Finite Group Databasebeta

← #29984

Subgroups — #29984

Order 256.

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

normal subgroups only

This group has 365 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#2868C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#2100C212yesmaximal
4#2095C212yesmaximal
5#2057C212yesmaximal
6#2024C212yesmaximal
7#1947C212yesmaximal
8#1831C211yesmaximal, characteristic
9#1803C212yesmaximal
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8—21no—
12C2×C4⋊Q8C2212yes—
13C2×C4×Q8C2211yescharacteristic
14C2×#107C2214yes—
15C2×C4⋊C8C2212yes—
16C2×#103C2214yes—
17#391C2211yescharacteristic
18#391C2211yescharacteristic
19#389C2211yescharacteristic
20#389C2211yescharacteristic
21#387—21no—
22#387—21no—
23#384—21no—
24#384—21no—
25#384C2212yes—
26#384C2212yes—
27C4×C4⋊C4—21no—
28C4×C4⋊C4C2211yescharacteristic
29#376—21no—
30C43C2211yescharacteristic
31#340C2214yes—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—22no—
36C22×Q8C2311yescharacteristic
37C22×C8C2314yes—
38C4⋊Q8—44no—
39C4⋊Q8—44no—
40C4⋊Q8—44no—
41C4×Q8—22no—
42C4×Q8Q814yescomplemented
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4C2314yes—
52C2×C4⋊C4C2311yescharacteristic
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2311yescharacteristic
55C2×C42—21no—
56C2×C42—21no—
57C2×C42D411yescharacteristic
58C2×C42D411yescharacteristic
59C2×C42D411yescharacteristic
60C2×C42D411yescharacteristic
61C2×C42C2311yescharacteristic
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#107—44no—
65C4⋊C8—24no—
66#103—44no—
67#95—21no—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74C2×Q8—21no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—44no—
78C2×Q8—44no—
79C2×Q8—44no—
80C2×Q8C2×Q812yes—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—24no—
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2411yescharacteristic
93C2×C8—24no—
94C2×C8—24no—
95C2×C8—24no—
96C4⋊C4—24no—
97C4⋊C4—21no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—24no—
101C4⋊C4—22no—
102C4⋊C4—44no—
103C4⋊C4—44no—
104C4⋊C4—44no—
105C4⋊C4—44no—
106C4⋊C4—42no—
107C4⋊C4—44no—
108C4⋊C4—42no—
109C4⋊C4C2×Q812yes—
110C42—24no—
111C42—22no—
112C42—22no—
113C42—22no—
114C42—22no—
115C42—22no—
116C42—42no—
117C42Q1614yes—
118C42C2×Q812yes—
119C42#4112yes—
120C23C22×D411yescharacteristic
121Q8—24no—
122Q8—42no—
123Q8—88no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—44no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4C22⋊Q812yes—
150C2×C4C22⋊Q812yes—
151C2×C4C2×Q1612yes—
152C2×C4C2×Q1612yes—
153C2×C4C2×Q1612yes—
154C2×C4AGL(1,ℤ/8)12yes—
155C2×C4AGL(1,ℤ/8)12yes—
156C2×C4#13712yes—
157C2×C4C22×Q811yescharacteristic
158C2×C4C2×#4111yescharacteristic
159C2×C4H(5,𝔽2)12yes—
160C2×C42−1+412yes—
161C8—44no—
162C22C2×C22⋊Q811yescharacteristic
163C22#53511yescharacteristic
164C22#53611yescharacteristic
165C22C22×Q1611yescharacteristic
166C22C2×AGL(1,ℤ/8)11yescharacteristic
167C22C2×AGL(1,ℤ/8)11yescharacteristic
168C22C2×#13711yescharacteristic
169C4—22no—
170C4—22no—
171C4—22no—
172C4—22no—
173C4—22no—
174C4—42no—
175C4—42no—
176C4—81no—
177C4—84no—
178C4—81no—
179C4#47714yes—
180C4#54212yes—
181C4#54412yes—
182C2#262811yescharacteristic
183C2C2×#47711yescharacteristic
184C2#307911yescharacteristic
185C2#320911yescharacteristic
186C2#321711yescharacteristic
187C2#323011yescharacteristic
188C2#323911yescharacteristic
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

Complements

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

NH ≅ G/Ntype
42. C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#28682
#22952
#21002
#20952
#20572
#20242
#19472
#18312
#18032