Finite Group Databasebeta

← #12723

Subgroups — #12723

Order 256.

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

normal subgroups only

This group has 347 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#3510C211yesmaximal, characteristic
2#2134C211yesmaximal, characteristic
3#2128C211yesmaximal, characteristic
4#2123C212yesmaximal, complemented
5#1827C212yesmaximal, complemented
6#559—22no—
7#556—22no—
8#551—24no—
9#542—22no—
10#536C2211yescharacteristic
11#535C2211yescharacteristic
12#533C411yescharacteristic, complemented
13#529—22no—
14C2×#124C411yescharacteristic, complemented
15#519C411yescharacteristic, complemented
16#518C411yescharacteristic, complemented
17C4×#41C2211yescharacteristic
18#413C2212yes—
19#410C2212yes—
20#355—22no—
21#353—22no—
22#352—24no—
23#339—22no—
24H(3,ℤ/4)—22no—
25C2×#41C2311yescharacteristic
26C22×Q8C2×C411yescharacteristic
27C22×D4C2×C411yescharacteristic
28C2×M16—22no—
29C4⋊Q8—21no—
30C4⋊Q8—21no—
31#127D412yes—
32#126—24no—
33#126—24no—
34#125—21no—
35#124—24no—
36#124—22no—
37#124—21no—
38#124—21no—
39#124—21no—
40#124D412yes—
41#123—24no—
42#123—24no—
43#123D414yes—
44C22⋊Q8—24no—
45C22⋊Q8—22no—
46C22⋊Q8—22no—
47#121—22no—
48#120—22no—
49C4×Q8—21no—
50C4×Q8—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×D4—24no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—21no—
57C4×D4—22no—
58#117—24no—
59#117—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—
74#41—42no—
75C2×Q8—22no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—42no—
79C2×Q8C2×D411yescharacteristic
80C2×D4—24no—
81C2×D4—21no—
82C2×D4—42no—
83C2×D4C22⋊C412yes—
84C2×D4C2×D411yescharacteristic
85C2×D4C2×D412yes—
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4C22⋊C412yes—
90C22×C4C22×C411yescharacteristic
91C22×C4C2×D412yes—
92M16—42no—
93M16—42no—
94C2×C8—42no—
95C4⋊C4—22no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—24no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C22⋊C4—24no—
106C22⋊C4—24no—
107C22⋊C4—24no—
108C22⋊C4—21no—
109C22⋊C4—21no—
110C22⋊C4—22no—
111C22⋊C4—24no—
112C22⋊C4—24no—
113C22⋊C4—42no—
114C22⋊C4C22⋊C412yes—
115C42—24no—
116C42—21no—
117C42—21no—
118C42—22no—
119C42—42no—
120C42C22⋊C412yes—
121C42C22⋊C411yescharacteristic
122C42C22⋊C411yescharacteristic
123C42C22⋊C412yes—
124C23—22no—
125C23—42no—
126C23—82no—
127C23C2×C22⋊C411yescharacteristic
128C23#12012yes—
129Q8—41no—
130Q8—42no—
131Q8—42no—
132Q8—82no—
133D4—42no—
134D4—41no—
135D4—42no—
136D4—42no—
137D4—84no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—82no—
149C2×C4—84no—
150C2×C4—81no—
151C2×C4—82no—
152C2×C4C2×C22⋊C411yescharacteristic
153C2×C4C2×C22⋊C411yescharacteristic
154C2×C4#12012yes—
155C8—82no—
156C22—21no—
157C22—22no—
158C22—42no—
159C22—41no—
160C22—44no—
161C22—42no—
162C22—162no—
163C22#37911yescharacteristic
164C4—22no—
165C4—21no—
166C4—42no—
167C4—41no—
168C4—42no—
169C4—41no—
170C4—42no—
171C4—42no—
172C4—162no—
173C4#41012yes—
174C2—21no—
175C2—42no—
176C2—41no—
177C2—82no—
178C2#178211yescharacteristic
subgroupG/H
center, socleC2#1782
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#535C22
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
4. #2123C2semidirect
5. #1827C2semidirect
12. #533C4semidirect
14. C2×#124C4semidirect
15. #519C4semidirect
16. #518C4semidirect

Maximal subgroups

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

subgroupindex
#35102
#21342
#21282
#21232
#18272