Finite Group Databasebeta

← #27985

Subgroups — #27985

Order 256.

365 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 119 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#2074C212yesmaximal, complemented
4#2072C212yesmaximal, complemented
5#1960C211yesmaximal, characteristic, complemented
6#1956C211yesmaximal, characteristic, complemented
7#1873C212yesmaximal
8#1872C212yesmaximal
9#1836C211yesmaximal, characteristic, complemented
10#1835C211yesmaximal, characteristic, complemented
11#1750C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22C2×#97C2211yescharacteristic
23C2×C4×C8C2211yescharacteristic
24#398—22no—
25#397—22no—
26#391C2212yes—
27#389—22no—
28#385—22no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32#382C2211yescharacteristic
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—22no—
45C4⋊Q8—22no—
46C22⋊Q8—24no—
47C22⋊Q8—24no—
48#117—22no—
49#117—22no—
50#117—42no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4D412yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D412yes—
66C2×C42—21no—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#103—42no—
70#103—42no—
71#103—42no—
72#103—42no—
73C22⋊C8—24no—
74#97—22no—
75C4×C8—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—24no—
80C24C2×D411yescharacteristic
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—82no—
98C22×C4C2×D412yes—
99C22×C4C2×D412yes—
100C22×C4C2×D411yescharacteristic
101C22×C4#4114yes—
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—42no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C22⋊C4—24no—
122C22⋊C4—42no—
123C22⋊C4—44no—
124C42—21no—
125C42—21no—
126C42—42no—
127C42#4112yes—
128C42#4112yes—
129C23—21no—
130C23—21no—
131C23—22no—
132C23—22no—
133C23#12412yes—
134C23C22×D411yescharacteristic
135Q8—42no—
136Q8—42no—
137Q8—42no—
138Q8—42no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—84no—
163C2×C4#12314yes—
164C2×C4#12412yes—
165C2×C4C2×#4111yescharacteristic
166C2×C4C2×#4111yescharacteristic
167C2×C4C2×#4112yes—
168C2×C4H(5,𝔽2)12yes—
169C8—42no—
170C8—42no—
171C22—24no—
172C22—22no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22C2×#12311yescharacteristic
177C22C2×#12411yescharacteristic
178C22#53511yescharacteristic
179C22#57712yes—
180C22#57812yes—
181C4—24no—
182C4—22no—
183C4—42no—
184C4—82no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—81no—
189C4—82no—
190C4#55312yes—
191C4#55312yes—
192C2—42no—
193C2#250411yescharacteristic
194C2#309712yes—
195C2#313712yes—
196C2#322211yescharacteristic
197C2#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
3. #2074C2semidirect
4. #2072C2semidirect
5. #1960C2semidirect
6. #1956C2semidirect
9. #1836C2semidirect
10. #1835C2semidirect

Maximal subgroups

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

subgroupindex
#26212
#23172
#20742
#20722
#19602
#19562
#18732
#18722
#18362
#18352
#17502