Finite Group Databasebeta

← #24577

Subgroups — #24577

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
1C2×#483C211yesmaximal, characteristic
2C22×#107C211yesmaximal, characteristic
3#2503C211yesmaximal, characteristic
4#2344C211yesmaximal, characteristic
5#2085C212yesmaximal, complemented
6#2068C212yesmaximal, complemented
7#1949C212yesmaximal, complemented
8#1928C212yesmaximal, complemented
9#1892C211yesmaximal, characteristic
10#1850C211yesmaximal, characteristic
11#1823C211yesmaximal, characteristic
12C23×C8C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#117—21no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#483—24no—
18C2×#107—24no—
19C2×#107C2212yes—
20C2×#107C2212yes—
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#400—22no—
29#391C2212yes—
30#389C2212yes—
31#387—22no—
32#382C2212yes—
33#381—21no—
34#381—21no—
35#381—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4C2212yes—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47C22×C8—22no—
48C22×C8—24no—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54#117—44no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4D414yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4D411yescharacteristic
71C2×C22⋊C4D411yescharacteristic
72C2×C42—22no—
73C2×C42C2312yes—
74#107—24no—
75#107—24no—
76#106—22no—
77#106—22no—
78C4⋊C8—24no—
79#103—24no—
80#103—22no—
81#103—22no—
82C22⋊C8—22no—
83#95—22no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C24C2×D411yescharacteristic
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4#4112yes—
115C22×C4C2411yescharacteristic
116C2×C8—28no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8#4114yes—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4C2×D412yes—
137C4⋊C4C2×D412yes—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C42—44no—
145C23—22no—
146C23—22no—
147C23#12112yes—
148C23#13512yes—
149C23AGL(1,ℤ/8)12yes—
150C23C22×D411yescharacteristic
151Q8—41no—
152Q8—41no—
153Q8—42no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—48no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—84no—
182C2×C4#12112yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C2×#4112yes—
185C2×C4C2×#4111yescharacteristic
186C2×C42−1+412yes—
187C8—24no—
188C8—41no—
189C8—41no—
190C22—24no—
191C22—24no—
192C22—22no—
193C22—22no—
194C22#46314yes—
195C22C2×#12111yescharacteristic
196C22#54712yes—
197C22C2×#13511yescharacteristic
198C22C2×AGL(1,ℤ/8)11yescharacteristic
199C22#57812yes—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—81no—
207C4—82no—
208C4—81no—
209C4#52511yescharacteristic
210C4#53611yescharacteristic
211C4#55212yes—
212C2—24no—
213C2#241311yescharacteristic
214C2C2×#46311yescharacteristic
215C2#304411yescharacteristic
216C2#333612yes—
217C2#334312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2085C2semidirect
6. #2068C2semidirect
7. #1949C2semidirect
8. #1928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C22×#1072
#25032
#23442
#20852
#20682
#19492
#19282
#18922
#18502
#18232