Finite Group Databasebeta

← #25650

Subgroups — #25650

Order 256.

527 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 139 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×#556C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic, complemented
5#2042C214yesmaximal
6#1984C211yesmaximal, characteristic, complemented
7#1981C211yesmaximal, characteristic, complemented
8#1863C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic
10#556—24no—
11#556—24no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#124C2212yes—
17C2×C22⋊Q8—21no—
18C2×C4×Q8C2212yes—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22#487—24no—
23#474—24no—
24C2×#107C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2214yes—
27C2×#103C2214yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×C4×C8C2211yescharacteristic
31#398—22no—
32#391—21no—
33#389C2214yes—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8D411yescharacteristic
39C22×Q8C2312yes—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—44no—
48C4⋊Q8D414yes—
49#126—216no—
50#124—24no—
51#124—24no—
52C22⋊Q8—28no—
53C4×Q8—24no—
54C4×Q8—24no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—22no—
58C4×D4D414yes—
59#117—28no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D412yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—24no—
78C4⋊C8—22no—
79#103—28no—
80#103—28no—
81#102—24no—
82#102—24no—
83C4×C8—22no—
84#95—21no—
85#95—24no—
86C24—21no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—24no—
92C2×Q8—24no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C2×D414yes—
97C2×D4—21no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4C2×D412yes—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—24no—
117C2×C8—22no—
118C4⋊C4—28no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—24no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—48no—
128C4⋊C4—42no—
129C4⋊C4C2×D412yes—
130C4⋊C4#4114yes—
131C22⋊C4—28no—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C42—28no—
135C42—22no—
136C42—22no—
137C42—42no—
138C42C2×D412yes—
139C42C2×D412yes—
140C23—22no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23—42no—
145C23C22×D411yescharacteristic
146Q8—28no—
147Q8—42no—
148Q8—42no—
149Q8—44no—
150Q8—84no—
151D4—22no—
152D4—22no—
153D4—42no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—48no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4#12012yes—
173C2×C4#12012yes—
174C2×C4#13512yes—
175C2×C4AGL(1,ℤ/8)12yes—
176C2×C4C22×D411yescharacteristic
177C2×C4C22×D411yescharacteristic
178C2×C4C2×#4112yes—
179C2×C42−1+412yes—
180C8—42no—
181C8—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—82no—
186C22—82no—
187C22C2×#12011yescharacteristic
188C22#54812yes—
189C22C2×#13511yescharacteristic
190C22C2×AGL(1,ℤ/8)11yescharacteristic
191C22#57812yes—
192C4—22no—
193C4—44no—
194C4—44no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—82no—
199C4#44914yes—
200C4#53211yescharacteristic
201C4#53611yescharacteristic
202C4#54712yes—
203C2—82no—
204C2#243211yescharacteristic
205C2C2×#44911yescharacteristic
206C2#301111yescharacteristic
207C2#336412yes—
208C2#336712yes—
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×#487C2semidirect
4. #2546C2semidirect
6. #1984C2semidirect
7. #1981C2semidirect
8. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4872
C2×#4742
#25462
#20422
#19842
#19812
#18632
C4×#1022