Finite Group Databasebeta

← #27953

Subgroups — #27953

Order 256.

421 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 147 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#2851C211yesmaximal, characteristic
2C2×#384C211yesmaximal, characteristic
3#2084C212yesmaximal, complemented
4#2083C212yesmaximal, complemented
5#1992C212yesmaximal, complemented
6#1853C212yesmaximal
7#1852C212yesmaximal
8#1847C212yesmaximal, complemented
9#1749C211yesmaximal, characteristic
10C2×#125C2211yescharacteristic
11C22×C4⋊C4—21no—
12C22×C4⋊C4—21no—
13C22×C4⋊C4C2212yes—
14C22×C42C2211yescharacteristic
15C2×#107C2214yes—
16C2×#106C2214yes—
17C2×C22⋊C8C2212yes—
18C2×#97C2212yes—
19#389C2211yescharacteristic
20#389C2211yescharacteristic
21#387—21no—
22#387—21no—
23#385—21no—
24#385—21no—
25#384—21no—
26#384—21no—
27#384C2212yes—
28#384C2212yes—
29#382C2211yescharacteristic
30#380C2211yescharacteristic
31#380C2211yescharacteristic
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35C23×C4—22no—
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C22×C8C2314yes—
40#125—24no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—44no—
51C2×C4⋊C4—44no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4C2314yes—
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C42—24no—
61C2×C42D411yescharacteristic
62C2×C42D411yescharacteristic
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65#107—44no—
66#106—44no—
67C22⋊C8Q818yes—
68#97—24no—
69#95—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75C24C2×D411yescharacteristic
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—24no—
89C22×C4—44no—
90C22×C4—44no—
91C22×C4—44no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×Q814yes—
98C22×C4C2411yescharacteristic
99C2×C8—24no—
100C2×C8—24no—
101C2×C8C2×Q818yes—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—24no—
108C4⋊C4—44no—
109C4⋊C4—44no—
110C4⋊C4—44no—
111C4⋊C4—44no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—44no—
116C4⋊C4—42no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C42—22no—
120C42—22no—
121C42—22no—
122C42#4114yes—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23C4⋊Q812yes—
128C23AGL(1,ℤ/8)12yes—
129C23#13712yes—
130C23C22×D411yescharacteristic
131C2×C4—28no—
132C2×C4—28no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—28no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—88no—
160C2×C4#12312yes—
161C2×C4#12312yes—
162C2×C4C4⋊Q812yes—
163C2×C4AGL(1,ℤ/8)12yes—
164C2×C4#13712yes—
165C2×C4C22×Q812yes—
166C2×C4C2×#4112yes—
167C2×C4H(5,𝔽2)12yes—
168C8—28no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22#50114yes—
176C22C2×#12311yescharacteristic
177C22C2×C4⋊Q811yescharacteristic
178C22#53511yescharacteristic
179C22C2×AGL(1,ℤ/8)11yescharacteristic
180C22C2×AGL(1,ℤ/8)11yescharacteristic
181C22C2×#13711yescharacteristic
182C22C2×#13711yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—22no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4—84no—
192C4—81no—
193C4#55414yes—
194C2—24no—
195C2#250211yescharacteristic
196C2#309211yescharacteristic
197C2#309211yescharacteristic
198C2C2×#50111yescharacteristic
199C2#316511yescharacteristic
200C2#321711yescharacteristic
201C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2084C2semidirect
4. #2083C2semidirect
5. #1992C2semidirect
8. #1847C2semidirect

Maximal subgroups

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

subgroupindex
#28512
C2×#3842
#20842
#20832
#19922
#18532
#18522
#18472
#17492