Finite Group Databasebeta

← #26299

Subgroups — #26299

Order 256.

367 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 123 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#2563C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2075C212yesmaximal, complemented
5#1990C211yesmaximal, characteristic, complemented
6#1987C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1825C212yesmaximal, complemented
11C4×#106C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×D4C2211yescharacteristic
14C2×#117C2212yes—
15C2×#107C2212yes—
16C2×#106C2211yescharacteristic
17C2×#106C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×C22⋊C8C2212yes—
20C2×C22⋊C8C2212yes—
21C2×C4×C8C2211yescharacteristic
22#393—22no—
23#390—21no—
24#390—21no—
25#388—22no—
26#384C2211yescharacteristic
27#382—21no—
28#380C2212yes—
29#380C2212yes—
30#380C2212yes—
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#340C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×Q8—21no—
42C22×D4D411yescharacteristic
43C23×C4C2312yes—
44C22×C8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2311yescharacteristic
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4×D4—22no—
50C4×D4—22no—
51#117—24no—
52#117—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D412yes—
68C2×C42—22no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42C2311yescharacteristic
72#107—42no—
73#106—22no—
74#106—22no—
75C4⋊C8—22no—
76C22⋊C8—24no—
77C22⋊C8—42no—
78C4×C8—22no—
79#95—22no—
80#95—21no—
81#95—21no—
82#95—22no—
83#95—22no—
84C24C2×D412yes—
85C2×Q8—42no—
86C2×Q8—81no—
87C2×Q8—81no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2411yescharacteristic
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—21no—
115C2×C8#4112yes—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—81no—
131C4⋊C4#4112yes—
132C22⋊C4—24no—
133C22⋊C4—44no—
134C22⋊C4—82no—
135C22⋊C4—82no—
136C22⋊C4—82no—
137C42—44no—
138C42—42no—
139C42—42no—
140C42#4112yes—
141C42#4112yes—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—22no—
146C23—22no—
147C23—22no—
148C23—22no—
149C23C22×D411yescharacteristic
150Q8—84no—
151D4—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4#12312yes—
180C2×C4#12312yes—
181C2×C4#13512yes—
182C2×C4#13712yes—
183C2×C4C2×#4111yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C4H(5,𝔽2)11yescharacteristic
188C2×C42−1+411yescharacteristic
189C8—22no—
190C8—41no—
191C8—42no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22C2×#12311yescharacteristic
200C22#54611yescharacteristic
201C22#54711yescharacteristic
202C22C2×#13511yescharacteristic
203C22C2×#13711yescharacteristic
204C22#57611yescharacteristic
205C22#57711yescharacteristic
206C4—22no—
207C4—42no—
208C4—42no—
209C4—81no—
210C4—82no—
211C4—81no—
212C4—82no—
213C4—81no—
214C4#48514yes—
215C4#52911yescharacteristic
216C4#54211yescharacteristic
217C4#55611yescharacteristic
218C4#55911yescharacteristic
219C2—82no—
220C2#244711yescharacteristic
221C2C2×#48511yescharacteristic
222C2#309611yescharacteristic
223C2#330511yescharacteristic
224C2#333111yescharacteristic
225C2#334011yescharacteristic
226C2#335711yescharacteristic
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. #2075C2semidirect
5. #1990C2semidirect
6. #1987C2semidirect
8. #1947C2semidirect
9. #1854C2semidirect
10. #1825C2semidirect
11. C4×#106C2semidirect

Maximal subgroups

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

subgroupindex
#25632
#23542
#20772
#20752
#19902
#19872
#19542
#19472
#18542
#18252
C4×#1062