Finite Group Databasebeta

← #29344

Subgroups — #29344

Order 256.

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

normal subgroups only

This group has 365 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#2317C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2043C212yesmaximal, complemented
5#1935C212yesmaximal, complemented
6#1933C212yesmaximal, complemented
7#1873C212yesmaximal
8#1872C212yesmaximal
9#1750C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C22⋊Q8C2212yes—
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C2×#103C2212yes—
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×C22⋊C8C2212yes—
20C2×#97C2211yescharacteristic
21C2×C4×C8C2211yescharacteristic
22#398—21no—
23#398—21no—
24#397—21no—
25#397—21no—
26#391C2212yes—
27#389C2212yes—
28#389C2212yes—
29#385—21no—
30#385—21no—
31#384—21no—
32#382—21no—
33#382C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8C2312yes—
40C23×C4—21no—
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—41no—
45C4⋊Q8—41no—
46C22⋊Q8—24no—
47C22⋊Q8—24no—
48#117—24no—
49#117—41no—
50#117—41no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4D411yescharacteristic
67C2×C22⋊C4D411yescharacteristic
68C2×C42—21no—
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#103—24no—
72#103—24no—
73#103—42no—
74#103—42no—
75C22⋊C8—24no—
76#97—41no—
77C4×C8—41no—
78#95—22no—
79#95—22no—
80#95—21no—
81#95—21no—
82#95—22no—
83#95—22no—
84C24C2×D411yescharacteristic
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—82no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4#4114yes—
106C22×C4C2411yescharacteristic
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C4⋊C4#4114yes—
127C22⋊C4—24no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—44no—
131C42—24no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—21no—
136C23—21no—
137C23—21no—
138C23—22no—
139C23—22no—
140C23#12412yes—
141C23C22×D411yescharacteristic
142Q8—42no—
143Q8—42no—
144Q8—42no—
145Q8—42no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
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—84no—
171C2×C4#12412yes—
172C2×C4C2×#4112yes—
173C2×C4C2×#4112yes—
174C2×C4H(5,𝔽2)11yescharacteristic
175C2×C4H(5,𝔽2)11yescharacteristic
176C8—42no—
177C8—42no—
178C22—22no—
179C22—24no—
180C22—42no—
181C22—41no—
182C22—41no—
183C22#52511yescharacteristic
184C22C2×#12411yescharacteristic
185C22#53411yescharacteristic
186C22#57712yes—
187C22#57812yes—
188C4—22no—
189C4—44no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—41no—
194C4—82no—
195C4—82no—
196C4—82no—
197C4#55312yes—
198C4#55312yes—
199C2—42no—
200C2#259111yescharacteristic
201C2#312012yes—
202C2#313712yes—
203C2#320011yescharacteristic
204C2#320211yescharacteristic
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
3. #2101C2semidirect
4. #2043C2semidirect
5. #1935C2semidirect
6. #1933C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#23172
#21012
#20432
#19352
#19332
#18732
#18722
#17502