Finite Group Databasebeta

← #29579

Subgroups — #29579

Order 256.

328 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 117 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#2859C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic
4#2095C211yesmaximal, characteristic
5#2084C211yesmaximal, characteristic
6#2072C211yesmaximal, characteristic
7#1944C212yesmaximal
8#1935C212yesmaximal
9#1918C212yesmaximal
10#1865C212yesmaximal
11#1745C211yesmaximal, characteristic
12C2×#125—21no—
13C2×C4×Q8C2211yescharacteristic
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#97C2212yes—
19#391C2211yescharacteristic
20#391C2211yescharacteristic
21#389—21no—
22#389—21no—
23#389C2211yescharacteristic
24#389C2212yes—
25#389C2212yes—
26#389C2211yescharacteristic
27#387—21no—
28#387—21no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382—21no—
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2211yescharacteristic
34#376—21no—
35C43C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#125—44no—
47C4×Q8—22no—
48C4×Q8—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C42—21no—
64C2×C42—21no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#107—24no—
73#106—24no—
74#103—24no—
75#103—24no—
76#97—24no—
77#95—21no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—22no—
86C2×Q8—21no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8#4112yes—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8#4114yes—
109C4⋊C4—24no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4#4112yes—
126C42—24no—
127C42—22no—
128C42—22no—
129C42—22no—
130C42—22no—
131C42—21no—
132C42—21no—
133C42—22no—
134C42—41no—
135C42—42no—
136C23C22×D411yescharacteristic
137Q8—24no—
138Q8—42no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4AGL(1,ℤ/8)12yes—
173C2×C4#13712yes—
174C2×C4#13712yes—
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4C2×#4112yes—
178C2×C42−1+412yes—
179C8—24no—
180C8—42no—
181C22#52511yescharacteristic
182C22#53211yescharacteristic
183C22#53611yescharacteristic
184C22C2×AGL(1,ℤ/8)11yescharacteristic
185C22C2×#13711yescharacteristic
186C22C2×#13711yescharacteristic
187C22#57511yescharacteristic
188C4—22no—
189C4—22no—
190C4—22no—
191C4—41no—
192C4—42no—
193C4—42no—
194C4—41no—
195C4—41no—
196C4—82no—
197C4—81no—
198C4—82no—
199C4—81no—
200C4#55212yes—
201C4#55212yes—
202C2#261111yescharacteristic
203C2#311211yescharacteristic
204C2#311611yescharacteristic
205C2#317111yescharacteristic
206C2#318511yescharacteristic
207C2#323611yescharacteristic
208C2#323911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#28592
#22952
#21002
#20952
#20842
#20722
#19442
#19352
#19182
#18652
#17452