Finite Group Databasebeta

← #28629

Subgroups — #28629

Order 256.

419 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 123 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×#497C211yesmaximal, characteristic
2C2×#488C211yesmaximal, characteristic
3#2642C211yesmaximal, characteristic, complemented
4#2326C211yesmaximal, characteristic
5#2056C212yesmaximal, complemented
6#2055C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal
11#1768C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#497—24no—
20#488—24no—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#400—22no—
30#391C2212yes—
31#391C2211yescharacteristic
32#391C2212yes—
33#390C2211yescharacteristic
34#388—22no—
35#385—22no—
36#384C2211yescharacteristic
37#376C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8C2312yes—
42C22×Q8C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C2×Q16—24no—
46C2×Q16—24no—
47C2×Q16—42no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55#125—22no—
56#125—22no—
57#124—22no—
58#124D414yes—
59C4×Q8—42no—
60C4×D4—42no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C4⋊C8—24no—
76#103—24no—
77#103—42no—
78#103—42no—
79#102—24no—
80#97—22no—
81C4×C8—22no—
82#95—22no—
83#95—24no—
84#95—22no—
85#95—22no—
86#95D412yes—
87#95D412yes—
88C24—21no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8C2×D412yes—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4C2×D412yes—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—82no—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2411yescharacteristic
115Q16—48no—
116SD16—48no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8#4114yes—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C4⋊C4—82no—
131C22⋊C4—28no—
132C22⋊C4—42no—
133C22⋊C4—82no—
134C42—22no—
135C42—22no—
136C42—42no—
137C42C2×D412yes—
138C42#4112yes—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—42no—
144C23C22×D411yescharacteristic
145Q8—24no—
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
150Q8—42no—
151D4—24no—
152D4—42no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—84no—
176C2×C4#12114yes—
177C2×C4C22×D411yescharacteristic
178C2×C4C2×#4112yes—
179C2×C4C2×#4111yescharacteristic
180C2×C4H(5,𝔽2)12yes—
181C8—24no—
182C8—42no—
183C22—44no—
184C22—42no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22C2×#12111yescharacteristic
189C22#53211yescharacteristic
190C22#53511yescharacteristic
191C22#57712yes—
192C22#57812yes—
193C4—24no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—42no—
198C4—81no—
199C4—81no—
200C4—82no—
201C4—82no—
202C4#54612yes—
203C4#55912yes—
204C2—82no—
205C2#254411yescharacteristic
206C2#304612yes—
207C2#317812yes—
208C2#322211yescharacteristic
209C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2642C2semidirect
5. #2056C2semidirect
7. #1992C2semidirect
8. #1956C2semidirect
9. #1878C2semidirect
11. #1768C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4882
#26422
#23262
#20562
#20552
#19922
#19562
#18782
#18772
#17682