Finite Group Databasebeta

← #25930

Subgroups — #25930

Order 256.

591 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 143 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, complemented
2C2×#486C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4#2548C211yesmaximal, characteristic, complemented
5#2025C214yesmaximal, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1879C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic, complemented
10C22×D8C2212yes—
11#554—24no—
12#554—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#127C2211yescharacteristic
15C2×#125C2212yes—
16C2×#124—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2212yes—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4—21no—
23#486—24no—
24#459—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#102C2214yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C4×C8C2211yescharacteristic
30#394—22no—
31#390—21no—
32#390C2214yes—
33C4×C4⋊C4C2211yescharacteristic
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8D411yescharacteristic
39C22×D4—21no—
40C22×D4—21no—
41C22×D4C2312yes—
42C22×D4C2311yescharacteristic
43C23×C4—21no—
44C23×C4—22no—
45C2×D8—24no—
46C2×D8—28no—
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50C4⋊Q8D414yes—
51#127—22no—
52#127—22no—
53#125—24no—
54#125—24no—
55#124—44no—
56C22⋊Q8—28no—
57C22⋊Q8—216no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×D4—24no—
61C4×D4—24no—
62C4×D4—22no—
63C4×D4D414yes—
64C2×C4⋊C4—28no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4D412yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C42—21no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80C4⋊C8—22no—
81#102—24no—
82#102—24no—
83#102—28no—
84C4×C8—22no—
85#95—21no—
86C24—21no—
87C24—22no—
88C2×Q8—22no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—82no—
93C2×D4—21no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—24no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—82no—
104C2×D4C2×D412yes—
105C2×D4#4114yes—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—28no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—84no—
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119D8—48no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—24no—
124C2×C8—22no—
125C4⋊C4—28no—
126C4⋊C4—28no—
127C4⋊C4—22no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4C2×D414yes—
135C4⋊C4C2×D412yes—
136C22⋊C4—28no—
137C22⋊C4—24no—
138C22⋊C4—48no—
139C22⋊C4—84no—
140C42—22no—
141C42—22no—
142C42—42no—
143C42C2×D412yes—
144C42C2×D412yes—
145C23—22no—
146C23—24no—
147C23—41no—
148C23—41no—
149C23—42no—
150C23—42no—
151C23—44no—
152C23—42no—
153C23—42no—
154C23—44no—
155C23C22×D411yescharacteristic
156Q8—44no—
157Q8—84no—
158D4—22no—
159D4—24no—
160D4—42no—
161D4—42no—
162D4—44no—
163D4—84no—
164D4—84no—
165D4#12414yescomplemented
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—48no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—88no—
182C2×C4#12012yes—
183C2×C4#12012yes—
184C2×C4#13512yes—
185C2×C4AGL(1,ℤ/8)12yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C22×D411yescharacteristic
188C2×C4C2×#4112yes—
189C2×C42−1+412yes—
190C8—42no—
191C8—42no—
192C22—24no—
193C22—44no—
194C22—44no—
195C22—42no—
196C22—42no—
197C22—84no—
198C22—84no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22C2×#12011yescharacteristic
203C22#54812yes—
204C22C2×#13511yescharacteristic
205C22C2×AGL(1,ℤ/8)11yescharacteristic
206C22#57612yes—
207C4—22no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—44no—
212C4—82no—
213C4#44914yes—
214C4C2×#12411yescharacteristic
215C4#53611yescharacteristic
216C4#54812yes—
217C2—44no—
218C2—162no—
219C2#243611yescharacteristic
220C2C2×#44911yescharacteristic
221C2#300811yescharacteristic
222C2#336112yes—
223C2#336612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
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×#554C2semidirect
2. C2×#486C2semidirect
4. #2548C2semidirect
5. #2025C2semidirect
6. #1987C2semidirect
7. #1984C2semidirect
8. #1879C2semidirect
9. C4×#102C2semidirect
165. D4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4862
C2×#4592
#25482
#20252
#19872
#19842
#18792
C4×#1022