Finite Group Databasebeta

← #10342

Subgroups — #10342

Order 256.

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

normal subgroups only

This group has 272 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#1823C212yesmaximal
3#1445C211yesmaximal, characteristic
4#1294C212yesmaximal
5#1280C211yesmaximal, characteristic
6C2×C22⋊Q8—21no—
7C22×C4⋊C4C2211yescharacteristic
8C2×C22⋊C8C2212yes—
9#398—22no—
10#395—22no—
11#380—21no—
12#380C2212yes—
13C2×#95—21no—
14C2×#95—21no—
15C2×#95C2211yescharacteristic
16C2×#95C2211yescharacteristic
17#340—22no—
18#340—22no—
19C22×Q8—21no—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C22⋊Q8—42no—
27C22⋊Q8—42no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4D412yes—
36C2×C4⋊C4D412yes—
37C2×C4⋊C4Q812yescomplemented
38C2×C22⋊C4—22no—
39C22⋊C8—22no—
40C22⋊C8—22no—
41#95—22no—
42#95—22no—
43#95—22no—
44#95—22no—
45#95—42no—
46#95—42no—
47C24C2×D411yescharacteristic
48C2×Q8—42no—
49C2×Q8—81no—
50C2×Q8—81no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
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×C4D812yes—
78C22×C4SD1612yes—
79C22×C4C2×D412yes—
80C22×C4C2×Q811yescharacteristic
81C22×C4#4112yes—
82C22×C4#4111yescharacteristic
83C2×C8—42no—
84C2×C8—42no—
85C2×C8—42no—
86C4⋊C4—24no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—44no—
91C4⋊C4—42no—
92C4⋊C4—81no—
93C4⋊C4—81no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23#12011yescharacteristic
101C23C22⋊Q811yescharacteristic
102C23#12411yescharacteristic
103C23C2×D811yescharacteristic
104C23C2×SD1611yescharacteristic
105C23AGL(1,ℤ/8)11yescharacteristic
106C23#13711yescharacteristic
107Q8—84no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—81no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4C22⋊Q812yes—
144C2×C4#12412yes—
145C8—82no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22#39311yescharacteristic
159C22#44711yescharacteristic
160C22#44811yescharacteristic
161C22#45511yescharacteristic
162C22#45611yescharacteristic
163C22#45811yescharacteristic
164C22#45811yescharacteristic
165C22#47411yescharacteristic
166C22#47611yescharacteristic
167C22#48611yescharacteristic
168C22#48911yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—42no—
173C4—82no—
174C4—82no—
175C4—82no—
176C4—82no—
177C2—21no—
178C2—21no—
179C2—21no—
180C2—21no—
181C2#160711yescharacteristic
182C2#160811yescharacteristic
183C2#161011yescharacteristic
184C2#161111yescharacteristic
185C2#202311yescharacteristic
186C2#202811yescharacteristic
187C2#203011yescharacteristic
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

Complements

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

NH ≅ G/Ntype
37. C2×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#26592
#18232
#14452
#12942
#12802