Finite Group Databasebeta

← #30309

Subgroups — #30309

Order 256.

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

normal subgroups only

This group has 343 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
1#2868C211yesmaximal, characteristic
2#2864C211yesmaximal, characteristic
3#2084C212yesmaximal
4#2083C212yesmaximal
5#2058C212yesmaximal
6#2057C212yesmaximal
7#1847C212yesmaximal
8#1836C212yesmaximal
9#1745C211yesmaximal, characteristic
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8—21no—
12C2×C4⋊Q8C2212yes—
13C2×#107C2214yes—
14C2×#106C2214yes—
15C2×#97C2212yes—
16#389C2211yescharacteristic
17#389C2211yescharacteristic
18#389C2211yescharacteristic
19#389C2211yescharacteristic
20#387—21no—
21#387—21no—
22#387—21no—
23#387—21no—
24#384—21no—
25#384—21no—
26#384C2212yes—
27#384C2212yes—
28#382C2211yescharacteristic
29#382C2211yescharacteristic
30C43C2211yescharacteristic
31#340C2214yes—
32#340C2214yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—22no—
36C22×C8C2314yes—
37C4⋊Q8—44no—
38C4⋊Q8—44no—
39C4⋊Q8—44no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4C2314yes—
50C2×C4⋊C4C2311yescharacteristic
51C2×C4⋊C4C2311yescharacteristic
52C2×C4⋊C4C2311yescharacteristic
53C2×C4⋊C4C2311yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42D411yescharacteristic
56C2×C42D411yescharacteristic
57C2×C42D411yescharacteristic
58C2×C42C2311yescharacteristic
59C2×C42C2311yescharacteristic
60C2×C42C2311yescharacteristic
61#107—44no—
62#106—44no—
63#97Q818yescomplemented
64#95—21no—
65#95—21no—
66#95—21no—
67#95—21no—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72C2×Q8—44no—
73C2×Q8—44no—
74C2×Q8—44no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—24no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2411yescharacteristic
87C2×C8—24no—
88C2×C8—24no—
89C2×C8C2×Q818yes—
90C4⋊C4—22no—
91C4⋊C4—24no—
92C4⋊C4—22no—
93C4⋊C4—24no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—44no—
97C4⋊C4—44no—
98C4⋊C4—44no—
99C4⋊C4—44no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—44no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C42—22no—
106C42—22no—
107C42—22no—
108C42—22no—
109C42—22no—
110C42—22no—
111C42C2×Q814yes—
112C23C22×D411yescharacteristic
113Q8—88no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—44no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—44no—
132C2×C4—44no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4C4⋊Q812yes—
142C2×C4C4⋊Q812yes—
143C2×C4AGL(1,ℤ/8)12yes—
144C2×C4AGL(1,ℤ/8)12yes—
145C2×C4#13712yes—
146C2×C4#13712yes—
147C2×C4C22×Q812yes—
148C2×C4H(5,𝔽2)12yes—
149C2×C42−1+412yes—
150C8—28no—
151C22C2×C4⋊Q811yescharacteristic
152C22#53511yescharacteristic
153C22#53611yescharacteristic
154C22C2×AGL(1,ℤ/8)11yescharacteristic
155C22C2×AGL(1,ℤ/8)11yescharacteristic
156C22C2×#13711yescharacteristic
157C22C2×#13711yescharacteristic
158C4—22no—
159C4—22no—
160C4—22no—
161C4—22no—
162C4—22no—
163C4—22no—
164C4—81no—
165C4—84no—
166C4—81no—
167C4—81no—
168C4—81no—
169C4#54414yes—
170C2#266011yescharacteristic
171C2#316511yescharacteristic
172C2#316511yescharacteristic
173C2#321711yescharacteristic
174C2#321811yescharacteristic
175C2#323811yescharacteristic
176C2#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
63. #97Q8semidirect

Maximal subgroups

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

subgroupindex
#28682
#28642
#20842
#20832
#20582
#20572
#18472
#18362
#17452