Finite Group Databasebeta

← #18197

Subgroups — #18197

Order 256.

587 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 129 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#3501C211yesmaximal, characteristic
2C2×#451C212yesmaximal, complemented
3#2409C211yesmaximal, characteristic
4#1605C214yesmaximal, complemented
5#1602C214yesmaximal, complemented
6#1520C212yesmaximal, complemented
7#1459C211yesmaximal, characteristic
8C23×Q8C2211yescharacteristic
9C22×Q16—22no—
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20#518—21no—
21C2×C4×D4C2211yescharacteristic
22#451—24no—
23#451C2218yes—
24C2×#103—22no—
25C2×C22⋊C8C2212yes—
26#395C2212yes—
27#393C2212yes—
28#391—21no—
29#386—21no—
30#385—21no—
31#380C2211yescharacteristic
32#328C2218yes—
33#324C2212yes—
34#323C2212yes—
35C22×Q8—21no—
36C22×Q8—21no—
37C22×Q8—22no—
38C22×Q8—22no—
39C22×Q8—44no—
40C22×Q8D412yes—
41C22×Q8C2312yes—
42C22×D4D411yescharacteristic
43C23×C4—21no—
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C2×Q16—28no—
47C2×Q16—48no—
48C22×C8—22no—
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8—24no—
52#125—22no—
53#125—24no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—24no—
64C22⋊Q8—44no—
65C22⋊Q8D414yes—
66C22⋊Q8C2314yes—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—24no—
70C4×D4—24no—
71C4×D4—22no—
72C4×D4—22no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4D412yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4D411yescharacteristic
86C2×C22⋊C4C2311yescharacteristic
87C2×C42D411yescharacteristic
88#103—28no—
89C22⋊C8—22no—
90C22⋊C8C2314yes—
91#95—22no—
92#95—22no—
93#95C2312yes—
94C24C2×D411yescharacteristic
95C24C2×D411yescharacteristic
96C2×Q8—24no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—28no—
101C2×Q8—24no—
102C2×Q8—44no—
103C2×Q8—42no—
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8—48no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—82no—
110C2×Q8C2×D414yes—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4Q1614yescomplemented
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D412yes—
132C22×C4C2411yescharacteristic
133Q16—416no—
134C2×C8—24no—
135C2×C8—44no—
136C4⋊C4—22no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C4⋊C4C2×D414yes—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—22no—
152C22⋊C4—24no—
153C22⋊C4—22no—
154C22⋊C4—44no—
155C22⋊C4—42no—
156C42—24no—
157C42—21no—
158C42—21no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23#12012yes—
166C23C2×Q1612yes—
167C23C2×Q1612yescomplemented
168C23#13712yes—
169C23C22×D411yescharacteristic
170Q8—28no—
171Q8—44no—
172Q8—44no—
173Q8—44no—
174Q8—44no—
175Q8—44no—
176Q8—82no—
177Q8—82no—
178D4—28no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4C2×Q1612yescomplemented
207C2×C4C22×D412yes—
208C8—44no—
209C22—21no—
210C22—22no—
211C22—21no—
212C22—22no—
213C22—22no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22#45114yes—
221C22C2×#12011yescharacteristic
222C22C22×Q1611yescharacteristic
223C22C2×#13711yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—41no—
228C4—44no—
229C4—44no—
230C4—41no—
231C4—44no—
232C4—82no—
233C4B(4,𝔽2)12yes—
234C2—21no—
235C2—21no—
236C2—24no—
237C2—42no—
238C2C2×#45111yescharacteristic
239C2#302211yescharacteristic
240C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#451C2semidirect
4. #1605C2semidirect
5. #1602C2semidirect
6. #1520C2semidirect
113. C2×D4Q16semidirect
167. C23C2×Q16semidirect
206. C2×C4C2×Q16semidirect

Maximal subgroups

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

subgroupindex
#35012
C2×#4512
#24092
#16052
#16022
#15202
#14592