Finite Group Databasebeta

← #16846758

Subgroups — #16846758

Order 3968.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#11758528C212yesmaximal, complemented
2C2×#11758425C212yesmaximal, complemented
3#14520954C211yesmaximal, characteristic, complemented
4#14520818C211yesmaximal, characteristic, complemented
5C31×#537C211yesmaximal, characteristic
6C2×#41×C31C2212yes—
7C22×Dic62C2211yescharacteristic
8C31×C22⋊Q8—22no—
9C31×#121—22no—
10C31×#121—22no—
11C2×C4⋊C4×C31C2211yescharacteristic
12#11758528—24no—
13#11758528—24no—
14C2×#92482—22no—
15C2×#92480—22no—
16C2×C4×Dic31C2211yescharacteristic
17C2×#92474C2212yes—
18#11758425—24no—
19C31⋊#95—22no—
20#41×C31—22no—
21#41×C31—22no—
22#41×C31—22no—
23#41×C31—22no—
24C2×Q8×C31D412yes—
25C2×D4×C31—22no—
26C2×D4×C31—22no—
27C2×D4×C31D412yes—
28C22×C4×C31—22no—
29C22×C4×C31D412yes—
30C22×C4×C31C2311yescharacteristic
31C22×Dic31—21no—
32C22×Dic31—21no—
33C2×Dic62—22no—
34C2×Dic62—22no—
35C2×Dic62—21no—
36C2×Dic62D412yescomplemented
37C4⋊C4×C31—22no—
38C4⋊C4×C31—22no—
39C22⋊C4×C31—24no—
40#92482—28no—
41#92480—28no—
42C4×Dic31—22no—
43#92474—22no—
44#92474D414yescomplemented
45C2×C31⋊C8—22no—
46C23×C31—22no—
47C23×C31C2×D411yescharacteristic
48Q8×C31—22no—
49Q8×C31—22no—
50D4×C31—22no—
51D4×C31—22no—
52D4×C31—42no—
53D4×C31—42no—
54D4×C31—82no—
55C2×C4×C31—22no—
56C2×C4×C31—22no—
57C2×C4×C31—42no—
58C2×C4×C31—42no—
59C2×C4×C31—42no—
60C2×C4×C31C2×D411yescharacteristic
61C2×C4×C31C2×D412yes—
62C2×C4×C31C2×D412yes—
63C2×C4×C31#4111yescharacteristic
64C2×Dic31—22no—
65C2×Dic31—22no—
66C2×Dic31—41no—
67C2×Dic31—41no—
68C2×Dic31—41no—
69C2×Dic31—41no—
70Dic62—24no—
71Dic62—42no—
72C31⋊C8—24no—
73#2007—311nomaximal, Sylow
74C22×C31—21no—
75C22×C31—21no—
76C22×C31—42no—
77C22×C31—42no—
78C22×C31—42no—
79C22×C31#12011yescharacteristic
80C22×C31#12111yescharacteristic
81C22×C31#12711yescharacteristic
82C4×C31—42no—
83C4×C31—42no—
84C4×C31#12012yes—
85C4×C31#12112yes—
86Dic31—42no—
87Dic31—42no—
88C2×#137—312no—
89#537D3111yescharacteristic
90C2×#104—312no—
91#391—311no—
92#343—311no—
93C2×C31—21no—
94C2×C31—21no—
95C2×C31—82no—
96C2×C31#39211yescharacteristic
97C2×C31#45511yescharacteristic
98C2×C31#45611yescharacteristic
99C2×#41C2×D3112yes—
100C22×Q8—311no—
101#137—624no—
102#137—624no—
103C2×Q16—622no—
104C2×SD16—622no—
105C2×M16—312no—
106C22⋊Q8—22no—
107#121—22no—
108#121—22no—
109C2×C4⋊C4C2×D3111yescharacteristic
110C2×C42—311no—
111#104—624no—
112#95—622no—
113C31#200711yescharacteristic, complemented, Sylow
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—312no—
119C2×Q8—621no—
120C2×Q8—622no—
121C2×Q8—622no—
122C2×Q8C22⋊D3112yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D3112yes—
126C22×C4—22no—
127C22×C4—621no—
128C22×C4—621no—
129C22×C4C22⋊D3112yes—
130C22×C4C22×D3111yescharacteristic
131Q16—628no—
132SD16—628no—
133M16—314no—
134M16—622no—
135C2×C8—622no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—622no—
140C23—22no—
141C23C2×C22⋊D3111yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—624no—
145Q8—1242no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—622no—
157C2×C4—622no—
158C2×C4—1241no—
159C2×C4—1241no—
160C2×C4—1241no—
161C2×C4—1241no—
162C2×C4D4×D3111yescharacteristic
163C2×C4D4×D3112yes—
164C2×C4#9249711yescharacteristic
165C2×C4C2×C22⋊D3112yes—
166C8—624no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1175851511yescharacteristic
173C22#1175851711yescharacteristic
174C22#1175853011yescharacteristic
175C4—42no—
176C4—42no—
177C4—1242no—
178C4—1242no—
179C4#1175851412yes—
180C4#1175851612yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#1452095711yescharacteristic
185C2#1452122711yescharacteristic
186C2#1452123211yescharacteristic
subgroupG/H
centerC22#11758530
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
Fitting, prime radicalC31×#537C2
socleC22×C31#120
hypercenter#537D31
hypocenterC31#2007
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#11758528C2semidirect
2. C2×#11758425C2semidirect
3. #14520954C2semidirect
4. #14520818C2semidirect
36. C2×Dic62D4semidirect
44. #92474D4semidirect
113. C31#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#117585282
C2×#117584252
#145209542
#145208182
C31×#5372
#200731