Finite Group Databasebeta

← #25690

Subgroups — #25690

Order 256.

536 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 123 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#3166C211yesmaximal, characteristic, complemented
3#3080C211yesmaximal, characteristic, complemented
4#2641C211yesmaximal, characteristic, complemented
5#2037C214yesmaximal, complemented
6#1995C211yesmaximal, characteristic, complemented
7#1965C211yesmaximal, characteristic, complemented
8#1856C214yesmaximal, complemented
9#1759C211yesmaximal, characteristic, complemented
10#559—22no—
11#550—22no—
12#546—24no—
13#540—22no—
14#535C2212yes—
15#533C2211yescharacteristic
16#532—22no—
17C2×#127C2211yescharacteristic
18C2×#125C2211yescharacteristic
19#518C2212yes—
20C4×#41C2211yescharacteristic
21#501—22no—
22#499—22no—
23#479—22no—
24#478—22no—
25#428C2212yes—
26#428C2214yes—
27#423C2211yescharacteristic
28C2×#104C2212yes—
29#417C2212yes—
30C2×#100C2214yes—
31C4×M16C2211yescharacteristic
32#394—24no—
33#390C2214yes—
34#376C2211yescharacteristic
35#339C2212yes—
36#339C2214yes—
37C2×#41—24no—
38C2×#41C2311yescharacteristic
39C22×D4D414yes—
40C22×D4C2312yes—
41C2×M16C2312yes—
42C2×M16C2312yes—
43C4⋊Q8D412yes—
44#127—22no—
45#127—21no—
46#127—21no—
47#127—22no—
48#125—22no—
49#125—21no—
50#125—21no—
51#125D412yes—
52#124—24no—
53#124—22no—
54#123—28no—
55C22⋊Q8—22no—
56#121—24no—
57#121—28no—
58#121—24no—
59#120—24no—
60C4×Q8—22no—
61C4×Q8D412yes—
62C4×D4—24no—
63C4×D4—22no—
64C4×D4—24no—
65C4×D4—24no—
66C4×D4D412yes—
67#117—21no—
68#117C2311yescharacteristic
69#117C2312yes—
70C2×C4⋊C4—22no—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—28no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—22no—
78#107—24no—
79#106—22no—
80#106—24no—
81C4⋊C8—22no—
82#104—24no—
83#103—22no—
84#102—22no—
85#100—28no—
86#97—21no—
87C4×C8—21no—
88#95—22no—
89C24C2×D414yes—
90#41—22no—
91#41—22no—
92#41—88no—
93C2×Q8—22no—
94C2×Q8C2×D411yescharacteristic
95C2×D4—21no—
96C2×D4—28no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—84no—
104C2×D4C2×D411yescharacteristic
105C22×C4—28no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4C2×D412yes—
111C22×C4C2411yescharacteristic
112M16—22no—
113M16—24no—
114M16#4114yescomplemented
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4C2×D412yes—
125C4⋊C4C2×D414yes—
126C22⋊C4—28no—
127C22⋊C4—28no—
128C22⋊C4—21no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—88no—
132C42—22no—
133C42—21no—
134C42—22no—
135C42—22no—
136C42—22no—
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C42C2×D412yes—
140C23—24no—
141C23—21no—
142C23—28no—
143C23—48no—
144C23C22×D411yescharacteristic
145Q8—21no—
146Q8—21no—
147Q8—44no—
148D4—21no—
149D4—21no—
150D4—48no—
151D4—42no—
152D4—42no—
153D4—42no—
154D4—44no—
155D4—84no—
156D4—88no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—88no—
168C2×C4#12012yes—
169C2×C4#12012yes—
170C2×C4C22×D411yescharacteristic
171C2×C4C22×D411yescharacteristic
172C2×C4C2×#4112yes—
173C2×C42−1+412yes—
174C8—24no—
175C8—42no—
176C22—22no—
177C22—41no—
178C22—44no—
179C22—42no—
180C22—48no—
181C22—88no—
182C22C2×#12011yescharacteristic
183C22#54712yes—
184C4—22no—
185C4—22no—
186C4—41no—
187C4—42no—
188C4—44no—
189C4—82no—
190C4—82no—
191C4#53211yescharacteristic
192C4#53611yescharacteristic
193C4#54812yes—
194C2—22no—
195C2—81no—
196C2—84no—
197C2#243211yescharacteristic
198C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3508C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3166C2semidirect
3. #3080C2semidirect
4. #2641C2semidirect
5. #2037C2semidirect
6. #1995C2semidirect
7. #1965C2semidirect
8. #1856C2semidirect
9. #1759C2semidirect
114. M16#41semidirect

Maximal subgroups

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

subgroupindex
#35082
#31662
#30802
#26412
#20372
#19952
#19652
#18562
#17592