Finite Group Databasebeta

← #29584211

Subgroups — #29584211

Order 15616.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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#21445956C211yesmaximal, characteristic
2#21444557C212yesmaximal
3#21444274C212yesmaximal
4#21443732C211yesmaximal, characteristic
5C61×#3255C211yesmaximal, characteristic
6#16839786C2211yescharacteristic
7#16839786C2212yes—
8C2×C4×#92418C2211yescharacteristic
9#16839526—22no—
10#16839508—22no—
11#16839508—22no—
12Q8⋊#92418—24no—
13C61×#533C2212yes—
14C61×#475—22no—
15C61×#470—22no—
16C61×#462—22no—
17C61×#423C2211yescharacteristic
18C2×#11758194—21no—
19C2×#11758194—22no—
20C22×#92418—21no—
21C4⋊#92418—22no—
22C4⋊#92418—22no—
23C4⋊#92418—22no—
24C4×#92418—21no—
25C4×#92418D412yes—
26C2×Q16×C61—22no—
27C2×M16×C61D412yes—
28C4⋊Q8×C61D412yes—
29C4⋊Q8×C61D412yes—
30#125×C61—22no—
31C22⋊Q8×C61—22no—
32C22⋊Q8×C61—22no—
33C4×Q8×C61—22no—
34C4×Q8×C61—22no—
35#117×C61—22no—
36C2×C42×C61C2311yescharacteristic
37#106×C61—22no—
38C4⋊C8×C61—22no—
39#103×C61—24no—
40C2×Q8×C61—22no—
41C2×Q8×C61—22no—
42C22×C4×C61C2×D411yescharacteristic
43C22×C4×C61#4112yes—
44Q16×C61—82no—
45M16×C61—42no—
46C2×C8×C61—22no—
47C4⋊C4×C61—22no—
48C4⋊C4×C61—22no—
49C4⋊C4×C61—22no—
50C4⋊C4×C61—22no—
51C4⋊C4×C61—42no—
52C4⋊C4×C61—42no—
53C22⋊C4×C61—42no—
54C42×C61—42no—
55C42×C61C2×D412yes—
56C42×C61C2×D411yescharacteristic
57C42×C61#4111yescharacteristic
58#11758194—42no—
59#11758194—82no—
60C2×#92418—22no—
61C2×#92418—41no—
62C2×#92418—41no—
63C2×#92418—41no—
64C2×#92418—42no—
65C23×C61#12411yescharacteristic
66Q8×C61—42no—
67Q8×C61—42no—
68Q8×C61—42no—
69Q8×C61—42no—
70C2×C4×C61—21no—
71C2×C4×C61—21no—
72C2×C4×C61—22no—
73C2×C4×C61—22no—
74C2×C4×C61—42no—
75C2×C4×C61—42no—
76C2×C4×C61#12011yescharacteristic
77C2×C4×C61#12111yescharacteristic
78C2×C4×C61#12112yes—
79C2×C4×C61#12312yes—
80C8×C61—42no—
81#92418—42no—
82#92418—81no—
83#92418—82no—
84#10535—611nomaximal, Sylow
85C22×C61—22no—
86C22×C61—21no—
87C22×C61#39411yescharacteristic
88C4×C61—22no—
89C4×C61—21no—
90C4×C61—82no—
91C4×C61—82no—
92C4×C61#45411yescharacteristic
93C4×C61#45511yescharacteristic
94C4×C61#47112yes—
95C4×C61#47312yes—
96#3255D6111yescharacteristic
97#1837—611no—
98#1561—612no—
99#1493—612no—
100#1309—611no—
101C2×C61—41no—
102C2×C61#204011yescharacteristic
103C2×C61#205212yes—
104#533C2×D6112yes—
105#475—22no—
106#470—22no—
107#462—22no—
108#423—611no—
109#423—612no—
110#423C2×D6111yescharacteristic
111C2×C4×C8—611no—
112#341—1222no—
113#332—1222no—
114#332—1222no—
115Q8⋊C8—1224no—
116C61#1053511yescharacteristic, complemented, Sylow
117C2×Q16—22no—
118C2×M16—1222no—
119C2×M16—1221no—
120C2×M16C22⋊D6112yes—
121C22×C8—1221no—
122C4⋊Q8C22⋊D6112yes—
123C4⋊Q8C22⋊D6112yes—
124#125—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129#117—22no—
130C2×C42C22×D6111yescharacteristic
131#106—22no—
132C4⋊C8—22no—
133C4⋊C8—1222no—
134C4⋊C8—1222no—
135C4⋊C8—1222no—
136#103—24no—
137C4×C8—612no—
138C4×C8—1221no—
139C2×Q8—22no—
140C2×Q8—22no—
141C22×C4#1175822412yes—
142C22×C4C2×C22⋊D6111yescharacteristic
143Q16—82no—
144M16—42no—
145M16—4882no—
146M16—2442no—
147C2×C8—22no—
148C2×C8—1222no—
149C2×C8—2441no—
150C2×C8—2442no—
151C2×C8—2441no—
152C2×C8—2441no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C42—42no—
161C42D4×D6111yescharacteristic
162C42#1175822411yescharacteristic
163C42C2×C22⋊D6112yes—
164C23#1451976411yescharacteristic
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—42no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#1451974611yescharacteristic
176C2×C4#1451974811yescharacteristic
177C2×C4#1451974912yes—
178C2×C4#1451975812yes—
179C8—42no—
180C8—4882no—
181C8—4881no—
182C8—2442no—
183C22—21no—
184C22—22no—
185C22#1683967611yescharacteristic
186C4—21no—
187C4—22no—
188C4—82no—
189C4—82no—
190C4#1683993511yescharacteristic
191C4#1683994011yescharacteristic
192C4#1684002712yes—
193C4#1684003512yes—
194C2—41no—
195C2#2144704511yescharacteristic
196C2#2144711512yes—
subgroupG/H
centerC22#16839676
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3255C2
socleC22×C61#394
prime radicalC23×C61#124
hypercenter#3255D61
hypocenterC61#10535
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
116. C61#10535semidirect

Maximal subgroups

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

subgroupindex
#214459562
#214445572
#214442742
#214437322
C61×#32552
#1053561