Finite Group Databasebeta

← #11906

Subgroups — #11906

Order 256.

220 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 57 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#2564C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1354C212yesmaximal, complemented
5#1353C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8#423—21no—
9C2×#100—21no—
10C2×C22⋊C8—21no—
11#400C2211yescharacteristic
12#399C411yescharacteristic
13#395C2211yescharacteristic
14#393C411yescharacteristic
15#387—21no—
16#386—21no—
17#385—21no—
18#380—21no—
19C2×#95C411yescharacteristic
20#343C2212yes—
21#324C2212yes—
22#323—21no—
23C22×Q8C2×C411yescharacteristic
24C22×D4D411yescharacteristic
25C23×C4—21no—
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C2×M16—22no—
29C22×C8—41no—
30C22⋊Q8—41no—
31C22⋊Q8—41no—
32C4×D4—41no—
33C4×D4—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4D411yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—41no—
46C4⋊C8—41no—
47#100—42no—
48C22⋊C8—22no—
49C22⋊C8—41no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C22⋊C411yescharacteristic
59C2×Q8—41no—
60C2×Q8—81no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4SD1612yes—
64C2×D4Q1612yes—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—81no—
76C22×C4—81no—
77C22×C4—81no—
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83M16—42no—
84M16—42no—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—82no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—42no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—81no—
99C42—21no—
100C42—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23#10312yes—
110C23#10312yes—
111C23C2×C22⋊C411yescharacteristic
112Q8—82no—
113D4—42no—
114D4—42no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—84no—
141C2×C4#9911yescharacteristic
142C2×C4#9911yescharacteristic
143C2×C4C2×SD1611yescharacteristic
144C2×C4C2×Q1611yescharacteristic
145C8—82no—
146C8—82no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—22no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22—81no—
159C22C2×#9911yescharacteristic
160C22C2×#10311yescharacteristic
161C22#42111yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—42no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—81no—
172C4—81no—
173C2—21no—
174C2—21no—
175C2—42no—
176C2—42no—
177C2#152011yescharacteristic
178C2#212111yescharacteristic
179C2#213211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1354C2semidirect
5. #1353C2semidirect

Maximal subgroups

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

subgroupindex
#25642
#18002
#14592
#13542
#13532