Finite Group Databasebeta

← #24747

Subgroups — #24747

Order 256.

484 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#464C212yesmaximal, complemented
2C2×#460C211yesmaximal, characteristic
3C2×#441C211yesmaximal, characteristic, complemented
4#2548C211yesmaximal, characteristic
5#2351C211yesmaximal, characteristic, complemented
6#2065C212yesmaximal, complemented
7#2042C212yesmaximal, complemented
8#1961C211yesmaximal, characteristic
9#1940C211yesmaximal, characteristic, complemented
10#1880C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×Q16C2211yescharacteristic
13C22×SD16C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C2×C4×Q8C2211yescharacteristic
19#464—28no—
20#460—24no—
21#441—24no—
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#97C2211yescharacteristic
32#394—21no—
33#394—21no—
34#390—22no—
35#390C2212yes—
36#389—22no—
37#389C2212yes—
38#382—21no—
39#382C2211yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C22×D4—21no—
47C22×D4—21no—
48C22×D4C2312yes—
49C2×Q16—22no—
50C2×Q16—22no—
51C2×Q16D414yescomplemented
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—24no—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58#127—22no—
59#127—22no—
60#124—24no—
61#124—42no—
62C4×Q8—22no—
63C4×Q8—24no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8—42no—
67C4×Q8D414yes—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D412yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C42—21no—
81C2×C42—22no—
82C2×C42—21no—
83C2×C42C2312yes—
84C2×C42C2311yescharacteristic
85#106—22no—
86C4⋊C8—24no—
87C4⋊C8—22no—
88#103—22no—
89#103—24no—
90#103—22no—
91#102—24no—
92#102—24no—
93#102—42no—
94#97—22no—
95#95—21no—
96#95—22no—
97#95—21no—
98#95D411yescharacteristic
99#95D411yescharacteristic
100C24—22no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8C2×D412yes—
110C2×Q8C2×D412yes—
111C2×Q8C2×D412yes—
112C2×Q8#4112yes—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—44no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—82no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133Q16—28no—
134SD16—48no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8C2×D412yes—
144C4⋊C4—24no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—42no—
154C4⋊C4C2×D412yes—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C42—24no—
160C42—22no—
161C42—22no—
162C42—44no—
163C42—42no—
164C42C2×D412yes—
165C42#4112yes—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23C22×D411yescharacteristic
174Q8—22no—
175Q8—22no—
176Q8—22no—
177Q8—22no—
178Q8—22no—
179Q8—41no—
180Q8—41no—
181Q8#12114yes—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—42no—
186D4—84no—
187D4—84no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4#12112yes—
210C2×C4#12112yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C2×C4H(5,𝔽2)11yescharacteristic
216C2×C42−1+411yescharacteristic
217C8—22no—
218C8—42no—
219C8—41no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22C2×#12111yescharacteristic
228C22D4211yescharacteristic
229C22#54811yescharacteristic
230C22#57512yes—
231C22#57712yes—
232C4—24no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—82no—
240C4—81no—
241C4C2×#12111yescharacteristic
242C4#53711yescharacteristic
243C4#54711yescharacteristic
244C4#55311yescharacteristic
245C2—162no—
246C2#241611yescharacteristic
247C2#304011yescharacteristic
248C2#304311yescharacteristic
249C2#327112yes—
250C2#337112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#464C2semidirect
3. C2×#441C2semidirect
5. #2351C2semidirect
6. #2065C2semidirect
7. #2042C2semidirect
9. #1940C2semidirect
10. #1880C2semidirect
11. #1805C2semidirect
51. C2×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4602
C2×#4412
#25482
#23512
#20652
#20422
#19612
#19402
#18802
#18052