Finite Group Databasebeta

← #10180

Subgroups — #10180

Order 256.

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

normal subgroups only

This group has 281 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#3206C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3#3036C211yesmaximal, characteristic
4#1549C211yesmaximal, characteristic, complemented
5#1480C211yesmaximal, characteristic, complemented
6#1460C211yesmaximal, characteristic, complemented
7#1301C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#492—21no—
13#487—21no—
14#476—22no—
15#476—22no—
16#463—22no—
17#461—22no—
18#461—21no—
19#459—21no—
20#448—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23C2×#106—21no—
24#423C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#417—21no—
27C2×#102—21no—
28#332—22no—
29D4⋊C8—22no—
30#323—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×SD16—24no—
39C2×D8—22no—
40C2×M16—21no—
41C22×C8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8D412yes—
46C4⋊Q8D411yescharacteristic
47#127D411yescharacteristic
48#124—21no—
49#121—21no—
50#121—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C22⋊C4—21no—
61C2×C42C2311yescharacteristic
62#106—24no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8D412yes—
68C4⋊C8D412yes—
69#103—22no—
70#103—24no—
71#102—24no—
72#102—22no—
73C22⋊C8—21no—
74C22⋊C8D412yes—
75C4×C8—21no—
76C4×C8—21no—
77C24—21no—
78#41—41no—
79#41—41no—
80#41—41no—
81#41—41no—
82C2×Q8—21no—
83C2×Q8—22no—
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×D4SD1612yescomplemented
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—82no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4#4111yescharacteristic
105SD16—44no—
106SD16—84no—
107D8—44no—
108M16—41no—
109M16—41no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—81no—
137C23—81no—
138C23#12111yescharacteristic
139Q8—44no—
140Q8—41no—
141Q8—41no—
142Q8—82no—
143D4—22no—
144D4—22no—
145D4—42no—
146D4—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—42no—
151D4—81no—
152D4—81no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—82no—
163C2×C4—81no—
164C2×C4#12011yescharacteristic
165C2×C4#12011yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12711yescharacteristic
170C2×C4C2×SD1611yescharacteristic
171C2×C4#13511yescharacteristic
172C2×C4AGL(1,ℤ/8)11yescharacteristic
173C2×C4#13711yescharacteristic
174C8—42no—
175C8—42no—
176C8—42no—
177C8—81no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—82no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22#39211yescharacteristic
189C22#46511yescharacteristic
190C22#46811yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—81no—
195C4—82no—
196C4—81no—
197C4#44811yescharacteristic
198C4#44911yescharacteristic
199C4#45011yescharacteristic
200C4#45211yescharacteristic
201C4#47112yes—
202C2—21no—
203C2—21no—
204C2—82no—
205C2—161no—
206C2#200011yescharacteristic
207C2#200811yescharacteristic
208C2#201911yescharacteristic
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. #3206C2semidirect
2. C2×#476C2semidirect
4. #1549C2semidirect
5. #1480C2semidirect
6. #1460C2semidirect
7. #1301C2semidirect
95. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
C2×#4762
#30362
#15492
#14802
#14602
#13012