Finite Group Databasebeta

← #25884

Subgroups — #25884

Order 256.

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

normal subgroups only

This group has 651 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×#475C211yesmaximal, characteristic, complemented
4#2547C211yesmaximal, characteristic
5#2035C214yesmaximal, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic
8#1787C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10C23×Q8C2212yes—
11D4×Q8—24no—
12D4×Q8—24no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124—21no—
17C2×C22⋊Q8—22no—
18C2×C22⋊Q8C2212yes—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22#487—24no—
23#475—24no—
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2214yes—
28C2×#103C2212yes—
29C2×#103C2214yes—
30C2×C22⋊C8C2214yes—
31C2×C4×C8C2211yescharacteristic
32#397—22no—
33#391—21no—
34#380C2214yes—
35C4×C4⋊C4C2211yescharacteristic
36#340C2212yes—
37C22×Q8—24no—
38C22×Q8—24no—
39C22×Q8—22no—
40C22×Q8—21no—
41C22×Q8—28no—
42C22×Q8—44no—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×D4D411yescharacteristic
46C23×C4—22no—
47C23×C4C2312yes—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—28no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55#124—44no—
56C22⋊Q8—216no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C4×Q8—22no—
60C4×Q8D414yes—
61C4×D4—28no—
62C4×D4—22no—
63C4×D4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4D412yes—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C2311yescharacteristic
79#106—24no—
80C4⋊C8—22no—
81#103—24no—
82#103—28no—
83#103—24no—
84#103—44no—
85C22⋊C8—44no—
86C4×C8—22no—
87#95—21no—
88#95—24no—
89C24C2×D412yes—
90C2×Q8—28no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—24no—
96C2×Q8—48no—
97C2×Q8—416no—
98C2×Q8—48no—
99C2×Q8—48no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8C2×D414yes—
104C2×Q8C2×D412yes—
105C2×Q8#4114yes—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4SD1614yes—
109C22×C4—21no—
110C22×C4—28no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—48no—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—24no—
129C4⋊C4—28no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4C2×D412yes—
141C22⋊C4—28no—
142C22⋊C4—24no—
143C22⋊C4—48no—
144C42—24no—
145C42—42no—
146C42C2×D412yes—
147C42C2×D412yes—
148C23—22no—
149C23—24no—
150C23—22no—
151C23—24no—
152C23C2×SD1614yes—
153C23C22×D411yescharacteristic
154Q8—22no—
155Q8—24no—
156Q8—28no—
157Q8—416no—
158Q8—42no—
159Q8—42no—
160Q8—84no—
161Q8#12414yescomplemented
162D4—28no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—416no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4#12012yes—
183C2×C4#12012yes—
184C2×C4C2×SD1612yes—
185C2×C4#13712yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C22×D411yescharacteristic
188C2×C4C2×#4112yes—
189C2×C4H(5,𝔽2)12yes—
190C8—42no—
191C8—42no—
192C22—24no—
193C22—28no—
194C22—44no—
195C22—42no—
196C22—42no—
197C22C2×#12011yescharacteristic
198C22#54612yes—
199C22C22×SD1611yescharacteristic
200C22C2×#13711yescharacteristic
201C22#57812yes—
202C4—24no—
203C4—22no—
204C4—42no—
205C4—44no—
206C4—42no—
207C4—82no—
208C4—82no—
209C4#44814yes—
210C4C2×#12411yescharacteristic
211C4#53511yescharacteristic
212C4#54612yes—
213C2—44no—
214C2#243511yescharacteristic
215C2C2×#44811yescharacteristic
216C2#301111yescharacteristic
217C2#329812yes—
218C2#332812yes—
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×#487C2semidirect
3. C2×#475C2semidirect
5. #2035C2semidirect
6. #1981C2semidirect
8. #1787C2semidirect
9. C4×#103C2semidirect
161. Q8#124semidirect

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×#4752
#25472
#20352
#19812
#19542
#17872
C4×#1032