Finite Group Databasebeta

← #18527

Subgroups — #18527

Order 256.

588 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 119 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#3481C211yesmaximal, characteristic, complemented
2#3151C211yesmaximal, characteristic, complemented
3#3038C212yesmaximal, complemented
4#1622C214yesmaximal, complemented
5#1620C212yesmaximal, complemented
6#1619C212yesmaximal, complemented
7#1560C211yesmaximal, characteristic, complemented
8#1489C212yesmaximal, complemented
9C2×AGL(1,ℤ/8)—22no—
10C2×AGL(1,ℤ/8)—22no—
11#559C2211yescharacteristic, complemented
12#554C2212yescomplemented
13#553—22no—
14#551—22no—
15#546—22no—
16D42C2211yescharacteristic, complemented
17#532C2211yescharacteristic
18C2×#127C2211yescharacteristic, complemented
19#529—22no—
20#525—22no—
21C4×#41C2212yes—
22C2×C4×D4—22no—
23#496—22no—
24#493C2212yes—
25#492C2212yescomplemented
26#460—22no—
27#460C2214yescomplemented
28#459—22no—
29#459C2214yes—
30#423C2212yescomplemented
31#418—22no—
32C4×M16C2211yescharacteristic, complemented
33#333C2212yescomplemented
34#331C2212yescomplemented
35Q8⋊C8C2214yescomplemented
36D4⋊C8C2214yescomplemented
37C2×#41D412yes—
38C2×#41D412yes—
39C22×D4—22no—
40C22×D4—21no—
41C22×D4—22no—
42C23×C4—22no—
43AGL(1,ℤ/8)—48no—
44AGL(1,ℤ/8)—48no—
45C2×SD16—24no—
46C2×SD16—24no—
47C2×D8—24no—
48C2×D8—24no—
49C2×M16—21no—
50C2×M16—22no—
51C4⋊Q8C2311yescharacteristic
52#127—22no—
53#127—21no—
54#127C2312yes—
55#127C2311yescharacteristic
56#126—24no—
57#125—22no—
58#124—22no—
59#124D412yes—
60#123—22no—
61#123—24no—
62#123—22no—
63C22⋊Q8—24no—
64C22⋊Q8—22no—
65#121—24no—
66#121—22no—
67#121—22no—
68#120—22no—
69#120—22no—
70C4×Q8D412yes—
71C4×Q8C2312yes—
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4D412yes—
80C4×D4C2312yes—
81#117—22no—
82#117—22no—
83#117D412yes—
84C2×C4⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C42—22no—
87C2×C42C2311yescharacteristic
88C4⋊C8—22no—
89C4⋊C8C2314yes—
90#102—24no—
91#102—24no—
92#97—21no—
93C4×C8C2312yes—
94C24—22no—
95C24—41no—
96#41—24no—
97#41—42no—
98#41—44no—
99C2×Q8C2×D412yes—
100C2×Q8C2×D412yes—
101C2×D4—24no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—22no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4C2×D412yes—
115C2×D4C2×D412yes—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—84no—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125SD16—48no—
126SD16—48no—
127D8—48no—
128D8—48no—
129M16—24no—
130M16—44no—
131C2×C8—22no—
132C2×C8—24no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4C2×D412yes—
139C4⋊C4C2×D412yes—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—22no—
148C42—22no—
149C42—22no—
150C42—22no—
151C42C2×D411yescharacteristic
152C42C2×D412yes—
153C42C2411yescharacteristic
154C23—22no—
155C23—22no—
156C23—48no—
157C23—42no—
158C23—84no—
159C23—82no—
160C23—82no—
161C23#12011yescharacteristic
162Q8—22no—
163Q8—22no—
164Q8—42no—
165D4—22no—
166D4—42no—
167D4—44no—
168D4—44no—
169D4—48no—
170D4—42no—
171D4—44no—
172D4—42no—
173D4—44no—
174D4—84no—
175D4—82no—
176D4—82no—
177D4AGL(1,ℤ/8)14yescomplemented
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4#12012yes—
193C2×C4#12011yescharacteristic
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D412yes—
196C8—24no—
197C8—44no—
198C22—22no—
199C22—24no—
200C22—21no—
201C22—42no—
202C22—42no—
203C22—44no—
204C22—42no—
205C22—88no—
206C22—84no—
207C22—82no—
208C22—164no—
209C22C2×#12011yescharacteristic
210C4—22no—
211C4—21no—
212C4—42no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4#45312yes—
217C4C2×AGL(1,ℤ/8)12yes—
218C4#57512yes—
219C2—41no—
220C2—44no—
221C2—82no—
222C2—162no—
223C2#300612yes—
224C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3481C2semidirect
2. #3151C2semidirect
3. #3038C2semidirect
4. #1622C2semidirect
5. #1620C2semidirect
6. #1619C2semidirect
7. #1560C2semidirect
8. #1489C2semidirect
11. #559C22semidirect
12. #554C22semidirect
16. D42C22semidirect
18. C2×#127C22semidirect
25. #492C22semidirect
27. #460C22semidirect
30. #423C22semidirect
32. C4×M16C22semidirect
33. #333C22semidirect
34. #331C22semidirect
35. Q8⋊C8C22semidirect
36. D4⋊C8C22semidirect
177. D4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#34812
#31512
#30382
#16222
#16202
#16192
#15602
#14892