Finite Group Databasebeta

← #10206

Subgroups — #10206

Order 256.

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

normal subgroups only

This group has 280 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#3246C211yesmaximal, characteristic
2#3229C211yesmaximal, characteristic, complemented
3#3152C211yesmaximal, characteristic
4#1988C211yesmaximal, characteristic, complemented
5#1986C211yesmaximal, characteristic, complemented
6#1502C211yesmaximal, characteristic
7#1283C211yesmaximal, characteristic, complemented
8C2×#137—22no—
9#533C2211yescharacteristic
10#532C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#498—21no—
13#497—22no—
14#494—21no—
15#492—21no—
16#487—21no—
17#483—22no—
18#476—21no—
19#475—22no—
20#475—21no—
21#465—22no—
22#461—21no—
23#460—21no—
24#431C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C4×M16C2211yescharacteristic
27C4×C4⋊C4C2211yescharacteristic
28#340—22no—
29#339—22no—
30#327—22no—
31C2×#41—21no—
32C22×Q8—21no—
33#137—44no—
34#137—44no—
35C2×Q16—24no—
36C2×SD16—24no—
37C2×SD16—22no—
38C2×M16D412yes—
39C22×C8D412yes—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8D412yes—
44C4⋊Q8D411yescharacteristic
45C4⋊Q8D411yescharacteristic
46C4⋊Q8D411yescharacteristic
47#127D411yescharacteristic
48#125—21no—
49#124—21no—
50C22⋊Q8—21no—
51C22⋊Q8—21no—
52#121—21no—
53#121—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56C4×D4—21no—
57C4×D4—21no—
58#117—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C42—21no—
63C2×C42C2311yescharacteristic
64#106—22no—
65#106—24no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69#103—22no—
70#103—24no—
71#102—22no—
72C22⋊C8D412yes—
73#97—21no—
74C4×C8—21no—
75C4×C8—21no—
76#95—21no—
77#41—42no—
78#41—42no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—21no—
82C2×Q8—22no—
83C2×Q8—41no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96Q16—48no—
97SD16—48no—
98SD16—82no—
99M16—22no—
100M16SD1614yescomplemented
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—42no—
105C4⋊C4—22no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
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—
123C42—42no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C23—41no—
129C23#12711yescharacteristic
130Q8—42no—
131Q8—44no—
132Q8—42no—
133Q8—41no—
134Q8—41no—
135Q8—82no—
136D4—42no—
137D4—41no—
138D4—41no—
139D4—81no—
140D4—81no—
141D4—81no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—82no—
155C2×C4#12011yescharacteristic
156C2×C4#12011yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4C2×SD1612yes—
162C2×C4AGL(1,ℤ/8)12yes—
163C8—24no—
164C8—42no—
165C8—42no—
166C22—21no—
167C22—21no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#39211yescharacteristic
172C22#49211yescharacteristic
173C22#49611yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—82no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—82no—
182C4#45012yes—
183C4#45511yescharacteristic
184C4#45611yescharacteristic
185C4#46012yes—
186C2—21no—
187C2—21no—
188C2—161no—
189C2#200711yescharacteristic
190C2#200911yescharacteristic
191C2#201411yescharacteristic
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
2. #3229C2semidirect
4. #1988C2semidirect
5. #1986C2semidirect
7. #1283C2semidirect
100. M16SD16semidirect

Maximal subgroups

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

subgroupindex
#32462
#32292
#31522
#19882
#19862
#15022
#12832