Finite Group Databasebeta

← #25929

Subgroups — #25929

Order 256.

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

normal subgroups only

This group has 631 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×D4×Q8C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic, complemented
5#2024C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1984C211yesmaximal, characteristic, complemented
8#1877C214yesmaximal
9C4×#102C211yesmaximal, characteristic
10C23×Q8—21no—
11C22×SD16C2212yes—
12D4×Q8—24no—
13D4×Q8—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2212yes—
21C2×C4×D4C2211yescharacteristic
22#487—24no—
23#461—24no—
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2214yes—
26C2×#103C2212yes—
27C2×#102C2212yes—
28C2×C4×C8C2211yescharacteristic
29#400—22no—
30#391—21no—
31#391C2214yes—
32#382—21no—
33C4×C4⋊C4C2211yescharacteristic
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—21no—
40C22×Q8—44no—
41C22×Q8—44no—
42C22×Q8D411yescharacteristic
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—22no—
46C23×C4—21no—
47C2×SD16—24no—
48C2×SD16—28no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—24no—
52C4⋊Q8—24no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8D414yes—
57C22⋊Q8—28no—
58C22⋊Q8—216no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×D4—24no—
62C4×D4—24no—
63C4×D4—22no—
64C4×D4D414yes—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4D412yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C42—21no—
77C2×C42C2312yes—
78C2×C42C2311yescharacteristic
79C4⋊C8—22no—
80#103—24no—
81#103—28no—
82#102—24no—
83C4×C8—22no—
84#95—21no—
85#95—24no—
86#95—24no—
87C24—21no—
88C2×Q8—28no—
89C2×Q8—22no—
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—24no—
96C2×Q8—44no—
97C2×Q8—432no—
98C2×Q8—44no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8SD1614yescomplemented
102C2×D4—21no—
103C2×D4—22no—
104C2×D4C2×D412yes—
105C2×D4#4114yes—
106C22×C4—28no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121SD16—48no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—24no—
126C2×C8—22no—
127C4⋊C4—28no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4C2×D414yes—
138C4⋊C4C2×D412yes—
139C22⋊C4—28no—
140C22⋊C4—24no—
141C42—22no—
142C42—22no—
143C42—42no—
144C42C2×D412yes—
145C42C2×D412yes—
146C23—22no—
147C23—24no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23C22×D411yescharacteristic
152Q8—28no—
153Q8—416no—
154Q8—48no—
155Q8—42no—
156Q8—42no—
157Q8—44no—
158D4—22no—
159D4—24no—
160D4#12414yes—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—416no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—48no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4#12012yes—
180C2×C4#12012yes—
181C2×C4C2×SD1614yes—
182C2×C4C2×SD1612yes—
183C2×C4AGL(1,ℤ/8)12yes—
184C2×C4C22×D411yescharacteristic
185C2×C4C22×D411yescharacteristic
186C2×C4C2×#4112yes—
187C2×C42−1+412yes—
188C8—42no—
189C8—42no—
190C22—24no—
191C22—44no—
192C22—44no—
193C22—42no—
194C22—42no—
195C22C2×#12011yescharacteristic
196C22#54812yes—
197C22C22×SD1611yescharacteristic
198C22C2×AGL(1,ℤ/8)11yescharacteristic
199C22#57812yes—
200C4—24no—
201C4—22no—
202C4—44no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—82no—
207C4#45014yes—
208C4C2×#12411yescharacteristic
209C4#53611yescharacteristic
210C4#54812yes—
211C2—44no—
212C2#243611yescharacteristic
213C2C2×#45011yescharacteristic
214C2#301111yescharacteristic
215C2#335912yes—
216C2#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. #2568C2semidirect
5. #2024C2semidirect
6. #1988C2semidirect
7. #1984C2semidirect
101. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4872
C2×#4612
#25682
#20242
#19882
#19842
#18772
C4×#1022