Finite Group Databasebeta

← #16847362

Subgroups — #16847362

Order 3968.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#14521613C211yesmaximal, characteristic
2#14521023C211yesmaximal, characteristic, complemented
3#14520837C214yesmaximal, complemented
4C31×#413C211yesmaximal, characteristic, complemented
5C2×#41×C31C2211yescharacteristic
6C22×D124C411yescharacteristic, complemented
7C2×M16×C31—21no—
8#11758524—21no—
9#11758517—21no—
10#11758514C412yescomplemented
11#11758514C2212yes—
12#11758507—21no—
13#11758507—21no—
14#11758503—21no—
15#11758503C411yescharacteristic, complemented
16C31×#101—21no—
17C31×#100C2212yes—
18#11758422—21no—
19#11758422C2212yes—
20C23×D31C2×C412yes—
21#41×C31—41no—
22#41×C31—41no—
23C2×Q8×C31D411yescharacteristic
24C2×D4×C31C2×C411yescharacteristic
25C2×D4×C31D411yescharacteristic
26C2×D4×C31C2311yescharacteristic
27C22×C4×C31C2×C411yescharacteristic
28C22×C4×C31D411yescharacteristic
29C22×C4×C31D411yescharacteristic
30C2×C22⋊D31—22no—
31C2×C22⋊D31—22no—
32C2×D124—21no—
33C2×D124—21no—
34C2×D124—41no—
35C2×D124—41no—
36M16×C31—22no—
37M16×C31—41no—
38C2×C8×C31—41no—
39C22⋊Dic31—22no—
40C22⋊Dic31C2×C412yes—
41(C2×D31)⋊C4—22no—
42(C2×D31)⋊C4—22no—
43C4⋊Dic31—21no—
44C4⋊Dic31—41no—
45C4×Dic31—21no—
46C4×Dic31—41no—
47C23×C31C22⋊C411yescharacteristic
48C23×C31C22×C411yescharacteristic
49C23×C31C2×D411yescharacteristic
50C22×D31—22no—
51C22×D31—42no—
52C22×D31—82no—
53Q8×C31—41no—
54D4×C31—41no—
55D4×C31—41no—
56D4×C31—41no—
57C2×C4×C31—21no—
58C2×C4×C31—21no—
59C2×C4×C31—21no—
60C2×C4×C31—21no—
61C2×C4×C31—21no—
62C2×C4×C31—21no—
63C2×C4×C31C22⋊C411yescharacteristic
64C2×C4×C31C22⋊C411yescharacteristic
65C2×C4×C31C22⋊C411yescharacteristic
66C2×C4×C31C2×D411yescharacteristic
67C22⋊D31—82no—
68C22⋊D31—82no—
69C2×Dic31—22no—
70C2×Dic31—42no—
71D124—42no—
72D124—41no—
73D124—41no—
74C8×C31—42no—
75#2122—311nomaximal, Sylow
76C22×C31—21no—
77C22×C31—21no—
78C22×C31—21no—
79C22×C31—41no—
80C22×C31—41no—
81C22×C31—41no—
82C22×C31C2×C22⋊C411yescharacteristic
83C2×D31—42no—
84C2×D31—42no—
85C2×D31—42no—
86C2×D31—84no—
87C4×C31—21no—
88C4×C31—21no—
89C4×C31—21no—
90C4×C31#9911yescharacteristic
91C4×C31#9911yescharacteristic
92Dic31—42no—
93Dic31—82no—
94#535—311no—
95#413D3111yescharacteristic, complemented
96#410—311no—
97AΣL(1,𝔽16)—314no—
98C2×C31—21no—
99C2×C31—41no—
100C2×C31—41no—
101C2×C31—41no—
102C2×C31C2×#9911yescharacteristic
103D31—82no—
104C2×#41C2×D3111yescharacteristic
105C22×D4—311no—
106C2×M16—21no—
107#127—621no—
108#124—621no—
109#121—621no—
110#121—621no—
111#120—312no—
112#120—312no—
113#117—311no—
114#117—621no—
115#101—21no—
116#100C2×D3112yes—
117#99—312no—
118#99—621no—
119C31#212211yescharacteristic, complemented, Sylow
120C24—312no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D3111yescharacteristic
124C2×D4—622no—
125C2×D4—622no—
126C2×D4—621no—
127C2×D4—621no—
128C2×D4—1241no—
129C2×D4—1241no—
130C2×D4C4×D3111yescharacteristic
131C2×D4D12411yescharacteristic
132C2×D4C22×D3111yescharacteristic
133C22×C4C4×D3111yescharacteristic
134C22×C4D12411yescharacteristic
135C22×C4C22⋊D3111yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—621no—
140C4⋊C4—1241no—
141C22⋊C4—312no—
142C22⋊C4—622no—
143C22⋊C4—622no—
144C22⋊C4—622no—
145C42—621no—
146C42—1241no—
147C23—622no—
148C23—1242no—
149C23—2482no—
150C23(C2×D31)⋊C411yescharacteristic
151C23C2×C4×D3111yescharacteristic
152C23C2×D12411yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—1242no—
158D4—1241no—
159D4—1241no—
160D4—2482no—
161D4—2482no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—622no—
169C2×C4—1242no—
170C2×C4(C2×D31)⋊C411yescharacteristic
171C2×C4(C2×D31)⋊C411yescharacteristic
172C2×C4(C2×D31)⋊C411yescharacteristic
173C2×C4C2×C22⋊D3111yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—1242no—
182C22—1242no—
183C22—1242no—
184C22—2484no—
185C22C2×(C2×D31)⋊C411yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—1242no—
190C4—2482no—
191C4#1175839411yescharacteristic
192C4#1175839411yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—2482no—
198C2C2×#1175839411yescharacteristic
subgroupG/H
centerC2C2×#11758394
commutatorC23×C31C22×C4
FrattiniC2×D4C22×D31
FittingC31×#413C2
socleC2×C31C2×#99
prime radical#14521613C2
cosocleC2×D4×C31C23
hypercenter#413D31
hypocenterC31#2122
LCSG ▷ C23×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #14521023C2semidirect
3. #14520837C2semidirect
4. C31×#413C2semidirect
6. C22×D124C4semidirect
10. #11758514C4semidirect
15. #11758503C4semidirect
95. #413D31semidirect
119. C31#2122semidirect

Maximal subgroups

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

subgroupindex
#145216132
#145210232
#145208372
C31×#4132
#212231