Finite Group Databasebeta

← #10044

Subgroups — #10044

Order 256.

233 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 49 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#3235C211yesmaximal, characteristic
2#3113C211yesmaximal, characteristic, complemented
3#3103C211yesmaximal, characteristic
4#1536C211yesmaximal, characteristic
5#1497C211yesmaximal, characteristic, complemented
6#1467C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#501—21no—
12#490—21no—
13#483—21no—
14#482—22no—
15#481—21no—
16#476—21no—
17#475—21no—
18#475—21no—
19#474—21no—
20#462—21no—
21#460—21no—
22#451—21no—
23#448—21no—
24#441—22no—
25C4×SD16—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28#432C2211yescharacteristic
29#431C2211yescharacteristic
30#329—22no—
31Q8⋊C8—22no—
32#324—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C2×Q16—22no—
36C2×SD16—22no—
37C4⋊Q8D411yescharacteristic
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#127D411yescharacteristic
41#125—21no—
42#124—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45#121—21no—
46#121—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×D4—21no—
54C4×D4—21no—
55#117—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#107—22no—
61#106—22no—
62#106—22no—
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68#103—22no—
69#103—22no—
70#103—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8D412yes—
78#97—21no—
79#97—21no—
80#97D412yes—
81C4×C8D412yes—
82#41—82no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—42no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—41no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4#4111yescharacteristic
99Q16—44no—
100SD16—44no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—41no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C42—41no—
121C42—41no—
122C42—41no—
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C23—41no—
128C23#12111yescharacteristic
129Q8—22no—
130Q8—22no—
131Q8—41no—
132Q8—41no—
133Q8—42no—
134Q8—42no—
135Q8—81no—
136D4—42no—
137D4—42no—
138D4—81no—
139D4—82no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—82no—
151C2×C4—81no—
152C2×C4#12011yescharacteristic
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12711yescharacteristic
158C8—42no—
159C8—42no—
160C8—42no—
161C8—81no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22#39211yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—42no—
175C4—81no—
176C4—81no—
177C4—81no—
178C4—81no—
179C4#47112yes—
180C2—41no—
181C2—161no—
182C2#200911yescharacteristic
183C2#201011yescharacteristic
184C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
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. #3113C2semidirect
5. #1497C2semidirect
6. #1467C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32352
#31132
#31032
#15362
#14972
#14672
#12872