Finite Group Databasebeta

← #29577555

Subgroups — #29577555

Order 15616.

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

normal subgroups only

This group has 422 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#21456199C211yesmaximal, characteristic
2#21444545C211yesmaximal, characteristic
3#21444369C212yesmaximal
4#21443683C212yesmaximal
5C61×#2381C211yesmaximal, characteristic
6#16840319C2211yescharacteristic
7#16840130—22no—
8#16840112C414yescomplemented
9#16839819C2212yes—
10#16839771—22no—
11C4×#11758194C2211yescharacteristic
12#16839511—22no—
13#16839498—24no—
14C2×C4⋊Q8×C61C2211yescharacteristic
15C61×#391—22no—
16C61×#387—24no—
17C61×#383C2212yes—
18C2×#11758194—21no—
19#14519777C2×C412yes—
20#14519769—21no—
21#14519756—22no—
22C4×Dic122—22no—
23#14519736—21no—
24C4⋊#92418—22no—
25#14519701—28no—
26C22⋊#92418—22no—
27#14519688—21no—
28#14519688—22no—
29C4×#92418C2×C412yes—
30C22×Q8×C61D412yes—
31C4⋊Q8×C61—22no—
32C4⋊Q8×C61C2×C412yes—
33C4⋊Q8×C61D412yes—
34C2×C4⋊C4×C61—22no—
35C2×C4⋊C4×C61—22no—
36C2×C42×C61—22no—
37C2×C42×C61C2311yescharacteristic
38#95×C61—24no—
39#95×C61—22no—
40#95×C61—22no—
41#95×C61D414yes—
42C2×Q8×C61—22no—
43C2×Q8×C61—42no—
44C2×Q8×C61—42no—
45C2×Q8×C61C22⋊C414yes—
46C22×C4×C61—22no—
47C22×C4×C61—24no—
48C22×C4×C61C2×D411yescharacteristic
49C22×C4×C61C2×D412yes—
50C2×Dic122—22no—
51C4⋊C4×C61—24no—
52C4⋊C4×C61—24no—
53C4⋊C4×C61—82no—
54C42×C61—82no—
55C42×C61C22×C411yescharacteristic
56C42×C61C2×D411yescharacteristic
57C42×C61#4112yes—
58C22⋊Dic61—41no—
59C4⋊Dic61—22no—
60C4⋊Dic61—22no—
61#11758196—42no—
62C4×Dic61—41no—
63#11758194—42no—
64C2×#92418—22no—
65C2×#92418—42no—
66C23×C61#12011yescharacteristic
67Q8×C61—44no—
68Q8×C61—82no—
69C2×C4×C61—22no—
70C2×C4×C61—24no—
71C2×C4×C61—42no—
72C2×C4×C61—42no—
73C2×C4×C61—42no—
74C2×C4×C61—48no—
75C2×C4×C61—44no—
76C2×C4×C61C2×C22⋊C411yescharacteristic
77C2×C4×C61C4×D412yes—
78C2×C4×C61#12112yes—
79C2×C4×C61#12311yescharacteristic
80C2×Dic61—42no—
81Dic122—82no—
82#92418—42no—
83#92418—82no—
84#9426—611nomaximal, Sylow
85C22×C61—22no—
86C22×C61#38611yescharacteristic
87C22×C61#45512yes—
88C4×C61—21no—
89C4×C61—21no—
90C4×C61—22no—
91C4×C61—22no—
92C4×C61—44no—
93C4×C61—84no—
94Dic61—82no—
95#3139—611no—
96#2381D6111yescharacteristic
97#1558—611no—
98#1515—612no—
99#1288—612no—
100C2×C61—22no—
101C2×C61#188511yescharacteristic
102C2×C61#190512yes—
103#533—611no—
104C2×C4⋊Q8C2×D6111yescharacteristic
105#491—1222no—
106#487—614no—
107#432—612no—
108#419—1222no—
109C4×M16—611no—
110#391—22no—
111#387—24no—
112#383C2×D6112yes—
113#333—1222no—
114#328—1224no—
115C61#942611yescharacteristic, complemented, Sylow
116C22×Q8C22⋊D6112yes—
117C2×M16—1221no—
118C4⋊Q8—22no—
119C4⋊Q8—612no—
120C4⋊Q8C4×D6112yes—
121C4⋊Q8D24412yes—
122#125—1221no—
123C22⋊Q8—1222no—
124C4×Q8—1222no—
125#117—1221no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C42—22no—
129C2×C42C22×D6111yescharacteristic
130C4⋊C8—1222no—
131#103—1228no—
132C22⋊C8—1222no—
133#97—1221no—
134#97—1222no—
135C4×C8—612no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95C22⋊D6114yes—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—1222no—
144C2×Q8#1175819814yes—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4C2×C22⋊D6111yescharacteristic
148C22×C4C2×C22⋊D6112yes—
149M16—2442no—
150C2×C8—1222no—
151C2×C8—2442no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—82no—
155C4⋊C4—1222no—
156C4⋊C4—1222no—
157C4⋊C4—2442no—
158C22⋊C4—2441no—
159C42—82no—
160C42—2441no—
161C42C2×C4×D6111yescharacteristic
162C42C2×D24411yescharacteristic
163C42#1175822212yes—
164C23#1451974611yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—4882no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—44no—
175C2×C4—2442no—
176C2×C4C2×#1175819811yescharacteristic
177C2×C4C4×C22⋊D6112yes—
178C2×C4#1451975112yes—
179C2×C4#1451976011yescharacteristic
180C8—4882no—
181C8—2442no—
182C22—22no—
183C22#1683964111yescharacteristic
184C22#1683994012yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—84no—
191C4—4882no—
192C2—22no—
193C2#2144620711yescharacteristic
194C2#2144631312yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2381C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#2381D61
hypocenterC61#9426
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
8. #16840112C4semidirect
115. C61#9426semidirect

Maximal subgroups

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

subgroupindex
#214561992
#214445452
#214443692
#214436832
C61×#23812
#942661