Finite Group Databasebeta

← #16847736

Subgroups — #16847736

Order 3968.

236 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 41 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
1#14521765C211yesmaximal, characteristic, complemented
2#14521222C212yesmaximal, complemented
3#14520848C211yesmaximal, characteristic, complemented
4#14520845C212yesmaximal, complemented
5C31×#456C211yesmaximal, characteristic, complemented
6C31×2−1+4C2211yescharacteristic
7C31×#137—21no—
8C31×C4⋊Q8C2211yescharacteristic
9#11758526—22no—
10#11758524—21no—
11#11758524C2212yes—
12#11758472—22no—
13#11758469—22no—
14C4×D124—21no—
15C31×#104—21no—
16C31×#101C2211yescharacteristic
17#11758425—22no—
18#11758424C2212yes—
19#41×C31—22no—
20#41×C31—21no—
21#41×C31—41no—
22C2×Q8×C31—41no—
23C2×Q8×C31D411yescharacteristic
24C2×Q8×C31C2311yescharacteristic
25C2×D124—21no—
26C2×D124—21no—
27C2×D124D412yes—
28C2×C4×D31—22no—
29Q16×C31—41no—
30SD16×C31—41no—
31M16×C31—21no—
32C4⋊C4×C31—22no—
33C42×C31D411yescharacteristic
34#92481—42no—
35#92479—42no—
36(C2×D31)⋊C4—22no—
37(C2×D31)⋊C4—22no—
38(C2×D31)⋊C4—22no—
39C4⋊Dic31—21no—
40C4×Dic31—21no—
41C4×Dic31D412yes—
42#92474—22no—
43C22×D31—22no—
44Q8×C31—22no—
45Q8×C31—21no—
46Q8×C31—41no—
47Q8×C31—41no—
48D4×C31—22no—
49D4×C31—21no—
50D4×C31—41no—
51C2×C4×C31—21no—
52C2×C4×C31—21no—
53C2×C4×C31—42no—
54C2×C4×C31—41no—
55C2×C4×C31C2×D412yes—
56C2×C4×C31C2×D411yescharacteristic
57C2×Dic31—22no—
58D124—42no—
59D124—42no—
60D124—81no—
61C4×D31—44no—
62C8×C31—41no—
63C31⋊C8—42no—
64#2204—311nomaximal, Sylow
65C22×C31—41no—
66C22×C31#12011yescharacteristic
67C2×D31—42no—
68C2×D31—42no—
69C2×D31—42no—
70C4×C31—22no—
71C4×C31—22no—
72C4×C31—21no—
73C4×C31—41no—
74C4×C31—41no—
75Dic31—42no—
76#555—311no—
77#456D3111yescharacteristic, complemented
78#454—312no—
79#356—311no—
80#355—312no—
81C2×C31—21no—
82C2×C31—81no—
83C2×C31B(4,𝔽2)11yescharacteristic
84D31—82no—
852−1+4C2×D3111yescharacteristic
86#137—21no—
87AGL(1,ℤ/8)—622no—
88C4⋊Q8C2×D3111yescharacteristic
89#124—312no—
90#124—621no—
91#121—622no—
92C4×D4—621no—
93#117—622no—
94#104—21no—
95#104—622no—
96#101—312no—
97#101C2×D3111yescharacteristic
98C31#220411yescharacteristic, complemented, Sylow
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D3111yescharacteristic
104C2×Q8C22×D3111yescharacteristic
105C2×D4—312no—
106C2×D4—621no—
107C2×D4—621no—
108C22×C4—622no—
109Q16—41no—
110SD16—41no—
111SD16—1242no—
112D8—1242no—
113M16—21no—
114M16—622no—
115C4⋊C4—22no—
116C4⋊C4—621no—
117C22⋊C4—622no—
118C22⋊C4—622no—
119C22⋊C4—622no—
120C42—312no—
121C42—621no—
122C42C22⋊D3111yescharacteristic
123C23—622no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—1242no—
132D4—1242no—
133D4—2481no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—622no—
139C2×C4—1244no—
140C2×C4D4×D3112yes—
141C2×C4C2×C22⋊D3111yescharacteristic
142C8—41no—
143C8—1242no—
144C22—41no—
145C22—1242no—
146C22—1242no—
147C22—1242no—
148C22#1175851411yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—1242no—
155C2—21no—
156C2—81no—
157C2—2482no—
158C2#1452123511yescharacteristic
subgroupG/H
centerC2#14521235
commutator, cosocleC2×Q8×C31C23
FrattiniC2×Q8C22×D31
FittingC31×#456C2
socleC2×C31B(4,𝔽2)
hypercenter#456D31
hypocenterC31#2204
LCSG ▷ C2×Q8×C31 ▷ C22×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #14521765C2semidirect
2. #14521222C2semidirect
3. #14520848C2semidirect
4. #14520845C2semidirect
5. C31×#456C2semidirect
77. #456D31semidirect
98. C31#2204semidirect

Maximal subgroups

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

subgroupindex
#145217652
#145212222
#145208482
#145208452
C31×#4562
#220431