Finite Group Databasebeta

← #12253

Subgroups — #12253

Order 256.

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

normal subgroups only

This group has 199 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#2763C211yesmaximal, characteristic
2#2062C211yesmaximal, characteristic
3#1602C211yesmaximal, characteristic
4#1602C211yesmaximal, characteristic
5#1351C211yesmaximal, characteristic
6#1351C211yesmaximal, characteristic
7#1326C211yesmaximal, characteristic
8#543C2211yescharacteristic
9#451—21no—
10#451—21no—
11#400—21no—
12#395—21no—
13#393C2211yescharacteristic
14#393C2211yescharacteristic
15#391—21no—
16#388—21no—
17#385—21no—
18#385—21no—
19#385C2211yescharacteristic
20#382—21no—
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×Q8D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×Q16—41no—
31C2×Q16—41no—
32C22⋊Q8—21no—
33C22⋊Q8—21no—
34C22⋊Q8—41no—
35C22⋊Q8—41no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—41no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4D411yescharacteristic
46C2×C22⋊C4C2311yescharacteristic
47C2×C42—21no—
48#103—41no—
49#103—41no—
50C22⋊C8—21no—
51C22⋊C8—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—41no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C24C2×D411yescharacteristic
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—42no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—81no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—81no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82Q16—82no—
83Q16—82no—
84C2×C8—41no—
85C2×C8—41no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—81no—
91C4⋊C4—81no—
92C4⋊C4—81no—
93C4⋊C4—81no—
94C4⋊C4—82no—
95C22⋊C4—21no—
96C22⋊C4—22no—
97C22⋊C4—41no—
98C22⋊C4—42no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—81no—
103C22⋊C4Q1612yescomplemented
104C42—42no—
105C23—21no—
106C23—21no—
107C23—41no—
108C23—41no—
109C23#12011yescharacteristic
110C23C2×Q1611yescharacteristic
111C23#13711yescharacteristic
112Q8—42no—
113Q8—42no—
114Q8—81no—
115Q8—81no—
116Q8—81no—
117Q8—81no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C8—81no—
150C8—81no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#45111yescharacteristic
161C22#45611yescharacteristic
162C22B(4,𝔽2)11yescharacteristic
163C4—42no—
164C4—41no—
165C4—42no—
166C4—41no—
167C4—81no—
168C4—81no—
169C4—82no—
170C4—81no—
171C4—81no—
172C4—81no—
173C4—81no—
174C2—21no—
175C2—21no—
176C2—41no—
177C2—41no—
178C2#160511yescharacteristic
179C2#219711yescharacteristic
180C2#219911yescharacteristic
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
103. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#27632
#20622
#16022
#13512
#13262