Finite Group Databasebeta

← #32459

Subgroups — #32459

Order 256.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#2487C211yesmaximal, characteristic
2#2486C211yesmaximal, characteristic
3#2088C212yesmaximal, complemented
4#2077C212yesmaximal, complemented
5#2077C212yesmaximal, complemented
6#1954C211yesmaximal, characteristic
7#1941C212yesmaximal, complemented
8#1853C212yesmaximal, complemented
9#1853C212yesmaximal, complemented
10C2×#123—21no—
11C2×C4×D4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C2×#106C2212yes—
15C2×#106C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C22⋊C8C2212yes—
19C2×C22⋊C8C2212yes—
20C2×C4×C8C2211yescharacteristic
21#400—21no—
22#399—21no—
23#398—21no—
24#397—21no—
25#389C2212yes—
26#389C2212yes—
27#386—21no—
28#385—21no—
29#381—21no—
30#380C2211yescharacteristic
31#380C2211yescharacteristic
32#380C2212yes—
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39C22×D4D411yescharacteristic
40C23×C4—21no—
41C23×C4—21no—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#123—44no—
47C4×D4—41no—
48C4×D4—41no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4D411yescharacteristic
67C2×C22⋊C4D411yescharacteristic
68C2×C42C2311yescharacteristic
69#106—24no—
70#106—42no—
71#106—42no—
72C4⋊C8—41no—
73C22⋊C8—24no—
74C22⋊C8—24no—
75C4×C8—41no—
76#95—22no—
77#95—22no—
78#95—21no—
79#95—21no—
80#95—22no—
81#95—22no—
82C24C2×D411yescharacteristic
83C24C2×D411yescharacteristic
84C2×D4—41no—
85C2×D4—41no—
86C2×D4SD1614yes—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—82no—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2411yescharacteristic
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8#4114yes—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—41no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—82no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C22⋊C4—82no—
134C42—21no—
135C42—21no—
136C23—21no—
137C23—21no—
138C23—22no—
139C23—22no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—22no—
144C23C2×SD1612yes—
145C23C2×SD1612yes—
146C23C22×D411yescharacteristic
147D4—44no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—84no—
175C2×C4—84no—
176C2×C4C2×SD1612yes—
177C2×C4C2×#4112yes—
178C2×C4H(5,𝔽2)11yescharacteristic
179C2×C4H(5,𝔽2)11yescharacteristic
180C2×C42−1+412yes—
181C8—24no—
182C8—42no—
183C22—24no—
184C22—24no—
185C22—22no—
186C22—22no—
187C22—42no—
188C22—42no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22#53411yescharacteristic
194C22#54712yes—
195C22C22×SD1611yescharacteristic
196C22#57511yescharacteristic
197C22#57712yes—
198C4—22no—
199C4—41no—
200C4—41no—
201C4—41no—
202C4—82no—
203C4—82no—
204C4—82no—
205C4—82no—
206C4#53912yes—
207C4#55911yescharacteristic
208C4#56311yescharacteristic
209C2—42no—
210C2—42no—
211C2#272411yescharacteristic
212C2#318711yescharacteristic
213C2#320011yescharacteristic
214C2#333512yes—
215C2#335112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2088C2semidirect
4. #2077C2semidirect
5. #2077C2semidirect
7. #1941C2semidirect
8. #1853C2semidirect
9. #1853C2semidirect

Maximal subgroups

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

subgroupindex
#24872
#24862
#20882
#20772
#19542
#19412
#18532