Finite Group Databasebeta

← #24016

Subgroups — #24016

Order 256.

638 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 135 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×#485C212yesmaximal, complemented
3#2311C211yesmaximal, characteristic
4#2036C212yesmaximal, complemented
5#2035C212yesmaximal, complemented
6#1872C212yesmaximal, complemented
7#1854C212yesmaximal, complemented
8#1787C212yesmaximal, complemented
9#1780C211yesmaximal, characteristic
10C22×#41C2211yescharacteristic
11C23×Q8C2211yescharacteristic
12#536—28no—
13C2×C4⋊Q8—21no—
14C2×#124—21no—
15C2×#123—21no—
16C2×#123—21no—
17C2×C22⋊Q8C2212yes—
18C2×C22⋊Q8C2212yes—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#485—28no—
23C2×#107C2212yes—
24C2×#106C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C22⋊C8C2212yes—
30C2×C22⋊C8C2212yes—
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#391—21no—
34#390—21no—
35#386—21no—
36#386—21no—
37#380C2212yes—
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41C2×#41—24no—
42C2×#41—24no—
43C2×#41—41no—
44C2×#41—41no—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—24no—
48C22×Q8—41no—
49C22×Q8—41no—
50C22×Q8D411yescharacteristic
51C22×Q8C2312yes—
52C22×D4D411yescharacteristic
53C22×D4D411yescharacteristic
54C22×D4D411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C22×C8C2312yes—
61C22×C8C2312yes—
62C4⋊Q8—28no—
63#124—28no—
64#123—28no—
65#123—28no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68C22⋊Q8—24no—
69C22⋊Q8D418yes—
70#117—24no—
71#117—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2312yes—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C42—21no—
87C2×C42—21no—
88#107—24no—
89#106—24no—
90#103—24no—
91#103—24no—
92#103—42no—
93#103—42no—
94C22⋊C8—24no—
95C22⋊C8—24no—
96C22⋊C8—41no—
97C22⋊C8—41no—
98#95—22no—
99#95—22no—
100C24C2×D411yescharacteristic
101C24C2×D411yescharacteristic
102C24C2×D411yescharacteristic
103#41—48no—
104#41—48no—
105C2×Q8—24no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—41no—
114C2×Q8—48no—
115C2×Q8—48no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8C2×D414yes—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—82no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D414yes—
150C22×C4C2×D411yescharacteristic
151C22×C4#4114yes—
152C22×C4C2411yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C4⋊C4—28no—
160C4⋊C4—28no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4C2×D414yes—
168C22⋊C4—24no—
169C22⋊C4—28no—
170C22⋊C4—28no—
171C22⋊C4—42no—
172C22⋊C4—82no—
173C22⋊C4—82no—
174C42—24no—
175C42—42no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23#12012yes—
190C23#12312yes—
191C23C22×D411yescharacteristic
192Q8—24no—
193Q8—24no—
194Q8—44no—
195Q8—48no—
196Q8—42no—
197Q8—42no—
198D4—44no—
199D4—44no—
200D4—44no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4#12012yes—
229C2×C4#12312yes—
230C2×C4C22×D412yes—
231C2×C4C2×#4112yes—
232C2×C4H(5,𝔽2)11yescharacteristic
233C2×C4H(5,𝔽2)11yescharacteristic
234C8—42no—
235C8—42no—
236C22—24no—
237C22—22no—
238C22—22no—
239C22—24no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22C2×#12011yescharacteristic
252C22C2×#12311yescharacteristic
253C22#53411yescharacteristic
254C22#57712yes—
255C22#57812yes—
256C4—22no—
257C4—22no—
258C4—41no—
259C4—44no—
260C4—44no—
261C4—82no—
262C4—82no—
263C4#54612yes—
264C4#54612yes—
265C2—42no—
266C2—42no—
267C2—81no—
268C2#240511yescharacteristic
269C2#301312yes—
270C2#309712yes—
271C2#320011yescharacteristic
272C2#320211yescharacteristic
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×#485C2semidirect
4. #2036C2semidirect
5. #2035C2semidirect
6. #1872C2semidirect
7. #1854C2semidirect
8. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4852
#23112
#20362
#20352
#18722
#18542
#17872
#17802