Finite Group Databasebeta

← #25835

Subgroups — #25835

Order 256.

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

normal subgroups only

This group has 541 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×#554C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic, complemented
5#2059C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1864C214yesmaximal
9C4×#102C211yesmaximal, characteristic
10#554—24no—
11#554—24no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—22no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2212yes—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4—21no—
22#498—24no—
23#474—24no—
24C2×#107C2212yes—
25C2×#106C2214yes—
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C4×C8C2211yescharacteristic
30#398—22no—
31#391—21no—
32#389C2214yes—
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8D411yescharacteristic
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C23×C4—22no—
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—44no—
49C4⋊Q8D414yes—
50#125—24no—
51#125—24no—
52C22⋊Q8—28no—
53C22⋊Q8—216no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—24no—
57C4×D4—24no—
58C4×D4—22no—
59C4×D4D414yes—
60C2×C4⋊C4—28no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4D412yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—28no—
79C4⋊C8—22no—
80#102—24no—
81#102Q818yescomplemented
82C4×C8—22no—
83#95—21no—
84#95—24no—
85C24—21no—
86C2×Q8—22no—
87C2×Q8—24no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—82no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4C2×D412yes—
97C2×D4C2×Q814yes—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—28no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—84no—
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—24no—
116C2×C8C2×Q814yes—
117C4⋊C4—28no—
118C4⋊C4—28no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—24no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—48no—
128C4⋊C4—42no—
129C4⋊C4C2×D414yes—
130C4⋊C4C2×D412yes—
131C4⋊C4C2×Q814yes—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C42—22no—
135C42—22no—
136C42—42no—
137C42C2×D412yes—
138C42C2×D412yes—
139C23—22no—
140C23—24no—
141C23—41no—
142C23—41no—
143C23—42no—
144C23C22×D411yescharacteristic
145Q8—44no—
146Q8—84no—
147Q8—84no—
148D4—22no—
149D4—24no—
150D4C4⋊Q814yescomplemented
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—48no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—88no—
170C2×C4#12012yes—
171C2×C4#12012yes—
172C2×C4#13512yes—
173C2×C4AGL(1,ℤ/8)12yes—
174C2×C4C22×D411yescharacteristic
175C2×C4C22×D411yescharacteristic
176C2×C4C22×Q812yes—
177C2×C42−1+412yes—
178C8—24no—
179C8—42no—
180C22—24no—
181C22—44no—
182C22—44no—
183C22—42no—
184C22—42no—
185C22C2×#12011yescharacteristic
186C22D4×Q812yes—
187C22C2×#13511yescharacteristic
188C22C2×AGL(1,ℤ/8)11yescharacteristic
189C22#57712yes—
190C4—22no—
191C4—44no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—44no—
196C4—82no—
197C4#44914yes—
198C4C2×C4⋊Q811yescharacteristic
199C4#53611yescharacteristic
200C4D4×Q812yes—
201C2—44no—
202C2#243411yescharacteristic
203C2C2×#44911yescharacteristic
204C2#300911yescharacteristic
205C2#338112yes—
206C2#338412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. C2×#498C2semidirect
4. #2546C2semidirect
5. #2059C2semidirect
6. #1987C2semidirect
7. #1984C2semidirect
81. #102Q8semidirect
150. D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4982
C2×#4742
#25462
#20592
#19872
#19842
#18642
C4×#1022