Finite Group Databasebeta

← #32260

Subgroups — #32260

Order 256.

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

normal subgroups only

This group has 374 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#2547C211yesmaximal, characteristic
2#2487C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2077C212yesmaximal, complemented
5#2058C212yesmaximal, complemented
6#1920C212yesmaximal, complemented
7#1869C211yesmaximal, characteristic
8#1853C212yesmaximal, complemented
9#1807C212yesmaximal, complemented
10C23×C8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#124—21no—
13C2×C4×D4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C2×#106C2212yes—
16C2×#106C2212yes—
17C2×#106C2212yes—
18C2×C22⋊C8C2212yes—
19C2×C22⋊C8C2211yescharacteristic
20#400—21no—
21#399—21no—
22#397—21no—
23#397—21no—
24#391—21no—
25#389C2212yes—
26#386—21no—
27#385—21no—
28#380C2212yes—
29#380C2212yes—
30#380C2211yescharacteristic
31#380C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8—24no—
45C22×C8—41no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—41no—
49C4⋊Q8—41no—
50#124—44no—
51C4×D4—41no—
52C4×D4—41no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D411yescharacteristic
68C2×C22⋊C4D411yescharacteristic
69C2×C42—21no—
70C2×C42C2311yescharacteristic
71#106—24no—
72#106—42no—
73#106—42no—
74C22⋊C8—24no—
75C22⋊C8—41no—
76#95—21no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—21no—
81#95—22no—
82C24C2×D411yescharacteristic
83C24C2×D411yescharacteristic
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8—42no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4SD1614yes—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
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—44no—
115C2×C8#4114yes—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—41no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—82no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—42no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C22⋊C4—82no—
134C42—21no—
135C42—21no—
136C42—42no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—22no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—22no—
145C23C2×SD1612yes—
146C23C2×SD1612yes—
147C23C22×D411yescharacteristic
148Q8—84no—
149D4—44no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
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—42no—
175C2×C4—42no—
176C2×C4—84no—
177C2×C4C2×SD1612yes—
178C2×C4C2×#4112yes—
179C2×C4H(5,𝔽2)11yescharacteristic
180C2×C4H(5,𝔽2)11yescharacteristic
181C2×C42−1+412yes—
182C8—24no—
183C8—42no—
184C22—24no—
185C22—24no—
186C22—22no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22#53411yescharacteristic
195C22#54712yes—
196C22C22×SD1611yescharacteristic
197C22#57511yescharacteristic
198C22#57712yes—
199C4—22no—
200C4—41no—
201C4—41no—
202C4—41no—
203C4—82no—
204C4—82no—
205C4—82no—
206C4—82no—
207C4#54212yes—
208C4#55111yescharacteristic
209C4#56311yescharacteristic
210C2—42no—
211C2—42no—
212C2#271711yescharacteristic
213C2#318711yescharacteristic
214C2#320011yescharacteristic
215C2#333512yes—
216C2#335012yes—
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. #2077C2semidirect
4. #2077C2semidirect
5. #2058C2semidirect
6. #1920C2semidirect
8. #1853C2semidirect
9. #1807C2semidirect

Maximal subgroups

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

subgroupindex
#25472
#24872
#20772
#20582
#19202
#18692
#18532
#18072