Finite Group Databasebeta

← #25634

Subgroups — #25634

Order 256.

541 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#557C211yesmaximal, characteristic
2C2×#494C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic
5#2016C214yesmaximal
6#1984C211yesmaximal, characteristic
7#1981C211yesmaximal, characteristic
8#1878C214yesmaximal
9C4×#103C211yesmaximal, characteristic
10C22×Q16C2214yes—
11C22×Q16C2212yes—
12#557—24no—
13#557—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×#125—22no—
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#494—24no—
24#462—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2214yes—
29C2×C4×C8C2211yescharacteristic
30#400—22no—
31#391—21no—
32#391C2214yes—
33#382—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#336C2214yes—
37C22×Q8D411yescharacteristic
38C22×Q8C2312yes—
39C22×Q8C2311yescharacteristic
40C22×Q8C2312yes—
41C2×Q16—24no—
42C2×Q16—28no—
43C2×Q16—28no—
44C2×Q16—44no—
45C2×Q16—44no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—24no—
49C4⋊Q8—24no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54#125—216no—
55#125—28no—
56C4×Q8—28no—
57C4×Q8—24no—
58C4×Q8—24no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8D414yes—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C42—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80C4⋊C8—22no—
81#103—24no—
82#103—24no—
83#103—44no—
84C4×C8—22no—
85#95—21no—
86#95—24no—
87#95—24no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—24no—
96C2×Q8—24no—
97C2×Q8—24no—
98C2×Q8—22no—
99C2×Q8—24no—
100C2×Q8Q1614yescomplemented
101C2×Q8C2×D414yes—
102C2×Q8C2×D412yes—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113Q16—416no—
114Q16—48no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—24no—
119C2×C8#4114yes—
120C4⋊C4—28no—
121C4⋊C4—28no—
122C4⋊C4—28no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4C2×D412yes—
134C42—28no—
135C42—24no—
136C42—22no—
137C42—22no—
138C42—42no—
139C42C2×D412yes—
140C42C2×D412yes—
141C23C22×D411yescharacteristic
142Q8—28no—
143Q8—22no—
144Q8—22no—
145Q8—28no—
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
150Q8—42no—
151Q8—44no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4#12012yes—
169C2×C4#12012yes—
170C2×C4C2×Q1614yes—
171C2×C4C2×Q1612yes—
172C2×C4#13712yes—
173C2×C4C22×D411yescharacteristic
174C2×C4C22×D411yescharacteristic
175C2×C4C2×#4112yes—
176C2×C4H(5,𝔽2)12yes—
177C8—24no—
178C8—42no—
179C22C2×#12011yescharacteristic
180C22#55012yes—
181C22C22×Q1611yescharacteristic
182C22C2×#13711yescharacteristic
183C22#57812yes—
184C4—24no—
185C4—22no—
186C4—42no—
187C4—44no—
188C4—42no—
189C4—42no—
190C4—82no—
191C4—82no—
192C4#45114yes—
193C4#53211yescharacteristic
194C4#53511yescharacteristic
195C4#54612yes—
196C2#243111yescharacteristic
197C2C2×#45111yescharacteristic
198C2#301111yescharacteristic
199C2#339512yes—
200C2#339812yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
100. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4942
C2×#4622
#25682
#20162
#19842
#19812
#18782
C4×#1032