Finite Group Databasebeta

← #25700

Subgroups — #25700

Order 256.

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

normal subgroups only

This group has 561 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×#555C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2547C211yesmaximal, characteristic, complemented
5#2034C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1854C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10#555—24no—
11#555—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124—21no—
15C2×#124—22no—
16C2×#121—21no—
17C2×#121C2212yes—
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117—21no—
21C2×#117C2212yes—
22#498—24no—
23#474—24no—
24C2×#107C2212yes—
25C2×#106C2214yes—
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2212yes—
28C2×#102C2212yes—
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×D4—22no—
40C22×D4D411yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4—21no—
43C23×C4C2312yes—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8D414yes—
50#124—216no—
51#124—44no—
52#121—28no—
53#121—24no—
54#121—24no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4D414yes—
59#117—28no—
60#117—24no—
61#117—24no—
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—21no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4D412yes—
74C2×C42—22no—
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—28no—
79C4⋊C8—22no—
80#102—24no—
81#102—24no—
82C22⋊C8—28no—
83C4×C8—22no—
84#95—21no—
85#95—24no—
86C24—21no—
87C24C2×D412yes—
88C2×Q8—24no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—82no—
93C2×D4—28no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—21no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—84no—
101C2×D4C2×D412yes—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—24no—
121C2×C8#4114yes—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4C2×D414yes—
132C4⋊C4C2×D412yes—
133C22⋊C4—28no—
134C22⋊C4—28no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—48no—
138C42—28no—
139C42—42no—
140C42C2×D412yes—
141C42C2×D412yes—
142C23—22no—
143C23—22no—
144C23—24no—
145C23—22no—
146C23—22no—
147C23—24no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23C22×D411yescharacteristic
153Q8—84no—
154Q8—84no—
155D4—22no—
156D4—22no—
157D4—42no—
158D4—44no—
159D4—88no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—48no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4#12012yes—
179C2×C4#12012yes—
180C2×C4#13512yes—
181C2×C4AGL(1,ℤ/8)12yes—
182C2×C4C22×D411yescharacteristic
183C2×C4C22×D411yescharacteristic
184C2×C4C2×#4112yes—
185C2×C42−1+412yes—
186C8—24no—
187C8—42no—
188C22—44no—
189C22—44no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22C2×#12011yescharacteristic
199C22#54712yes—
200C22C2×#13511yescharacteristic
201C22C2×AGL(1,ℤ/8)11yescharacteristic
202C22#57712yes—
203C4—22no—
204C4—42no—
205C4—42no—
206C4—44no—
207C4—82no—
208C4—82no—
209C4#44914yes—
210C4#53211yescharacteristic
211C4#53611yescharacteristic
212C4#54812yes—
213C2—82no—
214C2—82no—
215C2#243211yescharacteristic
216C2C2×#44911yescharacteristic
217C2#300911yescharacteristic
218C2#334012yes—
219C2#335412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#555C2
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×#498C2semidirect
3. C2×#474C2semidirect
4. #2547C2semidirect
5. #2034C2semidirect
6. #1987C2semidirect
7. #1954C2semidirect
8. #1854C2semidirect
9. C4×#102C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4982
C2×#4742
#25472
#20342
#19872
#19542
#18542
C4×#1022