Finite Group Databasebeta

← #25932

Subgroups — #25932

Order 256.

569 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 151 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×Q82C211yesmaximal, characteristic, complemented
2C2×#486C211yesmaximal, characteristic
3C2×#460C211yesmaximal, characteristic
4#2548C211yesmaximal, characteristic
5#2023C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1880C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10C22×SD16C2212yes—
11Q82—24no—
12Q82—24no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8—22no—
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124—21no—
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#486—24no—
24#460—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#102C2214yes—
28C2×#102C2212yes—
29C2×C4×C8C2211yescharacteristic
30#394—22no—
31#390—21no—
32#390C2214yes—
33C4×C4⋊C4C2211yescharacteristic
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—22no—
38C22×Q8—21no—
39C22×Q8D411yescharacteristic
40C22×Q8C2311yescharacteristic
41C22×D4—21no—
42C22×D4—21no—
43C22×D4C2312yes—
44C2×SD16—24no—
45C2×SD16—28no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—216no—
49C4⋊Q8—28no—
50C4⋊Q8—24no—
51C4⋊Q8—24no—
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54#127—22no—
55#127—22no—
56#124—44no—
57C4×Q8—28no—
58C4×Q8—24no—
59C4×Q8—24no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8D414yes—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4D412yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C42—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80C4⋊C8—22no—
81#103—24no—
82#102—24no—
83#102—28no—
84C4×C8—22no—
85#95—21no—
86C24—22no—
87C2×Q8—28no—
88C2×Q8—22no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—82no—
94C2×Q8—84no—
95C2×Q8SD1614yescomplemented
96C2×Q8C2×D412yes—
97C2×Q8#4114yes—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—82no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116SD16—48no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—24no—
121C2×C8—22no—
122C4⋊C4—28no—
123C4⋊C4—28no—
124C4⋊C4—24no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4C2×D414yes—
133C4⋊C4C2×D412yes—
134C22⋊C4—48no—
135C22⋊C4—84no—
136C42—28no—
137C42—24no—
138C42—22no—
139C42—22no—
140C42—42no—
141C42C2×D412yes—
142C42C2×D412yes—
143C23—42no—
144C23—44no—
145C23—42no—
146C23—42no—
147C23—44no—
148C23C22×D411yescharacteristic
149Q8—22no—
150Q8—24no—
151Q8—28no—
152Q8—84no—
153Q8—88no—
154Q8#12414yescomplemented
155D4—42no—
156D4—42no—
157D4—44no—
158D4—84no—
159D4—84no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4#12012yes—
175C2×C4#12012yes—
176C2×C4C2×SD1614yes—
177C2×C4C2×SD1612yes—
178C2×C4#13712yes—
179C2×C4C22×D411yescharacteristic
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4112yes—
182C2×C42−1+412yes—
183C8—42no—
184C8—42no—
185C22—84no—
186C22—84no—
187C22—82no—
188C22—82no—
189C22—82no—
190C22C2×#12011yescharacteristic
191C22#54812yes—
192C22C22×SD1611yescharacteristic
193C22C2×#13711yescharacteristic
194C22#57612yes—
195C4—24no—
196C4—24no—
197C4—22no—
198C4—42no—
199C4—44no—
200C4—42no—
201C4—82no—
202C4#44814yes—
203C4C2×#12411yescharacteristic
204C4#53611yescharacteristic
205C4#54812yes—
206C2—162no—
207C2#243611yescharacteristic
208C2C2×#44811yescharacteristic
209C2#300811yescharacteristic
210C2#336212yes—
211C2#337512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
1. C2×Q82C2semidirect
5. #2023C2semidirect
6. #1987C2semidirect
7. #1984C2semidirect
8. #1880C2semidirect
9. C4×#103C2semidirect
95. C2×Q8SD16semidirect
154. Q8#124semidirect

Maximal subgroups

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

subgroupindex
C2×Q822
C2×#4862
C2×#4602
#25482
#20232
#19872
#19842
#18802
C4×#1032