Finite Group Databasebeta

← #29577627

Subgroups — #29577627

Order 15616.

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

normal subgroups only

This group has 452 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#21456195C211yesmaximal, characteristic
2#21444557C211yesmaximal, characteristic, complemented
3#21444486C212yesmaximal
4#21443710C212yesmaximal, complemented
5C61×#2986C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7#16840128C414yes—
8#16840110—22no—
9#16839786C2212yes—
10#16839766—22no—
11C4×#11758194C2211yescharacteristic
12#16839508—22no—
13#16839505—22no—
14#16839505—22no—
15C61×#533C2211yescharacteristic
16C61×#446—22no—
17C8×Q8×C61—22no—
18C61×#433—22no—
19C61×#431C2212yes—
20C22×D244—21no—
21C22×D244—22no—
22C2×#11758194—21no—
23C2×#11758194—22no—
24#14519774—22no—
25#14519774—21no—
26#14519774C2×C412yes—
27C4⋊#92418—22no—
28C4⋊#92418—22no—
29#14519697—28no—
30#14519688C2×C412yes—
31C4×#92418—21no—
32C4⋊Q8×C61D412yes—
33#125×C61C2×C412yes—
34C22⋊Q8×C61—22no—
35C4×Q8×C61—22no—
36#117×C61D412yes—
37C2×C42×C61C2311yescharacteristic
38C4⋊C8×C61—24no—
39C4⋊C8×C61D412yes—
40C4⋊C8×C61D412yes—
41C22⋊C8×C61—22no—
42#97×C61—22no—
43C4×C8×C61—22no—
44C4×C8×C61—22no—
45C23×D61—41no—
46C2×Q8×C61—22no—
47C22×C4×C61C2×D411yescharacteristic
48C22×C4×C61#4112yes—
49C2×D244—22no—
50C2×D244—24no—
51C2×D244—44no—
52C2×D244—41no—
53C2×D244—42no—
54C2×D244—82no—
55C2×D244—82no—
56C2×C8×C61—22no—
57C2×C8×C61—22no—
58C4⋊C4×C61—24no—
59C4⋊C4×C61—22no—
60C4⋊C4×C61C22⋊C414yes—
61C22⋊C4×C61—22no—
62C42×C61—22no—
63C42×C61C22×C411yescharacteristic
64C42×C61C2×D411yescharacteristic
65C42×C61C2×D412yes—
66#11758194—42no—
67#11758194—82no—
68C2×#92418—22no—
69C2×#92418—42no—
70C23×C61#12311yescharacteristic
71C22×D61—42no—
72C22×D61—82no—
73C22×D61—84no—
74Q8×C61—42no—
75Q8×C61—42no—
76C2×C4×C61—22no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—24no—
80C2×C4×C61C2×C22⋊C411yescharacteristic
81C2×C4×C61C4×D412yes—
82C2×C4×C61#12011yescharacteristic
83C2×C4×C61#12112yes—
84D244—44no—
85D244—84no—
86D244—82no—
87D244—82no—
88D244—82no—
89C8×C61—42no—
90C8×C61—42no—
91#92418—42no—
92#92418—82no—
93#9438—611nomaximal, Sylow
94C22×C61—21no—
95C22×C61—22no—
96C22×C61#38611yescharacteristic
97C2×D61—82no—
98C2×D61—84no—
99C2×D61—164no—
100C4×C61—21no—
101C4×C61—22no—
102C4×C61—44no—
103C4×C61#45412yes—
104C4×C61#47112yes—
105C4×C61#47212yes—
106#3138—611no—
107#2986D6111yescharacteristic, complemented
108#1561—611no—
109#1545—612no—
110#1299—612no—
111C2×C61—41no—
112C2×C61#188811yescharacteristic
113C2×C61#190712yes—
114D61—162no—
115#533C2×D6111yescharacteristic
116C2×#127—611no—
117#490—614no—
118#486—1222no—
119#446—22no—
120C8×Q8—22no—
121#433—22no—
122#431C2×D6112yes—
123#423—612no—
124#418—1222no—
125C4×M16—611no—
126#332—1222no—
127#331—1222no—
128#331—1222no—
129C61#943811yescharacteristic, complemented, Sylow
130C22×D4—1222no—
131C22×D4—1221no—
132C2×M16—1221no—
133C2×M16—1222no—
134C4⋊Q8D24412yes—
135#127—612no—
136#127—1221no—
137#127—1222no—
138#125C4×D6112yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D6112yes—
142C2×C42C22×D6111yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—1222no—
145C4⋊C8—1222no—
146C4⋊C8C22⋊D6112yes—
147C4⋊C8C22⋊D6112yes—
148#102—1228no—
149C22⋊C8—22no—
150#97—22no—
151#97—612no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—1221no—
155C24—2441no—
156C2×Q8—22no—
157C2×D4—1222no—
158C2×D4—1224no—
159C2×D4—4882no—
160C2×D4—4882no—
161C2×D4—2444no—
162C2×D4—2441no—
163C2×D4—2442no—
164C22×C4#1175822212yes—
165C22×C4C2×C22⋊D6111yescharacteristic
166M16—4882no—
167M16—2442no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—1222no—
171C2×C8—2442no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4#1175819814yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D6111yescharacteristic
178C42C2×D24411yescharacteristic
179C42C2×C22⋊D6112yes—
180C23—4884no—
181C23—4882no—
182C23—2442no—
183C23#1451976011yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—4884no—
187D4—4882no—
188D4—4882no—
189D4—4882no—
190D4—2444no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×#1175819811yescharacteristic
196C2×C4C4×C22⋊D6112yes—
197C2×C4#1451974611yescharacteristic
198C2×C4#1451975112yes—
199C8—42no—
200C8—42no—
201C8—4882no—
202C8—2442no—
203C22—21no—
204C22—22no—
205C22—9764no—
206C22—4882no—
207C22—4884no—
208C22#1683964111yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1683993512yes—
213C4#1684002512yes—
214C4#1684003212yes—
215C2—41no—
216C2—9762no—
217C2#2144622211yescharacteristic
218C2#2144632712yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2986C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#2986D61
hypocenterC61#9438
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #21444557C2semidirect
4. #21443710C2semidirect
5. C61×#2986C2semidirect
107. #2986D61semidirect
129. C61#9438semidirect

Maximal subgroups

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

subgroupindex
#214561952
#214445572
#214444862
#214437102
C61×#29862
#943861