Finite Group Databasebeta

← #10024

Subgroups — #10024

Order 256.

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

normal subgroups only

This group has 256 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#3140C211yesmaximal, characteristic
2#3113C211yesmaximal, characteristic, complemented
3#2632C211yesmaximal, characteristic
4#1533C211yesmaximal, characteristic
5#1515C211yesmaximal, characteristic, complemented
6#1491C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#124C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#489—21no—
12#489—21no—
13#488—22no—
14#483—21no—
15#475—21no—
16#462—21no—
17#451—21no—
18#441—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#419—21no—
22#418—21no—
23C4×M16C2211yescharacteristic
24#399—21no—
25#395—21no—
26#394—21no—
27#393—21no—
28#388—22no—
29#383C2211yescharacteristic
30#329—22no—
31#328—22no—
32Q8⋊C8—22no—
33C22×Q8D411yescharacteristic
34C22×Q8D411yescharacteristic
35C22×D4D411yescharacteristic
36C2×Q16—22no—
37C2×M16—21no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#125—21no—
41#124—21no—
42#124—21no—
43#124—21no—
44#124—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D411yescharacteristic
57C2×C22⋊C4—22no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—22no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66#103—22no—
67#103—22no—
68#103—22no—
69#103—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—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—42no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101Q16—44no—
102M16—42no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—41no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—82no—
117C22⋊C4—44no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—82no—
124C42—41no—
125C42—41no—
126C42—41no—
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42#4111yescharacteristic
131C23—42no—
132C23—42no—
133C23—41no—
134C23—42no—
135C23#12011yescharacteristic
136Q8—22no—
137Q8—22no—
138Q8—41no—
139Q8—41no—
140Q8—42no—
141Q8—42no—
142Q8—81no—
143Q8—81no—
144D4—42no—
145D4—81no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4#12011yescharacteristic
162C2×C4#12011yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12711yescharacteristic
167C8—42no—
168C8—42no—
169C8—81no—
170C22—21no—
171C22—21no—
172C22—84no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#39211yescharacteristic
177C22#45512yes—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—42no—
185C4—82no—
186C4—81no—
187C4—81no—
188C4—81no—
189C4—81no—
190C2—22no—
191C2—161no—
192C2#200411yescharacteristic
193C2#200911yescharacteristic
194C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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. #3113C2semidirect
5. #1515C2semidirect
6. #1491C2semidirect
7. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#31402
#31132
#26322
#15332
#15152
#14912
#12882