Finite Group Databasebeta

← #10408

Subgroups — #10408

Order 256.

354 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 75 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#2541C211yesmaximal, characteristic
2#1957C211yesmaximal, characteristic, complemented
3#1947C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic, complemented
5C2×#331C211yesmaximal, characteristic, complemented
6C2×D4⋊C8C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×#127—21no—
9C2×#127C2211yescharacteristic
10C2×#121—21no—
11C2×#121—21no—
12C2×C4×D4C2211yescharacteristic
13C2×#107—21no—
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#102—21no—
17C2×#102—22no—
18C2×C4×C8C2211yescharacteristic
19#392—22no—
20#390—21no—
21#384C2211yescharacteristic
22C4×C4⋊C4C2211yescharacteristic
23#340—21no—
24#340—21no—
25#331—22no—
26D4⋊C8—22no—
27C22×D4—22no—
28C22×D4—22no—
29C22×D4—21no—
30C22×D4—21no—
31C22×D4D411yescharacteristic
32C22×D4D412yes—
33C23×C4—21no—
34C22×C8—21no—
35C22×C8—21no—
36C22×C8—21no—
37#127—21no—
38#127—21no—
39#127—21no—
40#127—21no—
41#127—42no—
42#127—42no—
43#121—42no—
44#121—42no—
45#121—42no—
46#121—42no—
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—21no—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62#107—42no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8D412yescomplemented
67#102—44no—
68#102—42no—
69C4×C8—21no—
70C4×C8—21no—
71#95—21no—
72C24—21no—
73C24—22no—
74C2×D4—21no—
75C2×D4—22no—
76C2×D4—21no—
77C2×D4—22no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—42no—
82C2×D4—42no—
83C2×D4—44no—
84C2×D4—44no—
85C2×D4—44no—
86C2×D4—44no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—82no—
100C2×D4—82no—
101C2×D4D812yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—42no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4D812yes—
134C4⋊C4SD1612yes—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C42—41no—
140C42—41no—
141C42C2×D411yescharacteristic
142C42#4111yescharacteristic
143C42#4111yescharacteristic
144C42#4111yescharacteristic
145C23—22no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23#12011yescharacteristic
160D4—22no—
161D4—22no—
162D4—44no—
163D4—41no—
164D4—41no—
165D4—82no—
166D4—88no—
167D4—88no—
168D4—81no—
169D4—81no—
170D4—81no—
171D4—81no—
172D4—81no—
173D4—81no—
174D4—81no—
175D4—81no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4#12111yescharacteristic
195C2×C4#12111yescharacteristic
196C2×C4#12111yescharacteristic
197C2×C4#12311yescharacteristic
198C2×C4#12311yescharacteristic
199C2×C4#12411yescharacteristic
200C2×C4C2×D811yescharacteristic
201C2×C4C2×D811yescharacteristic
202C2×C4C2×SD1611yescharacteristic
203C2×C4#13511yescharacteristic
204C2×C4AGL(1,ℤ/8)11yescharacteristic
205C2×C4AGL(1,ℤ/8)11yescharacteristic
206C2×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C4#13711yescharacteristic
208C8—42no—
209C8—42no—
210C8—81no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—82no—
219C22—82no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—82no—
225C22#39411yescharacteristic
226C22#44711yescharacteristic
227C22#44711yescharacteristic
228C22#45011yescharacteristic
229C22#45211yescharacteristic
230C22#45311yescharacteristic
231C22#45511yescharacteristic
232C4—21no—
233C4—21no—
234C4—42no—
235C4—42no—
236C4—81no—
237C4—161no—
238C4#45911yescharacteristic
239C4#46011yescharacteristic
240C4#46611yescharacteristic
241C4#46811yescharacteristic
242C4#48011yescharacteristic
243C4#48211yescharacteristic
244C4#48611yescharacteristic
245C4#49011yescharacteristic
246C2—82no—
247C2—162no—
248C2#161911yescharacteristic
249C2#163111yescharacteristic
250C2#163911yescharacteristic
251C2#164111yescharacteristic
252C2#203311yescharacteristic
253C2#203711yescharacteristic
254C2#204111yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radical#2541C2
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. #1957C2semidirect
3. #1947C2semidirect
4. #1802C2semidirect
5. C2×#331C2semidirect
6. C2×D4⋊C8C2semidirect
7. #1279C2semidirect
66. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#25412
#19572
#19472
#18022
C2×#3312
C2×D4⋊C82
#12792