Finite Group Databasebeta

← #25868

Subgroups — #25868

Order 256.

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

normal subgroups only

This group has 615 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×#491C211yesmaximal, characteristic, complemented
3C2×#488C211yesmaximal, characteristic, complemented
4#2629C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1959C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1788C212yesmaximal
10#1787C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14#536—24no—
15#536—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#123—22no—
22C2×C22⋊Q8—22no—
23C2×#117C2211yescharacteristic
24#491—24no—
25#488—24no—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×#97C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#398—22no—
38#396—22no—
39#389C2212yes—
40#384C2211yescharacteristic
41#380C2212yes—
42#336C2212yes—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—24no—
46C2×#41—42no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—21no—
50C22×Q8—24no—
51C22×Q8—42no—
52C22×Q8C2311yescharacteristic
53C22×Q8C2311yescharacteristic
54C22×Q8C2311yescharacteristic
55C22×D4—21no—
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C2311yescharacteristic
60C22×C8C2312yes—
61C22×C8C2312yes—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8D414yes—
66#125—22no—
67#125—22no—
68#124—22no—
69#124—22no—
70#124—22no—
71#124D414yes—
72#123—216no—
73C22⋊Q8—216no—
74#117—22no—
75#117—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4D412yes—
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2312yes—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4D412yes—
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90#103—24no—
91#103—24no—
92#103—24no—
93#103—42no—
94#103—42no—
95#103—42no—
96#102—24no—
97#102—42no—
98C22⋊C8—24no—
99#97—22no—
100C4×C8—22no—
101#95—22no—
102#95—22no—
103C24—21no—
104C24C2×D411yescharacteristic
105#41—48no—
106#41—48no—
107C2×Q8—24no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—48no—
118C2×Q8—48no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8C2×D412yes—
122C2×Q8C2×D412yes—
123C2×Q8C2×D412yes—
124C2×D4—24no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4C2×D412yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4C2×D411yescharacteristic
148C22×C4#4114yes—
149C22×C4C2411yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—28no—
157C4⋊C4—28no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—44no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C22⋊C4—28no—
167C22⋊C4—28no—
168C22⋊C4—24no—
169C22⋊C4—82no—
170C42—21no—
171C42—21no—
172C42C2×D412yes—
173C42C2×D412yes—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—42no—
183C23#12412yes—
184C23C22×D411yescharacteristic
185Q8—24no—
186Q8—24no—
187Q8—24no—
188Q8—48no—
189Q8—42no—
190Q8—42no—
191Q8—42no—
192D4—24no—
193D4—48no—
194D4—42no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4#12014yes—
222C2×C4#12412yes—
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4112yes—
226C2×C4H(5,𝔽2)12yes—
227C8—42no—
228C8—42no—
229C22—24no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—82no—
237C22—81no—
238C22—81no—
239C22C2×#12011yescharacteristic
240C22C2×#12411yescharacteristic
241C22#53511yescharacteristic
242C22#57712yes—
243C22#57812yes—
244C4—22no—
245C4—24no—
246C4—42no—
247C4—42no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—82no—
252C4#54612yes—
253C4#54612yes—
254C2—42no—
255C2—82no—
256C2#243511yescharacteristic
257C2#301312yes—
258C2#313712yes—
259C2#322111yescharacteristic
260C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#491C2semidirect
3. C2×#488C2semidirect
4. #2629C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #1959C2semidirect
8. #1956C2semidirect
10. #1787C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4912
C2×#4882
#26292
#20432
#20422
#19592
#19562
#17882
#17872
#17502