Finite Group Databasebeta

← #10199

Subgroups — #10199

Order 256.

243 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 63 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#3249C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic
3#3165C211yesmaximal, characteristic
4#1987C211yesmaximal, characteristic
5#1986C211yesmaximal, characteristic
6#1503C211yesmaximal, characteristic
7#1283C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#501—22no—
12#500—21no—
13#498—21no—
14#494—21no—
15#487—21no—
16#477—21no—
17#477—21no—
18#467—22no—
19#462—22no—
20#462—21no—
21#462—21no—
22#451—22no—
23#431C2211yescharacteristic
24#428—22no—
25C2×C4⋊C8C2211yescharacteristic
26C4×M16C2211yescharacteristic
27C4×C4⋊C4C2211yescharacteristic
28#340—22no—
29#339—22no—
30#328—22no—
31C22×Q8—21no—
32C2×Q16—24no—
33C2×Q16—22no—
34C2×M16D412yes—
35C22×C8D412yes—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8D412yes—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#125—21no—
45#125—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54#117—21no—
55#117—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61#107—24no—
62#107—22no—
63#106—24no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67#103—24no—
68#103—22no—
69#103—22no—
70C22⋊C8D412yes—
71#97—21no—
72C4×C8—21no—
73C4×C8—21no—
74#95—21no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—22no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—41no—
81C2×Q8—82no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87Q16—84no—
88Q16—82no—
89M16—22no—
90M16Q1614yescomplemented
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—42no—
95C4⋊C4—22no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—41no—
116C42—41no—
117C42—42no—
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C42C2×D411yescharacteristic
121C42#4111yescharacteristic
122C23#12711yescharacteristic
123Q8—42no—
124Q8—44no—
125Q8—41no—
126Q8—41no—
127Q8—41no—
128Q8—41no—
129Q8—81no—
130Q8—82no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—82no—
142C2×C4—81no—
143C2×C4#12011yescharacteristic
144C2×C4#12011yescharacteristic
145C2×C4#12011yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4C2×Q1612yes—
150C2×C4#13712yes—
151C8—24no—
152C8—42no—
153C8—42no—
154C22—21no—
155C22—21no—
156C22#39211yescharacteristic
157C22#49411yescharacteristic
158C22#49711yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—82no—
163C4—81no—
164C4—81no—
165C4—82no—
166C4—81no—
167C4—81no—
168C4#45112yes—
169C4#45511yescharacteristic
170C4#45611yescharacteristic
171C4#46212yes—
172C2—21no—
173C2—21no—
174C2#200711yescharacteristic
175C2#200811yescharacteristic
176C2#201611yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
90. M16Q16semidirect

Maximal subgroups

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

subgroupindex
#32492
#32092
#31652
#19872
#19862
#15032
#12832