Finite Group Databasebeta

← #10515

Subgroups — #10515

Order 256.

183 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 47 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#3246C211yesmaximal, characteristic
2#1832C211yesmaximal, characteristic, complemented
3#1552C211yesmaximal, characteristic, complemented
4#1551C211yesmaximal, characteristic
5#1506C211yesmaximal, characteristic, complemented
6#1505C211yesmaximal, characteristic
7#1290C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#476—21no—
11#475—21no—
12#465—22no—
13#461—21no—
14#460—21no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19#376C2211yescharacteristic
20#336—22no—
21#333—21no—
22#332—21no—
23#332—21no—
24#331—21no—
25#328—22no—
26#327—22no—
27C2×#41—21no—
28C2×SD16—22no—
29C22×C8—21no—
30C22×C8D412yes—
31C4⋊Q8D411yescharacteristic
32C4⋊Q8D411yescharacteristic
33C4⋊Q8D411yescharacteristic
34#127D411yescharacteristic
35#125—21no—
36#124—21no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39#121—21no—
40#121—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×D4—21no—
44C4×D4—21no—
45#117—21no—
46C2×C42—21no—
47C2×C42C2311yescharacteristic
48#106—22no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55#103—22no—
56#102—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59#97—21no—
60#97—21no—
61#95—21no—
62#95D412yes—
63#41—82no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×Q8—21no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—21no—
70C2×D4—41no—
71C22×C4—22no—
72C22×C4—41no—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4#4111yescharacteristic
76SD16—82no—
77C2×C8—22no—
78C2×C8—42no—
79C2×C8—42no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C42—41no—
95C42—41no—
96C42—41no—
97C42C2×D411yescharacteristic
98C42C2×D411yescharacteristic
99C42#4111yescharacteristic
100C42#4111yescharacteristic
101C23—41no—
102C23#12111yescharacteristic
103Q8—41no—
104Q8—41no—
105Q8—81no—
106Q8—81no—
107D4—41no—
108D4—41no—
109D4—81no—
110D4—81no—
111D4—82no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—82no—
123C2×C4#12011yescharacteristic
124C2×C4#12111yescharacteristic
125C2×C4#12111yescharacteristic
126C2×C4#12311yescharacteristic
127C2×C4#12311yescharacteristic
128C2×C4#12411yescharacteristic
129C8—42no—
130C8—81no—
131C8—81no—
132C8—81no—
133C22—21no—
134C22—21no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22#39411yescharacteristic
139C22#47112yes—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—21no—
144C4—21no—
145C4—82no—
146C4—81no—
147C4—81no—
148C4—81no—
149C4#45411yescharacteristic
150C4#45511yescharacteristic
151C2—22no—
152C2—161no—
153C2#204011yescharacteristic
154C2#204611yescharacteristic
155C2#205311yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1832C2semidirect
3. #1552C2semidirect
5. #1506C2semidirect
7. #1290C2semidirect

Maximal subgroups

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

subgroupindex
#32462
#18322
#15522
#15512
#15062
#15052
#12902