Finite Group Databasebeta

← #25874

Subgroups — #25874

Order 256.

653 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 653 conjugacy classes of subgroups — too many to draw. Only its 127 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#3496C211yesmaximal, characteristic
2#3142C212yesmaximal, complemented
3#2566C211yesmaximal, characteristic, complemented
4#2040C212yesmaximal, complemented
5#2039C212yesmaximal, complemented
6#1980C212yesmaximal, complemented
7#1792C212yesmaximal, complemented
8#1792C212yesmaximal, complemented
9C4×#100C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)C2212yes—
11#547—22no—
12#546—24no—
13D42—22no—
14#537C2212yes—
15#534—22no—
16#532C2212yes—
17C2×#125C2211yescharacteristic, complemented
18C2×#121—22no—
19C2×C4×D4C2211yescharacteristic
20#491—22no—
21#490—22no—
22#488—24no—
23C2×#104C2214yescomplemented
24C2×#104C2214yescomplemented
25#417C2214yescomplemented
26#417C2214yescomplemented
27C2×#100C2212yes—
28C4×M16C2212yescomplemented
29#396—22no—
30#390C2211yescharacteristic
31#390C2211yescharacteristic
32#385—22no—
33#384C2211yescharacteristic, complemented
34#382C2211yescharacteristic, complemented
35C4×C4⋊C4C2211yescharacteristic, complemented
36#343C2212yescomplemented
37H(5,𝔽2)—24no—
38H(5,𝔽2)—24no—
39H(5,𝔽2)—44no—
40C2×#41—24no—
41C2×#41C2314yes—
42C22×D4—24no—
43C22×D4—22no—
44C22×D4—24no—
45C22×D4C2311yescharacteristic
46C23×C4D412yes—
47C2×M16C2314yes—
48C4⋊Q8D412yes—
49#127D412yes—
50#125—22no—
51#125—22no—
52#124D414yes—
53#123—28no—
54C22⋊Q8—24no—
55#121—24no—
56#121—24no—
57#121—24no—
58#121—22no—
59#121—24no—
60#121—22no—
61#120—28no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4D412yes—
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#104—44no—
82#104—44no—
83#103—24no—
84#103—24no—
85#102—24no—
86#102—24no—
87#100—24no—
88#97—22no—
89C4×C8—22no—
90#95—21no—
91#95—22no—
92C24—24no—
93C24C2×D412yes—
94#41—24no—
95#41—24no—
96#41—24no—
97#41—24no—
98#41—48no—
99#41—48no—
100C2×Q8C2×D414yes—
101C2×D4—28no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—48no—
109C2×D4—48no—
110C2×D4—48no—
111C2×D4—48no—
112C2×D4—84no—
113C2×D4C2×D414yes—
114C2×D4#4114yes—
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—82no—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128M16—24no—
129M16—24no—
130C2×C8—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—82no—
142C22⋊C4—28no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—44no—
147C22⋊C4—82no—
148C42—22no—
149C42—22no—
150C42C2×D412yes—
151C42C2×D412yes—
152C23—22no—
153C23—22no—
154C23—24no—
155C23—42no—
156C23—42no—
157C23—48no—
158C23—84no—
159C23—84no—
160C23#12414yes—
161C23C22×D411yescharacteristic
162Q8—24no—
163Q8—24no—
164D4—28no—
165D4—24no—
166D4—24no—
167D4—44no—
168D4—416no—
169D4—44no—
170D4—44no—
171D4—82no—
172D4—84no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—44no—
179C2×C4—41no—
180C2×C4—48no—
181C2×C4—41no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4#12012yes—
194C2×C4#12011yescharacteristic
195C2×C4#12011yescharacteristic
196C2×C4C22×D412yes—
197C2×C4C2×#4112yes—
198C2×C4H(5,𝔽2)12yes—
199C8—44no—
200C22—21no—
201C22—21no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—44no—
207C22—44no—
208C22—44no—
209C22—82no—
210C22—168no—
211C22C2×#12011yescharacteristic
212C22C2×#12411yescharacteristic
213C22#53511yescharacteristic
214C4—22no—
215C4—21no—
216C4—21no—
217C4—42no—
218C4—44no—
219C4—42no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4#54614yes—
224C2—21no—
225C2—21no—
226C2—44no—
227C2—84no—
228C2#243511yescharacteristic
229C2#301811yescharacteristic
230C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3496C2
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. #3142C2semidirect
3. #2566C2semidirect
4. #2040C2semidirect
5. #2039C2semidirect
6. #1980C2semidirect
7. #1792C2semidirect
8. #1792C2semidirect
9. C4×#100C2semidirect
17. C2×#125C22semidirect
23. C2×#104C22semidirect
24. C2×#104C22semidirect
25. #417C22semidirect
26. #417C22semidirect
28. C4×M16C22semidirect
33. #384C22semidirect
34. #382C22semidirect
35. C4×C4⋊C4C22semidirect
36. #343C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#31422
#25662
#20402
#20392
#19802
#17922
C4×#1002