Finite Group Databasebeta

← #12855

Subgroups — #12855

Order 256.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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#2311C211yesmaximal, characteristic
2#1785C211yesmaximal, characteristic
3#1782C211yesmaximal, characteristic
4#1390C214yesmaximal, complemented
5C22×#41C2211yescharacteristic
6C2×#117C411yescharacteristic, complemented
7C22×C4⋊C4C411yescharacteristic, complemented
8C2×#101—21no—
9C2×#100C2212yes—
10C2×#99—21no—
11C2×#99C2212yes—
12#407—21no—
13#391—21no—
14#390—21no—
15#386C412yescomplemented
16#386C2212yes—
17#381—21no—
18#380—21no—
19#380—21no—
20C2×#41—24no—
21C2×#41—42no—
22C2×#41—42no—
23C22×Q8D411yescharacteristic
24C22×D4C2×C411yescharacteristic
25C22×D4D411yescharacteristic
26C22×D4C2311yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C412yes—
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C2×M16—22no—
32#117—42no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—42no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4C2×C412yes—
41C2×C42—21no—
42#101—42no—
43#100—24no—
44#99—24no—
45#99—42no—
46C22⋊C8—42no—
47#95—22no—
48#95—22no—
49#95—42no—
50C24C22⋊C411yescharacteristic
51C24C22×C411yescharacteristic
52C24C2×D411yescharacteristic
53#41—416no—
54C2×Q8—22no—
55C2×Q8—22no—
56C2×Q8—41no—
57C2×D4—22no—
58C2×D4—22no—
59C2×D4—22no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4#4114yes—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—82no—
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84M16—42no—
85M16—42no—
86C2×C8—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C22⋊C4—82no—
94C22⋊C4—82no—
95C42—42no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23C2×C22⋊C411yescharacteristic
110C23#11712yes—
111C23#12312yes—
112Q8—44no—
113D4—44no—
114D4—44no—
115D4—44no—
116C2×C4—28no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—84no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4#9912yes—
139C2×C4#9912yes—
140C2×C4#12312yes—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—42no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—41no—
155C22—41no—
156C22—41no—
157C22—81no—
158C22—81no—
159C22#38111yescharacteristic
160C22C2×#9911yescharacteristic
161C22C2×#9911yescharacteristic
162C4—22no—
163C4—22no—
164C4—22no—
165C4—41no—
166C4—82no—
167C4—82no—
168C4—162no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2—42no—
173C2—81no—
174C2#181911yescharacteristic
175C2#212211yescharacteristic
176C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC22×D4C23
prime radicalC22×#41C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1390C2semidirect
6. C2×#117C4semidirect
7. C22×C4⋊C4C4semidirect
15. #386C4semidirect

Maximal subgroups

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

subgroupindex
#23112
#17852
#17822
#13902