Finite Group Databasebeta

← #24529

Subgroups — #24529

Order 256.

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

normal subgroups only

This group has 483 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×#451C211yesmaximal, characteristic
2C22×#103C211yesmaximal, characteristic
3#2503C211yesmaximal, characteristic
4Q8×C22⋊C4C211yesmaximal, characteristic
5#2074C212yesmaximal, complemented
6#2056C212yesmaximal, complemented
7#1892C211yesmaximal, characteristic
8#1878C212yesmaximal, complemented
9#1850C211yesmaximal, characteristic
10#1823C211yesmaximal, characteristic
11#1805C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2211yescharacteristic
14C22×Q16C2211yescharacteristic
15C23×C8C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19#451—24no—
20C2×C4⋊C8C2212yes—
21C2×#103—24no—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28#400—22no—
29#391—22no—
30#391C2212yes—
31#391C2212yes—
32#382C2212yes—
33#381—21no—
34#381—21no—
35#380—21no—
36#380C2211yescharacteristic
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C22×Q8D414yes—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C22×Q8C2311yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—22no—
55C2×Q16—22no—
56C2×Q16—24no—
57C2×Q16—42no—
58C22×C8—22no—
59C22×C8—42no—
60C22×C8C2312yes—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C4×Q8—24no—
66C4×Q8—42no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4D411yescharacteristic
78C2×C22⋊C4D411yescharacteristic
79C2×C42—22no—
80C2×C42C2312yes—
81C4⋊C8—42no—
82#103—24no—
83#103—24no—
84#103—22no—
85#103—22no—
86#103—42no—
87C22⋊C8—22no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—22no—
95#95D412yes—
96C24C2×D411yescharacteristic
97C2×Q8—24no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—28no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—44no—
109C2×Q8—44no—
110C2×Q8—48no—
111C2×Q8C2×D412yes—
112C2×Q8C2×D412yes—
113C2×Q8#4114yes—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4#4112yes—
132C22×C4C2411yescharacteristic
133Q16—44no—
134Q16—44no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—44no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C42—22no—
157C42—22no—
158C42—44no—
159C23—22no—
160C23—22no—
161C23#12112yes—
162C23#13512yes—
163C23#13712yes—
164C23C22×D411yescharacteristic
165Q8—22no—
166Q8—24no—
167Q8—22no—
168Q8—22no—
169Q8—48no—
170Q8—42no—
171Q8—41no—
172Q8—41no—
173Q8—42no—
174Q8#12314yes—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—48no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4#12112yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4112yes—
204C2×C4C2×#4111yescharacteristic
205C2×C4H(5,𝔽2)12yes—
206C8—41no—
207C8—42no—
208C8—41no—
209C22—22no—
210C22—22no—
211C22—24no—
212C22—24no—
213C22#46414yes—
214C22C2×#12111yescharacteristic
215C22#54612yes—
216C22C2×#13511yescharacteristic
217C22C2×#13711yescharacteristic
218C22#57812yes—
219C4—22no—
220C4—24no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4—81no—
227C4—82no—
228C4C2×#12311yescharacteristic
229C4#53411yescharacteristic
230C4#55112yes—
231C2—24no—
232C2#241211yescharacteristic
233C2C2×#46411yescharacteristic
234C2#304411yescharacteristic
235C2#330012yes—
236C2#331212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2074C2semidirect
6. #2056C2semidirect
8. #1878C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
C22×#1032
#25032
Q8×C22⋊C42
#20742
#20562
#18922
#18782
#18502
#18232
#18052