Finite Group Databasebeta

← #26112

Subgroups — #26112

Order 256.

351 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 143 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
2Q8×C4⋊C4C211yesmaximal, characteristic
3#2100C212yesmaximal
4#2087C212yesmaximal
5#1947C211yesmaximal, characteristic
6#1939C211yesmaximal, characteristic
7#1863C212yesmaximal
8#1834C211yesmaximal, characteristic
9#1803C212yesmaximal
10#1803C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×#125—21no—
13C2×C4×Q8C2212yes—
14C2×C4×Q8C2211yescharacteristic
15C2×#107C2212yes—
16C2×C4⋊C8C2211yescharacteristic
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2211yescharacteristic
20C2×#103C2211yescharacteristic
21C2×C4×C8C2211yescharacteristic
22#398—22no—
23#389—21no—
24#389—22no—
25#389C2212yes—
26#389C2212yes—
27#384—21no—
28#384C2211yescharacteristic
29#384C2212yes—
30#384C2211yescharacteristic
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×C8C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2311yescharacteristic
45#125—44no—
46C4×Q8—22no—
47C4×Q8—24no—
48C4×Q8—42no—
49C4×Q8Q814yes—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—21no—
70C2×C42C2312yes—
71C2×C42C2311yescharacteristic
72#107—42no—
73C4⋊C8—22no—
74C4⋊C8—42no—
75#103—24no—
76#103—22no—
77#103—22no—
78C4×C8—22no—
79#95—22no—
80#95—21no—
81#95—22no—
82#95D412yes—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8C2×Q812yes—
87C2×Q8#4112yes—
88C2×Q8#4112yes—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4C2×D412yes—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2411yescharacteristic
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—21no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—44no—
121C4⋊C4—44no—
122C4⋊C4—44no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—81no—
126C4⋊C4—82no—
127C4⋊C4—81no—
128C4⋊C4—82no—
129C4⋊C4Q1614yes—
130C4⋊C4C2×Q812yes—
131C42—24no—
132C42—22no—
133C42—22no—
134C42—44no—
135C42—42no—
136C42C2×Q812yes—
137C42#4112yes—
138C23C22×D411yescharacteristic
139Q8—22no—
140Q8—22no—
141Q8—22no—
142Q8#12514yescomplemented
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4C22⋊Q812yes—
169C2×C4C22⋊Q812yes—
170C2×C4C2×Q1614yes—
171C2×C4C2×Q1612yes—
172C2×C4AGL(1,ℤ/8)12yes—
173C2×C4C22×Q811yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)11yescharacteristic
178C2×C42−1+411yescharacteristic
179C8—41no—
180C8—41no—
181C8—42no—
182C22C2×C22⋊Q811yescharacteristic
183C22#54611yescharacteristic
184C22#54811yescharacteristic
185C22C22×Q1611yescharacteristic
186C22C2×AGL(1,ℤ/8)11yescharacteristic
187C22#57711yescharacteristic
188C22#57711yescharacteristic
189C4—22no—
190C4—24no—
191C4—24no—
192C4—42no—
193C4—42no—
194C4—82no—
195C4—81no—
196C4—81no—
197C4—82no—
198C4#47714yes—
199C4C2×#12511yescharacteristic
200C4#54411yescharacteristic
201C4#55211yescharacteristic
202C4#55311yescharacteristic
203C2#244211yescharacteristic
204C2C2×#47711yescharacteristic
205C2#308411yescharacteristic
206C2#329911yescharacteristic
207C2#333211yescharacteristic
208C2#336311yescharacteristic
209C2#336811yescharacteristic
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
142. Q8#125semidirect

Maximal subgroups

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

subgroupindex
#25522
Q8×C4⋊C42
#21002
#20872
#19472
#19392
#18632
#18342
#18032
C4×#1032