Finite Group Databasebeta

← #11848

Subgroups — #11848

Order 256.

286 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 51 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#2406C211yesmaximal, characteristic
2#1821C211yesmaximal, characteristic, complemented
3C2×#324C211yesmaximal, characteristic, complemented
4#1354C212yesmaximal, complemented
5#1345C212yesmaximal, complemented
6C22×#41C411yescharacteristic
7C2×#123—21no—
8C22×C4⋊C4C2211yescharacteristic
9#407—21no—
10C2×C22⋊C8—21no—
11#395C411yescharacteristic
12#395C2211yescharacteristic
13#394C411yescharacteristic
14#394C2211yescharacteristic
15#386—21no—
16#385—21no—
17#385—21no—
18#381C411yescharacteristic
19#343—21no—
20#343C2212yes—
21#324—21no—
22#324C2212yes—
23C2×#41—41no—
24C2×#41—41no—
25C2×#41—44no—
26C22×Q8C2×C411yescharacteristic
27C22×D4—21no—
28C22×D4C2×C411yescharacteristic
29C23×C4—21no—
30C23×C4C2×C411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4D411yescharacteristic
33C2×M16—22no—
34C22×C8—41no—
35#123—44no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4—41no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4C2311yescharacteristic
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C22⋊C8—22no—
50C22⋊C8—41no—
51C22⋊C8—41no—
52C22⋊C8—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24—21no—
59C24C22⋊C411yescharacteristic
60#41—44no—
61#41—44no—
62#41—44no—
63#41—44no—
64C2×Q8—42no—
65C2×Q8—41no—
66C2×D4—24no—
67C2×D4—41no—
68C2×D4—42no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—42no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22×C411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94M16—42no—
95M16—42no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—82no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C23—22no—
111C23—21no—
112C23—21no—
113C23—22no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23#9911yescharacteristic
122C23#9911yescharacteristic
123C23C2×C22⋊C411yescharacteristic
124Q8—42no—
125Q8—42no—
126D4—44no—
127D4—42no—
128D4—42no—
129D4—44no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—48no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4#13512yes—
160C8—82no—
161C8—82no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22AΣL(1,𝔽16)12yes—
179C22C2×#9911yescharacteristic
180C22#41611yescharacteristic
181C22#42111yescharacteristic
182C4—22no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—81no—
190C4—81no—
191C2—21no—
192C2—21no—
193C2—22no—
194C2—41no—
195C2—82no—
196C2#151611yescharacteristic
197C2C2×AΣL(1,𝔽16)11yescharacteristic
198C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×#41C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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
2. #1821C2semidirect
3. C2×#324C2semidirect
4. #1354C2semidirect
5. #1345C2semidirect

Maximal subgroups

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

subgroupindex
#24062
#18212
C2×#3242
#13542
#13452