Finite Group Databasebeta

← #27596

Subgroups — #27596

Order 256.

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

normal subgroups only

This group has 397 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#2314C211yesmaximal, characteristic
2#2310C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1854C212yesmaximal
10#1818C211yesmaximal, characteristic
11#1787C212yesmaximal
12C23×Q8C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22#391—22no—
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#391C2211yescharacteristic
26#391C2211yescharacteristic
27#388—22no—
28#387—22no—
29#384C2211yescharacteristic
30#382C2211yescharacteristic
31#381—22no—
32#380C2212yes—
33C4×C4⋊C4C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2211yescharacteristic
37#336C2211yescharacteristic
38#336C2212yes—
39C22×Q8—22no—
40C22×Q8—22no—
41C22×Q8—21no—
42C22×Q8—24no—
43C22×Q8—42no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#117—24no—
51#117—42no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4D412yes—
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4D412yes—
61C2×C42—21no—
62C2×C42—22no—
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65#107—42no—
66#106—42no—
67#103—24no—
68#103—24no—
69#103—42no—
70#103—42no—
71C22⋊C8—24no—
72#95—22no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79C24C2×D411yescharacteristic
80C2×Q8—24no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—48no—
88C2×Q8—48no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8#4112yes—
92C2×Q8#4112yes—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—24no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4#4114yes—
109C22×C4C2411yescharacteristic
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—82no—
124C22⋊C4—24no—
125C22⋊C4—42no—
126C22⋊C4—82no—
127C42—22no—
128C42—22no—
129C42—42no—
130C42—82no—
131C23—21no—
132C23—21no—
133C23—22no—
134C23—22no—
135C23#12312yes—
136C23C22×D411yescharacteristic
137Q8—24no—
138Q8—24no—
139Q8—48no—
140Q8—42no—
141Q8—42no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—48no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4#12312yes—
170C2×C4C2×#4112yes—
171C2×C4C2×#4111yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4H(5,𝔽2)12yes—
174C8—42no—
175C8—42no—
176C22—24no—
177C22—22no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22C2×#12311yescharacteristic
182C22#52511yescharacteristic
183C22#53511yescharacteristic
184C22#57712yes—
185C22#57812yes—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—42no—
190C4—42no—
191C4—82no—
192C4—81no—
193C4—82no—
194C4—81no—
195C4#55112yes—
196C4#55112yes—
197C2—42no—
198C2#248211yescharacteristic
199C2#309712yes—
200C2#312112yes—
201C2#322111yescharacteristic
202C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2096C2semidirect
4. #2095C2semidirect
5. #2074C2semidirect
6. #2074C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect

Maximal subgroups

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

subgroupindex
#23142
#23102
#20962
#20952
#20742
#19282
#19272
#18542
#18182
#17872