Finite Group Databasebeta

← #25672

Subgroups — #25672

Order 256.

547 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 155 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×#533C211yesmaximal, characteristic
2C2×#498C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic
5#2035C214yesmaximal
6#1990C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1865C214yesmaximal, complemented
9C4×C22⋊C8C211yesmaximal, characteristic
10#533—24no—
11#533—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125—21no—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×#117—21no—
22C22×C42C2211yescharacteristic
23#498—24no—
24#487—24no—
25C2×#106C2214yes—
26C2×#103C2214yes—
27C2×#103C2214yes—
28C2×C22⋊C8C2212yes—
29C2×C4×C8C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#391—21no—
32#389—21no—
33#389C2212yes—
34#384C2211yescharacteristic
35#380—21no—
36#380C2212yes—
37#340C2214yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8—21no—
41C22×Q8C2312yes—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8D414yes—
52C4⋊Q8D414yes—
53#125—28no—
54C22⋊Q8—28no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—44no—
58C4×Q8—28no—
59C4×Q8—24no—
60C4×Q8—24no—
61#117—28no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#106—28no—
84#103—28no—
85#103—28no—
86C22⋊C8—24no—
87C4×C8—22no—
88C4×C8—22no—
89#95—22no—
90#95—22no—
91C24C2×D411yescharacteristic
92C2×Q8—24no—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—24no—
96C2×Q8—42no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8C2×D414yes—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4SD1614yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8#4114yes—
128C4⋊C4—28no—
129C4⋊C4—28no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4C2×D414yes—
142C22⋊C4—28no—
143C22⋊C4—44no—
144C42—28no—
145C42—22no—
146C42—22no—
147C42—21no—
148C42—21no—
149C42C2×D412yes—
150C42C2×D412yes—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23C2×SD1612yes—
158C23C22×D411yescharacteristic
159Q8—28no—
160Q8—44no—
161Q8—48no—
162Q8—84no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—28no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4#12012yes—
188C2×C4#12012yes—
189C2×C4C2×SD1612yes—
190C2×C4C2×SD1612yes—
191C2×C4#13512yes—
192C2×C4#13712yes—
193C2×C4#13712yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C2×#4112yes—
197C2×C42−1+412yes—
198C8—24no—
199C8—42no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22C2×#12011yescharacteristic
208C22#53211yescharacteristic
209C22#53611yescharacteristic
210C22C22×SD1611yescharacteristic
211C22C2×#13511yescharacteristic
212C22C2×#13711yescharacteristic
213C22C2×#13711yescharacteristic
214C4—22no—
215C4—41no—
216C4—44no—
217C4—41no—
218C4—44no—
219C4—82no—
220C4—82no—
221C4#44814yes—
222C4#45214yes—
223C4#54712yes—
224C4#54812yes—
225C2—42no—
226C2#243211yescharacteristic
227C2C2×#44811yescharacteristic
228C2C2×#45211yescharacteristic
229C2#316711yescharacteristic
230C2#318511yescharacteristic
231C2#322911yescharacteristic
232C2#323111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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×#487C2semidirect
6. #1990C2semidirect
7. #1956C2semidirect
8. #1865C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4982
C2×#4872
#28572
#20352
#19902
#19562
#18652
C4×C22⋊C82