×C4×C61D411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C2×M16×C61—22no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61—22no—
47C2×C4⋊C4×C61C2×C412yes—
48C2×C4⋊C4×C61D412yes—
49C22⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C24×C61C2×D411yescharacteristic
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—81no—
58C22×C4×C61—81no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D412yes—
64C22×C4×C61#4112yes—
65C22×Dic61—22no—
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—24no—
69C2×Dic122—22no—
70C2×Dic122—42no—
71M16×C61—44no—
72C2×C8×C61—42no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—24no—
75C22⋊Dic61—24no—
76C22⋊Dic61—44no—
77C22⋊Dic61—41no—
78C22⋊Dic61—41no—
79C4⋊Dic61—22no—
80C4⋊Dic61—82no—
81#11758196—42no—
82#11758196—82no—
83#11758196—82no—
84C4×Dic61—42no—
85C4×Dic61—82no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—41no—
89C23×C61C2×C22⋊C411yescharacteristic
90C23×C61#12011yescharacteristic
91C23×C61#12311yescharacteristic
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—81no—
103C2×C4×C61—81no—
104C2×C4×C61C4×D412yes—
105C2×C4×C61#12112yes—
106C2×Dic61—24no—
107C2×Dic61—42no—
108C2×Dic61—42no—
109C2×Dic61—44no—
110C2×Dic61—42no—
111C2×Dic61—44no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—42no—
115C2×Dic61—44no—
116Dic122—82no—
117C8×C61—82no—
118#9461—611nomaximal, Sylow
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—44no—
127C22×C61#38611yescharacteristic
128C22×C61#41012yes—
129C22×C61#45612yes—
130C4×C61—22no—
131C4×C61—22no—
132C4×C61—42no—
133C4×C61—42no—
134Dic61—44no—
135Dic61—84no—
136Dic61—81no—
137Dic61—81no—
138#2502—611no—
139#2313—611no—
140#1821D6111yescharacteristic
141#1511—612no—
142#1284—612no—
143C2×C61—22no—
144C2×C61—22no—
145C2×C61—41no—
146C2×C61#188611yescharacteristic
147C2×C61#189911yescharacteristic
148C2×C61#191111yescharacteristic
149C2×C22⋊Q8—611no—
150C22×C4⋊C4C2×D6111yescharacteristic
151#407C2×D6112yes—
152#398—1222no—
153#391—612no—
154#387—1224no—
155#385—1221no—
156#385—1221no—
157#384—612no—
158#381—612no—
159#380—1221no—
160C4×C22⋊C4—611no—
161C4×C22⋊C4—1221no—
162#343—22no—
163#328—1224no—
164C61#946111yescharacteristic, complemented, Sylow
165C22×Q8—611no—
166C23×C4D24411yescharacteristic
167C23×C4D24411yescharacteristic
168C23×C4C22×D6111yescharacteristic
169C2×M16—22no—
170C22⋊Q8—1222no—
171C22⋊Q8—1222no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—611no—
175C2×C4⋊C4—1222no—
176C2×C4⋊C4—1222no—
177C2×C4⋊C4—1222no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4C4×D6112yes—
180C2×C4⋊C4C22⋊D6112yes—
181C2×C22⋊C4—1221no—
182C2×C22⋊C4—1221no—
183C2×C22⋊C4—1222no—
184C2×C42—612no—
185C2×C42—1222no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188#95—614no—
189#95—1222no—
190#95—1222no—
191#95—1222no—
192#95—1222no—
193#95—1222no—
194#95—1221no—
195C24C2×D24411yescharacteristic
196C2×Q8—1222no—
197C2×Q8—2442no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—1222no—
206C22×C4—1221no—
207C22×C4—1221no—
208C22×C4—1224no—
209C22×C4#1175819812yes—
210C22×C4#1175819812yes—
211C22×C4C2×C4×D6111yescharacteristic
212C22×C4D4×D6112yes—
213C22×C4#1175822412yes—
214C22×C4C2×C22⋊D6111yescharacteristic
215M16—44no—
216C2×C8—42no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—1222no—
220C4⋊C4—4882no—
221C4⋊C4—4882no—
222C4⋊C4—4882no—
223C4⋊C4—2442no—
224C22⋊C4—1224no—
225C22⋊C4—2444no—
226C22⋊C4—2441no—
227C22⋊C4—2441no—
228C42—4882no—
229C42—2442no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×#1175819811yescharacteristic
234C23#1451974711yescharacteristic
235C23#1451976111yescharacteristic
236Q8—4882no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—1224no—
250C2×C4—2444no—
251C2×C4—2444no—
252C2×C4—2442no—
253C2×C4—2444no—
254C2×C4—2442no—
255C2×C4—2441no—
256C2×C4—2441no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4#1451973712yes—
260C2×C4#1451974912yes—
261C8—82no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1683963911yescharacteristic
271C22#1683973512yes—
272C22#1683994712yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—4884no—
278C4—4881no—
279C4—4881no—
280C4—2444no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2144621011yescharacteristic
285C2#2144627511yescharacteristic
286C2#2144634911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1821C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1821D61
hypocenterC61#9461
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
8. #16839667C4semidirect
12. #16839631C4semidirect
164. C61#9461semidirect

Maximal subgroups

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

subgroupindex
#214502932
#214484222
#214443532
#214436732
C61×#18212
#946161