Finite Group Databasebeta

← #9244

Subgroups — #9244

Order 256.

232 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#464C211yesmaximal, characteristic
2#2984C211yesmaximal, characteristic, complemented
3#1527C211yesmaximal, characteristic
4#1491C211yesmaximal, characteristic, complemented
5#1469C211yesmaximal, characteristic
6#1290C211yesmaximal, characteristic, complemented
7#1283C211yesmaximal, characteristic, complemented
8C22×Q16C411yescharacteristic
9C22×SD16C411yescharacteristic
10C2×#124C2211yescharacteristic
11C2×C4×Q8C2211yescharacteristic
12#464—21no—
13#464—21no—
14#464—21no—
15#464—21no—
16#446—21no—
17#446—22no—
18C8×Q8—21no—
19#432—21no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431—21no—
23#431C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C411yescharacteristic
26C2×#102C411yescharacteristic
27C4×M16C2211yescharacteristic
28#329—22no—
29Q8⋊C8—21no—
30Q8⋊C8—21no—
31#323—21no—
32#323—21no—
33C22×Q8C2×C411yescharacteristic
34C22×Q8C2×C411yescharacteristic
35C22×D4C2×C411yescharacteristic
36C2×Q16—21no—
37C2×Q16—21no—
38C2×Q16—42no—
39C2×SD16—21no—
40C2×SD16—21no—
41C2×SD16—42no—
42C2×M16—21no—
43C22×C8C2×C411yescharacteristic
44C22×C8C2×C411yescharacteristic
45#124—21no—
46#124—21no—
47#124—21no—
48#124—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8D411yescharacteristic
52C4×Q8D411yescharacteristic
53C4×Q8D411yescharacteristic
54C4×Q8D411yescharacteristic
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2×C411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61C4⋊C8—22no—
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—21no—
73C22⋊C8—21no—
74C22⋊C8D411yescharacteristic
75C22⋊C8D411yescharacteristic
76#97—22no—
77#97—21no—
78#97—21no—
79#97D412yes—
80C4×C8—22no—
81C4×C8—21no—
82C4×C8—21no—
83C24—21no—
84C2×Q8—21no—
85C2×Q8—22no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—41no—
89C2×Q8—41no—
90C2×Q8—41no—
91C2×Q8C22⋊C411yescharacteristic
92C2×Q8C22⋊C411yescharacteristic
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—41no—
96C2×D4—41no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4C22×C411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4111yescharacteristic
103Q16—42no—
104Q16—42no—
105SD16—42no—
106SD16—42no—
107M16—22no—
108M16—41no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4C22⋊C411yescharacteristic
124C4⋊C4C22⋊C411yescharacteristic
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—21no—
130C42—21no—
131C42—21no—
132C42—21no—
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42#4111yescharacteristic
137C23—41no—
138C23—41no—
139C23—41no—
140C23—81no—
141C23—81no—
142C23C4×D411yescharacteristic
143Q8—21no—
144Q8—21no—
145Q8—21no—
146Q8—42no—
147Q8—41no—
148Q8—41no—
149Q8—81no—
150Q8#10012yes—
151D4—41no—
152D4—41no—
153D4—81no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—81no—
168C2×C4C2×C22⋊C411yescharacteristic
169C2×C4C4×D411yescharacteristic
170C2×C4#12011yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12311yescharacteristic
174C8—41no—
175C8—41no—
176C8—42no—
177C8—42no—
178C8—41no—
179C8—41no—
180C8—81no—
181C22—21no—
182C22—21no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—161no—
187C22—161no—
188C22#38611yescharacteristic
189C22#44311yescharacteristic
190C22#44411yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—22no—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—81no—
201C4C2×#10011yescharacteristic
202C4#41311yescharacteristic
203C2—21no—
204C2—21no—
205C2—161no—
206C2#187611yescharacteristic
207C2#190011yescharacteristic
208C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×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. #2984C2semidirect
4. #1491C2semidirect
6. #1290C2semidirect
7. #1283C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
#29842
#15272
#14912
#14692
#12902
#12832