Finite Group Databasebeta

← #16846755

Subgroups — #16846755

Order 3968.

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

normal subgroups only

This group has 406 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×#11758485C212yesmaximal, complemented
2#14521630C211yesmaximal, characteristic, complemented
3C2×#11758413C212yesmaximal, complemented
4#14520950C211yesmaximal, characteristic, complemented
5C31×#343C211yesmaximal, characteristic, complemented
6C2×#92495C2212yes—
7C22×D124C2211yescharacteristic
8C2×M16×C31C2212yes—
9C2×C4⋊C4×C31C2211yescharacteristic
10#11758507—22no—
11C2×(C2×D31)⋊C4—22no—
12C2×C4×Dic31C2211yescharacteristic
13#11758485—24no—
14#11758485—24no—
15C2×D248—22no—
16C2×#92470—22no—
17#11758474—22no—
18#11758472—22no—
19#11758413—24no—
20C23×D31—21no—
21C22×C4×C31D412yes—
22C22×C4×C31C2311yescharacteristic
23C2×C22⋊D31—22no—
24C22×Dic31—21no—
25#92495—22no—
26#92495—22no—
27#92495—22no—
28#92495—22no—
29C2×D124—22no—
30C2×D124—22no—
31C2×D124—21no—
32C2×D124—22no—
33C2×D124D412yes—
34C2×D124D412yescomplemented
35C2×C4×D31—22no—
36C2×Dic62D412yes—
37M16×C31—22no—
38M16×C31D414yescomplemented
39C2×C8×C31—22no—
40C4⋊C4×C31—22no—
41(C2×D31)⋊C4—24no—
42(C2×D31)⋊C4—44no—
43C4⋊Dic31—22no—
44C4×Dic31—22no—
45D248—28no—
46#92470—28no—
47C23×C31C2×D411yescharacteristic
48C22×D31—22no—
49C22×D31—22no—
50C22×D31—42no—
51C22×D31—42no—
52C22×D31—41no—
53C22×D31—41no—
54C2×C4×C31—22no—
55C2×C4×C31—42no—
56C2×C4×C31C2×D412yes—
57C2×C4×C31C2×D411yescharacteristic
58C2×C4×C31C2×D412yes—
59C2×C4×C31#4111yescharacteristic
60C22⋊D31—42no—
61C22⋊D31—42no—
62C2×Dic31—22no—
63C2×Dic31—22no—
64C2×Dic31—41no—
65C2×Dic31—41no—
66D124—24no—
67D124—22no—
68D124—22no—
69D124—42no—
70D124—82no—
71C4×D31—42no—
72C4×D31—42no—
73Dic62—22no—
74Dic62—22no—
75C8×C31—24no—
76#2006—311nomaximal, Sylow
77C22×C31—21no—
78C22×C31—21no—
79C22×C31#12011yescharacteristic
80C22×C31#12111yescharacteristic
81C22×C31#12711yescharacteristic
82C2×D31—44no—
83C2×D31—42no—
84C2×D31—42no—
85C2×D31—42no—
86C2×D31—42no—
87C2×D31—82no—
88C2×D31—81no—
89C2×D31—81no—
90C4×C31—42no—
91C4×C31#12012yes—
92C4×C31#12112yes—
93Dic31—42no—
94Dic31—42no—
95C2×AGL(1,ℤ/8)—312no—
96#537—311no—
97C2×#104—312no—
98#390—311no—
99#343D3111yescharacteristic, complemented
100C2×C31—21no—
101C2×C31—21no—
102C2×C31#39211yescharacteristic
103C2×C31#45311yescharacteristic
104C2×C31#45411yescharacteristic
105D31—82no—
106D31—82no—
107C2×#41—312no—
108C22×D4—311no—
109AGL(1,ℤ/8)—624no—
110AGL(1,ℤ/8)—624no—
111C2×SD16—622no—
112C2×D8—622no—
113C2×M16C2×D3112yes—
114C22⋊Q8—622no—
115#121—622no—
116#121—622no—
117C2×C4⋊C4C2×D3111yescharacteristic
118C2×C22⋊C4—622no—
119C2×C42—311no—
120#104—624no—
121C31#200611yescharacteristic, complemented, Sylow
122C24—621no—
123#41—622no—
124#41—622no—
125#41—622no—
126#41—622no—
127C2×Q8—312no—
128C2×D4—312no—
129C2×D4—312no—
130C2×D4—622no—
131C2×D4—622no—
132C2×D4—622no—
133C2×D4—622no—
134C2×D4—621no—
135C22×C4—621no—
136C22×C4—622no—
137C22×C4D12412yes—
138C22×C4C22×D3111yescharacteristic
139SD16—628no—
140D8—628no—
141M16—22no—
142M16D12414yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—622no—
146C22⋊C4—624no—
147C22⋊C4—1244no—
148C42—622no—
149C23—622no—
150C23—622no—
151C23—1242no—
152C23—1242no—
153C23—1241no—
154C23—1241no—
155C23C2×D12411yescharacteristic
156Q8—622no—
157Q8—622no—
158D4—622no—
159D4—624no—
160D4—622no—
161D4—1242no—
162D4—1242no—
163D4—1242no—
164D4—2482no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—622no—
168C2×C4—622no—
169C2×C4—1241no—
170C2×C4—1241no—
171C2×C4—1242no—
172C2×C4—1242no—
173C2×C4C2×D12412yes—
174C2×C4D4×D3111yescharacteristic
175C2×C4D4×D3112yes—
176C2×C4#9249911yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—1244no—
181C22—1242no—
182C22—1242no—
183C22—1242no—
184C22—1242no—
185C22—2482no—
186C22—2481no—
187C22—2481no—
188C22#1175845111yescharacteristic
189C22#1175845911yescharacteristic
190C22#1175847211yescharacteristic
191C4—42no—
192C4—1242no—
193C4—1242no—
194C4#1175845912yes—
195C4#1175847212yes—
196C2—21no—
197C2—21no—
198C2—2482no—
199C2—2482no—
200C2#1452095911yescharacteristic
201C2#1452121911yescharacteristic
202C2#1452122411yescharacteristic
subgroupG/H
centerC22#11758459
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
FittingC31×#343C2
socleC22×C31#120
hypercenter#343D31
hypocenterC31#2006
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×#11758485C2semidirect
2. #14521630C2semidirect
3. C2×#11758413C2semidirect
4. #14520950C2semidirect
5. C31×#343C2semidirect
34. C2×D124D4semidirect
38. M16×C31D4semidirect
99. #343D31semidirect
121. C31#2006semidirect
142. M16D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117584852
#145216302
C2×#117584132
#145209502
C31×#3432
#200631