Finite Group Databasebeta

← #29577625

Subgroups — #29577625

Order 15616.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454921C211yesmaximal, characteristic, complemented
3#21444488C212yesmaximal
4#21443708C212yesmaximal, complemented
5C61×#1561C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7#16840128C414yes—
8#16840110—22no—
9#16839893—22no—
10Q8×#92418—22no—
11#16839823—22no—
12#16839815C2212yes—
13#16839766—22no—
14C4×#11758194C2211yescharacteristic
15#16839505—22no—
16#16839505—22no—
17C61×#533C2211yescharacteristic
18C61×#423C2212yes—
19C61×#332—22no—
20C22×D244—21no—
21C22×D244—22no—
22C2×#11758194—21no—
23#14519774—22no—
24#14519774—21no—
25#14519774C2×C412yes—
26C4⋊#92418—24no—
27C4⋊#92418D412yes—
28C4⋊#92418D412yes—
29#14519697—28no—
30C22⋊#92418—22no—
31#14519688—22no—
32#14519688C2×C412yes—
33C4×#92418—22no—
34C4×#92418—21no—
35C4×#92418—22no—
36C2×M16×C61—22no—
37C4⋊Q8×C61D412yes—
38#125×C61C2×C412yes—
39C22⋊Q8×C61—22no—
40C4×Q8×C61—22no—
41#117×C61D412yes—
42C2×C42×C61C2311yescharacteristic
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—
56M16×C61—82no—
57C2×C8×C61—42no—
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—
67C2×#92418—22no—
68C2×#92418—22no—
69C2×#92418—22no—
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—82no—
90#92418—42no—
91#92418—42no—
92#92418—42no—
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#2986—611no—
108#1561D6111yescharacteristic, complemented
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—1222no—
120C8×Q8—1222no—
121#433—1222no—
122#431—612no—
123#423C2×D6112yes—
124#418—1222no—
125C4×M16—611no—
126#332—22no—
127#331—1222no—
128#331—1222no—
129C61#943811yescharacteristic, complemented, Sylow
130C22×D4—1221no—
131C22×D4—1222no—
132C2×M16—22no—
133C2×M16—1221no—
134C4⋊Q8C22⋊D6112yes—
135#127—612no—
136#127—1222no—
137#127—1221no—
138#125C4×D6112yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117D24412yes—
142C2×C42C22×D6111yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—612no—
146C4⋊C8—612no—
147C4⋊C8—1224no—
148#102—1228no—
149C22⋊C8—1222no—
150#97—612no—
151#97—1222no—
152C4×C8—1222no—
153C4×C8—1222no—
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×C4C2×D24411yescharacteristic
165C22×C4#1175822412yes—
166M16—82no—
167M16—2442no—
168C2×C8—42no—
169C2×C8—1222no—
170C2×C8—1222no—
171C2×C8—1222no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4#1175819814yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D6111yescharacteristic
178C42D4×D6112yes—
179C42C2×C22⋊D6111yescharacteristic
180C23—4884no—
181C23—4882no—
182C23—2442no—
183C23#1451976111yescharacteristic
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×C4#1451973712yes—
197C2×C4#1451974711yescharacteristic
198C2×C4#1451974912yes—
199C8—82no—
200C8—2442no—
201C8—2442no—
202C8—2442no—
203C22—21no—
204C22—22no—
205C22—9764no—
206C22—4882no—
207C22—4884no—
208C22#1683963911yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1683993712yes—
213C4#1684002712yes—
214C4#1684003012yes—
215C2—41no—
216C2—9762no—
217C2#2144622011yescharacteristic
218C2#2144632512yes—
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1561C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1561D61
hypocenterC61#9438
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #21454921C2semidirect
4. #21443708C2semidirect
5. C61×#1561C2semidirect
108. #1561D61semidirect
129. C61#9438semidirect

Maximal subgroups

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

subgroupindex
#214561952
#214549212
#214444882
#214437082
C61×#15612
#943861