Finite Group Databasebeta

← #9350

Subgroups — #9350

Order 256.

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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#3040C211yesmaximal, characteristic
2#2971C211yesmaximal, characteristic, complemented
3#1527C211yesmaximal, characteristic, complemented
4#1488C211yesmaximal, characteristic, complemented
5#1469C211yesmaximal, characteristic, complemented
6#1307C211yesmaximal, characteristic, complemented
7#1299C211yesmaximal, characteristic, complemented
8C2×#137C411yescharacteristic
9C2×AGL(1,ℤ/8)C411yescharacteristic
10C2×#124C2211yescharacteristic
11C4×#41C2211yescharacteristic
12#464—21no—
13#464—21no—
14#463—21no—
15#463—21no—
16C8×Q8—21no—
17#436—21no—
18#436—22no—
19#432—21no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24#419C411yescharacteristic
25#418C411yescharacteristic
26C4×M16—21no—
27C4×M16C2211yescharacteristic
28#329—21no—
29#329—21no—
30Q8⋊C8—21no—
31D4⋊C8—21no—
32#323—22no—
33C2×#41C2×C411yescharacteristic
34C22×Q8C2×C411yescharacteristic
35C22×D4C2×C411yescharacteristic
36#137—42no—
37AGL(1,ℤ/8)—42no—
38C2×Q16—21no—
39C2×SD16—21no—
40C2×SD16—21no—
41C2×D8—21no—
42C2×M16—22no—
43C2×M16—21no—
44C2×M16C2×C411yescharacteristic
45C2×M16C2×C411yescharacteristic
46#124—21no—
47#124—21no—
48#124—21no—
49#124—21no—
50C4×Q8D411yescharacteristic
51C4×Q8D411yescharacteristic
52C4×D4—21no—
53C4×D4—21no—
54C4×D4D411yescharacteristic
55C4×D4D411yescharacteristic
56#117—21no—
57#117C2×C411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62C4⋊C8—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68#103—21no—
69#103—21no—
70#102—21no—
71#102—21no—
72C22⋊C8—22no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8D412yes—
76#97—21no—
77#97—21no—
78#97D411yescharacteristic
79#97D411yescharacteristic
80C4×C8—22no—
81C4×C8—21no—
82C4×C8—21no—
83C24—21no—
84#41—22no—
85#41—41no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—41no—
89C2×Q8—41no—
90C2×Q8C22⋊C411yescharacteristic
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4C22⋊C411yescharacteristic
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4C22×C411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4#4111yescharacteristic
104Q16—42no—
105SD16—42no—
106SD16—42no—
107D8—42no—
108M16—42no—
109M16—44no—
110M16—41no—
111M16—41no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—21no—
117C2×C8—41no—
118C2×C8—41no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4C22⋊C411yescharacteristic
122C4⋊C4C22⋊C411yescharacteristic
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C42—21no—
131C42—21no—
132C42—21no—
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42#4111yescharacteristic
137C23—21no—
138C23—42no—
139C23—41no—
140C23—81no—
141C23—81no—
142C23C4×D411yescharacteristic
143Q8—21no—
144Q8—41no—
145Q8—41no—
146Q8—81no—
147Q8#10012yes—
148D4—21no—
149D4—21no—
150D4—42no—
151D4—41no—
152D4—41no—
153D4—81no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4C2×C22⋊C411yescharacteristic
167C2×C4C4×D411yescharacteristic
168C2×C4#12011yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12311yescharacteristic
172C8—22no—
173C8—42no—
174C8—41no—
175C8—41no—
176C8—41no—
177C8—81no—
178C8—81no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—162no—
189C22#38611yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—22no—
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—81no—
199C4C2×#10011yescharacteristic
200C4#41311yescharacteristic
201C2—41no—
202C2—81no—
203C2—161no—
204C2#188111yescharacteristic
205C2#190011yescharacteristic
206C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×AGL(1,ℤ/8)C4
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. #2971C2semidirect
3. #1527C2semidirect
4. #1488C2semidirect
5. #1469C2semidirect
6. #1307C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#30402
#29712
#15272
#14882
#14692
#13072
#12992