Finite Group Databasebeta

← #25836

Subgroups — #25836

Order 256.

493 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 159 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×#498C211yesmaximal, characteristic
3C2×#477C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic
5#2059C214yesmaximal
6#1988C211yesmaximal, characteristic
7#1984C211yesmaximal, characteristic
8#1865C214yesmaximal
9C4×#103C211yesmaximal, characteristic
10#557—24no—
11#557—24no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125—22no—
16C2×#125C2212yes—
17C2×C4×Q8—21no—
18C2×C4×Q8C2212yes—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21#498—24no—
22#477—24no—
23C2×#107C2212yes—
24C2×#106C2214yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×C4×C8C2211yescharacteristic
29#395—22no—
30#391—21no—
31#389C2214yes—
32#384—21no—
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8D411yescharacteristic
40C22×Q8C2311yescharacteristic
41C22×C8C2312yes—
42C22×C8C2312yes—
43C4⋊Q8—28no—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8D414yes—
48#125—216no—
49#125—24no—
50#125—24no—
51C4×Q8—28no—
52C4×Q8—24no—
53C4×Q8—24no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D412yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C42—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—28no—
78C4⋊C8—22no—
79#103—24no—
80#103Q818yescomplemented
81C4×C8—22no—
82#95—21no—
83#95—24no—
84C2×Q8—22no—
85C2×Q8—24no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—21no—
89C2×Q8—22no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—82no—
94C2×Q8C2×D412yes—
95C2×Q8C2×Q814yes—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2411yescharacteristic
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—24no—
110C2×C8C2×Q814yes—
111C4⋊C4—28no—
112C4⋊C4—28no—
113C4⋊C4—28no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—42no—
124C4⋊C4—84no—
125C4⋊C4C2×D414yes—
126C4⋊C4C2×D412yes—
127C4⋊C4C2×Q814yes—
128C42—28no—
129C42—24no—
130C42—22no—
131C42—22no—
132C42—42no—
133C42C2×D412yes—
134C42C2×D412yes—
135C23C22×D411yescharacteristic
136Q8—22no—
137Q8—24no—
138Q8—44no—
139Q8—48no—
140Q8—84no—
141Q8C4⋊Q814yes—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4#12012yes—
161C2×C4#12012yes—
162C2×C4#13512yes—
163C2×C4#13712yes—
164C2×C4C22×D411yescharacteristic
165C2×C4C22×D411yescharacteristic
166C2×C4C22×Q812yes—
167C2×C42−1+412yes—
168C8—24no—
169C8—42no—
170C22C2×#12011yescharacteristic
171C22D4×Q812yes—
172C22C2×#13511yescharacteristic
173C22C2×#13711yescharacteristic
174C22#57712yes—
175C4—24no—
176C4—22no—
177C4—44no—
178C4—42no—
179C4—44no—
180C4—42no—
181C4—42no—
182C4—82no—
183C4#45214yes—
184C4C2×C4⋊Q811yescharacteristic
185C4#53611yescharacteristic
186C4D4×Q812yes—
187C2#243411yescharacteristic
188C2C2×#45211yescharacteristic
189C2#301011yescharacteristic
190C2#338112yes—
191C2#338712yes—
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
80. #103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4982
C2×#4772
#25672
#20592
#19882
#19842
#18652
C4×#1032