Finite Group Databasebeta

← #10509

Subgroups — #10509

Order 256.

180 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 47 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#1989C211yesmaximal, characteristic, complemented
3#1573C211yesmaximal, characteristic, complemented
4#1551C211yesmaximal, characteristic
5#1502C211yesmaximal, characteristic
6#1470C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#125C2211yescharacteristic
10#501—21no—
11#490—21no—
12#481—21no—
13#480—21no—
14#478—21no—
15#478—21no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#431C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C4×C4⋊C4C2211yescharacteristic
21#340—21no—
22#340—21no—
23#335—21no—
24#334—21no—
25#333—21no—
26#331—21no—
27#327—22no—
28#324—22no—
29C2×#41—21no—
30C22×C8—21no—
31C4⋊Q8D411yescharacteristic
32#127D411yescharacteristic
33#125—21no—
34#125—21no—
35#125—21no—
36#125—21no—
37#124—21no—
38#121—21no—
39#121—21no—
40C4×D4—21no—
41C4×D4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42C2311yescharacteristic
49#107—21no—
50#107—21no—
51#106—21no—
52#106—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8D411yescharacteristic
58C4⋊C8D411yescharacteristic
59#102—21no—
60#102—21no—
61#102—21no—
62#102—21no—
63C22⋊C8—21no—
64C22⋊C8—21no—
65C22⋊C8D412yes—
66#97—21no—
67#97—21no—
68C4×C8—21no—
69#95—21no—
70#41—81no—
71#41—81no—
72C2×Q8—21no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—41no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4#4111yescharacteristic
84C2×C8—21no—
85C2×C8—21no—
86C2×C8—22no—
87C2×C8—42no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C42—81no—
105C42C2×D411yescharacteristic
106C42C2×D411yescharacteristic
107C42#4111yescharacteristic
108C42#4111yescharacteristic
109C23—41no—
110C23#12111yescharacteristic
111Q8—81no—
112D4—41no—
113D4—41no—
114D4—81no—
115D4—81no—
116D4—81no—
117D4—81no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—82no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4#12011yescharacteristic
134C2×C4#12111yescharacteristic
135C2×C4#12111yescharacteristic
136C2×C4#12311yescharacteristic
137C2×C4#12311yescharacteristic
138C2×C4#12411yescharacteristic
139C8—42no—
140C8—41no—
141C8—41no—
142C8—81no—
143C8—81no—
144C22—21no—
145C22—21no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22#39411yescharacteristic
150C22#47111yescharacteristic
151C22#47211yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—82no—
158C4—81no—
159C4—81no—
160C4—81no—
161C4#45411yescharacteristic
162C4#45611yescharacteristic
163C2—21no—
164C2—21no—
165C2—161no—
166C2#203811yescharacteristic
167C2#205111yescharacteristic
168C2#205311yescharacteristic
subgroupG/H
center, socleC22#394
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
2. #1989C2semidirect
3. #1573C2semidirect
6. #1470C2semidirect
7. #1285C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#19892
#15732
#15512
#15022
#14702
#12852