Finite Group Databasebeta

← #10019

Subgroups — #10019

Order 256.

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

normal subgroups only

This group has 270 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#3142C211yesmaximal, characteristic, complemented
2#3109C211yesmaximal, characteristic
3#2633C211yesmaximal, characteristic, complemented
4#1534C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic, complemented
6#1490C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#125C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#491—21no—
12#490—21no—
13#488—22no—
14#481—21no—
15#476—21no—
16#459—21no—
17#447—21no—
18#442—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#417—21no—
22#417—21no—
23C4×M16C2211yescharacteristic
24#399—21no—
25#398—21no—
26#397—21no—
27#394—21no—
28#385—22no—
29#383C2211yescharacteristic
30#330—22no—
31#327—22no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×D8—22no—
37C2×M16—21no—
38C4⋊Q8D411yescharacteristic
39#127D411yescharacteristic
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44#124—21no—
45#121—21no—
46#121—21no—
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C42C2311yescharacteristic
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66#103—22no—
67#103—22no—
68#102—22no—
69#102—22no—
70#102—22no—
71#102—22no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74#97—21no—
75#97—21no—
76#97D412yes—
77C4×C8D412yes—
78#95—22no—
79#95—21no—
80#95—21no—
81#95D412yes—
82C24—21no—
83#41—44no—
84C2×Q8—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—42no—
91C2×D4—41no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—82no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103D8—44no—
104M16—42no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—82no—
121C22⋊C4—44no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—82no—
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42#4111yescharacteristic
131C23—22no—
132C23—42no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—41no—
137C23#12011yescharacteristic
138Q8—42no—
139D4—22no—
140D4—22no—
141D4—41no—
142D4—41no—
143D4—42no—
144D4—42no—
145D4—81no—
146D4—82no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4#12011yescharacteristic
165C2×C4#12011yescharacteristic
166C2×C4#12111yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12711yescharacteristic
170C8—42no—
171C8—42no—
172C8—81no—
173C22—21no—
174C22—21no—
175C22—44no—
176C22—42no—
177C22—82no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22#39211yescharacteristic
184C22#45412yes—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—82no—
192C4—81no—
193C4—81no—
194C4—81no—
195C4—81no—
196C2—22no—
197C2—82no—
198C2—161no—
199C2#200311yescharacteristic
200C2#200811yescharacteristic
201C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3109C2
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
1. #3142C2semidirect
3. #2633C2semidirect
4. #1534C2semidirect
5. #1514C2semidirect
6. #1490C2semidirect
7. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#31422
#31092
#26332
#15342
#15142
#14902
#12882