Finite Group Databasebeta

← #9340

Subgroups — #9340

Order 256.

240 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 59 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#3217C211yesmaximal, characteristic
2#2995C211yesmaximal, characteristic, complemented
3#1965C211yesmaximal, characteristic, complemented
4#1555C211yesmaximal, characteristic, complemented
5#1529C211yesmaximal, characteristic, complemented
6#1307C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8C2×#127C2211yescharacteristic
9#529C2211yescharacteristic
10#496—21no—
11#490—21no—
12#460—21no—
13#459—21no—
14#450C412yes—
15#447C412yes—
16#446—21no—
17#436—21no—
18#435—21no—
19#435—21no—
20#433—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#424—21no—
25#423C2211yescharacteristic
26C2×#100—21no—
27C2×#100—21no—
28C4×M16C2211yescharacteristic
29#331—21no—
30#331—21no—
31#330—21no—
32#329—21no—
33C22×D4—21no—
34C22×D4—21no—
35C22×D4C2×C412yes—
36C2×SD16—22no—
37C2×D8—22no—
38C2×M16—21no—
39C2×M16—21no—
40C2×M16D411yescharacteristic
41C2×M16D411yescharacteristic
42C22×C8—21no—
43#127—21no—
44#127—21no—
45#127—21no—
46#127—21no—
47#126—21no—
48#125D411yescharacteristic
49#124D411yescharacteristic
50#123—21no—
51C22⋊Q8C2×C411yescharacteristic
52#121C2×C411yescharacteristic
53C4×Q8D411yescharacteristic
54C4×D4—21no—
55C4×D4D411yescharacteristic
56#117—21no—
57C2×C42C2311yescharacteristic
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8D411yescharacteristic
65C4⋊C8D411yescharacteristic
66#102—22no—
67#102—22no—
68#100—42no—
69#100—42no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8C2×C412yes—
75#97—21no—
76#97—21no—
77#97—21no—
78#97—21no—
79#97—21no—
80C4×C8—21no—
81C4×C8—21no—
82C24—22no—
83C2×Q8C22⋊C411yescharacteristic
84C2×D4—21no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—82no—
96C2×D4—82no—
97C2×D4C22⋊C411yescharacteristic
98C22×C4—21no—
99C22×C4C22×C411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4#4111yescharacteristic
102SD16—82no—
103D8—82no—
104M16—22no—
105M16—41no—
106M16—41no—
107M16—41no—
108M16—81no—
109C2×C8—22no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—81no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4C22⋊C411yescharacteristic
122C4⋊C4C22⋊C411yescharacteristic
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—21no—
127C22⋊C4—21no—
128C42—21no—
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42#4111yescharacteristic
133C23—21no—
134C23—42no—
135C23—42no—
136C23—42no—
137C23—82no—
138C23—82no—
139C23C4×D411yescharacteristic
140Q8—41no—
141D4—41no—
142D4—42no—
143D4—42no—
144D4—82no—
145D4—81no—
146D4—81no—
147D4—82no—
148D4—82no—
149D4—81no—
150D4—81no—
151D4—81no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×C22⋊C411yescharacteristic
162C2×C4C4×D411yescharacteristic
163C2×C4#12011yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12311yescharacteristic
167C8—42no—
168C8—42no—
169C8—41no—
170C8—41no—
171C8—41no—
172C8—41no—
173C8—81no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22—162no—
184C22—162no—
185C22#38611yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—41no—
192C4—41no—
193C4—41no—
194C4#45311yescharacteristic
195C4#45411yescharacteristic
196C2—41no—
197C2—81no—
198C2—162no—
199C2#188311yescharacteristic
200C2#190311yescharacteristic
201C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3217C2
LCSG ▷ C22×C4 ▷ 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. #2995C2semidirect
3. #1965C2semidirect
4. #1555C2semidirect
5. #1529C2semidirect
6. #1307C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32172
#29952
#19652
#15552
#15292
#13072
#12912