Finite Group Databasebeta

← #29574367

Subgroups — #29574367

Order 15616.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 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#21448993C211yesmaximal, characteristic
2#21445985C212yesmaximal, complemented
3C2×#16839502C211yesmaximal, characteristic
4#21443701C212yesmaximal, complemented
5C61×#1959C211yesmaximal, characteristic
6C2×#14519710—22no—
7C2×#14519708—22no—
8C2×#14519688C2212yes—
9#16839669—21no—
10#16839655—22no—
11#16839642—21no—
12#16839637—22no—
13#16839628C2212yes—
14#16839502C414yescomplemented
15C2×#124×C61C2211yescharacteristic
16C2×#103×C61—21no—
17C2×#102×C61—21no—
18C2×#97×C61C2211yescharacteristic
19C61×#384C2211yescharacteristic
20C23×Dic61—21no—
21C22×#92418—22no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61D412yes—
24C2×C4⋊Dic61D412yes—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—21no—
27C2×C4×Dic61—21no—
28#14519710—44no—
29#14519708—44no—
30#14519688—22no—
31#14519688C2×C414yescomplemented
32#14519684—22no—
33#14519684—22no—
34#14519684—22no—
35C22×Q8×C61D411yescharacteristic
36C22×D4×C61D411yescharacteristic
37C22×C8×C61—21no—
38#124×C61—22no—
39#124×C61C2×C412yes—
40#124×C61Q812yes—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61—21no—
43C2×C22⋊C4×C61—22no—
44C2×C42×C61C2311yescharacteristic
45#103×C61—42no—
46#102×C61—42no—
47#97×C61—22no—
48C24×C61—21no—
49C2×Q8×C61—21no—
50C2×Q8×C61—41no—
51C2×Q8×C61—41no—
52C2×Q8×C61C4⋊C412yes—
53C2×D4×C61—21no—
54C2×D4×C61—41no—
55C2×D4×C61—41no—
56C2×D4×C61C4⋊C412yes—
57C22×C4×C61—21no—
58C22×C4×C61—41no—
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D412yes—
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—82no—
64C22×Dic61—81no—
65C2×C8×C61—22no—
66C2×C8×C61—41no—
67C2×C8×C61—41no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61—41no—
70C4⋊C4×C61—42no—
71C4⋊C4×C61—42no—
72C22⋊C4×C61—24no—
73C22⋊C4×C61—24no—
74C42×C61C22×C411yescharacteristic
75C42×C61C2×Q811yescharacteristic
76C42×C61#4112yes—
77C22⋊Dic61—82no—
78C22⋊Dic61—81no—
79C4⋊Dic61—42no—
80C4⋊Dic61—42no—
81C4⋊Dic61—81no—
82C4×Dic61—81no—
83C2×#92418—24no—
84C2×#92418—24no—
85C2×#92418—24no—
86C23×C61—22no—
87C23×C61—42no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61—42no—
91C23×C61#12011yescharacteristic
92Q8×C61—42no—
93Q8×C61—81no—
94D4×C61—42no—
95D4×C61—81no—
96C2×C4×C61—22no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—82no—
104C2×C4×C61—81no—
105C2×C4×C61C2×C4⋊C411yescharacteristic
106C2×C4×C61C4×D412yes—
107C2×C4×C61C22⋊Q812yes—
108C2×C4×C61#12311yescharacteristic
109C2×Dic61—42no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—84no—
116C8×C61—42no—
117#92418—28no—
118#8910—611nomaximal, Sylow
119C22×C61—42no—
120C22×C61—44no—
121C22×C61—81no—
122C22×C61—81no—
123C22×C61—82no—
124C22×C61#38511yescharacteristic
125C22×C61#45312yes—
126C22×C61#45311yescharacteristic
127C22×C61#45511yescharacteristic
128C22×C61#45612yes—
129C4×C61—21no—
130C4×C61—21no—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—42no—
134C4×C61—161no—
135Dic61—82no—
136Dic61—82no—
137#2379—611no—
138#1959D6111yescharacteristic
139#1847—612no—
140C2×#329—611no—
141#1295—612no—
142C2×C61—82no—
143C2×C61#165511yescharacteristic
144C2×C61#166011yescharacteristic
145C2×C61#166112yes—
146C2×C61#185611yescharacteristic
147C2×C61#185912yes—
148C2×#124C2×D6111yescharacteristic
149C2×#107—1222no—
150C2×#106—1222no—
151C2×#103—21no—
152C2×#102—21no—
153C2×#97—612no—
154C2×#97C2×D6111yescharacteristic
155#391—1221no—
156#388—1222no—
157#386—1221no—
158#385—1222no—
159#384C2×D6111yescharacteristic
160#382—612no—
161#329—614no—
162C61#891011yescharacteristic, complemented, Sylow
163C22×Q8C22⋊D6111yescharacteristic
164C22×D4C22⋊D6111yescharacteristic
165C23×C4—1221no—
166C22×C8—21no—
167C22×C8—1222no—
168#124—22no—
169#124Dic12212yes—
170#124C4×D6112yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—612no—
174C2×C4⋊C4—612no—
175C2×C4⋊C4—1221no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—1222no—
178C2×C22⋊C4—1221no—
179C2×C42—1221no—
180C2×C42C22×D6111yescharacteristic
181#107—2444no—
182#106—2444no—
183#103—42no—
184#102—42no—
185#97—22no—
186#97—614no—
187#97—1222no—
188#95—1222no—
189#95—1222no—
190#95—1222no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8#1175819612yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4#1175819612yes—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—1222no—
203C22×C4—1222no—
204C22×C4—4882no—
205C22×C4—4881no—
206C22×C4D4×D6112yes—
207C22×C4C2×C22⋊D6111yescharacteristic
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—1224no—
212C2×C8—1224no—
213C2×C8—1224no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—4881no—
219C4⋊C4—2442no—
220C4⋊C4—2442no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—4881no—
224C22⋊C4—4882no—
225C42—4881no—
226C42C2×Dic12211yescharacteristic
227C42C2×C4×D6111yescharacteristic
228C42#1175822412yes—
229C23—22no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23#1451974511yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—81no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—4884no—
249C2×C4—2442no—
250C2×C4—2442no—
251C2×C4—2442no—
252C2×C4—2442no—
253C2×C4—2442no—
254C2×C4—2442no—
255C2×C4C2×#1175819611yescharacteristic
256C2×C4#1451973712yes—
257C2×C4#1451975412yes—
258C2×C4#1451975811yescharacteristic
259C8—42no—
260C8—1228no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—82no—
265C22—81no—
266C22#1683963411yescharacteristic
267C22#1683992812yes—
268C22#1683992911yescharacteristic
269C22#1683993911yescharacteristic
270C22#1683994312yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—161no—
277C4—4882no—
278C4—4882no—
279C2—82no—
280C2#2144510711yescharacteristic
281C2#2144512811yescharacteristic
282C2#2144513512yes—
283C2#2144602811yescharacteristic
284C2#2144604312yes—
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1959C2
socleC23×C61#120
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1959D61
hypocenterC61#8910
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. #21445985C2semidirect
4. #21443701C2semidirect
14. #16839502C4semidirect
31. #14519688C2×C4semidirect
162. C61#8910semidirect

Maximal subgroups

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

subgroupindex
#214489932
#214459852
C2×#168395022
#214437012
C61×#19592
#891061