Finite Group Databasebeta

← #28294

Subgroups — #28294

Order 256.

369 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 135 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#2330C211yesmaximal, characteristic
3#2327C211yesmaximal, characteristic
4#2068C212yesmaximal
5#2066C212yesmaximal
6#1981C212yesmaximal
7#1949C212yesmaximal
8#1843C211yesmaximal, characteristic
9#1803C212yesmaximal
10C2×C4⋊Q8C2211yescharacteristic
11C2×#125C2211yescharacteristic
12C2×C4×Q8C2212yes—
13#479—28no—
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×C4⋊C8C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24#391C2212yes—
25#389—22no—
26#389C2212yes—
27#387—22no—
28#387—22no—
29#387—22no—
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32C4×C4⋊C4C2212yes—
33#376C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36C22×Q8C2312yes—
37C22×C8C2312yes—
38C22×C8C2312yes—
39C4⋊Q8—22no—
40C4⋊Q8Q814yes—
41#125—22no—
42#125D414yes—
43C4×Q8—24no—
44C4×Q8—24no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4C2312yes—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C42—22no—
54C2×C42—22no—
55C2×C42C2311yescharacteristic
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58#107—24no—
59#106—24no—
60C4⋊C8—24no—
61C4⋊C8—24no—
62C4⋊C8—41no—
63C4⋊C8—41no—
64#103—24no—
65#103—24no—
66#103—42no—
67#103—42no—
68#95—22no—
69#95—22no—
70#95—22no—
71#95—22no—
72#95—22no—
73#95D412yes—
74#95D412yes—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8—22no—
79C2×Q8—22no—
80C2×Q8C2×Q814yes—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D412yes—
88C22×C4C2×D412yes—
89C22×C4C2411yescharacteristic
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—22no—
96C4⋊C4—28no—
97C4⋊C4—28no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—44no—
104C4⋊C4—82no—
105C4⋊C4—82no—
106C4⋊C4C2×D414yes—
107C42—21no—
108C42—21no—
109C42—44no—
110C42—82no—
111C42C2×D412yes—
112C42C2×Q812yes—
113C42#4112yes—
114C42#4112yes—
115C23C22×D411yescharacteristic
116Q8—24no—
117Q8—24no—
118Q8—42no—
119Q8—42no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4#12114yes—
141C2×C4C22⋊Q814yes—
142C2×C4C22×D411yescharacteristic
143C2×C4C22×Q811yescharacteristic
144C2×C4C2×#4111yescharacteristic
145C2×C4C2×#4111yescharacteristic
146C2×C4H(5,𝔽2)11yescharacteristic
147C2×C42−1+411yescharacteristic
148C8—42no—
149C8—42no—
150C22C2×#12111yescharacteristic
151C22C2×C22⋊Q811yescharacteristic
152C22#53711yescharacteristic
153C22#57712yes—
154C22#57812yes—
155C4—24no—
156C4—24no—
157C4—41no—
158C4—41no—
159C4—44no—
160C4—44no—
161C4—82no—
162C4—82no—
163C4D4×Q811yescharacteristic
164C4#55011yescharacteristic
165C4#55411yescharacteristic
166C4#55611yescharacteristic
167C2#253811yescharacteristic
168C2#304612yes—
169C2#308212yes—
170C2#326011yescharacteristic
171C2#326111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ 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
#23302
#23272
#20682
#20662
#19812
#19492
#18432
#18032