Finite Group Databasebeta

← #25936

Subgroups — #25936

Order 256.

517 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 151 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×Q82C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic
5#2024C214yesmaximal
6#1987C211yesmaximal, characteristic
7#1984C211yesmaximal, characteristic
8#1878C214yesmaximal
9C4×#103C211yesmaximal, characteristic
10C22×Q16C2212yes—
11Q82—24no—
12Q82—24no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8—22no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4×Q8—21no—
19C2×C4×Q8C2212yes—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22#487—24no—
23#462—24no—
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2214yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×C4×C8C2211yescharacteristic
29#400—22no—
30#391—21no—
31#391C2214yes—
32#382—21no—
33C4×C4⋊C4C2211yescharacteristic
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—22no—
38C22×Q8—21no—
39C22×Q8D411yescharacteristic
40C22×Q8C2312yes—
41C22×Q8C2311yescharacteristic
42C2×Q16—24no—
43C2×Q16—28no—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—216no—
47C4⋊Q8—28no—
48C4⋊Q8—24no—
49C4⋊Q8—24no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54C4×Q8—28no—
55C4×Q8—24no—
56C4×Q8—24no—
57C4×Q8—22no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×Q8D414yes—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C42—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77C4⋊C8—22no—
78#103—24no—
79#103—24no—
80#103—28no—
81C4×C8—22no—
82#95—21no—
83#95—24no—
84#95—24no—
85C2×Q8—28no—
86C2×Q8—22no—
87C2×Q8—24no—
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—24no—
94C2×Q8—24no—
95C2×Q8—82no—
96C2×Q8—84no—
97C2×Q8Q1614yescomplemented
98C2×Q8C2×D412yes—
99C2×Q8#4114yes—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4C2×D412yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110Q16—48no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—24no—
115C2×C8—22no—
116C4⋊C4—28no—
117C4⋊C4—28no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4C2×D414yes—
129C4⋊C4C2×D412yes—
130C42—28no—
131C42—24no—
132C42—22no—
133C42—22no—
134C42—42no—
135C42C2×D412yes—
136C42C2×D412yes—
137C23C22×D411yescharacteristic
138Q8—22no—
139Q8—24no—
140Q8—28no—
141Q8—42no—
142Q8—42no—
143Q8—44no—
144Q8—84no—
145Q8—88no—
146Q8#12414yes—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—44no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#12012yes—
164C2×C4#12012yes—
165C2×C4C2×Q1614yes—
166C2×C4C2×Q1612yes—
167C2×C4#13712yes—
168C2×C4C22×D411yescharacteristic
169C2×C4C22×D411yescharacteristic
170C2×C4C2×#4112yes—
171C2×C42−1+412yes—
172C8—42no—
173C8—42no—
174C22C2×#12011yescharacteristic
175C22#54812yes—
176C22C22×Q1611yescharacteristic
177C22C2×#13711yescharacteristic
178C22#57712yes—
179C4—24no—
180C4—24no—
181C4—22no—
182C4—42no—
183C4—44no—
184C4—42no—
185C4—82no—
186C4—82no—
187C4#45114yes—
188C4C2×#12411yescharacteristic
189C4#53611yescharacteristic
190C4#54812yes—
191C2#243611yescharacteristic
192C2C2×#45111yescharacteristic
193C2#300911yescharacteristic
194C2#336312yes—
195C2#336912yes—
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
97. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×Q822
C2×#4872
C2×#4622
#25682
#20242
#19872
#19842
#18782
C4×#1032