Finite Group Databasebeta

← #29871

Subgroups — #29871

Order 256.

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

normal subgroups only

This group has 341 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
1C2×#479C212yesmaximal
2#2859C211yesmaximal, characteristic
3#2295C211yesmaximal, characteristic
4#2085C212yesmaximal
5#2084C211yesmaximal, characteristic
6#2083C211yesmaximal, characteristic
7#2074C212yesmaximal
8#2072C211yesmaximal, characteristic
9#2072C211yesmaximal, characteristic
10#1989C212yesmaximal
11#1831C211yesmaximal, characteristic
12C2×#125C2212yes—
13C2×C4×Q8C2211yescharacteristic
14#479—28no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20#391C2211yescharacteristic
21#391C2211yescharacteristic
22#389—21no—
23#389—21no—
24#389—21no—
25#389—21no—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#387—21no—
29#387—21no—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32#382C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35#376—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—24no—
48#125—24no—
49C4×Q8—22no—
50C4×Q8D414yes—
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⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—21no—
66C2×C42—21no—
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—24no—
75#106—24no—
76C4⋊C8—24no—
77#103—24no—
78#103—24no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—22no—
88C2×Q8—21no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8C2×D412yes—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C4⋊C4—24no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4C2×D412yes—
128C42—24no—
129C42—22no—
130C42—22no—
131C42—22no—
132C42—22no—
133C42—41no—
134C42—41no—
135C42—42no—
136C42C2×D412yes—
137C42#4112yes—
138C23C22×D411yescharacteristic
139Q8—24no—
140Q8—42no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4#12112yes—
174C2×C4#12112yes—
175C2×C4#13512yes—
176C2×C4#13712yes—
177C2×C4C22×D411yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C42−1+412yes—
180C2×C42−1+412yes—
181C8—42no—
182C8—42no—
183C22C2×#12111yescharacteristic
184C22#53611yescharacteristic
185C22#53611yescharacteristic
186C22C2×#13511yescharacteristic
187C22C2×#13711yescharacteristic
188C22#57511yescharacteristic
189C22#57511yescharacteristic
190C4—22no—
191C4—22no—
192C4—22no—
193C4—22no—
194C4—42no—
195C4—41no—
196C4—41no—
197C4—42no—
198C4—81no—
199C4—81no—
200C4—82no—
201C4—82no—
202C4#53612yes—
203C4#54112yes—
204C2#262111yescharacteristic
205C2#303511yescharacteristic
206C2#303711yescharacteristic
207C2#323111yescharacteristic
208C2#323211yescharacteristic
209C2#323611yescharacteristic
210C2#324011yescharacteristic
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
C2×#4792
#28592
#22952
#20852
#20842
#20832
#20742
#20722
#19892
#18312