Finite Group Databasebeta

← #16846142

Subgroups — #16846142

Order 3968.

360 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 105 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×#92474C211yesmaximal, characteristic
2C2×#11758501C211yesmaximal, characteristic
3C2×#11758398C212yesmaximal, complemented
4#14520811C212yesmaximal, complemented
5C31×#380C211yesmaximal, characteristic
6C23×C4×C31C2211yescharacteristic
7C22×Dic62C2211yescharacteristic
8C2×C4⋊C4×C31C2212yes—
9C2×C22⋊C4×C31—21no—
10C2×C22⋊Dic31—21no—
11C2×C4⋊Dic31C2211yescharacteristic
12#11758501—22no—
13#11758501C414yescomplemented
14C2×Dic31⋊C4—21no—
15C2×#92474—22no—
16C2×#92474—24no—
17C22×C31⋊C8C2212yes—
18C31×#95—21no—
19#11758398—24no—
20C24×C31D411yescharacteristic
21C22×C4×C31—22no—
22C22×C4×C31—22no—
23C22×C4×C31—22no—
24C22×C4×C31C2×C412yes—
25C22×C4×C31D411yescharacteristic
26C22×C4×C31D412yes—
27C22×C4×C31C2311yescharacteristic
28C22×Dic31—21no—
29C22×Dic31—21no—
30C2×Dic62—22no—
31C2×Dic62—22no—
32C2×Dic62—21no—
33C2×Dic62C2×C412yes—
34C4⋊C4×C31—22no—
35C4⋊C4×C31—22no—
36C22⋊C4×C31—42no—
37C22⋊Dic31—24no—
38C4⋊Dic31—21no—
39C4⋊Dic31C2×C412yes—
40Dic31⋊C4—24no—
41#92474—28no—
42C2×C31⋊C8—22no—
43C2×C31⋊C8—22no—
44C2×C31⋊C8D414yes—
45C23×C31—22no—
46C23×C31—21no—
47C23×C31—21no—
48C23×C31—22no—
49C23×C31C22⋊C412yes—
50C23×C31C2×D411yescharacteristic
51C2×C4×C31—22no—
52C2×C4×C31—24no—
53C2×C4×C31—22no—
54C2×C4×C31—22no—
55C2×C4×C31—42no—
56C2×C4×C31—42no—
57C2×C4×C31—42no—
58C2×C4×C31C22⋊C412yes—
59C2×C4×C31C22×C411yescharacteristic
60C2×C4×C31C2×D411yescharacteristic
61C2×C4×C31C2×D412yes—
62C2×C4×C31#4112yes—
63C2×Dic31—22no—
64C2×Dic31—22no—
65C2×Dic31—42no—
66C2×Dic31—41no—
67C2×Dic31—41no—
68Dic62—24no—
69Dic62—41no—
70Dic62—41no—
71C31⋊C8—24no—
72#1894—311nomaximal, Sylow
73C22×C31—22no—
74C22×C31—22no—
75C22×C31—22no—
76C22×C31—42no—
77C22×C31—42no—
78C22×C31C2×C22⋊C411yescharacteristic
79C22×C31#12112yes—
80C22×C31AGL(1,ℤ/8)11yescharacteristic
81C22×C31#13711yescharacteristic
82C22×C31#13712yes—
83C4×C31—22no—
84C4×C31—82no—
85C4×C31C4×D412yes—
86C4×C31#12011yescharacteristic
87C4×C31#12311yescharacteristic
88Dic31—42no—
89Dic31—42no—
90C22×M16—311no—
91C2×C22⋊Q8—311no—
92C2×#103—312no—
93#380D3111yescharacteristic
94#340—312no—
95C2×C31—42no—
96C2×C31#38611yescharacteristic
97C2×C31#41711yescharacteristic
98C2×C31#41911yescharacteristic
99C2×C31#46812yes—
100C2×C31#47012yes—
101C22×Q8—311no—
102C23×C4C2×D3111yescharacteristic
103C2×M16—624no—
104C2×M16—622no—
105C22×C8—312no—
106C22⋊Q8—314no—
107C22⋊Q8—622no—
108C2×C4⋊C4—311no—
109C2×C4⋊C4—621no—
110C2×C4⋊C4C2×D3112yes—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—621no—
113#103—624no—
114#95—21no—
115C31#189411yescharacteristic, complemented, Sylow
116C24C22⋊D3111yescharacteristic
117C2×Q8—312no—
118C2×Q8—621no—
119C2×Q8—622no—
120C2×Q8—622no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—621no—
125C22×C4—621no—
126C22×C4C4×D3112yes—
127C22×C4D12412yes—
128C22×C4C22⋊D3111yescharacteristic
129C22×C4C22×D3111yescharacteristic
130M16—628no—
131C2×C8—314no—
132C2×C8—622no—
133C2×C8—622no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—312no—
137C4⋊C4—624no—
138C4⋊C4—621no—
139C22⋊C4—42no—
140C22⋊C4—624no—
141C23—21no—
142C23—22no—
143C23—21no—
144C23—22no—
145C23(C2×D31)⋊C412yes—
146C23C2×C22⋊D3111yescharacteristic
147Q8—624no—
148Q8—1241no—
149Q8—1241no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—622no—
158C2×C4—622no—
159C2×C4—1241no—
160C2×C4—1242no—
161C2×C4—1241no—
162C2×C4(C2×D31)⋊C412yes—
163C2×C4C2×C4×D3111yescharacteristic
164C2×C4C2×D12411yescharacteristic
165C2×C4D4×D3112yes—
166C2×C4#9249712yes—
167C8—624no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×(C2×D31)⋊C411yescharacteristic
174C22#1175850911yescharacteristic
175C22#1175851612yes—
176C22#1175851911yescharacteristic
177C22#1175852812yes—
178C4—22no—
179C4—82no—
180C4—1242no—
181C4—1242no—
182C4(C22⋊D31)⋊C412yes—
183C4#1175845911yescharacteristic
184C4#1175846311yescharacteristic
185C2—42no—
186C2#1452092611yescharacteristic
187C2#1452104811yescharacteristic
188C2#1452105711yescharacteristic
189C2#1452129312yes—
190C2#1452130512yes—
subgroupG/H
centerC23C2×C22⋊D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#380C2
socleC23×C31C2×D4
prime radicalC24×C31D4
cosocleC22×C4×C31C23
hypercenter#380D31
hypocenterC31#1894
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#11758398C2semidirect
4. #14520811C2semidirect
13. #11758501C4semidirect
115. C31#1894semidirect

Maximal subgroups

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

subgroupindex
C22×#924742
C2×#117585012
C2×#117583982
#145208112
C31×#3802
#189431