Finite Group Databasebeta

← #16846019

Subgroups — #16846019

Order 3968.

404 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic31C211yesmaximal, characteristic
2C2×#11758423C212yesmaximal, complemented
3C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic
4#14520811C212yesmaximal, complemented
5C2×C31×C22⋊Q8C211yesmaximal, characteristic
6C22×Q8×C31C2211yescharacteristic
7C23×C4×C31C2211yescharacteristic
8C23×Dic31—21no—
9C31×C22⋊Q8—22no—
10C31×C22⋊Q8C414yescomplemented
11C2×C4⋊C4×C31—21no—
12C2×C4⋊C4×C31C2211yescharacteristic
13C2×C22⋊C4×C31—21no—
14C2×C4⋊Dic31—21no—
15C2×C4⋊Dic31—24no—
16C2×C4⋊Dic31C2212yes—
17C22×C31⋊C8C2212yes—
18#11758423—24no—
19C22⋊(C31⋊C8)—22no—
20C24×C31D411yescharacteristic
21C2×Q8×C31—21no—
22C2×Q8×C31—22no—
23C2×Q8×C31—22no—
24C2×Q8×C31C2×C412yescomplemented
25C22×C4×C31—22no—
26C22×C4×C31—22no—
27C22×C4×C31—21no—
28C22×C4×C31—21no—
29C22×C4×C31C2×C412yes—
30C22×C4×C31D411yescharacteristic
31C22×C4×C31D412yes—
32C22×C4×C31C2311yescharacteristic
33C22×Dic31—24no—
34C22×Dic31—22no—
35C22×Dic31—42no—
36C22×Dic31—42no—
37C4⋊C4×C31—24no—
38C4⋊C4×C31—21no—
39C4⋊C4×C31C2×C412yescomplemented
40C22⋊C4×C31—24no—
41C4⋊Dic31—24no—
42C4⋊Dic31—22no—
43C4⋊Dic31D414yes—
44C2×C31⋊C8—22no—
45C2×C31⋊C8—22no—
46C2×C31⋊C8—22no—
47C23×C31—21no—
48C23×C31—21no—
49C23×C31—21no—
50C23×C31—21no—
51C23×C31C22⋊C412yes—
52C23×C31SD1612yes—
53C23×C31Q1612yes—
54C23×C31C2×D411yescharacteristic
55Q8×C31—24no—
56Q8×C31—42no—
57C2×C4×C31—21no—
58C2×C4×C31—21no—
59C2×C4×C31—24no—
60C2×C4×C31—24no—
61C2×C4×C31—22no—
62C2×C4×C31—22no—
63C2×C4×C31—41no—
64C2×C4×C31—41no—
65C2×C4×C31—41no—
66C2×C4×C31—41no—
67C2×C4×C31C22⋊C412yes—
68C2×C4×C31C22×C411yescharacteristic
69C2×C4×C31C2×D412yes—
70C2×C4×C31C2×D411yescharacteristic
71C2×C4×C31#4112yes—
72C2×Dic31—24no—
73C2×Dic31—48no—
74C2×Dic31—42no—
75C2×Dic31—42no—
76C31⋊C8—42no—
77#1872—311nomaximal, Sylow
78C22×C31—22no—
79C22×C31—22no—
80C22×C31—22no—
81C22×C31—22no—
82C22×C31—22no—
83C22×C31—22no—
84C22×C31#10314yescomplemented
85C22×C31C2×C22⋊C411yescharacteristic
86C22×C31#12011yescharacteristic
87C22×C31#12311yescharacteristic
88C22×C31C2×SD1611yescharacteristic
89C22×C31C2×Q1611yescharacteristic
90C22×C31AGL(1,ℤ/8)11yescharacteristic
91C22×C31#13711yescharacteristic
92C4×C31—22no—
93C4×C31—42no—
94C4×C31—42no—
95C4×C31C4×D412yes—
96C4×C31#12112yes—
97Dic31—44no—
98C2×C22⋊Q8D3111yescharacteristic
99C22×C4⋊C4—311no—
100C2×#103—312no—
101C2×C22⋊C8—311no—
102#340—312no—
103C2×C31—24no—
104C2×C31#38611yescharacteristic
105C2×C31C2×#10311yescharacteristic
106C2×C31#41711yescharacteristic
107C2×C31#45011yescharacteristic
108C2×C31#45111yescharacteristic
109C2×C31#48311yescharacteristic
110C2×C31#48411yescharacteristic
111C22×Q8C2×D3111yescharacteristic
112C23×C4—621no—
113C23×C4C2×D3111yescharacteristic
114C22×C8—312no—
115C22⋊Q8—22no—
116C22⋊Q8Dic3114yescomplemented
117C2×C4⋊C4—21no—
118C2×C4⋊C4—312no—
119C2×C4⋊C4—624no—
120C2×C4⋊C4—621no—
121C2×C4⋊C4C2×D3111yescharacteristic
122C2×C22⋊C4—21no—
123#103—624no—
124C22⋊C8—622no—
125C31#187211yescharacteristic, complemented, Sylow
126C24C22⋊D3111yescharacteristic
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8C2×Dic3112yescomplemented
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—624no—
136C22×C4—622no—
137C22×C4—1242no—
138C22×C4—1242no—
139C22×C4C2×Dic3112yes—
140C22×C4C22⋊D3111yescharacteristic
141C22×C4C22⋊D3112yes—
142C22×C4C22×D3111yescharacteristic
143C2×C8—622no—
144C2×C8—622no—
145C2×C8—622no—
146C4⋊C4—24no—
147C4⋊C4—21no—
148C4⋊C4—314no—
149C4⋊C4—624no—
150C4⋊C4—622no—
151C4⋊C4C2×Dic3112yescomplemented
152C22⋊C4—24no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23#9248112yes—
158C23#9248212yes—
159C23C22⋊Dic3112yes—
160C23C2×C22⋊D3111yescharacteristic
161Q8—24no—
162Q8—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—624no—
174C2×C4—1248no—
175C2×C4—1242no—
176C2×C4—1242no—
177C2×C4C22⋊Dic3112yes—
178C2×C4D4×D3112yes—
179C2×C4#9249712yes—
180C2×C4C22×Dic3111yescharacteristic
181C2×C4C2×C22⋊D3111yescharacteristic
182C8—1242no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22#1175842314yescomplemented
190C22#1175851211yescharacteristic
191C22#1175851411yescharacteristic
192C22C2×#9248111yescharacteristic
193C22C2×#9248211yescharacteristic
194C22#1175852611yescharacteristic
195C22#1175852811yescharacteristic
196C22C2×C22⋊Dic3111yescharacteristic
197C4—22no—
198C4—42no—
199C4—42no—
200C4—1244no—
201C4D4×Dic3112yes—
202C4#1175851612yes—
203C2—24no—
204C2#1452092911yescharacteristic
205C2C2×#1175842311yescharacteristic
206C2#1452105011yescharacteristic
207C2#1452119911yescharacteristic
208C2#1452120411yescharacteristic
209C2#1452137711yescharacteristic
210C2#1452138211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC2×C31×C22⋊Q8C2
socleC23×C31C2×D4
prime radicalC24×C31D4
cosocleC22×C4×C31C23
hypercenterC2×C22⋊Q8D31
hypocenterC31#1872
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11758423C2semidirect
4. #14520811C2semidirect
10. C31×C22⋊Q8C4semidirect
24. C2×Q8×C31C2×C4semidirect
39. C4⋊C4×C31C2×C4semidirect
84. C22×C31#103semidirect
116. C22⋊Q8Dic31semidirect
125. C31#1872semidirect
130. C2×Q8C2×Dic31semidirect
151. C4⋊C4C2×Dic31semidirect
189. C22#11758423semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic312
C2×#117584232
C2×C22⋊(C31⋊C8)2
#145208112
C2×C31×C22⋊Q82
#187231