Finite Group Databasebeta

← #18309

Subgroups — #18309

Order 256.

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

normal subgroups only

This group has 584 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#3449C211yesmaximal, characteristic
2#3227C212yesmaximal, complemented
3#2628C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1613C214yesmaximal, complemented
6#1517C212yesmaximal, complemented
7#1458C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9#537—21no—
10#537C2212yes—
11#536—22no—
12#535—22no—
13#534—21no—
14#534C2212yes—
15#532—21no—
16#532C2212yes—
17C2×C4⋊Q8C2211yescharacteristic, complemented
18#525—24no—
19C2×#117C2211yescharacteristic
20#498—22no—
21#486—22no—
22#481C2218yescomplemented
23#476—24no—
24#431C2212yescomplemented
25#400C2212yescomplemented
26#399C2212yes—
27#395—21no—
28#393—21no—
29#389—21no—
30#382C2211yescharacteristic, complemented
31#380—21no—
32#327C2218yescomplemented
33#324C2212yescomplemented
34#323C2212yes—
35C2×#41—22no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—44no—
40C2×#41—44no—
41C22×Q8—21no—
42C22×Q8—21no—
43C22×D4—21no—
44C22×D4—22no—
45C23×C4—22no—
46C23×C4D411yescharacteristic
47C4⋊Q8—21no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8D412yes—
51#127—21no—
52#127D412yes—
53#124—22no—
54#124—22no—
55#123—24no—
56#123—24no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59#121—24no—
60#121—24no—
61#121—22no—
62#121—22no—
63#121—22no—
64#121C2314yes—
65#120—24no—
66#120—24no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4D414yes—
71#117—22no—
72#117—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4D411yescharacteristic
81C2×C22⋊C4C2311yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84#106—28no—
85C4⋊C8—22no—
86#102—28no—
87C22⋊C8C2314yes—
88C4×C8—22no—
89#95—22no—
90#95—21no—
91#95—22no—
92#95C2312yes—
93C24—22no—
94C24C2×D411yescharacteristic
95#41—44no—
96#41—48no—
97#41—48no—
98#41—44no—
99#41—44no—
100#41—44no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—44no—
104C2×Q8—42no—
105C2×Q8—82no—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—24no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4C2×D414yes—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—48no—
126C22×C4SD1614yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133C2×C8—24no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—24no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—82no—
144C4⋊C4C2×D414yes—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—22no—
150C22⋊C4—24no—
151C22⋊C4—82no—
152C42—21no—
153C42—21no—
154C42C2×D412yes—
155C42C2×D412yes—
156C23—21no—
157C23—21no—
158C23—24no—
159C23—41no—
160C23—41no—
161C23—44no—
162C23—82no—
163C23—82no—
164C23C2×SD1612yes—
165C23C22×D411yescharacteristic
166Q8—44no—
167Q8—44no—
168Q8—84no—
169D4—44no—
170D4—44no—
171D4—48no—
172D4—44no—
173D4—44no—
174D4—44no—
175D4—82no—
176D4—82no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—24no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—48no—
188C2×C4—44no—
189C2×C4—48no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4#12012yes—
199C2×C4#12011yescharacteristic
200C2×C4#12011yescharacteristic
201C2×C4C2×SD1612yes—
202C2×C4C2×SD1612yes—
203C2×C4AGL(1,ℤ/8)12yes—
204C2×C4C22×D412yes—
205C8—44no—
206C22—21no—
207C22—21no—
208C22—22no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—44no—
214C22—44no—
215C22—81no—
216C22—88no—
217C22C2×#12011yescharacteristic
218C22C22×SD1611yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—44no—
224C4—41no—
225C4—41no—
226C4—82no—
227C4—82no—
228C4#45012yes—
229C4#45012yes—
230C2—21no—
231C2—21no—
232C2—42no—
233C2—84no—
234C2C2×#45011yescharacteristic
235C2#302111yescharacteristic
236C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3227C2semidirect
3. #2628C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1517C2semidirect
7. #1458C2semidirect
17. C2×C4⋊Q8C22semidirect
22. #481C22semidirect
24. #431C22semidirect
25. #400C22semidirect
30. #382C22semidirect
32. #327C22semidirect
33. #324C22semidirect

Maximal subgroups

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

subgroupindex
#34492
#32272
#26282
#16162
#16132
#15172
#14582