Finite Group Databasebeta

← #10436

Subgroups — #10436

Order 256.

265 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 73 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#2538C211yesmaximal, characteristic
2#1834C211yesmaximal, characteristic, complemented
3#1803C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic
5C2×Q8⋊C8C211yesmaximal, characteristic, complemented
6C2×D4⋊C8C211yesmaximal, characteristic
7#1296C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8—21no—
9C2×C4×Q8C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11C2×C4⋊C8C2211yescharacteristic
12C2×C4⋊C8C2211yescharacteristic
13C2×#103—21no—
14C2×#102—21no—
15C2×C4×C8C2211yescharacteristic
16#396—22no—
17#390—21no—
18#389—21no—
19#384C2211yescharacteristic
20#384C2211yescharacteristic
21#380—21no—
22#340—21no—
23#340—21no—
24#340—22no—
25Q8⋊C8—22no—
26D4⋊C8—22no—
27C22×Q8D411yescharacteristic
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C22×C8—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22⋊Q8—42no—
34C22⋊Q8—42no—
35C4×Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×D4—21no—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D412yes—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—22no—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62#103—42no—
63#102—42no—
64C4×C8—21no—
65C4×C8D412yes—
66#95—22no—
67C24—21no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8SD1612yes—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4Q1612yes—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C2×C8—22no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4—42no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C22⋊C4—44no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—82no—
124C42—41no—
125C42—41no—
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C42#4111yescharacteristic
129C42#4111yescharacteristic
130C23—22no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23#12011yescharacteristic
138Q8—22no—
139Q8—22no—
140Q8—41no—
141Q8—41no—
142D4—22no—
143D4—22no—
144D4—41no—
145D4—41no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12311yescharacteristic
171C2×C4#12311yescharacteristic
172C2×C4#12411yescharacteristic
173C2×C4C2×SD1611yescharacteristic
174C2×C4C2×Q1611yescharacteristic
175C2×C4#13511yescharacteristic
176C2×C4#13511yescharacteristic
177C2×C4AGL(1,ℤ/8)11yescharacteristic
178C2×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4#13711yescharacteristic
181C8—42no—
182C8—81no—
183C8—81no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22#39411yescharacteristic
192C22#44911yescharacteristic
193C22#44911yescharacteristic
194C22#45011yescharacteristic
195C22#45111yescharacteristic
196C22#45411yescharacteristic
197C22#45511yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—82no—
202C4—81no—
203C4—81no—
204C4—161no—
205C4#46511yescharacteristic
206C4#46711yescharacteristic
207C4#46811yescharacteristic
208C4#46911yescharacteristic
209C4#48111yescharacteristic
210C4#48211yescharacteristic
211C4#48411yescharacteristic
212C4#48511yescharacteristic
213C2—82no—
214C2#162811yescharacteristic
215C2#163711yescharacteristic
216C2#164211yescharacteristic
217C2#164411yescharacteristic
218C2#203411yescharacteristic
219C2#203611yescharacteristic
220C2#204011yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1834C2semidirect
3. #1803C2semidirect
5. C2×Q8⋊C8C2semidirect
7. #1296C2semidirect

Maximal subgroups

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

subgroupindex
#25382
#18342
#18032
#18022
C2×Q8⋊C82
C2×D4⋊C82
#12962