Finite Group Databasebeta

← #12714

Subgroups — #12714

Order 256.

487 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 73 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#3487C211yesmaximal, characteristic
2#2134C211yesmaximal, characteristic, complemented
3#2128C211yesmaximal, characteristic, complemented
4#2120C212yesmaximal, complemented
5#1796C212yesmaximal, complemented
6C2×2−1+4C2211yescharacteristic
7C22×#41C411yescharacteristic, complemented
8#550—22no—
9#550—22no—
10#548—22no—
11#548—22no—
12#535—22no—
13#535C411yescharacteristic, complemented
14#535C2211yescharacteristic
15#525—22no—
16#525C411yescharacteristic, complemented
17#525C2211yescharacteristic
18#519C411yescharacteristic, complemented
19#421—22no—
20#419—22no—
21#413C2212yes—
22#410C2212yes—
23#355—22no—
24#353—22no—
25#352—22no—
26AΣL(1,𝔽16)—22no—
272−1+4—44no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—22no—
31C2×#41—21no—
32C2×#41—44no—
33C2×#41C2×C411yescharacteristic
34C2×#41D412yes—
35C2×#41C2311yescharacteristic
36C22×Q8—21no—
37C22×Q8C2×C411yescharacteristic
38C22×Q8D412yes—
39C22×D4—21no—
40C22×D4C2×C411yescharacteristic
41C23×C4—21no—
42C23×C4C2×C411yescharacteristic
43C2×M16—22no—
44#127—24no—
45#127—21no—
46#127D412yes—
47#124—24no—
48#124—21no—
49#124D412yes—
50#123—21no—
51#123D412yes—
52C22⋊Q8—24no—
53C22⋊Q8—21no—
54C22⋊Q8—21no—
55#121—24no—
56#121—24no—
57#121—24no—
58#121—21no—
59#121—22no—
60#121—21no—
61#121—22no—
62#120—21no—
63#120—22no—
64#120—22no—
65#120D412yes—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×D4—24no—
69C4×D4—24no—
70C4×D4—22no—
71C4×D4—22no—
72#117—22no—
73#117—22no—
74#117C2×C411yescharacteristic
75#117C2×C411yescharacteristic
76#104—44no—
77#103—44no—
78#101—22no—
79#100—22no—
80#99—22no—
81#99—22no—
82C24—21no—
83C24C22⋊C411yescharacteristic
84#41—24no—
85#41—42no—
86#41—42no—
87#41—42no—
88#41—48no—
89#41—48no—
90#41—44no—
91#41—44no—
92#41—44no—
93C2×Q8—24no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—24no—
97C2×Q8—42no—
98C2×Q8—44no—
99C2×Q8C22⋊C411yescharacteristic
100C2×Q8C22⋊C412yes—
101C2×Q8C2×D411yescharacteristic
102C2×D4—24no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—22no—
107C2×D4—24no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—44no—
111C2×D4—42no—
112C2×D4C22⋊C412yes—
113C2×D4C22⋊C411yescharacteristic
114C2×D4C2×D411yescharacteristic
115C2×D4C2×D412yes—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4C22⋊C411yescharacteristic
126C22×C4C22×C411yescharacteristic
127C22×C4C2×D412yes—
128M16—42no—
129M16—42no—
130C2×C8—42no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—42no—
137C22⋊C4—22no—
138C22⋊C4—24no—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C22⋊C4—22no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—42no—
145C22⋊C4C22⋊C412yes—
146C42—24no—
147C42—21no—
148C42—42no—
149C42C22⋊C412yes—
150C23—21no—
151C23—22no—
152C23—21no—
153C23—42no—
154C23—41no—
155C23—81no—
156C23—82no—
157C23C2×C22⋊C411yescharacteristic
158C23#12012yes—
159Q8—24no—
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—44no—
164Q8#9914yes—
165D4—22no—
166D4—42no—
167D4—41no—
168D4—44no—
169D4—42no—
170D4—48no—
171D4—42no—
172D4—42no—
173D4—84no—
174D4—82no—
175D4—84no—
176D4—82no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—48no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4C2×C22⋊C411yescharacteristic
195C2×C4C2×C22⋊C411yescharacteristic
196C2×C4#12012yes—
197C8—82no—
198C22—21no—
199C22—22no—
200C22—42no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—44no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—82no—
210C22—84no—
211C22#37911yescharacteristic
212C4—22no—
213C4—24no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—162no—
220C4C2×#9912yes—
221C2—21no—
222C2—42no—
223C2—41no—
224C2—81no—
225C2—81no—
226C2—82no—
227C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#3487C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2134C2semidirect
3. #2128C2semidirect
4. #2120C2semidirect
5. #1796C2semidirect
7. C22×#41C4semidirect
13. #535C4semidirect
16. #525C4semidirect
18. #519C4semidirect

Maximal subgroups

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

subgroupindex
#34872
#21342
#21282
#21202
#17962