Finite Group Databasebeta

← #29577570

Subgroups — #29577570

Order 15616.

508 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 95 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#21456276C211yesmaximal, characteristic
2#21449000C211yesmaximal, characteristic, complemented
3#21444367C212yesmaximal
4#21443685C212yesmaximal, complemented
5C61×#1558C211yesmaximal, characteristic, complemented
6C2×#14519808—22no—
7#16840312C2211yescharacteristic
8#16840154—22no—
9#16840141C412yescomplemented
10#16840136C412yescomplemented
11C4×#11758194C2211yescharacteristic
12#16839669—22no—
13#16839648—24no—
14#16839630C2212yes—
15#16839497—24no—
16C2×C4⋊Q8×C61C2211yescharacteristic
17C61×#432C2212yes—
18C61×#333—22no—
19C2×#11758222—21no—
20#14519808—44no—
21#14519808—44no—
22C2×#11758202—24no—
23C2×#11758201—24no—
24C2×#11758194—21no—
25#14519777C2×C411yescharacteristic
26#14519774C2×C411yescharacteristic
27#14519765—21no—
28#14519751—22no—
29C4×D244—22no—
30C2×C4⋊Dic61—22no—
31C2×C4×Dic61—22no—
32#14519688—21no—
33C4×#92418C2×C412yes—
34#14519684—24no—
35#14519684—22no—
36#14519684—22no—
37#14519684D414yes—
38C22×Q8×C61D412yes—
39C4⋊Q8×C61—22no—
40C4⋊Q8×C61C2×C412yes—
41C4⋊Q8×C61D412yes—
42C2×C4⋊C4×C61—22no—
43C2×C42×C61C2311yescharacteristic
44C4⋊C8×C61—22no—
45C22⋊C8×C61—22no—
46#97×C61—22no—
47C2×Q8×C61—22no—
48C2×Q8×C61—42no—
49C2×Q8×C61—42no—
50C2×Q8×C61C22⋊C414yes—
51C22×C4×C61—22no—
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61C2×D412yes—
54C2×C22⋊D61—41no—
55C22×Dic61—24no—
56#11758222—42no—
57#11758222—42no—
58C2×D244—21no—
59C2×D244—22no—
60C2×C4×D61—41no—
61C2×Dic122—21no—
62C2×C8×C61—42no—
63C4⋊C4×C61—24no—
64C4⋊C4×C61—24no—
65C42×C61C22×C411yescharacteristic
66C42×C61C2×D411yescharacteristic
67C42×C61#4112yes—
68#11758202—48no—
69#11758201—48no—
70#11758198—42no—
71C4⋊Dic61—22no—
72C4⋊Dic61—82no—
73C4×Dic61—82no—
74#11758194—24no—
75C2×#92418—22no—
76C23×C61#12011yescharacteristic
77C22×D61—41no—
78Q8×C61—44no—
79Q8×C61—82no—
80C2×C4×C61—22no—
81C2×C4×C61—24no—
82C2×C4×C61—42no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61C2×C22⋊C411yescharacteristic
86C2×C4×C61C4×D412yes—
87C2×C4×C61#12112yes—
88C2×C4×C61#12311yescharacteristic
89C22⋊D61—82no—
90C2×Dic61—48no—
91C2×Dic61—41no—
92C2×Dic61—44no—
93D244—42no—
94D244—82no—
95C4×D61—81no—
96C4×D61—81no—
97Dic122—42no—
98C8×C61—82no—
99#92418—24no—
100#9429—611nomaximal, Sylow
101C22×C61—22no—
102C22×C61#38611yescharacteristic
103C22×C61#45312yes—
104C2×D61—81no—
105C2×D61—82no—
106C4×C61—21no—
107C4×C61—21no—
108C4×C61—22no—
109C4×C61—22no—
110C4×C61—44no—
111Dic61—84no—
112Dic61—81no—
113#3152—611no—
114#2381—611no—
115#1558D6111yescharacteristic, complemented
116#1514—612no—
117#1288—612no—
118C2×C61—22no—
119C2×C61#188711yescharacteristic
120C2×C61#190512yes—
121D61—161no—
122C2×#137—1222no—
123#532—611no—
124C2×C4⋊Q8C2×D6111yescharacteristic
125#497—1222no—
126#494—612no—
127#492—612no—
128#432C2×D6112yes—
129C4×M16—611no—
130#391—1222no—
131#387—1224no—
132#383—612no—
133#333—22no—
134#327—1224no—
135C61#942911yescharacteristic, complemented, Sylow
136C2×#41—1221no—
137C22×Q8D24412yes—
138#137—2444no—
139#137—2444no—
140C2×Q16—1224no—
141C2×SD16—1224no—
142C2×M16—1221no—
143C4⋊Q8—22no—
144C4⋊Q8—611no—
145C4⋊Q8C4×D6112yes—
146C4⋊Q8C22⋊D6112yes—
147#127—611no—
148#124—1221no—
149#121—1222no—
150C4×D4—1222no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—1222no—
153C2×C42—1222no—
154C2×C42C22×D6111yescharacteristic
155C4⋊C8—22no—
156C22⋊C8—22no—
157#97—22no—
158#97—1221no—
159C4×C8—612no—
160#95—614no—
161#95—1224no—
162#95—1222no—
163#95—1222no—
164#41—2442no—
165#41—2442no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—1221no—
170C2×Q8#1175819814yes—
171C2×D4—1221no—
172C2×D4—1222no—
173C2×D4—2441no—
174C22×C4—22no—
175C22×C4—1224no—
176C22×C4—2441no—
177C22×C4C2×D24411yescharacteristic
178C22×C4D4×D6112yes—
179Q16—2448no—
180SD16—2448no—
181M16—1224no—
182C2×C8—42no—
183C2×C8—1222no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—1222no—
187C4⋊C4—4882no—
188C22⋊C4—2442no—
189C42—4882no—
190C42C2×C4×D6111yescharacteristic
191C42#1175822412yes—
192C42C2×C22⋊D6111yescharacteristic
193C23—2441no—
194C23#1451974711yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—2442no—
198D4—4882no—
199D4—4882no—
200D4—2442no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—4881no—
207C2×C4—4881no—
208C2×C4—2448no—
209C2×C4—2441no—
210C2×C4—2444no—
211C2×C4C2×#1175819811yescharacteristic
212C2×C4#1451973712yes—
213C2×C4#1451974912yes—
214C2×C4#1451976111yescharacteristic
215C8—82no—
216C8—1224no—
217C22—22no—
218C22—4881no—
219C22—4882no—
220C22#1683963911yescharacteristic
221C22#1683993212yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—4884no—
228C4—4881no—
229C2—22no—
230C2—9761no—
231C2#2144621511yescharacteristic
232C2#2144631112yes—
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1558C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1558D61
hypocenterC61#9429
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #21449000C2semidirect
4. #21443685C2semidirect
5. C61×#1558C2semidirect
9. #16840141C4semidirect
10. #16840136C4semidirect
115. #1558D61semidirect
135. C61#9429semidirect

Maximal subgroups

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

subgroupindex
#214562762
#214490002
#214443672
#214436852
C61×#15582
#942961