Finite Group Databasebeta

← #16847361

Subgroups — #16847361

Order 3968.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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#14521033C211yesmaximal, characteristic, complemented
2#14521023C211yesmaximal, characteristic, complemented
3#14520836C214yesmaximal, complemented
4C31×#535C211yesmaximal, characteristic
5C2×#41×C31C2211yescharacteristic
6C22×D4×C31C411yescharacteristic, complemented
7C31×#127—21no—
8C31×#124—21no—
9C31×#121—21no—
10C31×#121—21no—
11C31×#120C412yescomplemented
12C31×#120C2212yes—
13C31×#117C411yescharacteristic, complemented
14#11758503—21no—
15C2×#92474—21no—
16#11758424—21no—
17#11758422—21no—
18#11758422C2212yes—
19#11758421C2212yes—
20C24×C31C2×C412yes—
21#41×C31—41no—
22#41×C31—41no—
23C2×Q8×C31D411yescharacteristic
24C2×D4×C31—22no—
25C2×D4×C31—22no—
26C2×D4×C31—21no—
27C2×D4×C31—21no—
28C2×D4×C31—41no—
29C2×D4×C31—41no—
30C2×D4×C31C2×C411yescharacteristic
31C2×D4×C31D411yescharacteristic
32C2×D4×C31C2311yescharacteristic
33C22×C4×C31C2×C411yescharacteristic
34C22×C4×C31D411yescharacteristic
35C22×C4×C31D411yescharacteristic
36C4⋊C4×C31—21no—
37C22⋊C4×C31—22no—
38C22⋊C4×C31—22no—
39C22⋊C4×C31C2×C412yes—
40C42×C31—21no—
41C22⋊Dic31—22no—
42C4⋊Dic31—41no—
43C4×Dic31—41no—
44#92474—22no—
45#92474—41no—
46C2×C31⋊C8—41no—
47C23×C31—22no—
48C23×C31—42no—
49C23×C31—82no—
50C23×C31C22⋊C411yescharacteristic
51C23×C31C22×C411yescharacteristic
52C23×C31C2×D411yescharacteristic
53Q8×C31—41no—
54D4×C31—41no—
55D4×C31—41no—
56D4×C31—41no—
57D4×C31—42no—
58D4×C31—41no—
59D4×C31—41no—
60D4×C31—82no—
61D4×C31—82no—
62C2×C4×C31—21no—
63C2×C4×C31—21no—
64C2×C4×C31—21no—
65C2×C4×C31—21no—
66C2×C4×C31—21no—
67C2×C4×C31—21no—
68C2×C4×C31—22no—
69C2×C4×C31C22⋊C411yescharacteristic
70C2×C4×C31C22⋊C411yescharacteristic
71C2×C4×C31C22⋊C411yescharacteristic
72C2×C4×C31C2×D411yescharacteristic
73C2×Dic31—42no—
74C31⋊C8—42no—
75#2122—311nomaximal, Sylow
76C22×C31—21no—
77C22×C31—21no—
78C22×C31—21no—
79C22×C31—41no—
80C22×C31—41no—
81C22×C31—41no—
82C22×C31—42no—
83C22×C31—42no—
84C22×C31—42no—
85C22×C31—84no—
86C22×C31C2×C22⋊C411yescharacteristic
87C4×C31—21no—
88C4×C31—21no—
89C4×C31—21no—
90C4×C31—42no—
91C4×C31#9911yescharacteristic
92C4×C31#9911yescharacteristic
93Dic31—82no—
94#535D3111yescharacteristic
95#413—311no—
96#410—311no—
97AΣL(1,𝔽16)—314no—
98C2×C31—21no—
99C2×C31—41no—
100C2×C31—41no—
101C2×C31—41no—
102C2×C31—82no—
103C2×C31C2×#9911yescharacteristic
104C2×#41C2×D3111yescharacteristic
105C22×D4Dic3111yescharacteristic, complemented
106C2×M16—621no—
107#127—21no—
108#124—21no—
109#121—21no—
110#121—21no—
111#120Dic3112yescomplemented
112#120C2×D3112yes—
113#117—621no—
114#117Dic3111yescharacteristic, complemented
115#101—621no—
116#100—312no—
117#99—312no—
118#99—621no—
119C31#212211yescharacteristic, complemented, Sylow
120C24C2×Dic3112yes—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D3111yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic3111yescharacteristic
131C2×D4C22⋊D3111yescharacteristic
132C2×D4C22×D3111yescharacteristic
133C22×C4C2×Dic3111yescharacteristic
134C22×C4C22⋊D3111yescharacteristic
135C22×C4C22⋊D3111yescharacteristic
136M16—622no—
137M16—1241no—
138C2×C8—1241no—
139C4⋊C4—21no—
140C4⋊C4—1241no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—622no—
144C22⋊C4C2×Dic3112yes—
145C42—21no—
146C42—1241no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23C22⋊Dic3111yescharacteristic
151C23C22×Dic3111yescharacteristic
152C23C2×C22⋊D3111yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—82no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—1242no—
170C2×C4C22⋊Dic3111yescharacteristic
171C2×C4C22⋊Dic3111yescharacteristic
172C2×C4C22⋊Dic3111yescharacteristic
173C2×C4C2×C22⋊D3111yescharacteristic
174C8—1242no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊Dic3111yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—2482no—
191C4#1175842211yescharacteristic
192C4#1175842211yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#1175842211yescharacteristic
subgroupG/H
centerC2C2×#11758422
commutatorC23×C31C22×C4
FrattiniC2×D4C22×D31
Fitting, prime radicalC31×#535C2
socleC2×C31C2×#99
cosocleC2×D4×C31C23
hypercenter#535D31
hypocenterC31#2122
LCSG ▷ C23×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #14521033C2semidirect
2. #14521023C2semidirect
3. #14520836C2semidirect
6. C22×D4×C31C4semidirect
11. C31×#120C4semidirect
13. C31×#117C4semidirect
105. C22×D4Dic31semidirect
111. #120Dic31semidirect
114. #117Dic31semidirect
119. C31#2122semidirect

Maximal subgroups

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

subgroupindex
#145210332
#145210232
#145208362
C31×#5352
#212231