Finite Group Databasebeta

← #10241

Subgroups — #10241

Order 256.

295 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 61 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#3227C211yesmaximal, characteristic, complemented
2#3036C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic, complemented
4#1549C211yesmaximal, characteristic, complemented
5#1502C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic, complemented
7#1303C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#498—21no—
13#486—21no—
14#481—22no—
15#476—22no—
16#463—22no—
17#461—21no—
18#459—21no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#417—21no—
23#393—22no—
24#390—21no—
25#382—21no—
26#380—22no—
27C4×C4⋊C4C2211yescharacteristic
28#332—22no—
29#327—22no—
30D4⋊C8—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35AGL(1,ℤ/8)—42no—
36AGL(1,ℤ/8)—42no—
37C2×SD16—22no—
38C2×D8—22no—
39C2×M16—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8D411yescharacteristic
45#127D411yescharacteristic
46#124—21no—
47#121—21no—
48#121—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D412yes—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—24no—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63#106—24no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8D412yes—
67C4⋊C8D412yes—
68#103—22no—
69#102—22no—
70#102—24no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75#95—24no—
76#95D412yes—
77C24—21no—
78#41—41no—
79#41—41no—
80#41—41no—
81#41—41no—
82C2×Q8—21no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×Q8—82no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C2×D4SD1612yes—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—44no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106SD16—44no—
107D8—44no—
108M16—41no—
109M16—41no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—41no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—42no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—84no—
127C42—41no—
128C42—41no—
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42#4111yescharacteristic
133C23—22no—
134C23—42no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23#12011yescharacteristic
140Q8—41no—
141Q8—41no—
142Q8—82no—
143Q8—82no—
144D4—22no—
145D4—22no—
146D4—41no—
147D4—41no—
148D4—41no—
149D4—42no—
150D4—41no—
151D4—81no—
152D4—81no—
153D4—81no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4#12011yescharacteristic
169C2×C4#12011yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12711yescharacteristic
174C2×C4C2×SD1611yescharacteristic
175C2×C4#13511yescharacteristic
176C2×C4AGL(1,ℤ/8)11yescharacteristic
177C2×C4#13711yescharacteristic
178C8—42no—
179C8—42no—
180C8—81no—
181C22—21no—
182C22—21no—
183C22—44no—
184C22—42no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22#39211yescharacteristic
192C22#44811yescharacteristic
193C22#44911yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—82no—
198C4—81no—
199C4—81no—
200C4—82no—
201C4#45011yescharacteristic
202C4#45211yescharacteristic
203C4#45412yes—
204C4#46511yescharacteristic
205C4#46811yescharacteristic
206C2—21no—
207C2—21no—
208C2—82no—
209C2—161no—
210C2#200011yescharacteristic
211C2#200311yescharacteristic
212C2#200811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3036C2
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. #3227C2semidirect
3. #2563C2semidirect
4. #1549C2semidirect
5. #1502C2semidirect
6. #1478C2semidirect
7. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#30362
#25632
#15492
#15022
#14782
#13032