Finite Group Databasebeta

← #13377

Subgroups — #13377

Order 256.

324 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 45 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#2804C211yesmaximal, characteristic, complemented
2#1792C211yesmaximal, characteristic
3#1790C212yesmaximal
4#1393C211yesmaximal, characteristic, complemented
5#1390C212yesmaximal, complemented
6C2×H(5,𝔽2)C2211yescharacteristic
7C2×#104—21no—
8#417—21no—
9C2×#100C2211yescharacteristic
10C2×#99—22no—
11C2×#99C2212yes—
12#400—21no—
13#393—22no—
14#391—21no—
15#390C2211yescharacteristic
16#386—22no—
17#386C2212yes—
18#385—22no—
19H(5,𝔽2)—22no—
20H(5,𝔽2)—21no—
21H(5,𝔽2)—42no—
22H(5,𝔽2)Q812yescomplemented
23C2×#41—21no—
24C2×#41—41no—
25C22×Q8—21no—
26C22×D4—22no—
27C22×D4—41no—
28C22×D4C2311yescharacteristic
29C23×C4D412yes—
30C2×M16—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4D411yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—42no—
38C2×C22⋊C4D412yes—
39C2×C42D411yescharacteristic
40#104—42no—
41#103—41no—
42#102—41no—
43#100—22no—
44#99—22no—
45#99—22no—
46#99—44no—
47#95—22no—
48#95—22no—
49#95—21no—
50#95—22no—
51#95—42no—
52C24—22no—
53C24C2×D412yes—
54#41—22no—
55#41—22no—
56#41—42no—
57#41—41no—
58#41—44no—
59#41—42no—
60#41—41no—
61C2×Q8—21no—
62C2×Q8—42no—
63C2×Q8—81no—
64C2×Q8—81no—
65C2×D4—21no—
66C2×D4—24no—
67C2×D4—24no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—44no—
75C2×D4—41no—
76C2×D4—42no—
77C2×D4C2×Q811yescharacteristic
78C2×D4#4111yescharacteristic
79C2×D4#4112yes—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—41no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—82no—
89C22×C4C2×D411yescharacteristic
90M16—41no—
91M16—41no—
92C2×C8—41no—
93C4⋊C4—41no—
94C4⋊C4—81no—
95C4⋊C4—81no—
96C22⋊C4—44no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—82no—
101C22⋊C4—84no—
102C42—41no—
103C23—22no—
104C23—22no—
105C23—24no—
106C23—42no—
107C23—42no—
108C23—42no—
109C23—41no—
110C23—82no—
111C23—82no—
112C23#12011yescharacteristic
113C23C22⋊Q812yes—
114C23#12412yes—
115Q8—22no—
116Q8—41no—
117Q8—84no—
118D4—22no—
119D4—41no—
120D4—42no—
121D4—41no—
122D4—41no—
123D4—44no—
124D4—44no—
125D4—42no—
126D4—42no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—24no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4C22⋊Q811yescharacteristic
151C2×C4#12411yescharacteristic
152C8—81no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—82no—
163C22—84no—
164C22—81no—
165C22—81no—
166C22—162no—
167C22#39311yescharacteristic
168C22B(4,𝔽2)11yescharacteristic
169C22B(4,𝔽2)11yescharacteristic
170C4—21no—
171C4—21no—
172C4—42no—
173C4—42no—
174C4—82no—
175C4—81no—
176C4—82no—
177C4—162no—
178C2—21no—
179C2—21no—
180C2—42no—
181C2—42no—
182C2—82no—
183C2#202911yescharacteristic
184C2#219811yescharacteristic
185C2#219911yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23
prime radicalC2×H(5,𝔽2)C22
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2804C2semidirect
4. #1393C2semidirect
5. #1390C2semidirect
22. H(5,𝔽2)Q8semidirect

Maximal subgroups

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

subgroupindex
#28042
#17922
#17902
#13932
#13902