Finite Group Databasebeta

← #26447

Subgroups — #26447

Order 256.

343 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 139 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#2552C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic
3#2088C212yesmaximal
4#2059C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1989C212yesmaximal
7#1988C211yesmaximal, characteristic
8#1948C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1921C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2212yes—
14C2×C4×Q8—21no—
15C2×#107C2211yescharacteristic
16C2×#107C2211yescharacteristic
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C4×C8C2211yescharacteristic
21#398—22no—
22#391—21no—
23#389—21no—
24#389C2212yes—
25#389C2212yes—
26#387—22no—
27#384—21no—
28#384C2211yescharacteristic
29C4×C4⋊C4C2212yes—
30C4×C4⋊C4C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8—21no—
42C22×C8C2312yes—
43C22×C8C2311yescharacteristic
44C22×C8C2311yescharacteristic
45C4⋊Q8—22no—
46C4⋊Q8Q814yes—
47#125—24no—
48#125—42no—
49C4×Q8—44no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—22no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42C2312yes—
69C2×C42C2311yescharacteristic
70#107—22no—
71#107Q814yescomplemented
72#106—42no—
73C4⋊C8—22no—
74C4⋊C8—42no—
75C4×C8—22no—
76#95—21no—
77#95—22no—
78#95—21no—
79#95—22no—
80#95—22no—
81#95D412yes—
82C2×Q8—24no—
83C2×Q8—42no—
84C2×Q8—42no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4C2×D412yes—
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2411yescharacteristic
96C2×C8—21no—
97C2×C8—21no—
98C2×C8—21no—
99C2×C8—22no—
100C2×C8—21no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—21no—
104C2×C8C2×Q812yes—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—44no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—42no—
120C4⋊C4—81no—
121C4⋊C4—82no—
122C4⋊C4C2×Q812yes—
123C4⋊C4C2×Q812yes—
124C4⋊C4C2×Q814yes—
125C4⋊C4#4112yes—
126C42—22no—
127C42—22no—
128C42—44no—
129C42—42no—
130C42—42no—
131C42C2×Q812yes—
132C42#4112yes—
133C23C22×D411yescharacteristic
134Q8—48no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4C22⋊Q812yes—
162C2×C4C22⋊Q812yes—
163C2×C4#13512yes—
164C2×C4AGL(1,ℤ/8)12yes—
165C2×C4C22×Q811yescharacteristic
166C2×C4C22×Q811yescharacteristic
167C2×C4C2×#4111yescharacteristic
168C2×C4C2×#4111yescharacteristic
169C2×C4H(5,𝔽2)11yescharacteristic
170C2×C42−1+411yescharacteristic
171C8—22no—
172C8—41no—
173C8—42no—
174C22C2×C22⋊Q811yescharacteristic
175C22#54611yescharacteristic
176C22D4×Q811yescharacteristic
177C22C2×#13511yescharacteristic
178C22C2×AGL(1,ℤ/8)11yescharacteristic
179C22#57711yescharacteristic
180C22#57811yescharacteristic
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4—42no—
187C4—42no—
188C4—81no—
189C4—82no—
190C4#47814yes—
191C4#53311yescharacteristic
192C4#54211yescharacteristic
193C4#55711yescharacteristic
194C4Q8211yescharacteristic
195C2#245111yescharacteristic
196C2C2×#47811yescharacteristic
197C2#308211yescharacteristic
198C2#330411yescharacteristic
199C2#333011yescharacteristic
200C2#338311yescharacteristic
201C2#338411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
71. #107Q8semidirect

Maximal subgroups

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

subgroupindex
#25522
#23652
#20882
#20592
#19902
#19892
#19882
#19482
#19392
#19212
C4×#1072