Finite Group Databasebeta

← #23487

Subgroups — #23487

Order 256.

709 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×#536C211yesmaximal, characteristic
2C2×#427C211yesmaximal, characteristic, complemented
3C2×#419C211yesmaximal, characteristic, complemented
4C22×#103C211yesmaximal, characteristic, complemented
5C2×#408C211yesmaximal, characteristic
6#2311C211yesmaximal, characteristic
7#2036C214yesmaximal, complemented
8#2035C214yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C23×Q8C2211yescharacteristic
12C22×M16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14#536—24no—
15#536—24no—
16C2×C4⋊Q8—21no—
17C2×#124—21no—
18C2×#123—22no—
19C2×C22⋊Q8C2214yes—
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#427—24no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26#419—24no—
27C2×#103—24no—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2214yes—
31#408—24no—
32C2×C22⋊C8C2212yes—
33C2×C22⋊C8C2212yes—
34#391—21no—
35#390—21no—
36#386—22no—
37#380C2214yes—
38#340C2214yes—
39C2×#41—22no—
40C2×#41—22no—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C22×Q8—21no—
46C22×Q8—22no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—22no—
51C22×Q8D414yes—
52C22×Q8D411yescharacteristic
53C22×Q8C2312yes—
54C22×D4D412yes—
55C22×D4D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4C2312yes—
59C23×C4C2311yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8D414yes—
66C22×C8C2312yes—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—28no—
70#124—28no—
71#123—216no—
72C22⋊Q8—28no—
73C22⋊Q8—28no—
74#117—22no—
75#117—22no—
76#117—22no—
77#117D414yes—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—24no—
89C2×C42—21no—
90C2×C42—21no—
91#107—24no—
92#106—24no—
93#103—28no—
94#103—24no—
95#103—24no—
96C22⋊C8—24no—
97C22⋊C8—24no—
98#95—24no—
99C24C2×D412yes—
100C24C2×D411yescharacteristic
101#41—44no—
102#41—44no—
103#41—44no—
104#41—44no—
105C2×Q8—24no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—28no—
111C2×Q8—28no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—42no—
117C2×Q8—44no—
118C2×Q8—42no—
119C2×Q8—44no—
120C2×Q8C2×D414yes—
121C2×D4—24no—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—82no—
142C22×C4C2×D412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4#4112yes—
148C22×C4#4112yes—
149C22×C4#4112yes—
150C22×C4C2411yescharacteristic
151M16—44no—
152C2×C8—28no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8C2×D412yes—
161C2×C8C2×D412yes—
162C4⋊C4—216no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4C2×D414yes—
170C22⋊C4—216no—
171C22⋊C4—24no—
172C22⋊C4—42no—
173C22⋊C4—84no—
174C42—24no—
175C42—42no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—24no—
182C23—24no—
183C23#12012yes—
184C23#12112yes—
185C23#12312yes—
186C23C22×D411yescharacteristic
187Q8—24no—
188Q8—24no—
189Q8—42no—
190Q8—42no—
191Q8—48no—
192Q8—44no—
193D4—48no—
194D4—44no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4#12012yes—
216C2×C4#12114yes—
217C2×C4#12114yes—
218C2×C4#12112yes—
219C2×C4#12314yes—
220C2×C4#12312yes—
221C2×C4#12414yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4111yescharacteristic
226C2×C4C2×#4111yescharacteristic
227C2×C4C2×#4111yescharacteristic
228C8—22no—
229C8—22no—
230C8—42no—
231C22—24no—
232C22—24no—
233C22—22no—
234C22—22no—
235C22—44no—
236C22—42no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22#39414yes—
241C22C2×#12011yescharacteristic
242C22C2×#12111yescharacteristic
243C22C2×#12311yescharacteristic
244C22#57712yes—
245C22#57812yes—
246C4—42no—
247C4—44no—
248C4—44no—
249C4—82no—
250C4—82no—
251C4#39414yes—
252C4C2×#12111yescharacteristic
253C4C2×#12111yescharacteristic
254C4C2×#12311yescharacteristic
255C4C2×#12411yescharacteristic
256C2—24no—
257C2—82no—
258C2C2×#39411yescharacteristic
259C2#301312yes—
260C2#305711yescharacteristic
261C2#305911yescharacteristic
262C2#309712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#427C2semidirect
3. C2×#419C2semidirect
4. C22×#103C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4272
C2×#4192
C22×#1032
C2×#4082
#23112
#20362
#20352
#17362