Finite Group Databasebeta

← C16×GO+(4,𝔽2) #11797459

Subgroups — C16×GO+(4,𝔽2) #11797459

Order 1152.

272 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C8×GO+(4,𝔽2)C211yesmaximal, characteristic
2D32×C16C212yesmaximal, complemented
3C16×#157C211yesmaximal, characteristic, complemented
4#10590022C211yesmaximal, characteristic, complemented
5D32⋊C16C212yesmaximal, complemented
6C4×GO+(4,𝔽2)C411yescharacteristic
7D32×C8C2212yes—
8C8×#157C2211yescharacteristic
9#63764C2211yescharacteristic
10#63732C411yescharacteristic
11D32⋊C8C412yes—
12C16×C3⋊D3C2211yescharacteristic
13C3×D3×C16—22no—
14#63542C2212yes—
15D3×C3⋊C16—22no—
16C2×D3×C16—62no—
17C2×GO+(4,𝔽2)C811yescharacteristic
18C4×D32C2×C412yes—
19C4×#157C2×C411yescharacteristic
20#3787C2×C411yescharacteristic
21#3773C811yescharacteristic
22D32⋊C4C812yes—
23C8×C3⋊D3C2311yescharacteristic
24C3×D3×C8—22no—
25#3710C2×C412yes—
26D3×C3⋊C8—22no—
27#3708—21no—
28C32×C16D411yescharacteristic, complemented
29C3⋊(C3⋊C16)D411yescharacteristic, complemented
30C3×C3⋊C16—22no—
31D4×C16—91nomaximal, Sylow
32C2×D3×C8—62no—
33C2×C3×C16—62no—
34C2×C3⋊C16—62no—
35D3×C16—22no—
36D3×C16—62no—
37D3×C16—62no—
38D3×C16—62no—
39C2×D32C2×C812yes—
40C2×#157C2×C811yescharacteristic
41C2×#157C2×C811yescharacteristic
42GO+(4,𝔽2)C1614yesdirect product factor
43C4×C3⋊D3C22×C411yescharacteristic
44C3×C4×D3—22no—
45#624C2×C812yes—
46D3×Dic3—22no—
47#622—21no—
48C32×C8C2×D411yescharacteristic
49C3⋊(C3⋊C8)#4111yescharacteristic
50C3×C3⋊C8—22no—
51C22×C16—92no—
52C8×D4—91no—
53C4⋊C16—91no—
54C22⋊C16—92no—
55C4×C16—91no—
56C2×C4×D3—62no—
57C2×C3×C8—62no—
58C2×C3⋊C8—62no—
59D3×C8—22no—
60D3×C8—62no—
61D3×C8—62no—
62D3×C8—62no—
63C3×C16—22no—
64C3×C16—62no—
65C3⋊C16—22no—
66C3⋊C16—62no—
67C2×C3⋊D3C22×C811yescharacteristic
68C2×C3×D3—22no—
69D32—22no—
70D32C2×C1614yescomplemented
71#157—21no—
72#157C2×C1612yescomplemented
73C32×C4C4×D411yescharacteristic
74C3⋊Dic3#13111yescharacteristic
75C3×Dic3—22no—
76C22×C8—92no—
77C4×D4—91no—
78C2×C16—62no—
79C2×C16—92no—
80C2×C16—91no—
81C2×C16—91no—
82C2×C16—182no—
83C4⋊C8—91no—
84C22⋊C8—92no—
85C4×C8—91no—
86C22×D3—62no—
87C2×C3×C4—62no—
88C2×Dic3—62no—
89C4×D3—22no—
90C4×D3—62no—
91C4×D3—62no—
92C4×D3—62no—
93C3×C8—22no—
94C3×C8—62no—
95C3⋊C8—22no—
96C3⋊C8—62no—
97C2×C32C8×D411yescharacteristic
98C3⋊D3C22×C1611yescharacteristic
99C3⋊D3#50411yescharacteristic
100C3×D3—24no—
101C2×D4—91no—
102C22×C4—92no—
103C2×C8—62no—
104C2×C8—92no—
105C2×C8—91no—
106C2×C8—91no—
107C2×C8—182no—
108C4⋊C4—91no—
109C22⋊C4—92no—
110C42—91no—
111C16—62no—
112C16—91no—
113C16—181no—
114C16GO+(4,𝔽2)11yescharacteristic, direct product factor
115C22×C3—62no—
116C2×D3—22no—
117C2×D3—64no—
118C2×D3—62no—
119C2×D3—64no—
120C3×C4—22no—
121C3×C4—62no—
122Dic3—22no—
123Dic3—62no—
124C32D4×C1611yescharacteristic, complemented, Sylow
125C23—92no—
126D4—94no—
127C2×C4—62no—
128C2×C4—92no—
129C2×C4—91no—
130C2×C4—91no—
131C2×C4—91no—
132C2×C4—182no—
133C8—62no—
134C8—91no—
135C8—181no—
136C8C2×GO+(4,𝔽2)11yescharacteristic
137C2×C3—22no—
138C2×C3—64no—
139D3—24no—
140D3—62no—
141D3—62no—
142C22—62no—
143C22—94no—
144C22—91no—
145C22—182no—
146C4—62no—
147C4—92no—
148C4—91no—
149C4—181no—
150C4C4×GO+(4,𝔽2)11yescharacteristic
151C3—22no—
152C2—64no—
153C2—91no—
154C2—91no—
155C2C8×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC16GO+(4,𝔽2)
commutatorC3⋊D3C22×C16
FrattiniC8C2×GO+(4,𝔽2)
FittingC32×C16D4
socleC2×C32C8×D4
prime radicalC2×GO+(4,𝔽2)C8
cosocleC8×C3⋊D3C23
hypocenterC32D4×C16
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C16

Complements

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

NH ≅ G/Ntype
2. D32×C16C2semidirect
3. C16×#157C2semidirect
4. #10590022C2semidirect
5. D32⋊C16C2semidirect
28. C32×C16D4semidirect
29. C3⋊(C3⋊C16)D4semidirect
42. GO+(4,𝔽2)C16direct (N × H)
70. D32C2×C16semidirect
72. #157C2×C16semidirect
114. C16GO+(4,𝔽2)direct (N × H)
124. C32D4×C16semidirect

Maximal subgroups

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

subgroupindex
C8×GO+(4,𝔽2)2
D32×C162
C16×#1572
#105900222
D32⋊C162
D4×C169