Finite Group Databasebeta

← #29911

Subgroups — #29911

Order 256.

353 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 115 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#2855C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2097C212yesmaximal, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2072C211yesmaximal, characteristic, complemented
8#2072C211yesmaximal, characteristic, complemented
9#1873C212yesmaximal
10#1854C212yesmaximal
11#1817C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×#117C2212yes—
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×C22⋊C8C2212yes—
20#391C2211yescharacteristic
21#391C2211yescharacteristic
22#389C2211yescharacteristic
23#389C2211yescharacteristic
24#387—21no—
25#387—21no—
26#385—21no—
27#384C2211yescharacteristic
28#384C2211yescharacteristic
29#382C2212yes—
30#381—21no—
31#380—21no—
32#380—21no—
33#380C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C22⋊Q8—24no—
50#117—42no—
51#117—42no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C42—24no—
68C2×C42—22no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#106—42no—
75#103—42no—
76#103—42no—
77C22⋊C8—24no—
78#95—22no—
79#95—22no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87C24C2×D411yescharacteristic
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4#4114yes—
113C22×C4C2411yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—44no—
137C22⋊C4—42no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—44no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23#12312yes—
150C23C22×D411yescharacteristic
151Q8—42no—
152Q8—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4#12312yes—
188C2×C4#13512yes—
189C2×C4#13712yes—
190C2×C4C2×#4112yes—
191C2×C4H(5,𝔽2)11yescharacteristic
192C2×C4H(5,𝔽2)11yescharacteristic
193C2×C42−1+412yes—
194C8—42no—
195C8—42no—
196C22—22no—
197C22—22no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22C2×#12311yescharacteristic
204C22#53511yescharacteristic
205C22#53611yescharacteristic
206C22C2×#13511yescharacteristic
207C22C2×#13711yescharacteristic
208C22#57511yescharacteristic
209C22#57511yescharacteristic
210C4—22no—
211C4—22no—
212C4—22no—
213C4—41no—
214C4—41no—
215C4—81no—
216C4—81no—
217C4—82no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4#53912yes—
222C4#53912yes—
223C2—42no—
224C2#262311yescharacteristic
225C2#309111yescharacteristic
226C2#309511yescharacteristic
227C2#321211yescharacteristic
228C2#321611yescharacteristic
229C2#323111yescharacteristic
230C2#324011yescharacteristic
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
3. #2101C2semidirect
4. #2097C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect
7. #2072C2semidirect
8. #2072C2semidirect

Maximal subgroups

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

subgroupindex
#28552
#22852
#21012
#20972
#20842
#20832
#20722
#18732
#18542
#18172