Finite Group Databasebeta

← #10022

Subgroups — #10022

Order 256.

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

normal subgroups only

This group has 288 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, complemented
2#3110C211yesmaximal, characteristic, complemented
3#2631C211yesmaximal, characteristic
4#1533C211yesmaximal, characteristic, complemented
5#1515C211yesmaximal, characteristic, complemented
6#1490C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#124C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#489—21no—
12#489—21no—
13#488—22no—
14#484—21no—
15#474—21no—
16#461—21no—
17#450—21no—
18#440—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#419—21no—
22#418—21no—
23C4×M16C2211yescharacteristic
24#397—21no—
25#393—21no—
26#392—21no—
27#392—21no—
28#386—22no—
29#383C2211yescharacteristic
30#329—22no—
31#328—22no—
32D4⋊C8—22no—
33C22×Q8D411yescharacteristic
34C22×D4—22no—
35C22×D4D411yescharacteristic
36C22×D4D411yescharacteristic
37C23×C4—21no—
38C2×SD16—22no—
39C2×M16—21no—
40C4⋊Q8D411yescharacteristic, complemented
41C4⋊Q8D411yescharacteristic, complemented
42#125—21no—
43#124—21no—
44#124—21no—
45#124—21no—
46#124—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55#117—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C42C2311yescharacteristic
63#107—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66#103—22no—
67#103—22no—
68#103—22no—
69#102—22no—
70#102—22no—
71#102—22no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74#97—21no—
75#97—21no—
76#97D412yescomplemented
77C4×C8D412yescomplemented
78#95—22no—
79#95—21no—
80#95—21no—
81#95D412yes—
82C24—21no—
83C24—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—42no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4—44no—
96C2×D4—82no—
97C2×D4—82no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—82no—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107SD16—44no—
108M16—42no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—41no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—82no—
128C22⋊C4—82no—
129C42—41no—
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42C2×D411yescharacteristic
133C42#4111yescharacteristic
134C23—22no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23—42no—
144C23#12011yescharacteristic
145Q8—42no—
146Q8—42no—
147Q8—81no—
148Q8—81no—
149D4—22no—
150D4—22no—
151D4—41no—
152D4—41no—
153D4—42no—
154D4—81no—
155D4—88no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4#12011yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12711yescharacteristic
176C8—42no—
177C8—42no—
178C8—81no—
179C22—21no—
180C22—21no—
181C22—44no—
182C22—42no—
183C22—82no—
184C22—82no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22#39211yescharacteristic
192C22#45512yes—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—82no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4—81no—
204C2—22no—
205C2—82no—
206C2—161no—
207C2#200411yescharacteristic
208C2#200811yescharacteristic
209C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#2631C2
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. #3140C2semidirect
2. #3110C2semidirect
4. #1533C2semidirect
5. #1515C2semidirect
6. #1490C2semidirect
7. #1288C2semidirect
40. C4⋊Q8D4semidirect
41. C4⋊Q8D4semidirect
76. #97D4semidirect
77. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#31402
#31102
#26312
#15332
#15152
#14902
#12882