Finite Group Databasebeta

← #25705

Subgroups — #25705

Order 256.

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

normal subgroups only

This group has 529 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×#500C211yesmaximal, characteristic, complemented
3C2×#475C211yesmaximal, characteristic, complemented
4#2547C211yesmaximal, characteristic
5#2036C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic
8#1854C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10#556—24no—
11#556—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#126—22no—
15C2×#124—21no—
16C2×#124—21no—
17C2×C22⋊Q8C2212yes—
18C2×C4×Q8—21no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2212yes—
22#500—24no—
23#475—24no—
24C2×#107C2214yes—
25C2×#106C2212yes—
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C22⋊C8C2214yes—
30C2×C4×C8C2211yescharacteristic
31#397—22no—
32#391—21no—
33#380C2214yes—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8C2311yescharacteristic
40C22×D4D411yescharacteristic
41C23×C4C2312yes—
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8D414yes—
48#126—216no—
49#124—28no—
50#124—44no—
51C22⋊Q8—24no—
52C22⋊Q8—24no—
53C4×Q8—28no—
54C4×Q8—22no—
55C4×Q8D414yes—
56C4×D4—22no—
57C4×D4—22no—
58#117—24no—
59#117—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4D412yes—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77#107—28no—
78#106—24no—
79C4⋊C8—22no—
80#103—24no—
81#103—24no—
82C22⋊C8—28no—
83C4×C8—22no—
84#95—21no—
85#95—24no—
86C24C2×D412yes—
87C2×Q8—24no—
88C2×Q8—21no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C2×D412yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—22no—
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#4114yes—
118C4⋊C4—28no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4C2×D412yes—
130C4⋊C4C2×D414yes—
131C22⋊C4—28no—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—48no—
135C42—28no—
136C42—24no—
137C42—42no—
138C42C2×D412yes—
139C42C2×D412yes—
140C23—22no—
141C23—24no—
142C23—22no—
143C23—22no—
144C23—24no—
145C23C22×D411yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—42no—
149Q8—48no—
150Q8—84no—
151D4—44no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4#12012yes—
172C2×C4#12012yes—
173C2×C4#13512yes—
174C2×C4AGL(1,ℤ/8)12yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4112yes—
178C2×C42−1+412yes—
179C8—24no—
180C8—42no—
181C22—44no—
182C22—44no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22C2×#12011yescharacteristic
187C22#54712yes—
188C22C2×#13511yescharacteristic
189C22C2×AGL(1,ℤ/8)11yescharacteristic
190C22#57812yes—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—44no—
195C4—42no—
196C4—82no—
197C4—82no—
198C4#44914yes—
199C4#53211yescharacteristic
200C4#53611yescharacteristic
201C4#54812yes—
202C2—82no—
203C2#243211yescharacteristic
204C2C2×#44911yescharacteristic
205C2#301111yescharacteristic
206C2#334212yes—
207C2#335312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#500C2semidirect
3. C2×#475C2semidirect
5. #2036C2semidirect
6. #1988C2semidirect
8. #1854C2semidirect
9. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#5002
C2×#4752
#25472
#20362
#19882
#19542
#18542
C4×#1032