Finite Group Databasebeta

← #12722

Subgroups — #12722

Order 256.

395 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 69 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#3508C211yesmaximal, characteristic
2#2134C211yesmaximal, characteristic, complemented
3#2128C211yesmaximal, characteristic, complemented
4#2122C212yesmaximal, complemented
5#1827C212yesmaximal, complemented
6#559—22no—
7#550—22no—
8#546—24no—
9#540—22no—
10#535C2211yescharacteristic
11#535C2211yescharacteristic
12#533C411yescharacteristic, complemented
13#532—22no—
14C2×#127C411yescharacteristic, complemented
15#518C411yescharacteristic, complemented
16#518C411yescharacteristic, complemented
17C4×#41C2211yescharacteristic
18#413C2212yes—
19#410C2212yes—
20#355—22no—
21#353—22no—
22AΣL(1,𝔽16)—24no—
23#339—22no—
24H(3,ℤ/4)—22no—
25C2×#41—24no—
26C2×#41C2311yescharacteristic
27C22×D4—22no—
28C22×D4C2×C411yescharacteristic
29C22×D4C2×C411yescharacteristic
30C2×M16—22no—
31C4⋊Q8—21no—
32#127—22no—
33#127—21no—
34#127—21no—
35#127—21no—
36#127D412yes—
37#125—21no—
38#124—24no—
39#124—21no—
40#124D412yes—
41#123—24no—
42#123—24no—
43C22⋊Q8—22no—
44#121—24no—
45#121—24no—
46#121—24no—
47#121—22no—
48#121—22no—
49#120—22no—
50#120D414yes—
51C4×Q8—21no—
52C4×Q8—22no—
53C4×D4—24no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—21no—
57C4×D4—22no—
58C4×D4—22no—
59C4×D4—22no—
60#117—22no—
61#117C2×C411yescharacteristic
62#117C2×C411yescharacteristic
63#117C2×C411yescharacteristic
64#117D412yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C42C2×C411yescharacteristic
68C2×C42D412yes—
69#101—22no—
70#100—22no—
71#99—22no—
72#99—22no—
73C24—21no—
74C24C22⋊C412yes—
75#41—42no—
76#41—88no—
77C2×Q8—22no—
78C2×Q8C2×D411yescharacteristic
79C2×D4—24no—
80C2×D4—24no—
81C2×D4—24no—
82C2×D4—24no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—42no—
87C2×D4—42no—
88C2×D4—84no—
89C2×D4C22⋊C412yes—
90C2×D4C2×D411yescharacteristic
91C2×D4C2×D412yes—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—24no—
95C22×C4—24no—
96C22×C4C22×C411yescharacteristic
97C22×C4C2×D412yes—
98M16—42no—
99M16—42no—
100C2×C8—42no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C22⋊C4—24no—
110C22⋊C4—24no—
111C22⋊C4—21no—
112C22⋊C4—21no—
113C22⋊C4—22no—
114C22⋊C4—24no—
115C22⋊C4—24no—
116C22⋊C4—42no—
117C22⋊C4C22⋊C412yes—
118C42—21no—
119C42—21no—
120C42—22no—
121C42—42no—
122C42C22⋊C412yes—
123C42C22⋊C411yescharacteristic
124C42C22⋊C411yescharacteristic
125C42C22⋊C412yes—
126C23—22no—
127C23—22no—
128C23—42no—
129C23—42no—
130C23—82no—
131C23—82no—
132C23C2×C22⋊C411yescharacteristic
133C23#12012yes—
134Q8—41no—
135Q8—44no—
136D4—42no—
137D4—41no—
138D4—42no—
139D4—42no—
140D4—48no—
141D4—42no—
142D4—42no—
143D4—84no—
144D4—84no—
145D4—84no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—84no—
157C2×C4—84no—
158C2×C4—82no—
159C2×C4C2×C22⋊C411yescharacteristic
160C2×C4C2×C22⋊C411yescharacteristic
161C2×C4#12012yes—
162C8—82no—
163C22—22no—
164C22—21no—
165C22—42no—
166C22—41no—
167C22—44no—
168C22—44no—
169C22—42no—
170C22—42no—
171C22—162no—
172C22—162no—
173C22#37911yescharacteristic
174C4—22no—
175C4—22no—
176C4—21no—
177C4—42no—
178C4—41no—
179C4—42no—
180C4—42no—
181C4—162no—
182C4#41012yes—
183C2—21no—
184C2—42no—
185C2—41no—
186C2—82no—
187C2—82no—
188C2#178211yescharacteristic
subgroupG/H
center, socleC2#1782
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2134C2semidirect
3. #2128C2semidirect
4. #2122C2semidirect
5. #1827C2semidirect
12. #533C4semidirect
14. C2×#127C4semidirect
15. #518C4semidirect
16. #518C4semidirect

Maximal subgroups

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

subgroupindex
#35082
#21342
#21282
#21222
#18272