Finite Group Databasebeta

← #13296

Subgroups — #13296

Order 256.

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

normal subgroups only

This group has 308 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
1#2409C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic
3C4×#99C211yesmaximal, characteristic
4#1390C214yesmaximal
5C23×Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C2×C4×D4C2211yescharacteristic
8#423—21no—
9C2×#100C2212yes—
10C2×#99C2212yes—
11#395—21no—
12#393—21no—
13#391—21no—
14#386C2212yes—
15#385C412yescomplemented
16#380C411yescharacteristic, complemented
17C4×C22⋊C4—21no—
18#343—21no—
19#342—21no—
20C22×Q8—21no—
21C22×Q8—21no—
22C22×Q8—41no—
23C22×Q8—41no—
24C22×Q8—44no—
25C22×D4C2311yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C412yes—
28C23×C4D411yescharacteristic
29C2×M16—22no—
30C22⋊Q8—44no—
31C4×D4—41no—
32C4×D4—41no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4C2×C412yes—
39C2×C22⋊C4C2×C411yescharacteristic
40C2×C22⋊C4D411yescharacteristic
41C2×C42—41no—
42C2×C42D411yescharacteristic
43C4⋊C8—41no—
44C4⋊C8—41no—
45#100—24no—
46#99—22no—
47#99Q814yescomplemented
48#95—21no—
49#95—22no—
50#95—22no—
51#95—41no—
52C24C22×C411yescharacteristic
53C24C2×D411yescharacteristic
54C2×Q8—24no—
55C2×Q8—42no—
56C2×Q8—44no—
57C2×Q8—42no—
58C2×Q8—44no—
59C2×Q8—48no—
60C2×Q8—44no—
61C2×Q8—41no—
62C2×Q8—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4C2×Q812yes—
66C2×D4#4112yes—
67C22×C4—22no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—42no—
74C22×C4—44no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C2×D411yescharacteristic
83M16—42no—
84M16—42no—
85C2×C8—42no—
86C4⋊C4—41no—
87C4⋊C4—42no—
88C4⋊C4—44no—
89C22⋊C4—24no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—44no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C42—21no—
96C42—21no—
97C42—82no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23C2×C22⋊C411yescharacteristic
107C23C4×Q812yes—
108C23C22⋊Q812yes—
109Q8—44no—
110Q8—48no—
111Q8—44no—
112D4—42no—
113D4—42no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—24no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—44no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—44no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—81no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4#9911yescharacteristic
139C2×C4#9911yescharacteristic
140C2×C4#12411yescharacteristic
141C2×C4C4⋊Q811yescharacteristic
142C8—82no—
143C22—21no—
144C22—21no—
145C22—22no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22#39111yescharacteristic
156C22C2×#9911yescharacteristic
157C22#41011yescharacteristic
158C4—21no—
159C4—21no—
160C4—21no—
161C4—44no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—82no—
166C4—84no—
167C4AΣL(1,𝔽16)12yes—
168C2—21no—
169C2—21no—
170C2—42no—
171C2—42no—
172C2#199611yescharacteristic
173C2C2×AΣL(1,𝔽16)11yescharacteristic
174C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C23×Q8C4semidirect
15. #385C4semidirect
16. #380C4semidirect
47. #99Q8semidirect

Maximal subgroups

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

subgroupindex
#24092
#18002
C4×#992
#13902