Finite Group Databasebeta

← #25682

Subgroups — #25682

Order 256.

544 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 131 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×#540C211yesmaximal, characteristic
2C2×#499C211yesmaximal, characteristic, complemented
3C2×#490C211yesmaximal, characteristic, complemented
4#2629C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1992C211yesmaximal, characteristic, complemented
8#1957C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12#540—24no—
13#540—24no—
14C2×#127C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#123—22no—
18C2×#121—22no—
19C2×#121C2212yes—
20C2×C4×D4C2212yes—
21C2×#117C2211yescharacteristic
22#499—24no—
23#490—24no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#398—22no—
35#396—22no—
36#389C2212yes—
37#384C2211yescharacteristic
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×D4—22no—
44C22×D4—22no—
45C22×D4C2312yes—
46C23×C4—22no—
47C23×C4C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52#127—22no—
53#127D414yes—
54#125—22no—
55#125—22no—
56#125—22no—
57#125D414yes—
58#123—216no—
59#121—216no—
60#121—24no—
61#121—24no—
62C4×D4—24no—
63C4×D4—24no—
64#117—22no—
65#117—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4D412yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4D412yes—
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—24no—
81#106—24no—
82#102—24no—
83#102—24no—
84#102—24no—
85#102—24no—
86C22⋊C8—22no—
87C22⋊C8—22no—
88#97—22no—
89C4×C8—22no—
90#95—22no—
91#95—22no—
92C24—22no—
93C24C2×D411yescharacteristic
94C2×D4—28no—
95C2×D4—28no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—82no—
102C2×D4—82no—
103C2×D4—82no—
104C2×D4—82no—
105C2×D4C2×D414yes—
106C22×C4—22no—
107C22×C4—28no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—82no—
117C22×C4—82no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2411yescharacteristic
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8#4112yes—
133C2×C8#4112yes—
134C4⋊C4—28no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—44no—
142C4⋊C4—82no—
143C4⋊C4—82no—
144C4⋊C4C2×D414yes—
145C22⋊C4—28no—
146C22⋊C4—28no—
147C22⋊C4—24no—
148C22⋊C4—82no—
149C42—21no—
150C42—21no—
151C42C2×D412yes—
152C42C2×D412yes—
153C23—21no—
154C23—24no—
155C23—21no—
156C23—21no—
157C23—22no—
158C23—22no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23C22×D411yescharacteristic
166D4—24no—
167D4—24no—
168D4—42no—
169D4—42no—
170D4—88no—
171D4—88no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—88no—
194C2×C4#12014yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C42−1+411yescharacteristic
200C2×C42−1+411yescharacteristic
201C8—22no—
202C8—22no—
203C8—41no—
204C8—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—44no—
209C22—42no—
210C22—44no—
211C22—42no—
212C22—44no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22C2×#12011yescharacteristic
218C22#53211yescharacteristic
219C22#53611yescharacteristic
220C22#57612yes—
221C22#57712yes—
222C4—24no—
223C4—41no—
224C4—42no—
225C4—44no—
226C4—82no—
227C4—82no—
228C4#54711yescharacteristic
229C4#54711yescharacteristic
230C4#54811yescharacteristic
231C4#54811yescharacteristic
232C2—81no—
233C2—84no—
234C2#243211yescharacteristic
235C2#301212yes—
236C2#317411yescharacteristic
237C2#317611yescharacteristic
238C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#540C2
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×#499C2semidirect
3. C2×#490C2semidirect
4. #2629C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1992C2semidirect
8. #1957C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#4992
C2×#4902
#26292
#20342
#20332
#19922
#19572
#18642
#18622
#17502