Finite Group Databasebeta

← #27902

Subgroups — #27902

Order 256.

344 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 115 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#2621C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2036C212yesmaximal
4#2035C212yesmaximal
5#1960C211yesmaximal, characteristic, complemented
6#1956C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1847C211yesmaximal, characteristic, complemented
10#1835C211yesmaximal, characteristic, complemented
11#1750C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×C22⋊C8C2212yes—
23C2×#97C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25#398—22no—
26#397—22no—
27#391C2211yescharacteristic
28#391C2211yescharacteristic
29#388—22no—
30#387—22no—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#382C2211yescharacteristic
34#380C2212yes—
35C4×C4⋊C4C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×Q8C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C22⋊Q8—42no—
47C22⋊Q8—42no—
48#117—22no—
49#117—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4D412yes—
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4D412yes—
63C2×C42—22no—
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—42no—
67#106—42no—
68#103—24no—
69#103—24no—
70#103—42no—
71#103—42no—
72C22⋊C8—24no—
73#97—22no—
74C4×C8—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4C2×D412yes—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4C2411yescharacteristic
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8#4114yes—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—44no—
120C4⋊C4—82no—
121C22⋊C4—24no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—82no—
125C42—21no—
126C42—21no—
127C42—82no—
128C42#4112yes—
129C42#4112yes—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—22no—
134C23—22no—
135C23C22×D411yescharacteristic
136Q8—42no—
137Q8—42no—
138Q8—42no—
139Q8—42no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4#12314yes—
167C2×C4C2×#4111yescharacteristic
168C2×C4C2×#4112yes—
169C2×C4C2×#4111yescharacteristic
170C2×C4H(5,𝔽2)12yes—
171C8—24no—
172C8—42no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22C2×#12311yescharacteristic
179C22#53211yescharacteristic
180C22#53511yescharacteristic
181C22#57712yes—
182C22#57812yes—
183C4—24no—
184C4—42no—
185C4—41no—
186C4—82no—
187C4—81no—
188C4—82no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4#55312yes—
193C4#55512yes—
194C2—81no—
195C2#250011yescharacteristic
196C2#309712yes—
197C2#317812yes—
198C2#322111yescharacteristic
199C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
5. #1960C2semidirect
6. #1956C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect
9. #1847C2semidirect
10. #1835C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#23142
#20362
#20352
#19602
#19562
#19282
#19272
#18472
#18352
#17502