Finite Group Databasebeta

← #10324

Subgroups — #10324

Order 256.

455 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 61 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#3189C211yesmaximal, characteristic, complemented
2#2848C212yesmaximal
3#1779C211yesmaximal, characteristic
4#1520C212yesmaximal, complemented
5#1289C211yesmaximal, characteristic, complemented
6C24×C4C2211yescharacteristic
7C2×C22⋊Q8—22no—
8C2×C22⋊Q8C2212yes—
9#484—22no—
10#467—22no—
11#451—22no—
12C2×C22⋊C8—22no—
13C2×C22⋊C8C2211yescharacteristic
14C2×C22⋊C8C2211yescharacteristic
15#385—22no—
16#385—24no—
17#380—22no—
18#380C2212yes—
19#379—22no—
20#328—24no—
21C25D411yescharacteristic
22C22×Q8D412yes—
23C23×C4—22no—
24C23×C4—21no—
25C23×C4—22no—
26C23×C4—21no—
27C23×C4—44no—
28C23×C4D411yescharacteristic
29C23×C4C2311yescharacteristic
30C2×Q16—22no—
31C22×C8—21no—
32C22×C8D412yes—
33C22⋊Q8—22no—
34C22⋊Q8—22no—
35C22⋊Q8—22no—
36C22⋊Q8—22no—
37C22⋊Q8—48no—
38C2×C4⋊C4—24no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4D412yes—
42C2×C22⋊C4—24no—
43C2×C22⋊C4—24no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—22no—
46#107—22no—
47#103—24no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8—44no—
51#95—24no—
52#95D414yes—
53C24—22no—
54C24—21no—
55C24—21no—
56C24—22no—
57C24—42no—
58C24—42no—
59C24Q1612yescomplemented
60C24C2×D411yescharacteristic
61C2×Q8—22no—
62C2×Q8—22no—
63C2×Q8—44no—
64C22×C4—24no—
65C22×C4—28no—
66C22×C4—22no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—24no—
70C22×C4—22no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—44no—
74C22×C4—42no—
75C22×C4—44no—
76C22×C4—42no—
77C22×C4—44no—
78C22×C4—42no—
79C22×C4—41no—
80C22×C4—42no—
81C22×C4—41no—
82C22×C4C2×D412yes—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D412yes—
85C22×C4#4111yescharacteristic
86Q16—82no—
87C2×C8—22no—
88C2×C8—42no—
89C2×C8—42no—
90C2×C8—41no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4—44no—
94C4⋊C4—48no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C22⋊C4—48no—
98C22⋊C4—48no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C23—24no—
104C23—24no—
105C23—22no—
106C23—22no—
107C23—24no—
108C23—22no—
109C23—41no—
110C23—44no—
111C23—41no—
112C23—42no—
113C23—41no—
114C23—42no—
115C23—41no—
116C23—44no—
117C23—48no—
118C23—41no—
119C23—42no—
120C23—41no—
121C23#12012yes—
122C23#12111yescharacteristic
123C23C2×Q1611yescharacteristic
124C23#13511yescharacteristic
125C23#13712yes—
126Q8—44no—
127Q8—82no—
128C2×C4—24no—
129C2×C4—22no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—48no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—44no—
147C2×C4#12011yescharacteristic
148C2×C4#12112yes—
149C2×C4#12711yescharacteristic
150C8—42no—
151C8—81no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—21no—
156C22—28no—
157C22—24no—
158C22—41no—
159C22—41no—
160C22—42no—
161C22—44no—
162C22—44no—
163C22—44no—
164C22—42no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22#39211yescharacteristic
171C22#45112yes—
172C22#45212yes—
173C22#46711yescharacteristic
174C22#47011yescharacteristic
175C4—21no—
176C4—22no—
177C4—42no—
178C4—84no—
179C4—82no—
180C4—82no—
181C4B(4,𝔽2)12yes—
182C2—22no—
183C2—22no—
184C2—21no—
185C2—21no—
186C2—44no—
187C2#200111yescharacteristic
188C2#201011yescharacteristic
189C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3189C2semidirect
4. #1520C2semidirect
5. #1289C2semidirect
59. C24Q16semidirect

Maximal subgroups

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

subgroupindex
#31892
#28482
#17792
#15202
#12892