Finite Group Databasebeta

← #10126

Subgroups — #10126

Order 256.

227 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 49 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#3234C211yesmaximal, characteristic
2#3216C211yesmaximal, characteristic
3#3166C211yesmaximal, characteristic, complemented
4#1497C211yesmaximal, characteristic, complemented
5#1484C211yesmaximal, characteristic
6#1470C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#501—21no—
13#501—21no—
14#499—22no—
15#497—21no—
16#490—21no—
17#489—21no—
18#481—21no—
19#480—21no—
20#478—22no—
21#462—21no—
22#460—21no—
23#452—22no—
24#432C2211yescharacteristic
25#432C2211yescharacteristic
26#431C2211yescharacteristic
27#428—21no—
28#428—21no—
29C4×M16C2211yescharacteristic
30Q8⋊C8—22no—
31D4⋊C8—22no—
32#324—22no—
33C2×#41—21no—
34C2×Q16—22no—
35C2×SD16—22no—
36C2×M16—21no—
37C4⋊Q8D411yescharacteristic
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#127D411yescharacteristic
41#125—21no—
42#125—21no—
43#125—21no—
44#125—21no—
45#125—21no—
46#124—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49#121—21no—
50#121—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×D4—21no—
54C4×D4—21no—
55#117—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C42C2311yescharacteristic
61#107—22no—
62#107—22no—
63#107—22no—
64#106—22no—
65#106—22no—
66#106—22no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D412yes—
70#103—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77#97—21no—
78#97—21no—
79#97—21no—
80C4×C8—21no—
81C4×C8D412yes—
82#41—82no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—41no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4#4111yescharacteristic
96Q16—82no—
97SD16—82no—
98M16—24no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—41no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—41no—
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C23—41no—
128C23#12111yescharacteristic
129Q8—41no—
130Q8—41no—
131Q8—42no—
132Q8—42no—
133D4—42no—
134D4—42no—
135D4—42no—
136D4—82no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—82no—
147C2×C4—81no—
148C2×C4#12011yescharacteristic
149C2×C4#12011yescharacteristic
150C2×C4#12011yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12111yescharacteristic
153C2×C4#12711yescharacteristic
154C8—24no—
155C8—42no—
156C8—42no—
157C8—81no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22#39211yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—81no—
171C4—81no—
172C4—81no—
173C4—81no—
174C4—81no—
175C4#45612yes—
176C2—41no—
177C2—161no—
178C2#200511yescharacteristic
179C2#200811yescharacteristic
180C2#201711yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
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
3. #3166C2semidirect
4. #1497C2semidirect
6. #1470C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#32162
#31662
#14972
#14842
#14702
#12912