Finite Group Databasebeta

← #18308

Subgroups — #18308

Order 256.

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

normal subgroups only

This group has 594 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#3226C212yesmaximal, complemented
3#2621C211yesmaximal, characteristic, complemented
4#1610C214yesmaximal, complemented
5#1607C214yesmaximal, 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×C22⋊Q8—21no—
20C2×#117C2211yescharacteristic
21#500—22no—
22#486—22no—
23#480C2218yescomplemented
24#474—24no—
25#431C2212yescomplemented
26#398—21no—
27#398C2212yescomplemented
28#397—21no—
29#397C2212yes—
30#391—21no—
31#382C2211yescharacteristic, complemented
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—
59C22⋊Q8—44no—
60#121—24no—
61#121—24no—
62#121—22no—
63#121—22no—
64#121—22no—
65#121C2314yes—
66#120—24no—
67#120—24no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4D414yes—
72#117—22no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—22no—
81C2×C22⋊C4D411yescharacteristic
82C2×C22⋊C4C2311yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85#107—28no—
86C4⋊C8—22no—
87#102—28no—
88C22⋊C8C2314yes—
89C4×C8—22no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95C2312yes—
94C24—22no—
95C24C2×D411yescharacteristic
96#41—44no—
97#41—48no—
98#41—48no—
99#41—44no—
100#41—44no—
101#41—44no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—44no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—24no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4C2×D414yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—48no—
129C22×C4D814yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136C2×C8—24no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—24no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4C2×D414yes—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—22no—
150C22⋊C4—24no—
151C22⋊C4—22no—
152C22⋊C4—24no—
153C22⋊C4—44no—
154C42—21no—
155C42—21no—
156C42C2×D412yes—
157C42C2×D412yes—
158C23—21no—
159C23—21no—
160C23—24no—
161C23—41no—
162C23—41no—
163C23—44no—
164C23—82no—
165C23—82no—
166C23C2×D812yes—
167C23C22×D411yescharacteristic
168Q8—44no—
169Q8—44no—
170Q8—82no—
171Q8—82no—
172D4—44no—
173D4—44no—
174D4—48no—
175D4—44no—
176D4—44no—
177D4—44no—
178D4—82no—
179D4—82no—
180D4—82no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—48no—
191C2×C4—44no—
192C2×C4—48no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4#12012yes—
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4C2×D812yes—
205C2×C4C2×D812yes—
206C2×C4AGL(1,ℤ/8)12yes—
207C2×C4C22×D412yes—
208C8—44no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—81no—
219C22—88no—
220C22C2×#12011yescharacteristic
221C22C22×D811yescharacteristic
222C22C2×AGL(1,ℤ/8)11yescharacteristic
223C4—22no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—41no—
228C4—41no—
229C4—82no—
230C4—82no—
231C4#44712yes—
232C4#44712yes—
233C2—21no—
234C2—21no—
235C2—42no—
236C2—84no—
237C2C2×#44711yescharacteristic
238C2#302211yescharacteristic
239C2#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. #3226C2semidirect
3. #2621C2semidirect
4. #1610C2semidirect
5. #1607C2semidirect
6. #1517C2semidirect
7. #1458C2semidirect
17. C2×C4⋊Q8C22semidirect
23. #480C22semidirect
25. #431C22semidirect
27. #398C22semidirect
31. #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
#32262
#26212
#16102
#16072
#15172
#14582