Finite Group Databasebeta

← #26446

Subgroups — #26446

Order 256.

355 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 147 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#2551C211yesmaximal, characteristic
2#2369C211yesmaximal, characteristic
3#2087C212yesmaximal
4#2057C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1987C212yesmaximal
7#1987C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1921C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#107C2211yescharacteristic
16C2×#107C2212yes—
17C2×#107C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C4×C8C2211yescharacteristic
21#395—22no—
22#391—22no—
23#389—21no—
24#389—21no—
25#389C2212yes—
26#389C2212yes—
27#384C2211yescharacteristic
28C4×C4⋊C4—21no—
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×Q8—22no—
43C22×C8C2312yes—
44C22×C8C2311yescharacteristic
45C22×C8C2311yescharacteristic
46C4⋊Q8—24no—
47C4⋊Q8—22no—
48C4⋊Q8—42no—
49C4⋊Q8—44no—
50C4⋊Q8Q814yescomplemented
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
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—21no—
66C2×C42—21no—
67C2×C42—22no—
68C2×C42C2312yes—
69C2×C42C2311yescharacteristic
70#107—22no—
71#107—42no—
72#107Q814yescomplemented
73C4⋊C8—22no—
74C4⋊C8—42no—
75C4×C8—22no—
76#95—22no—
77#95—22no—
78#95—21no—
79#95—21no—
80#95—22no—
81#95D412yes—
82C2×Q8—24no—
83C2×Q8—44no—
84C2×Q8—44no—
85C2×Q8—81no—
86C2×Q8—81no—
87C2×Q8—82no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4C2×D412yes—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D412yes—
97C22×C4C2×D411yescharacteristic
98C22×C4C2411yescharacteristic
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—22no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8C2×Q812yes—
108C4⋊C4—24no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—81no—
122C4⋊C4—82no—
123C4⋊C4Q1614yes—
124C4⋊C4C2×Q812yes—
125C4⋊C4C2×Q812yes—
126C4⋊C4C2×Q814yes—
127C4⋊C4#4112yes—
128C42—22no—
129C42—22no—
130C42—42no—
131C42—42no—
132C42—82no—
133C42C2×Q812yes—
134C42#4112yes—
135C23C22×D411yescharacteristic
136Q8—84no—
137Q8—88no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4C22⋊Q812yes—
164C2×C4C22⋊Q812yes—
165C2×C4C2×Q1614yes—
166C2×C4C2×Q1612yes—
167C2×C4AGL(1,ℤ/8)12yes—
168C2×C4C22×Q811yescharacteristic
169C2×C4C22×Q811yescharacteristic
170C2×C4C2×#4111yescharacteristic
171C2×C4C2×#4111yescharacteristic
172C2×C4H(5,𝔽2)11yescharacteristic
173C2×C42−1+411yescharacteristic
174C8—22no—
175C8—41no—
176C8—42no—
177C22C2×C22⋊Q811yescharacteristic
178C22#54611yescharacteristic
179C22D4×Q811yescharacteristic
180C22C22×Q1611yescharacteristic
181C22C2×AGL(1,ℤ/8)11yescharacteristic
182C22#57611yescharacteristic
183C22#57711yescharacteristic
184C4—22no—
185C4—24no—
186C4—44no—
187C4—42no—
188C4—42no—
189C4—42no—
190C4—42no—
191C4—81no—
192C4—82no—
193C4#47714yes—
194C4#53311yescharacteristic
195C4#54211yescharacteristic
196C4#55711yescharacteristic
197C4Q8211yescharacteristic
198C2#245111yescharacteristic
199C2C2×#47711yescharacteristic
200C2#308111yescharacteristic
201C2#329911yescharacteristic
202C2#332511yescharacteristic
203C2Q8×Q1611yescharacteristic
204C2#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
50. C4⋊Q8Q8semidirect
72. #107Q8semidirect

Maximal subgroups

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

subgroupindex
#25512
#23692
#20872
#20572
#19902
#19872
#19472
#19392
#19212
C4×#1072