Finite Group Databasebeta

← #12252

Subgroups — #12252

Order 256.

226 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 37 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#2677C211yesmaximal, characteristic
2#2062C211yesmaximal, characteristic
3#1614C211yesmaximal, characteristic
4#1602C211yesmaximal, characteristic
5#1352C211yesmaximal, characteristic
6#1351C211yesmaximal, characteristic
7#1326C211yesmaximal, characteristic
8#543C2211yescharacteristic
9C2×C4⋊Q8—21no—
10C2×C22⋊Q8—21no—
11#484—21no—
12#451—21no—
13#399C2211yescharacteristic
14#398—21no—
15#395—21no—
16#393C2211yescharacteristic
17#388—21no—
18#387—21no—
19#385—21no—
20#385C2211yescharacteristic
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×Q16—41no—
31C4⋊Q8—44no—
32C22⋊Q8—21no—
33C22⋊Q8—21no—
34C22⋊Q8—41no—
35C22⋊Q8—41no—
36C22⋊Q8—44no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—41no—
44C2×C4⋊C4D411yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4D411yescharacteristic
50C2×C22⋊C4C2311yescharacteristic
51C2×C42—21no—
52#107—41no—
53#103—41no—
54#103—41no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—41no—
62#95D411yescharacteristic
63#95D411yescharacteristic
64C24C2×D411yescharacteristic
65C2×Q8—21no—
66C2×Q8—21no—
67C2×Q8—42no—
68C2×Q8—42no—
69C2×Q8—42no—
70C2×Q8—41no—
71C2×Q8—81no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—42no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85Q16—82no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—24no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—82no—
100C22⋊C4—21no—
101C22⋊C4—22no—
102C22⋊C4—41no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4Q1612yescomplemented
109C42—42no—
110C23—21no—
111C23—21no—
112C23—41no—
113C23—41no—
114C23#12011yescharacteristic
115C23C2×Q1611yescharacteristic
116C23#13711yescharacteristic
117Q8—42no—
118Q8—81no—
119Q8—84no—
120Q8—81no—
121Q8—81no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C8—81no—
154C8—81no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—81no—
164C22#45111yescharacteristic
165C22#45611yescharacteristic
166C22B(4,𝔽2)11yescharacteristic
167C4—42no—
168C4—42no—
169C4—41no—
170C4—42no—
171C4—42no—
172C4—41no—
173C4—81no—
174C4—82no—
175C4—81no—
176C4—81no—
177C4—81no—
178C2—21no—
179C2—21no—
180C2—41no—
181C2—41no—
182C2#160511yescharacteristic
183C2#219611yescharacteristic
184C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
108. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#26772
#20622
#16142
#16022
#13522
#13512
#13262