Finite Group Databasebeta

← #10406

Subgroups — #10406

Order 256.

310 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 65 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#2659C211yesmaximal, characteristic
2#1872C212yesmaximal
3C2×#328C212yesmaximal
4#1444C211yesmaximal, characteristic
5#1280C211yesmaximal, characteristic
6C2×C22⋊Q8C2212yes—
7C22×C4⋊C4—21no—
8C22×C4⋊C4C2211yescharacteristic
9C2×#103—22no—
10C2×C22⋊C8C2212yes—
11#398—22no—
12#395—22no—
13#380—21no—
14#380—21no—
15C2×#95—21no—
16C2×#95C2211yescharacteristic
17C2×#95C2211yescharacteristic
18#340—22no—
19#328—24no—
20C22×Q8D412yes—
21C23×C4—21no—
22C23×C4—21no—
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C22⋊Q8—22no—
28C22⋊Q8—22no—
29C22⋊Q8—22no—
30C22⋊Q8—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4—42no—
39C2×C4⋊C4—42no—
40C2×C4⋊C4—42no—
41C2×C4⋊C4D412yes—
42C2×C22⋊C4—22no—
43#103—44no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#95—22no—
47#95—22no—
48#95—22no—
49#95—21no—
50#95—42no—
51#95D412yes—
52C24C2×D411yescharacteristic
53C2×Q8—22no—
54C2×Q8—22no—
55C2×Q8—42no—
56C2×Q8—42no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4SD1612yes—
84C22×C4Q1612yes—
85C22×C4C2×D412yes—
86C22×C4C2×D411yescharacteristic
87C22×C4#4112yes—
88C22×C4#4111yescharacteristic
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—42no—
92C4⋊C4—24no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—42no—
96C4⋊C4—48no—
97C4⋊C4—42no—
98C4⋊C4—42no—
99C4⋊C4—81no—
100C4⋊C4—81no—
101C4⋊C4—81no—
102C4⋊C4—81no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23#12011yescharacteristic
110C23#12111yescharacteristic
111C23#12411yescharacteristic
112C23C2×SD1611yescharacteristic
113C23C2×Q1611yescharacteristic
114C23#13711yescharacteristic
115C23#13711yescharacteristic
116Q8—44no—
117Q8—82no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4#12112yes—
154C2×C4#12312yes—
155C8—82no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22#39411yescharacteristic
169C22#44811yescharacteristic
170C22#45111yescharacteristic
171C22#45311yescharacteristic
172C22#45611yescharacteristic
173C22B(4,𝔽2)11yescharacteristic
174C22#45811yescharacteristic
175C22#46111yescharacteristic
176C22#46211yescharacteristic
177C22#48711yescharacteristic
178C22#49111yescharacteristic
179C4—22no—
180C4—22no—
181C4—44no—
182C4—42no—
183C4—42no—
184C4—82no—
185C4—82no—
186C4—82no—
187C2—21no—
188C2—21no—
189C2—21no—
190C2—21no—
191C2#159611yescharacteristic
192C2#160511yescharacteristic
193C2#161111yescharacteristic
194C2#161411yescharacteristic
195C2#203911yescharacteristic
196C2#204311yescharacteristic
197C2#204411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#26592
#18722
C2×#3282
#14442
#12802