Finite Group Databasebeta

← #26298

Subgroups — #26298

Order 256.

413 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#2547C211yesmaximal, characteristic
2D4×C4⋊C4C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2077C212yesmaximal, complemented
5#1990C211yesmaximal, characteristic, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic
8#1948C211yesmaximal, characteristic, complemented
9#1853C212yesmaximal, complemented
10#1824C212yesmaximal, complemented
11C4×#106C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124—21no—
14C2×C4×D4—21no—
15C2×C4×D4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C2×#106C2211yescharacteristic
18C2×#106C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C22⋊C8C2212yes—
22C2×C22⋊C8C2212yes—
23C2×C4×C8C2211yescharacteristic
24#397—22no—
25#391—21no—
26#389—21no—
27#385—22no—
28#384C2211yescharacteristic
29#380C2212yes—
30#380C2212yes—
31#380C2212yes—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×Q8—21no—
42C22×D4D411yescharacteristic
43C23×C4—22no—
44C23×C4C2312yes—
45C22×C8C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50#124—44no—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4—44no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D412yes—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42C2311yescharacteristic
74#106—22no—
75#106—22no—
76#106—42no—
77C4⋊C8—22no—
78C22⋊C8—24no—
79C22⋊C8—42no—
80C4×C8—22no—
81#95—21no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95—22no—
86C24C2×D412yes—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—42no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4SD1614yes—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—82no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—21no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8#4112yes—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—81no—
137C4⋊C4#4112yes—
138C22⋊C4—24no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—44no—
142C42—42no—
143C42—42no—
144C42#4112yes—
145C42#4112yes—
146C23—22no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23C2×SD1614yes—
153C23C22×D411yescharacteristic
154Q8—84no—
155D4—28no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—88no—
184C2×C4#12312yes—
185C2×C4#12312yes—
186C2×C4C2×SD1612yes—
187C2×C4AGL(1,ℤ/8)12yes—
188C2×C4C2×#4111yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4H(5,𝔽2)11yescharacteristic
193C2×C42−1+411yescharacteristic
194C8—22no—
195C8—41no—
196C8—42no—
197C22—24no—
198C22—24no—
199C22—24no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22C2×#12311yescharacteristic
205C22#54611yescharacteristic
206C22#54711yescharacteristic
207C22C22×SD1611yescharacteristic
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22#57711yescharacteristic
210C22#57811yescharacteristic
211C4—22no—
212C4—42no—
213C4—42no—
214C4—81no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4—82no—
219C4#48114yes—
220C4#52911yescharacteristic
221C4#54211yescharacteristic
222C4#55611yescharacteristic
223C4#55911yescharacteristic
224C2—44no—
225C2#244711yescharacteristic
226C2C2×#48111yescharacteristic
227C2#309711yescharacteristic
228C2#329811yescharacteristic
229C2#332411yescharacteristic
230C2#333511yescharacteristic
231C2#335311yescharacteristic
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
3. #2077C2semidirect
4. #2077C2semidirect
5. #1990C2semidirect
6. #1988C2semidirect
8. #1948C2semidirect
9. #1853C2semidirect
10. #1824C2semidirect
11. C4×#106C2semidirect

Maximal subgroups

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

subgroupindex
#25472
D4×C4⋊C42
#20772
#19902
#19882
#19542
#19482
#18532
#18242
C4×#1062