Finite Group Databasebeta

← C2×#1354 #11720

Subgroups — C2×#1354 #11720

Order 256.

326 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 71 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×#395C211yesmaximal, characteristic
2C2×#343C211yesmaximal, characteristic
3C2×#324C211yesmaximal, characteristic
4#1354C214yesmaximal, direct product factor
5C23×Q8C411yescharacteristic
6C22×M16—21no—
7C22×C4⋊C4—21no—
8C22×C4⋊C4C2211yescharacteristic
9C2×C22⋊C8—21no—
10#395—21no—
11#395—21no—
12#395C412yes—
13#395C2212yes—
14C2×#95—21no—
15C2×#95C411yescharacteristic
16#343—21no—
17#343C2212yes—
18#324—21no—
19#324C2212yes—
20C22×Q8—21no—
21C22×Q8—44no—
22C22×Q8—41no—
23C22×Q8—41no—
24C22×Q8C2×C412yes—
25C23×C4—21no—
26C23×C4—21no—
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—24no—
31C2×M16—22no—
32C2×M16—41no—
33C2×M16—42no—
34C22×C8—41no—
35C22×C8—41no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4—41no—
40C2×C4⋊C4—44no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4C2311yescharacteristic
45C22⋊C8—22no—
46C22⋊C8—41no—
47#95—22no—
48#95—21no—
49#95—22no—
50#95C2×C412yes—
51C24C22⋊C411yescharacteristic
52C2×Q8—44no—
53C2×Q8—44no—
54C2×Q8—42no—
55C2×Q8—42no—
56C2×Q8—48no—
57C2×Q8—44no—
58C2×Q8—44no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—24no—
63C22×C4—24no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4SD1612yes—
81C22×C4Q1612yes—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85M16—24no—
86M16—44no—
87M16—42no—
88C2×C8—44no—
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—82no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—44no—
97C4⋊C4—44no—
98C23—22no—
99C23—22no—
100C23#9911yescharacteristic
101C23#9911yescharacteristic
102C23#10312yes—
103C23#10412yes—
104C23C2×C22⋊C411yescharacteristic
105Q8—44no—
106Q8—44no—
107Q8—48no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—24no—
117C2×C4—24no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—42no—
123C2×C4—48no—
124C2×C4—42no—
125C2×C4—44no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—44no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—81no—
137C2×C4#10312yes—
138C2×C4C2×SD1611yescharacteristic
139C2×C4C2×Q1611yescharacteristic
140C8—44no—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—24no—
145C22—24no—
146C22—21no—
147C22—21no—
148C22#32812yes—
149C22#32812yes—
150C22AΣL(1,𝔽16)12yes—
151C22#35612yes—
152C22C2×#9911yescharacteristic
153C22C2×#10311yescharacteristic
154C22C2×#10411yescharacteristic
155C4—21no—
156C4—22no—
157C4—21no—
158C4—44no—
159C4—41no—
160C4—44no—
161C4—41no—
162C4—41no—
163C4—41no—
164C2—21no—
165C2—22no—
166C2—21no—
167C2#135414yesdirect product factor
168C2C2×#32811yescharacteristic
169C2C2×AΣL(1,𝔽16)11yescharacteristic
170C2C2×#35611yescharacteristic
subgroupG/H
center, socleC23#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1354C2direct (N × H)
167. C2#1354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3952
C2×#3432
C2×#3242
#13542