Finite Group Databasebeta

← #16846760

Subgroups — #16846760

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×#11758486C212yesmaximal, complemented
2#14521630C211yesmaximal, characteristic
3C2×#11758413C212yesmaximal, complemented
4#14520954C211yesmaximal, characteristic, complemented
5C31×#343C211yesmaximal, characteristic, complemented
6C2×#92495C2212yes—
7C22×Dic62C2211yescharacteristic
8C2×M16×C31C2212yes—
9C2×C4⋊C4×C31C2211yescharacteristic
10#11758507—22no—
11C2×C4×Dic31C2211yescharacteristic
12#11758486—24no—
13#11758486—24no—
14C2×Dic124—22no—
15C2×#92470—22no—
16#11758474—22no—
17#11758472—22no—
18C31⋊#95—22no—
19#11758413—24no—
20C22×C4×C31D412yes—
21C22×C4×C31C2311yescharacteristic
22C2×C22⋊D31—22no—
23C22×Dic31—21no—
24C22×Dic31—21no—
25#92495—22no—
26#92495—22no—
27#92495—22no—
28#92495—22no—
29C2×D124—22no—
30C2×D124D412yes—
31C2×C4×D31—22no—
32C2×Dic62—22no—
33C2×Dic62—22no—
34C2×Dic62—21no—
35C2×Dic62D412yes—
36C2×Dic62D412yescomplemented
37M16×C31—22no—
38M16×C31D414yescomplemented
39C2×C8×C31—22no—
40C4⋊C4×C31—22no—
41(C2×D31)⋊C4—24no—
42C4⋊Dic31—22no—
43C4×Dic31—22no—
44Dic124—28no—
45#92470—28no—
46C23×C31C2×D411yescharacteristic
47C22×D31—22no—
48C2×C4×C31—22no—
49C2×C4×C31—42no—
50C2×C4×C31C2×D412yes—
51C2×C4×C31C2×D411yescharacteristic
52C2×C4×C31C2×D412yes—
53C2×C4×C31#4111yescharacteristic
54C22⋊D31—42no—
55C22⋊D31—42no—
56C2×Dic31—22no—
57C2×Dic31—22no—
58C2×Dic31—22no—
59C2×Dic31—41no—
60C2×Dic31—41no—
61C2×Dic31—41no—
62C2×Dic31—41no—
63D124—22no—
64D124—22no—
65D124—82no—
66C4×D31—42no—
67C4×D31—42no—
68Dic62—24no—
69Dic62—22no—
70Dic62—22no—
71Dic62—42no—
72C8×C31—24no—
73#2007—311nomaximal, Sylow
74C22×C31—21no—
75C22×C31—21no—
76C22×C31#12011yescharacteristic
77C22×C31#12111yescharacteristic
78C22×C31#12711yescharacteristic
79C2×D31—42no—
80C2×D31—42no—
81C2×D31—42no—
82C4×C31—42no—
83C4×C31#12012yes—
84C4×C31#12112yes—
85Dic31—42no—
86Dic31—42no—
87Dic31—42no—
88C2×#137—312no—
89#537—311no—
90C2×#104—312no—
91#391—311no—
92#343D3111yescharacteristic, complemented
93C2×C31—21no—
94C2×C31—21no—
95C2×C31#39211yescharacteristic
96C2×C31#45511yescharacteristic
97C2×C31#45611yescharacteristic
98D31—82no—
99C2×#41—312no—
100C22×Q8—311no—
101#137—624no—
102#137—624no—
103C2×Q16—622no—
104C2×SD16—622no—
105C2×M16C2×D3112yes—
106C22⋊Q8—622no—
107#121—622no—
108#121—622no—
109C2×C4⋊C4C2×D3111yescharacteristic
110C2×C42—311no—
111#104—624no—
112#95—622no—
113C31#200711yescharacteristic, complemented, Sylow
114#41—622no—
115#41—622no—
116#41—622no—
117#41—622no—
118C2×Q8—312no—
119C2×Q8—312no—
120C2×Q8—621no—
121C2×Q8—622no—
122C2×Q8—622no—
123C2×D4—312no—
124C2×D4—622no—
125C2×D4—622no—
126C22×C4—621no—
127C22×C4—621no—
128C22×C4—622no—
129C22×C4D12412yes—
130C22×C4C22×D3111yescharacteristic
131Q16—628no—
132SD16—628no—
133M16—22no—
134M16D12414yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—622no—
138C22⋊C4—624no—
139C42—622no—
140C23—622no—
141C23C2×D12411yescharacteristic
142Q8—624no—
143Q8—622no—
144Q8—622no—
145Q8—1242no—
146D4—622no—
147D4—622no—
148D4—1242no—
149D4—1242no—
150D4—2482no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—622no—
154C2×C4—622no—
155C2×C4—622no—
156C2×C4—1241no—
157C2×C4—1241no—
158C2×C4—1241no—
159C2×C4—1241no—
160C2×C4—1242no—
161C2×C4—1242no—
162C2×C4C2×D12412yes—
163C2×C4D4×D3111yescharacteristic
164C2×C4D4×D3112yes—
165C2×C4#9249911yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—1242no—
170C22—1242no—
171C22—1242no—
172C22#1175845111yescharacteristic
173C22#1175845911yescharacteristic
174C22#1175847211yescharacteristic
175C4—42no—
176C4—1242no—
177C4—1242no—
178C4—1242no—
179C4#1175845912yes—
180C4#1175847212yes—
181C2—21no—
182C2—21no—
183C2—2482no—
184C2#1452095911yescharacteristic
185C2#1452122911yescharacteristic
186C2#1452123411yescharacteristic
subgroupG/H
centerC22#11758459
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
FittingC31×#343C2
socleC22×C31#120
prime radical#14521630C2
hypercenter#343D31
hypocenterC31#2007
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#11758486C2semidirect
3. C2×#11758413C2semidirect
4. #14520954C2semidirect
5. C31×#343C2semidirect
36. C2×Dic62D4semidirect
38. M16×C31D4semidirect
92. #343D31semidirect
113. C31#2007semidirect
134. M16D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117584862
#145216302
C2×#117584132
#145209542
C31×#3432
#200731