Finite Group Databasebeta

← #28164

Subgroups — #28164

Order 256.

452 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 129 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
2C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#124C211yesmaximal, characteristic
4#2072C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic
6#2016C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic
8#1878C212yesmaximal, complemented
9#1877C212yesmaximal
10#1831C211yesmaximal, characteristic, complemented
11#1805C212yesmaximal, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#464—28no—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25#391C2212yes—
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#388—21no—
31#388—21no—
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C4×C22⋊C4—21no—
37C4×C22⋊C4—21no—
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C22×Q8C2312yes—
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—42no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—24no—
57#124—22no—
58#124D414yes—
59C4×Q8—24no—
60C4×Q8—24no—
61C4×Q8—42no—
62C4×D4—42no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C4⋊C8—24no—
87#103—24no—
88#103—42no—
89#103—42no—
90#102—24no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—22no—
96#95—22no—
97C24—21no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8C2×D412yes—
107C2×Q8#4114yes—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4C2×D412yes—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2411yescharacteristic
127Q16—48no—
128SD16—48no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—42no—
157C42—44no—
158C42C2×D412yes—
159C42#4112yes—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23C22×D411yescharacteristic
168Q8—24no—
169Q8—24no—
170Q8—24no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174D4—24no—
175D4—42no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—84no—
205C2×C4#12112yes—
206C2×C4#12112yes—
207C2×C4#13512yes—
208C2×C4#13712yes—
209C2×C4#13712yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C2×#4112yes—
212C2×C4C2×#4111yescharacteristic
213C2×C4H(5,𝔽2)12yes—
214C8—42no—
215C8—42no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22C2×#12111yescharacteristic
224C22#52511yescharacteristic
225C22#53511yescharacteristic
226C22C2×#13511yescharacteristic
227C22C2×#13711yescharacteristic
228C22C2×#13711yescharacteristic
229C22#57511yescharacteristic
230C4—22no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—41no—
236C4—42no—
237C4—44no—
238C4—81no—
239C4—81no—
240C4—82no—
241C4#54612yes—
242C4#55312yes—
243C2—82no—
244C2#253511yescharacteristic
245C2#303511yescharacteristic
246C2#303911yescharacteristic
247C2#310811yescharacteristic
248C2#311711yescharacteristic
249C2#321011yescharacteristic
250C2#321511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊Q8C2semidirect
4. #2072C2semidirect
6. #2016C2semidirect
8. #1878C2semidirect
10. #1831C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C4×C4⋊Q82
C4×#1242
#20722
#20712
#20162
#20132
#18782
#18772
#18312
#18052